Medicine, Faculty of
Population and Public Health (SPPH), School of
DSpace
UBCV
Kafi, Mohammad Abdullah Heel
Master of Science - MSc
University of British Columbia
In epidemiological research, the association between treatment and outcome may vary across values of a third variable, termed an “effect modifier”. Studying effect modifiers can help to identify subgroups for which treatment is beneficial or harmful. Standard approaches to assessing the impact of an effect modifier include subgroup analysis or adding an interaction term between treatment and the effect modifier in the regression analysis. Various propensity score (PS) based methods, also have been proposed, but it is unclear whether the PS should be derived based on the full dataset or within subgroups, or how these PS’s should then be used (matching, weights, stratification, covariate adjustment). We explored the performance of various PS-based approaches using simulation to compare how well the approach works under various scenarios. We performed comprehensive Monte Carlo simulation studies considering various scenarios, plasmode simulation, and empirical data analysis to evaluate the performance of all possible combinations of effect modifier approaches within all propensity score methods for various scenarios. We found that, for various simulation scenarios, there were comparable differences among various effect modifier approaches. For example, under simulation scenarios such as the model with additivity, mild non-linearity, moderate non-additivity, the subgroup-specific PS approach produced higher bias than the overall PS approach. PS matching approaches performed better than PS weighting methods in the model’s scenarios with mild non-linearity, mild non-additivity. In Monte Carlo simulation with interaction, the useful interaction approach had more downward bias, lower mean squared error than the overall PS, and subgroup-specific approach, in various scenarios such as additive, mild-linearity, moderate non-linearity. Our findings demonstrate that, under different scenarios, different effect modifiers approaches can be performed better than others. Therefore, this study will help the researcher to identify the appropriate approaches to deal with effect modifier with the appropriate PS methods under a given practical setting.
EXAMINING EFFECTMODIFICATION WITHINPROPENSITY SCORE METHODSbyMohammad Abdullah Heel KafiBSc, University of Dhaka, 2010MSc, University of Dhaka, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Population and Public Health)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2021© Mohammad Abdullah Heel Kafi 2021The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the thesisentitled:Examining Effect Modification within Propensity Score Meth-odssubmitted by Mohammad Abdullah Heel Kafi in partial fulfillmentof the requirements for the degree of Master of Science in Populationand Public Health.Examining Committee:Mohammad Ehsanul Karim, Assistant Professor, School of Populationand Public Health, UBCSupervisorHubert Wong, Associate Professor, School of Population and PublicHealth, UBCSupervisory Committee MemberWei Zhang, Assistant Professor, School of Population and Public Health,UBCSupervisory Committee MemberAbstractIn epidemiological research, the association between treatment and outcomemay vary across values of a third variable, termed an “effect modifier”.Studying effect modifiers can help to identify subgroups for which treat-ment is beneficial or harmful. Standard approaches to assessing the impactof an effect modifier include subgroup analysis or adding an interaction termbetween treatment and the effect modifier in the regression analysis. Var-ious propensity score (PS) based methods, also have been proposed, butit is unclear whether the PS should be derived based on the full datasetor within subgroups, or how these PS’s should then be used (matching,weights, stratification, covariate adjustment). We explored the performanceof various PS-based approaches using simulation to compare how well theapproach work under various scenarios. We performed comprehensive MonteCarlo simulation studies considering various scenarios, plasmode simulationand empirical data analysis to evaluate the performance of all possible com-binations of effect modifier approaches within all propensity score methodsfor various scenarios. We found that, for various simulation scenarios, therewere comparable differences among various effect modifier approaches. Forexample, under simulation scenarios such as the model with additivity, mildnon-linearity, moderate non-additivity, the subgroup-specific PS approachproduced higher bias than the overall PS approach. PS matching approachesperformed better than PS weighting methods in the model’s scenarios withmild non-linearity, mild non-additivity. In Monte Carlo simulation with in-teraction, the useful interaction approach had more downward bias, lowermean squared error than the overall PS and subgroup-specific approach,in various scenarios such as additive, mild-linearity, moderate non-linearity.Our findings demonstrate that, under different scenarios, different effectmodifiers approaches can be performed better than others. Therefore, thisstudy will help the researcher to identify the appropriate approaches to dealwith effect modifier with appropriate PS method under a given practicalsetting.iiiLay SummaryIn epidemiological study when the association treatment and outcome ismodified by a third variable, that variable is known as “effect modifier”.Studying treatment effect modifiers is essential to identify subgroups withdifferential treatment benefits. In observational study researchers widelyused the propensity score analysis. To address effect modifier within propen-sity score framework many combinations of propensity score and effect modi-fication methods are possible. However, there is a paucity of comprehensiveinvestigation of which methods are appropriate under a practical setting.This study demonstrates a comprehensive simulation study with variousscenarios and empirical data analysis to evaluate performance of various ap-proaches and provide guidelines to the researchers to address effect modifierwithin the analysis of propensity score methods.ivPrefaceI wrote this dissertation under the direction and guidance of Professor Dr.Mohammad Ehsanul Karim. I have done all the simulations, data analysisand written all the chapters.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background of the effect modifier . . . . . . . . . . . . . . . 11.2 Overview of Propensity score . . . . . . . . . . . . . . . . . . 31.2.1 Potential outcomes framework . . . . . . . . . . . . . 31.2.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . 61.2.3 Propensity score analysis . . . . . . . . . . . . . . . . 71.2.4 Estimation of propensity score . . . . . . . . . . . . . 81.2.5 Propensity score implementation . . . . . . . . . . . . 91.2.6 Propensity score matching . . . . . . . . . . . . . . . 91.2.7 Propensity score stratification . . . . . . . . . . . . . 111.2.8 Propensity score weighting . . . . . . . . . . . . . . . 121.2.9 Propensity score used as covariate . . . . . . . . . . . 131.2.10 Assessment of the covariate balance after implemen-tation of Propensity score . . . . . . . . . . . . . . . . 131.3 Literature review on Propensity score approaches to handleeffect modifier . . . . . . . . . . . . . . . . . . . . . . . . . . 141.4 Knowledge gap . . . . . . . . . . . . . . . . . . . . . . . . . . 211.5 Objectives of the thesis . . . . . . . . . . . . . . . . . . . . . 22vi2 Propensity Score Methods to Deal with Effect Modifier . 232.1 Propensity score matching approach . . . . . . . . . . . . . . 252.2 Propensity score weighting approach . . . . . . . . . . . . . . 302.2.1 Methods proposed in the literature . . . . . . . . . . 302.2.2 New methods we proposed . . . . . . . . . . . . . . . 332.3 Propensity score stratification approach . . . . . . . . . . . . 352.3.1 Methods proposed in the literature . . . . . . . . . . 352.3.2 New method we proposed . . . . . . . . . . . . . . . . 362.4 Propensity score covariate adjustment approach . . . . . . . 372.4.1 Methods proposed in the literature . . . . . . . . . . 372.4.2 New methods we proposed . . . . . . . . . . . . . . . 392.5 Features of the various methods to handle effect modifier inPropensity score context . . . . . . . . . . . . . . . . . . . . 413 Motivating Example Right Heart Catheterization Study . 453.1 Right Heart Catheterization study . . . . . . . . . . . . . . . 453.1.1 Analysis of the Right Heart Catheterization (RHC)dataset . . . . . . . . . . . . . . . . . . . . . . . . . . 463.2 Presence of effect modifier in the Right Heart Catheterization(RHC) study . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.3 Analysis of effect modifier in the Right Heart Catheterization(RHC) study . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 Two Data Generation Mechanisms for Simulation . . . . . 724.1 Monte Carlo simulation . . . . . . . . . . . . . . . . . . . . . 724.1.1 Monte Carlo simulation without considering interac-tion between subgroup variable and a covariate . . . 724.1.2 Simulation settings . . . . . . . . . . . . . . . . . . . 724.1.3 Data generation . . . . . . . . . . . . . . . . . . . . . 724.1.4 Scenarios of simulation study without considering in-teraction between subgroup variable and a covariate . 734.1.5 Monte Carlo simulation: considering an interactionbetween subgroup variable and a covariate . . . . . . 764.1.6 Scenarios of simulation study with considering inter-action between subgroup variable and a covariate . . 764.2 Plasmode simulation . . . . . . . . . . . . . . . . . . . . . . . 784.3 Performance measures of simulation study . . . . . . . . . . 825 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 845.1 Monte Carlo simulation without considering interaction be-tween subgroup variable and a covariate . . . . . . . . . . . 845.1.1 Monte Carlo simulation result for scenario A: modelwith additive effects . . . . . . . . . . . . . . . . . . 845.1.2 Monte Carlo simulation result for scenario B : mildnon-linearity . . . . . . . . . . . . . . . . . . . . . . 905.1.3 Monte Carlo simulation result for scenario C : mod-erate non-linearity . . . . . . . . . . . . . . . . . . . 955.1.4 Monte Carlo simulation result for scenario D : mildnon-additivity . . . . . . . . . . . . . . . . . . . . . . 1005.1.5 Monte Carlo simulation result for scenario E : mildnon-additivity and non-linearity . . . . . . . . . . . . 1055.1.6 Monte Carlo simulation result for scenario F: moder-ate non-additivity . . . . . . . . . . . . . . . . . . . . 1105.1.7 Monte Carlo simulation result for scenario G: mod-erate non-additivity and non-linearity . . . . . . . . . 1155.2 Monte Carlo simulation by considering an interaction betweensubgroup variable and a covariate . . . . . . . . . . . . . . . 1205.2.1 Monte Carlo simulation result for scenario A1 : ad-ditive effects with an interaction between subgroupvariable and a covariate . . . . . . . . . . . . . . . . . 1205.2.2 Monte Carlo simulation result for scenario B1 : mildnon-linearity with an interaction between subgroupvariable and a covariate . . . . . . . . . . . . . . . . . 1265.2.3 Monte Carlo simulation result for scenario C1 : mod-erate non-linearity with an interaction between sub-group variable and a covariate . . . . . . . . . . . . . 1315.2.4 Monte Carlo simulation result for scenario D1 : mildnon-additivity with an interaction between subgroupvariable and a covariate . . . . . . . . . . . . . . . . . 1365.2.5 Monte Carlo simulation result for scenario E1 : mildnon-additivity and non-linearity with an interactionbetween subgroup variable and a covariate . . . . . . 1415.2.6 Monte Carlo simulation result for scenario F1: mod-erate non-additivity with an interaction between sub-group variable and a covariate . . . . . . . . . . . . . 1465.2.7 Monte Carlo simulation result for scenario G1: mod-erate non-additivity and non-linearity with an inter-action between subgroup variable and a covariate . . 1515.3 Plasmode simulation result . . . . . . . . . . . . . . . . . . . 1565.3.1 Plasmode simulation result: crude analysis . . . . . . 1565.3.2 Plasmode simulation result from adjusted analysis . . 1615.4 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . 1665.4.1 Propensity score weighting Methods: Overlap weights 1665.4.2 Application of machine learning methods in the es-timation of the propensity score for each of the effectmodifier methods . . . . . . . . . . . . . . . . . . . . 1715.4.3 Results from the Monte Carlo simulation without in-teraction for scenario “A” after applying the decisiontree method in the PS estimation . . . . . . . . . . . 1715.4.4 Results from the Monte Carlo simulation without in-teraction for scenario “G” after applying the decisiontree method in the PS estimation . . . . . . . . . . . 1755.4.5 Results from the Monte Carlo simulation with inter-action for scenario “A1” after applying the decisiontree method in the PS estimation: . . . . . . . . . . . 1795.4.6 Results from the Monte Carlo simulation with inter-action for scenario “G1” after applying the decisiontree method in the PS estimation: . . . . . . . . . . . 1835.4.7 Plasmode simulation for reduced sample size . . . . . 1875.4.8 Plasmode simulation result from adjusted analysis forreduced sample size . . . . . . . . . . . . . . . . . . . 1895.4.9 Plasmode simulation result when exposure prevalenceis low . . . . . . . . . . . . . . . . . . . . . . . . . . . 1905.4.10 Plasmode simulation result when outcome prevalenceis low . . . . . . . . . . . . . . . . . . . . . . . . . . . 1936 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . 1956.1 Summary of the findings of study . . . . . . . . . . . . . . . 1956.2 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . 2076.3 Strengths of the study . . . . . . . . . . . . . . . . . . . . . . 2106.4 Limitations of the study . . . . . . . . . . . . . . . . . . . . . 2116.5 Future direction . . . . . . . . . . . . . . . . . . . . . . . . . 2126.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216List of Tables1.1 Potential outcome framework to define treatment effect . . . 41.2 Potential outcome framework to define treatment effect basedon observed outcome . . . . . . . . . . . . . . . . . . . . . . . 52.1 Propensity score effect modifier methods . . . . . . . . . . . . 242.2 Propensity score matching methods to handle effect modifier 412.3 Propensity score weighting methods to handle effect modifier 422.4 Propensity score stratification methods to handle effect modifier 432.5 Propensity score covariate adjustment methods to handle ef-fect modifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.1 Crude association between treatment Right Heart Catheteri-zation (RHC) and subsequent death . . . . . . . . . . . . . . 473.2 Unadjusted and adjusted effect of Right heart catheterizationtreatment on death . . . . . . . . . . . . . . . . . . . . . . . . 483.3 Crude and adjusted association between treatment Right HeartCatheterization and subsequent death for male and female inthe Right Heart Catheterization study . . . . . . . . . . . . . 493.4 Crude and adjusted treatment effect from propensity scoreeffect modifier matching methods M1-M6 . . . . . . . . . . . 593.5 Crude and adjusted treatment effect from propensity scoreeffect modifier weighting methods W1-W6 . . . . . . . . . . . 633.6 Crude and adjusted treatment effect after performing propen-sity score effect modifier methods S1-S3 . . . . . . . . . . . . 683.7 Crude and adjusted treatment effect after performing propen-sity effect modifier methods C1-C6 . . . . . . . . . . . . . . . 715.1 Results from the Monte Carlo simulation result for scenarioA: additive effects . . . . . . . . . . . . . . . . . . . . . . . . 865.2 Results from the Monte Carlo simulation result for scenarioB : mild non-linearity . . . . . . . . . . . . . . . . . . . . . . 91x5.3 Results from the Monte Carlo simulation result for scenarioC : moderate non-linearity . . . . . . . . . . . . . . . . . . . . 965.4 Results from the Monte Carlo simulation result for scenarioD : mild non-additivity . . . . . . . . . . . . . . . . . . . . . . 1015.5 Results from the Monte Carlo simulation result for scenarioE : mild non-additivity and non-linearity . . . . . . . . . . . . 1065.6 Results from the Monte Carlo simulation result for scenarioF: moderate non-additivity . . . . . . . . . . . . . . . . . . . 1115.7 Results from the Monte Carlo simulation result for scenarioG: moderate non-additivity and non-linearity . . . . . . . . . 1165.8 Results from the Monte Carlo simulation result for scenarioA1: additive effects . . . . . . . . . . . . . . . . . . . . . . . . 1225.9 Results from the Monte Carlo simulation result for scenarioB1 : mild non-linearity . . . . . . . . . . . . . . . . . . . . . 1275.10 Results from the Monte Carlo simulation result for scenarioC1 : moderate non-linearity . . . . . . . . . . . . . . . . . . . 1325.11 Results from the Monte Carlo simulation result for scenarioD1 : mild non-additivity . . . . . . . . . . . . . . . . . . . . . 1375.12 Results from the Monte Carlo simulation result for scenarioE1 : mild non-additivity and non-linearity . . . . . . . . . . . 1425.13 Results from the Monte Carlo simulation result for scenarioF1: moderate non-additivity . . . . . . . . . . . . . . . . . . . 1475.14 Results from the Monte Carlo simulation result for scenarioG1: moderate non-additivity and non-linearity . . . . . . . . 1525.15 Monte Carlo simulation result without interaction from over-lap weighting method . . . . . . . . . . . . . . . . . . . . . . 1685.16 Monte Carlo simulation result with interaction from overlapweighting method . . . . . . . . . . . . . . . . . . . . . . . . . 1705.17 Monte Carlo simulation without interaction for scenario A . . 1725.18 Monte Carlo simulation without interaction for scenario G . . 1765.19 Monte Carlo simulation with interaction for scenario A1 . . . 1805.20 Monte Carlo simulation with interaction for scenario G1 . . . 1845.21 Plasmode simulation result (crude) for reduced sample size . 1885.22 Plasmode simulation result (adjusted) for reduced sample size 1905.23 Plasmode simulation result with low exposure prevalence . . 1925.24 Plasmode simulation result with low outcome prevalence . . . 1946.1 Summary from Monte Carlo simulation without consideringan interaction between subgroup variable and a covariate . . 2036.2 Summary from Monte Carlo simulation by considering an in-teraction between subgroup variable and a covariate . . . . . 204List of Figures3.1 Covariate balance plot after performing PS effect modifiermethod M1 based on the levels of subgroup variable . . . . . 513.2 Covariate balance plot after performing PS effect modifiermethod M2 based on the levels of subgroup variable . . . . . 523.3 Covariate balance plot after performing PS effect modifiermethod M3 based on the levels of subgroup variable . . . . . 533.4 Covariate balance plot after performing PS effect modifiermethod M4 for Male . . . . . . . . . . . . . . . . . . . . . . . 543.5 Covariate balance plot after performing PS effect modifiermethod M4 for Female . . . . . . . . . . . . . . . . . . . . . 543.6 Covariate balance plot after performing PS effect modifiermethod M5 based on the levels of subgroup variable . . . . . 553.7 Covariate balance plot after performing PS effect modifiermethod M6 based on the levels of subgroup variable . . . . . 563.8 Distribution of standardized mean difference for all PS match-ing approach for Male . . . . . . . . . . . . . . . . . . . . . . 573.9 Distribution of standardized mean difference for all PS match-ing approach for Female . . . . . . . . . . . . . . . . . . . . . 573.10 Distribution of standardized mean difference for PS weightingapproach for Male . . . . . . . . . . . . . . . . . . . . . . . . 623.11 Distribution of standardized mean difference for all PS match-ing approach for Female . . . . . . . . . . . . . . . . . . . . . 633.12 Distribution of standardized mean difference for PS stratifi-cation approach method S1 for Male . . . . . . . . . . . . . . 643.13 Distribution of standardized mean difference for PS stratifi-cation approach method S1 for Female . . . . . . . . . . . . 653.14 Distribution of standardized mean difference for PS stratifi-cation approach method S2 for Male . . . . . . . . . . . . . . 653.15 Distribution of standardized mean difference for PS stratifi-cation approach method S2 for Female . . . . . . . . . . . . 66xiii3.16 Distribution of standardized mean difference for PS stratifi-cation approach method S3 for Male . . . . . . . . . . . . . . 673.17 Distribution of standardized mean difference for PS stratifi-cation approach method S3 for Female . . . . . . . . . . . . 674.1 DAG for the simulation study . . . . . . . . . . . . . . . . . 735.1 Bias for each of the approaches to handle effect modifier forsimulation scenario A : additive effects for subgroup labelMale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.2 Bias for each of the approaches to handle effect modifier forsimulation scenario A : additive effects for subgroup labelfemale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.3 Mean squared error for each of the approaches to handle ef-fect modifier for simulation scenario A : additive effects forsubgroup label Male . . . . . . . . . . . . . . . . . . . . . . . 885.4 Mean squared error for each of the approaches to handle ef-fect modifier for simulation scenario A : additive effects forsubgroup label female . . . . . . . . . . . . . . . . . . . . . . 885.5 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Male for simulation scenarioA : additive effects . . . . . . . . . . . . . . . . . . . . . . . . 895.6 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Female for simulation sce-nario A : additive effects . . . . . . . . . . . . . . . . . . . . 895.7 Bias for each of the approaches to handle effect modifier forsimulation scenario B : mild non-linearity for subgroup labelMale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.8 Bias for each of the approaches to handle effect modifier forsimulation scenario B : mild non-linearity for subgroup labelfemale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.9 Mean squared error for each of the approaches to handle ef-fect modifier for simulation scenario B : mild non-linearity forsubgroup label Male . . . . . . . . . . . . . . . . . . . . . . . 935.10 Mean squared error for each of the approaches to handle ef-fect modifier for simulation scenario B : mild non-linearity forsubgroup label female . . . . . . . . . . . . . . . . . . . . . . 935.11 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Male for simulation scenarioB : mild non-linearity . . . . . . . . . . . . . . . . . . . . . . 945.12 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Female for simulation sce-nario B : mild non-linearity . . . . . . . . . . . . . . . . . . . 945.13 Bias for each of the approaches to handle effect modifier forsimulation scenario C : moderate non-linearity for subgrouplabel Male . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.14 Bias for each of the approaches to handle effect modifier forsimulation scenario C : moderate non-linearity for subgrouplabel female . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.15 Mean squared error for each of the approaches to handle effectmodifier for simulation scenario C : moderate non-linearity forsubgroup label Male . . . . . . . . . . . . . . . . . . . . . . . 985.16 Mean squared error for each of the approaches to handle effectmodifier for simulation scenario C : moderate non-linearity forsubgroup label female . . . . . . . . . . . . . . . . . . . . . . 985.17 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Male for simulation scenarioC : moderate non-linearity . . . . . . . . . . . . . . . . . . . . 995.18 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Female for simulation sce-nario C : moderate non-linearity . . . . . . . . . . . . . . . . 995.19 Bias for each of the approaches to handle effect modifier forsimulation scenario D : mild non-additivity for subgroup labelMale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.20 Bias for each of the approaches to handle effect modifier forsimulation scenario D : mild non-additivity for subgroup labelfemale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.21 Mean squared error for each of the approaches to handle effectmodifier for simulation scenario D : mild non-additivity forsubgroup label Male . . . . . . . . . . . . . . . . . . . . . . . 1035.22 Mean squared error for each of the approaches to handle effectmodifier for simulation scenario D : mild non-additivity forsubgroup label female . . . . . . . . . . . . . . . . . . . . . . 1035.23 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Male for simulation scenarioD : mild non-additivity . . . . . . . . . . . . . . . . . . . . . 1045.24 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Female for simulation sce-nario D : mild non-additivity . . . . . . . . . . . . . . . . . . 1045.25 Bias for each of the approaches to handle effect modifier forsimulation scenario E : mild non-additivity and non-linearityfor subgroup label Male . . . . . . . . . . . . . . . . . . . . . 1075.26 Bias for each of the approaches to handle effect modifier forsimulation scenario E : mild non-additivity and non-linearityfor subgroup label female . . . . . . . . . . . . . . . . . . . . 1075.27 Mean squared error for each of the approaches to handle effectmodifier for simulation scenario E : mild non-additivity andnon-linearity for subgroup label Male . . . . . . . . . . . . . 1085.28 Mean squared error for each of the approaches to handle effectmodifier for simulation scenario E : mild non-additivity andnon-linearity for subgroup label female . . . . . . . . . . . . 1085.29 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Male for simulation scenarioE : mild non-additivity and non-linearity . . . . . . . . . . . 1095.30 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Female for simulation sce-nario E : mild non-additivity and non-linearity . . . . . . . . 1095.31 Bias for each of the approaches to handle effect modifier forsimulation scenario F: moderate non-additivity for subgrouplabel Male . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.32 Bias for each of the approaches to handle effect modifier forsimulation scenario F: moderate non-additivity for subgrouplabel female . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.33 Mean squared error for each of the approaches to handle effectmodifier for simulation scenario F: moderate non-additivityfor subgroup label Male . . . . . . . . . . . . . . . . . . . . . 1135.34 Mean squared error for each of the approaches to handle effectmodifier for simulation scenario F: moderate non-additivityfor subgroup label female . . . . . . . . . . . . . . . . . . . . 1135.35 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Male for simulation scenarioF: moderate non-additivity . . . . . . . . . . . . . . . . . . . 1145.36 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Female for simulation sce-nario F: moderate non-additivity . . . . . . . . . . . . . . . . 1145.37 Bias for each of the approaches to handle effect modifierfor simulation scenario G: moderate non-additivity and non-linearity for subgroup label Male . . . . . . . . . . . . . . . . 1175.38 Bias for each of the approaches to handle effect modifierfor simulation scenario G: moderate non-additivity and non-linearity for subgroup label female . . . . . . . . . . . . . . . 1175.39 Mean squared error for each of the approaches to handle effectmodifier for simulation scenario G: moderate non-additivityand non-linearity for subgroup label Male . . . . . . . . . . . 1185.40 Mean squared error for each of the approaches to handle ef-fect modifier for simulation scenario A : additive effects forsubgroup label female . . . . . . . . . . . . . . . . . . . . . . 1185.41 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Male for simulation scenarioG: moderate non-additivity and non-linearity . . . . . . . . . 1195.42 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Female for simulation sce-nario G: moderate non-additivity and non-linearity . . . . . 1195.43 Bias for each of the approaches to handle effect modifier forsimulation scenario A1 : additive effects for subgroup labelMale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235.44 Bias for each of the approaches to handle effect modifier forsimulation scenario A1 : additive effects for subgroup labelfemale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235.45 Mean squared error for each of the approaches to handle ef-fect modifier for simulation scenario A1 : additive effects forsubgroup label Male . . . . . . . . . . . . . . . . . . . . . . . 1245.46 Mean squared error for each of the approaches to handle ef-fect modifier for simulation scenario A1 : additive effects forsubgroup label female . . . . . . . . . . . . . . . . . . . . . . 1245.47 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Male for simulation scenarioA1 : additive effects . . . . . . . . . . . . . . . . . . . . . . . 1255.48 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Female for simulation sce-nario A1 : additive effects . . . . . . . . . . . . . . . . . . . . 1255.49 Bias for each of the approaches to handle effect modifier forsimulation scenario B1 : mild non-linearity for subgroup labelMale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.50 Bias for each of the approaches to handle effect modifier forsimulation scenario B1 : mild non-linearity for subgroup labelfemale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.51 Mean squared error for each of the approaches to handle effectmodifier for simulation scenario B1 : mild non-linearity forsubgroup label Male . . . . . . . . . . . . . . . . . . . . . . . 1295.52 Mean squared error for each of the approaches to handle effectmodifier for simulation scenario B1 : mild non-linearity forsubgroup label female . . . . . . . . . . . . . . . . . . . . . . 1295.53 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Male for simulation scenarioB1 : mild non-linearity . . . . . . . . . . . . . . . . . . . . . 1305.54 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Female for simulation sce-nario B1 : mild non-linearity . . . . . . . . . . . . . . . . . . 1305.55 Bias for each of the approaches to handle effect modifier forsimulation scenario C1 : moderate non-linearity for subgrouplabel Male . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.56 Bias for each of the approaches to handle effect modifier forsimulation scenario C1 : moderate non-linearity for subgrouplabel female . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.57 Mean squared error for each of the approaches to handle effectmodifier for simulation scenario C1 : moderate non-linearityfor subgroup label Male . . . . . . . . . . . . . . . . . . . . . 1345.58 Mean squared error for each of the approaches to handle effectmodifier for simulation scenario C1 : moderate non-linearityfor subgroup label female . . . . . . . . . . . . . . . . . . . . 1345.59 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Male for simulation scenarioC1 : moderate non-linearity . . . . . . . . . . . . . . . . . . 1355.60 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Female for simulation sce-nario C1 : moderate non-linearity . . . . . . . . . . . . . . . 1355.61 Bias for each of the approaches to handle effect modifier forsimulation scenario D1 : mild non-additivity for subgrouplabel Male . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.62 Bias for each of the approaches to handle effect modifier forsimulation scenario D1 : mild non-additivity for subgrouplabel female . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.63 Mean squared error for each of the approaches to handle effectmodifier for simulation scenario D1 : mild non-additivity forsubgroup label Male . . . . . . . . . . . . . . . . . . . . . . . 1395.64 Mean squared error for each of the approaches to handle effectmodifier for simulation scenario D1 : mild non-additivity forsubgroup label female . . . . . . . . . . . . . . . . . . . . . . 1395.65 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Male for simulation scenarioD1 : mild non-additivity . . . . . . . . . . . . . . . . . . . . 1405.66 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Female for simulation sce-nario D1 : mild non-additivity . . . . . . . . . . . . . . . . . 1405.67 Bias for each of the approaches to handle effect modifierfor simulation scenario E1 : mild non-additivity and non-linearity for subgroup label Male . . . . . . . . . . . . . . . . 1435.68 Bias for each of the approaches to handle effect modifierfor simulation scenario E1 : mild non-additivity and non-linearity for subgroup label female . . . . . . . . . . . . . . . 1435.69 Mean squared error for each of the approaches to handle effectmodifier for simulation scenario E1 : mild non-additivity andnon-linearity for subgroup label Male . . . . . . . . . . . . . 1445.70 Mean squared error for each of the approaches to handle effectmodifier for simulation scenario E1 : mild non-additivity andnon-linearity for subgroup label female . . . . . . . . . . . . 1445.71 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Male for simulation scenarioE1 : mild non-additivity and non-linearity . . . . . . . . . . 1455.72 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Female for simulation sce-nario E1 : mild non-additivity and non-linearity . . . . . . . 1455.73 Bias for each of the approaches to handle effect modifier forsimulation scenario F1: moderate non-additivity for subgrouplabel Male . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1485.74 Bias for each of the approaches to handle effect modifier forsimulation scenario F1: moderate non-additivity for subgrouplabel female . . . . . . . . . . . . . . . . . . . . . . . . . . . 1485.75 Mean squared error for each of the approaches to handle effectmodifier for simulation scenario F1: moderate non-additivityfor subgroup label Male . . . . . . . . . . . . . . . . . . . . . 1495.76 Mean squared error for each of the approaches to handle effectmodifier for simulation scenario F1: moderate non-additivityfor subgroup label female . . . . . . . . . . . . . . . . . . . . 1495.77 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Male for simulation scenarioF1: moderate non-additivity . . . . . . . . . . . . . . . . . . 1505.78 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Female for simulation sce-nario F1: moderate non-additivity . . . . . . . . . . . . . . . 1505.79 Bias for each of the approaches to handle effect modifier forsimulation scenario G1: moderate non-additivity and non-linearity for subgroup label Male . . . . . . . . . . . . . . . . 1535.80 Bias for each of the approaches to handle effect modifier forsimulation scenario G1: moderate non-additivity and non-linearity for subgroup label female . . . . . . . . . . . . . . . 1535.81 Mean squared error for each of the approaches to handle effectmodifier for simulation scenario G1: moderate non-additivityand non-linearity for subgroup label Male . . . . . . . . . . . 1545.82 Mean squared error for each of the approaches to handle effectmodifier for simulation scenario G1: moderate non-additivityand non-linearity for subgroup label female . . . . . . . . . . 1545.83 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Male for simulation scenarioG1: moderate non-additivity and non-linearity . . . . . . . . 1555.84 Coverage probability for each of the approaches to handleeffect modifier for subgroup label Female for simulation sce-nario G1: moderate non-additivity and non-linearity . . . . . 1555.85 Bias for each of the approaches to handle effect modifier forplasmode simulation (crude) for subgroup label Male . . . . 1585.86 Bias for each of the approaches to handle effect modifier forplasmode simulation (crude) for subgroup label Female . . . 1585.87 Mean squared error for each of the approaches to handle effectmodifier for plasmode simulation (crude) for subgroup labelMale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1595.88 Mean squared error for each of the approaches to handle effectmodifier for plasmode simulation (crude) for subgroup labelFemale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1595.89 Coverage probability for each of the approaches to handleeffect modifier for plasmode simulation (crude) for subgrouplabel Male . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1605.90 Coverage probability for each of the approaches to handleeffect modifier for plasmode simulation (crude) for subgrouplabel Female . . . . . . . . . . . . . . . . . . . . . . . . . . . 1605.91 Bias for each of the approaches to handle effect modifier forplasmode simulation (adjusted) for subgroup label Male . . . 1635.92 Bias for each of the approaches to handle effect modifier forplasmode simulation (adjusted) for subgroup label Female . 1635.93 Mean squared error for each of the approaches to handle ef-fect modifier for plasmode simulation (adjusted) for subgrouplabel Male . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1645.94 Mean squared error for each of the approaches to handle ef-fect modifier for plasmode simulation (adjusted) for subgrouplabel Female . . . . . . . . . . . . . . . . . . . . . . . . . . . 1645.95 Coverage probability for each of the approaches to handle ef-fect modifier for plasmode simulation (adjusted) for subgrouplabel Male . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1655.96 Coverage probability for each of the approaches to handle ef-fect modifier for plasmode simulation (adjusted) for subgrouplabel Female . . . . . . . . . . . . . . . . . . . . . . . . . . . 1655.97 Bias for each of the approaches to handle effect modifier forsimulation scenario A : additive effects for subgroup labelMale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1735.98 Bias for each of the approaches to handle effect modifier forsimulation scenario A : additive effects for subgroup labelfemale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1735.99 Mean squared error for each of the approaches to handle ef-fect modifier for simulation scenario A : additive effects forsubgroup label Male . . . . . . . . . . . . . . . . . . . . . . . 1745.100Mean squared error for each of the approaches to handle ef-fect modifier for simulation scenario A : additive effects forsubgroup label female . . . . . . . . . . . . . . . . . . . . . . 1745.101Bias for each of the approaches to handle effect modifierfor simulation scenario G: moderate non-additivity and non-linearity for subgroup label Male . . . . . . . . . . . . . . . . 1775.102Bias for each of the approaches to handle effect modifierfor simulation scenario G: moderate non-additivity and non-linearity for subgroup label female . . . . . . . . . . . . . . . 1775.103Mean squared error for each of the approaches to handle effectmodifier for simulation scenario G: moderate non-additivityand non-linearity for subgroup label Male . . . . . . . . . . . 1785.104Mean squared error for each of the approaches to handle effectmodifier for simulation scenario G: moderate non-additivityand non-linearity for subgroup label female . . . . . . . . . . 1785.105Bias for each of the approaches to handle effect modifier forsimulation scenario A1 : additive effects for subgroup labelMale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1815.106Bias for each of the approaches to handle effect modifier forsimulation scenario A1 : additive effects for subgroup labelfemale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1815.107Mean squared error for each of the approaches to handle ef-fect modifier for simulation scenario A1 : additive effects forsubgroup label Male . . . . . . . . . . . . . . . . . . . . . . . 1825.108Mean squared error for each of the approaches to handle ef-fect modifier for simulation scenario A1 : additive effects forsubgroup label female . . . . . . . . . . . . . . . . . . . . . . 1825.109Bias for each of the approaches to handle effect modifier forsimulation scenario G1: moderate non-additivity and non-linearity for subgroup label Male . . . . . . . . . . . . . . . . 1855.110Bias for each of the approaches to handle effect modifier forsimulation scenario G1: moderate non-additivity and non-linearity for subgroup label female . . . . . . . . . . . . . . . 1855.111Mean squared error for each of the approaches to handle effectmodifier for simulation scenario G1: moderate non-additivityand non-linearity for subgroup label Male . . . . . . . . . . . 1865.112Mean squared error for each of the approaches to handle effectmodifier for simulation scenario G1: moderate non-additivityand non-linearity for subgroup label female . . . . . . . . . . 186AcknowledgementsIt is my pleasure to express my deepest appreciation to my supervisor Dr.Mohammad Ehsanul Karim who supported me for the development of thework. At every step of analysis, write up he helped me to accomplish thiswork. Besides this, he provided me financial assistance so that, I couldpursue my study and research work smoothly. In any case of emergency,problems I got prompt support from him. Without his extra care on me Icould not complete this work.I also would like to thank the funding agency Dr. Karim’s NaturalSciences and Engineering Research Council of Canada (NSERC) DiscoveryGrant and NSERC Accelerator Supplement for funding this project.I also would like to appreciate other committee members of my thesis Dr.Hubert Wong and Dr. Wei Zhang for their valuable input to the developmentof this work. Dr. Hubert provided comprehensive idea in the modificationof analysis of the simulation study.I also would like to thank Dr. Jason Sutherland for examining my thesisand providing valuable suggestion for further improvement. I would likethank Dr. Anne Gadermann for chairing my thesis defense.I also would like to appreciate all the people from SPPH, specially Emilyfor her continuous support. I also would like to appreciate other collaboratorof my work Md. Belal Hossain and Hanna Frank for their support at anydifficulty.Finally, I want to thank my family members for their encouragementand support of doing this work smoothly.xxiiiChapter 1Introduction1.1 Background of the effect modifierOne of the primary goals in epidemiological research is estimation of thetreatment effect. This refers to any changes in the value of the outcomevariable due to an intervention. For example, a real-world population understudy invariably has heterogeneous characteristics, such as age, sex, diseaseseverity, comorbidity status, and geographical area of residence. These vari-ables modify the relationship between treatment and outcome and are thusreferred to as effect modifiers.Researchers have traditionally estimated the treatment effect in the pres-ence of effect modifiers, both in randomized controlled trials (RCT) andobservational studies, by assuming similarity in treatment effect across thestudy participants’ characteristics. Although this underlying assumptionworks well for some treatments, in others treatment effects may vary consid-erably within subgroups. Sometimes, due to heterogeneity among patients,the underlying treatment may be beneficial to some patients, but harmfulfor others [1]. Estimation of treatment effects at the subgroup level is thusgaining importance in epidemiological research [2, 3]. Identifying groupsof individuals for whom treatment is useful versus harmful has motivatedresearchers to explore patient heterogeneity in the clinical research contextand to strive for a refined understanding of effect modifiers in pursuit ofpersonalized treatments [4, 5].Depending on the nature of modification of treatment effect by the thirdvariable, two types of effect modifiers are available, such as,Qualitative effect modifier:When in the presence of an effect modifier, the nature or direction ofthe treatment’s effect varies at different levels of effect modifiers, then it iscalled a qualitative effect modifier [6]. For example, in the study [7], if therisk of death was higher for older patients (odds ratio (OR) of 1.7) versus1lower for younger patients (OR 0.6), then age was determined to have actedas a qualitative effect modifier in the study.Quantitative effect modifier:When the nature or direction of the association between treatment andoutcome does not change but differs in the strength depending on the effectmodifier’s levels [6], then that effect modifier is the quantitative effect modi-fier. For example, in the study of [7], the risk of death was higher for women;that is, the odds ratio (OR) was 1.62 for females. In contrast, for males, itwas 1.23, then the gender was acted as a quantitative effect modifier in thatstudy.In the [4] the authors pointed out the importance of exploring the causal-ity in the presence effect modifier in an epidemiological study. Through ran-domized controlled trial (RCT), drawing causality is possible because, underthis trial, treatment group and control has only difference in treatment as-signment having identical covariates structure. In this case, by comparingthe treatment and control group, one can quickly get the treatment’s ca-sual effect. In RCTs and observational studies, two common approaches arefollowed in the data analysis phase to address effect modifiers:• Subgroup analysis approach:Under the subgroup analysis approach, depending on the effect mod-ifier’s levels, the dataset is stratified and the treatment effect is thenestimated within each stratum. Finally, the stratum-specific treat-ment effects are pooled to get the pooled treatment effect. However,two important caveats should be noted. First, when many subgroupsare present, the stratum-specific analysis of data is not feasible. Toovercome this problem of subgroup-analysis approach, researchers havecome to employ the regression method approach.• Regression analysis approach:In the regression analysis approach, an interaction term between treat-ment and effect modifier is added in the model as a way to addressthe effect modifier. Compared to subgroup analysis, the regressionapproach works well in the case of large numbers of effect modifiers.Moreover, one can adjust for confounding bias, in addition to han-dling effect modifiers using this approach. However, when there aremany effect modifiers in a study, there is a risk of running into modelconvergence issues due to having a smaller sample size in any level ofsubgroup variables.Due to limited resources, high cost, and ethical considerations, it is notalways feasible to conduct an RCT [8]. To overcome the limitations of RCTs,researchers may use observational data. Due to improvements in the stor-age of patients’ electric health records, many observational databases havebeen developed. For this reason, the utilization of observational data todraw causality is gaining both in popularity and consequent scrutiny amongresearchers. One of the significant problems in observational studies is thatof having confounding bias in the treatment effect estimation. To removethe confounding bias, stratification and regression methods are applied inthe observational study. However, the Propensity score (PS) method devel-oped by [9] has gained popularity [10–15] over the past two decades to infercausality of treatment effect from observational data when certain assump-tions are met. Confounding adjustment like the regression and propensityscore methods provide the opportunity to evaluate covariate balance diag-nostics. Despite the use of PS methods having gained in popularity over thepast two decades, a paucity of literature exists to address the role of effectmodifiers within the PS context. Before discussing the different techniquesto handle effect modifiers within the PS context, we present an overview ofthe PS method in the next section.1.2 Overview of Propensity scoreTo mimic some of the RCT study characteristics, such as treatment andcontrol groups exhibiting covariate balance, the PS method is widely usedin observational studies. Before discussing the propensity score method,we have illustrated the idea of a potential framework and the assumptionsneeded to draw causality from an observational study. The concept of thepotential framework will help in understanding the role of PS in an obser-vational study to draw the causal inference.1.2.1 Potential outcomes frameworkIn epidemiological research, causality means the ability to estimate or pre-dict the effect of treatment on individuals or groups of people [16]. Hume[17] formed the sufficient and necessary condition for causality as, “Cause isan event followed by another (effect),” and “Without the first event (cause),the second (effect) would never happen.” These definitions of causality wereformalized in the statistical context by Splawa-Neyman, Dabrowska, andSpeed [18] and Rubin [19] with introducing the term “potential outcome.”Let us consider, the treatment indicator A which indicates treatmentif A = 1 and control if A = 0. Y1i is the potential outcome value for anyindividual i, if the individual i received the treatment and Y0i is the potentialoutcome if the same individual did not receive the treatment. Under theNeyman-Rubin causal model, the comparison between these two quantitiesrepresented by the following formula:TEi = Y1i − Y0i (1.1)For any individual i is known as the causal effect of treatment “A.” Depend-ing on the research objective, this comparison can be in various forms suchas difference, ratio, or average. For example, the mean difference of poten-tial outcomes, E(YA=1) and E(YA=0), is known as the causal risk difference.If the difference deviates from zero, then there is a causal effect of treatmentA.Although in the Neyman-Rubin causal model assumes that each indi-vidual i has two potential outcomes Yia, in reality, each individual can beobserved in only one state at any time point. The other potential outcomesthat can not be realized for any individual is defined as “counterfactuals.”In the Table 1.1 the potential outcome framework has been illustrated. Eachrow in Table 1.1 corresponds to a potential outcome for an individual i inboth treated and untreated contexts and the difference between these is thecausal effect. The marginal outcome is the average causal effect of “A”calculated without conditioning on covariates.Table 1.1: Potential outcome framework to define treatment effectIndividuals Treatment (A = 0 ) Treatment (A = 1) Outcome (YA=0) Outcome (YA=1) Causal effect: (YA=1-YA=0)1 0 1 YA=0,1 YA=1,1 YA=1,1 - YA=0,12 0 1 YA=0,2 YA=1,2 YA=1,2 - YA=0,23 0 1 YA=0,3 YA=1,3 YA=1,3 - YA=0,34 0 1 YA=0,4 YA=1,4 YA=1,4 - YA=0,45 0 1 YA=0,5 YA=1,5 YA=1,5 - YA=0,56 0 1 YA=0,6 YA=1,6 YA=1,6 - YA=0,67 0 1 YA=0,7 YA=1,7 YA=1,7 - YA=0,78 0 1 YA=0,8 YA=1,8 YA=1,8 - YA=0,89 0 1 YA=0,9 YA=1,9 YA=1,9 - YA=0,910 0 1 YA=0,10 YA=1,10 YA=1,10 - YA=0,10Average treatment effect E(YA=0) E(YA=1) E(YA=1) - E(YA=0)Determining causality of treatment in epidemiological research has tra-ditionally been the domain of RCTs that have been held up as the goldstandard method. Through randomization in a study, the bias in covariatelevels between exposed and unexposed to the treatment groups could beeradicated. However, adaptation of randomized trials is not always possiblein epidemiological research. For example, if we want to explore the causaleffect of depression on substance use, we cannot randomly assign people todepression. If we had an observational study with groups of depressed ver-sus non-depressed people with corresponding data on their substance usehistory, then it would be possible to explore the causal effect of depression.Moreover, to get a causal interpretation from the observational data, weneed some assumptions which will mimic the counterfactual scenario like arandomized trial.To get a causal interpretation from observational data, Rubin [19] furtherextended the potential outcome framework [19–21] by considering that, forindividual i and treatment indicator Ai, the outcome variable Yi would be,Yi = AiY1i + (1−Ai)Y0i (1.2)The above equation described the situation that, for any individual i thevalue of Yi is possible to observe in only one state– either the state of havingbeen exposed to the treatment or that of not not having been exposed tothe treatment. Therefore, individual TEi cannot be identified if they hadmissing outcomes in any of the states. For example, consider that we have agroup of people and they were either exposed to the treatment or not Table1.2.Table 1.2: Potential outcome framework to define treatment effect based onobserved outcomeIndividuals Treatment (A = 0 ) Treatment (A = 1) Outcome (YA=0) Outcome (YA=1) Causal effect: (YA=1-YA=0)1 0 YA=0,12 0 YA=0,23 0 YA=0,34 0 YA=0,45 0 YA=0,56 1 YA=1,67 1 YA=1,78 1 YA=1,89 1 YA=1,910 1 YA=1,10Average treatment effect E(Y |A = 0) E(Y |A = 1) E(Y |A = 1)− E(Y |A = 0)So, for each individual i the outcome cannot be observed at two statessimultaneously. Instead, to get the treatment effect, we can use averagecausal effect E(YA = 1)− E(YA = 0) here.The average treatment effect may also be estimated for a continuousoutcome Y if the individuals from two groups have similar characteristics.For example, if participants have similar age, sex, or other relevant variablesfor the treatment indicator A = 1 and A = 0 as,Average Treatment Effect (ATE) [22]=E[Yi1 − Yi0] (1.3)The ATE is a measurement of the difference in mean outcomes betweenunits ascribed to the treatment versus control groups. Besides ATE, some-times we need to calculate the effect of treatments on individuals being inthe treated group, which is defined as,Average treatment effect on the treated (ATT) [22] =E[Yi1 − Yi0|A = 1] (1.4)Similarly, we can calculate the causal risk ratio, say, for binary outcomeY as P (YA=1 = 1)/P (YA=0 = 1) or by using the conditional causal risk ratioP (Y = 1|A = 1)/P (Y = 1|A = 0). Here, the concept of condition arisedfrom the fact that, in reality some individuals will get the treatment whileothers will not in a population. In an observational study, to render thetreated and controlled groups comparable, some assumptions are needed tobe fulfilled which will be discussed in the next section.1.2.2 AssumptionsAs unobserved counterfactuals are needed to be estimated to get the causaleffect, the following assumptions should be met.• Exchangeability: This assumption states that, if we change the treat-ment status of the people under study (for example, change the non-exposed group to the exposed group) then the magnitude or directionof the treatment effect would not change, that is YiA ⊥ Ai [23].• Positivity: This assumption means that, there should be common sup-port or overlap in covariate distribution and comparable treatment andcontrol groups. That is, for all xX, 0 < P (Ai = 1|X) < 1. At leastone individual will be each of the treatment and non-exposure to thetreatment group for each of the covariate strata [23].• SUTVA (Stable Unit Treatment Value Assumption): This assumptionmeans that the treatment choice for one individual does not changethe outcome of another individual in the population [24].• Consistency: The consistency assumption states that the treatmentthat is assigned to one individual will not have a differential versionwhen applied to another individual in the population [25]. That is,all the subjects included in the treatment will get the similar ver-sion/doses of treatment.With the assumptions mentioned above, observational data can be con-sidered a conditional realization of randomized experiments. Here, condi-tional means the conditional on the measured covariates. A feasible solutionis to condition on the covariates X to get the covariate balance between thetreated and untreated group. But in the observational study, sometimes thisapproach does not work when X has a large dimension or X is a continuousvariable. A more popular method to overcome this problem of dimensional-ity of X is to use the Propensity Score (PS) method developed by Rosenbaumand Rubin Rosenbaum and Rubin [9]. In contrast to conditioning on multi-ple covariates or large dimensional covariates, PS combines all the covariatesinto a single number and make easier to get covariate balance. In the nextsection, we will discuss PS analysis and its implementation procedure.1.2.3 Propensity score analysisPropensity score is defined as the conditional probability of assigning anindividual to a particular treatment given a set of covariates, assuming nounmeasured covariates. Let us consider we have a set of covariates X andtreatment indicator A (binary). Then PS is defined as,ps(X) = P (A = 1 | X) (1.5)The value of ps is between 0 and 1. Those individuals who have alow probability of receiving the treatment, have a ps close to 0 whereas,individuals who have a higher chance to get treatment have a ps closer to1. In RCTs the PS is known because due to randomness in the treatmentassignment they have known probability of getting treatment. For example,if we assign the treatment to a randomly chosen subject based on a coin toss,we can assign the treatment with probability = 0.5 if the result is heads.In observational studies we do not have a known PS, because treatment isnot assigned randomly. We need to estimate the PS in that case by usingstatistical regression methods or machine learning methods.One of the benefits of PS is that, it combines all the covariate informa-tion into a single number and make it easier to check the covariate balanceachieved in treated and control group compared to using a large set of co-variates. Rosenbaum and Rubin [9], under the strong ignorability assump-tion, proved that if potential outcomes YA=0 and YA=1 are independent oftreatment assignment given observed covariates X, then they are also inde-pendent of treatment assignment given the propensity score, and treatmentassignment is independent of covariates given the propensity score.Some of the benefits of the use of PS are as follows:• In observational data, we can a use propensity score to infer counter-factuals to draw causality, when randomized experiment is not possible[15].• In observational data, we need to deal with a large set of covariates,which increases the dimensionality problem to get the conditional ef-fect of treatment. Using PS, in this case, reduces the number of co-variates needed to control for so that dimensionality problems can besolved and increase the chances of getting balance between treated anduntreated individuals.• To calculate a propensity score, the outcome model and PS model areseparately fitted. In this way, adjustments are performed indepen-dently of the outcome.1.2.4 Estimation of propensity scoreA commonly used method to estimate the PS is to apply the logistic regres-sion method [15] as this a well-understood technique and its implementationin standard software packages are available. When we have a binary out-come we can use the logistic regression to determine PS. However, in thelogistic regression approach, the assumptions related to variable selection,the functional form of the outcome and exposure, distribution of covariates,and interactions are required. Failing to meet these assumptions leads tobiased effect estimates [26, 27].Alternatively, researchers nowadays use various machine learning meth-ods to estimate the PS [26–29]. Some standard machine learning methodsare classification and regression trees (CARTs), neural networks, randomforests, Generalized Boosted modeling [27, 30, 31]. Setoguchi [30] conducteda simulation study to explore the comparison among various machine learn-ing algorithms with logistic regression in terms of PS matching for severalscenarios. They found that compared to traditional logistic regression inmany scenarios, machine learning algorithms (neural networks) performedwell. Machine learning methods can also handle high-dimensional data.After getting the PS using statistical methods (e.g., regression, machinelearning methods), the estimated PS needs to be evaluated whether it ad-equately achieved the covariate balance. Ho, Imai, King, and Stuart [32]suggested that, after estimating propensity and implementing the estimatedPS, it is important to evaluate the balancing property of the PS. The au-thors emphasized that, instead of relying on one measure to assess balance,it is good to check measures of balance. In the next section, we will reviewtopics related to the the implementation of PS.1.2.5 Propensity score implementationAfter estimating the PS score, we can implement the PS using matching,stratification, weighting, and covariate adjustment [33–37].1.2.6 Propensity score matchingIn PS matching, after calculating PS based on this each treated subject ismatched to one or more controls to estimate ATT [22]. If the parameterof interest is ATE, weighting (discussed in section 1.2.8) would be useful.PS matching is the most used PS implementation method [38–40]. Matchedgroups of treated and untreated subjects have similar PS values [9]. Todetermine the similarity or closeness for PS matching, several distance mea-sures are considered, such as, for covariate X, variance-covariance matrixamong X is S, the distance measure Dij between individual i and j is,• For Exact matching:Dij ={0 if Xi = Xj∞ if Xi 6= Xj , (1.6)where Xi and Xj are the value of a covariate for ith and jth individual.• Matching using Mahalanobis distance:Dij = (Xi −Xj)′S−1(Xi −Xj) (1.7)where S is the covariance matrix.• Distance based on difference between individual propensity score:Dij = |psi − psj | (1.8)where psi and psj is the propensity score for individual i and j.• Generalization of Mahalanobis distance:Dij =√(Xi −Xj)(S−1/2)′WS−1/2(Xi −Xj) (1.9)Where the weight matrix W is a k× k positive definite matrix and S1/2is a Cholesky decomposition of S, the variance-covariance matrix of X .To do PS matching, generally two procedures such as, matching ratioand matching algorithm, are followed. Common matching ratio includesone-to-one, one-to-many, variable-ratio, or caliper [41]. On the other hand,matching algorithms include greedy matching, optimal matching [42], andgenetic matching [43, 44]. These matching algorithm can be done eitherwith or without replacement, full matching, exact or caliper matching [41].Among various matching algorithms, greedy matching is commonly used,including nearest neighborhood matching (NN) and caliper matching.• NN matching can be implemented either with replacement or withoutreplacement way. When the NN matching is done in such a way that,a subject from the treated group can be matched to the closest subjectfrom the control group given that the control subject can be matchedmore than once, then this is NN matching with replacement. Onthe other hand, if under NN matching each match unit consists ofhaving exactly one subject from treatment group and another one fromcontrol group then this is called NN matching without replacement.Under the NN without replacement paradigm, once a control subjectis matched with a treated subject then that control subject will notbe considered again for further matching. To measure the closeness ofthe treated and control subjects, the Mahalanobis distance metric canbe used.Caliper matching considered a maximum distance usually instandard deviation units, within which if subjects falls are acceptableto be considered as matched sample. For continuous outcome [45]recommended the value of caliper as 0.2 of the standard deviation,whereas 0.1 is also a recommended value of caliper.• When matching an algorithm consider that, for every treated subjectthere is at least one matched control subject then this is called fullmatching. To do this matching researchers considered a 1:K matchingscheme [41].• Sekhon [44] proposed the genetic matching algorithm. Under this al-gorithm, generalization of Mahalanobis distance is used to measuredistance. This matching is an iterative procedure of checking covari-ate balance that optimizes a loss function. The loss functions includepaired t-tests, Kolmogorov-Smirnov tests, or researchers defined lossfunction.PS matching is a widely used implementation method of PS, but somematching methods discards the data (called the unmatched data) aftermatching. By using matching approach we can get the balanced treatedand untreated groups in terms of covariate distribution [37]. This makes itpossible to get treatment effect from the observational data.1.2.7 Propensity score stratificationIn the PS stratification process after estimating the PS, individuals are parti-tioned according to their estimated PS into mutually exclusive strata. Afterestimating PS, individuals involved in the study are ranked according totheir PS. Then subjects are stratified usually into 5 or 10 groups. Rosen-baum and Rubin [46] demonstrated that, when estimating linear treatment,stratifying on the quintiles of PS eradicate the 90% bias due to measuredconfounders.Within each stratum the treated and control subjects will have similardistribution of covariates and comparison between these two groups of sub-jects can be done [9]. After getting the stratum specific treatment effectATEs can be combined to get the weighted treatment effect. The stratumspecific treatment effect is defined as,ATEs = E(Ysi1 − Ysi0), (1.10)and combined weighted treatment effect is defined as,ATE =∑sNsNATEs, s = 1, 2, ..S (1.11)To calculate ATT a modification of ATE is needed in calculation ofstratum-specific weights of the treated subjects [22]. After that, we needto calculate the pooled estimate of variance by using the stratum-specifictreatment effect [46].Using stratification approach we can use the whole data and also we cancheck covariate balance within each stratum. However, there is a chance ofnot getting enough overlap and possibility to have smaller sample in somestrata [37].1.2.8 Propensity score weightingUnder propensity score matching and stratification approach we have torely on subjects having similar PS values. Moreover, in some PS matchingmethods there is a chance of discarding a large number of unmatched samplesfrom the data. On the other hand, PS weighting is another way of PSimplementation where no subjects are needed to be discarded.Propensity score weighting can be used is two ways such as weight-ing by odds [47] and another more commonly used approach is inverseprobability-of-treatment weighting [48]. Under the inverse probability-of-treatment weighting (IPTW) to get ATE, subjects from the treated groupwill receive weight by the inverse of the PS and control subjects will receiveinverse of one minus the PS. That is the weight wi for individual i will beas follows for individual propensity score psi and treatment Ai,wi =Aipsi+1−Ai1− psi , (1.12)and theATE =1NN∑i=1AiYipsi− 1NN∑i=1(1−Ai)Yi1− psi . (1.13)For ATT, the weight wi and ATT will be,wi = Ai +(1−Ai)psi1− psi , (1.14)and theATT =1NN∑i=1AiYi − 1NN∑i=1(1−Ai)psiYi1− psi . (1.15)The PS weighting approach uses all the individuals in the data so thatgeneralizability of the study findings can be improved. However, in somecases, there is a possibility to have extreme weights; in this case, truncationor stabilized weights may be used.1.2.9 Propensity score used as covariatePropensity score can be used as a covariate in the outcome regression be-sides the treatment variable [9]. Using PS as a covariate in the model, wecan get a parsimonious outcome model, and another advantage is that sep-aration of design stage and analysis stage is maintained. To apply the PSimplementation as a covariate, the PS model should have correct specifica-tions [37, 49]. Besides this, additional assumption is needed to implementthis method is that, namely that the relationship between estimated PS andoutcome variable is correctly modeled [15].Using PS as a covariate we can use the entire dataset, however, un-der this approach adjustment happens without considering the similarity ofsubjects from control and treatment, which increases the a chance of havingincomparable treated and untreated subjects [37].1.2.10 Assessment of the covariate balance afterimplementation of Propensity scoreFor each of the above-mentioned implementations of PS, it is necessary toevaluate whether the required covariate balance is achieved or not. If enoughcovariate balance is not achieved, we can re-estimate the PS by updating themodel’s covariates. We can correct the model by adding higher-order termsin the model or generating interactions between covariates and check thecovariate balance until acceptable levels of covariate balance on importantconfounding variables is achieved [46, 50].There are some graphical procedures and quantitative methods to eval-uate the covariate balance. Under graphical approach, commonly quantile-quantile (QQ) plot for continuous covariates and bar plot for categoricalvariables are used. In the QQ plot, the quantiles of PS of treated and un-treated displayed and better covaraite balance will achieve when the pointslie on the 45-degree indicator line. On the other hand, for categorical vari-ables, if overlapping bars are observed in the bar plot, then the covariatebalance is assured.For quantitative assessment of covariate balance the most commonlyused approach is the standardized difference [51] . For each covariate, thedifference between mean or proportion is standardized by the pooled stan-dard deviations. If we have a continuous covariates, the standardized differ-ence can be calculated as,d =x¯A=1 − x¯A=0√s2A=1+s2A=02(1.16)where x¯A=1 and x¯A=0 is the sample mean of covariates in the treatedand controlled groups respectively, s2A=1 and s2A=0 is the sample variance ofthe covariates in the treated and controlled groups respectively.And for binary variates the standardized difference is defined as,d =pˆA=1 − pˆA=0√pˆA=1(1−pˆA=1)+pˆA=0(1−pˆA=0))2(1.17)where pˆA=1 and pˆA=0 is the prevalence of dichotomous variable in thetreated and controlled groups respectively.In literature, various values of the standard difference have been rec-ommended. Austin [45] proposed that a strict standardized value below 0.1indicates covariate balanced between treated and controlled groups, whereasother flexible criteria has been proposed suggesting that the standardizeddifference should be less than 0.25 standard deviation [12, 52]Additional quantitative measures include variance ratio Rubin [53] and,distances between the mean and maximum values in the QQ (quantile-quantile) plots. Variance ratio measures the ratio of variances of treatedand untreated groups proposed by Rubin [53]. The value of closer to 1 ofvariance ratio indicates an acceptable balance, although the variance ratiobetween 0.5 to 2 was suggested as a fair balance [53]. Linden and Samuels[54] suggested variance ratio greater than 2 indicates imbalance. Besidesthese, Ho [32] suggested the difference between the mean and maximumvalues in the empirical QQ plot as an assessment of covariate balance.1.3 Literature review on Propensity scoreapproaches to handle effect modifierIn addition to the subgroup and regression approach, several studies havediscussed various other approaches to handle effect modifiers [55–61] in thecontext of PS methods. In these studies, researchers explored approachesbesides subgroup and regression approach to address the analysis of effectmodifier in different scenarios under various PS implementation methodssuch as matching, stratification, weighting, and covariate adjustment. Letus considered the outcome Y as a continuous variable, treatment indicatorA as binary (for treatment and control the A=1 and A=0 respectively),covariates Ci both continuous and categorical, and the effect modifier isgender G (M=male, F=Female).First study• One of the earliest study regarding effect modification within PS con-text was conducted by Rassen et al.[55]. In this study,• Approaches considered for handling effect modifier: The au-thors discussed two approaches such as use of overall cohort PS andsubgroup-specific PS approach.Under the overall cohort PS approach, at the first stage of propensityscore method that is in the estimation of PS, the subgroup variablewas considered as a covariate along with other covariates in the studyas follows,logit(Pr(A = 1 | C,G)) = β0 +β1C1 +β2C2 + ...+βkCk+βgG (1.18)The authors mentioned that as overall cohort propensity score ap-proach. Then used the PS to match (NN matching) within subgroupvariable.On other hand, under the subgroup-specific PS approach, the authorsat first divided the dataset according to the levels of subgroup vari-able. Then the calculated separate propensity scores similar to thesubgroup analysis that follows, for a set of covariates Ci’s as a mea-sured confounders estimation of PS for males and females separately,logit(Pr(A = 1 | G = M,C)) = α0 + α1C1 + ...+ αkCk (1.19)andlogit(Pr(A = 1 | G = F,C)) = α0 + α1C1 + ...+ αkCk (1.20)• Simulation and data analysis: The authors performed simulationand empirical data analysis. In their simulation study, they consid-ered some scenarios such as, rare outcomes, small subgroups, strongconfounder/treatment interactions and strong confounder/confounderinteractions.• Findings: Based on the simulation and empirical data analysis, theymentioned that, when the size of the subgroup was large (exceeds to1000) with common outcome, then use of overall cohort worked wellto determine inference within subgroup level.• Gap in the study: In reality sometimes getting a large sample sizeat each level of subgroup variable would be difficult and for more thanone effect modifiers this might be more difficult. Moreover, Green etal. [59] pointed out that, if the outcome has substantial differences atdifferent levels of subgroup variable, then getting correct specificationfor overall cohort PS that could be applicable to draw inference withinsubgroup is difficult. The authors in the Rassen et al. [55] did not con-sider the implication of ignoring the subgroup variable during match-ing or exact matching on subgroup variable using overall PS. In themodel performance they evaluated the difference of effect size betweenoverall cohort approach and subgroup specific approach. They did notprovide any performance measures such as bias, mean squared error,coverage probability to evaluate their simulation result. More exten-sive simulation for different scenarios and other performance measureswere needed to validate their claims.Second study• Kreif et al. [56] and Radice et al. [57] similar to the previous studyconsidered overall cohort PS and subgroup specific PS in their study.• Approaches considered for handling effect modifier: In the pre-vious study by Rassen et al. [55] the authors considered the propensityscore matching method, whereas Kreif et al. [56] and Radice et al. [57]considered propensity score matching (one-to-one matching), inverseprobability treatment weighting (IPTW) ang genetic matching (GM).They applied these propensity score methods under the overall co-hort PS approach same as equation (1.18) and subgroup-specific PSapproach same as equations (1.19 and 1.20).• Simulation and data analysis: The authors undertook empiricaland simulation study by varying scenarios. The scenarios they con-sidered were, (i) estimation of subgroup and overall PS and applyPS matching, IPTW and genetic matching; (ii) Adding a non-linearterm in the PS estimation model; (iii) Adding confounders related tothe outcome in addition to the interaction term of the effect modi-fiers. To evaluate simulation performance for different scenarios, theauthors provided covariate balance measures using weighted standard-ized mean differences, bias and root mean squared error at each of thesubgroup level.• Findings of the study: The authors found that, compared to theoverall PS approach the subgroup-specific approach produced lowerbias and, lower mean squared error. They also found that, compared topropensity score matching method propensity score weighting methodproduced lower bias, mean squared error. In case of model misspeci-fication the genetic matching method performed well in terms of pro-ducing lower bias and, mean squared error.• Gap in the study: In their study, the estimated the treatment effectat subgroup level by adding an interaction term in the model. Al-though adding an interaction term in model is one approach to handleeffect modification, generally interaction test has low power to detectvariation present on the basis of subgroup variable level [62]. Besidesthis, statistical interaction test does not represent the true assessmentof biological interaction. Moreover, during simulation in this studythe authors did not explore the performance of their methods in moregeneral scenarios of model specification such as model non-additivity,moderate additivity.Third study• Girman et al. [58] considered overall cohort PS and subgroup specificPS to address effect modifier in their study.• Approaches considered for handling effect modifier: To ad-dress effect modifiers the authors used an overall PS approach sameas equation (1.18) and subgroup-specific same as equations (1.19 and1.20). Then the authors performed the PS matching and stratificationmethods.• Simulation and data analysis: The authors did an empirical dataanalysis in their study.• Findings of the study: They found that, in their data analysis boththe overall PS approach and subgroup-specific PS approach producedbetter covariate balance. They also found that, compared to propen-sity score stratification method propensity score matching method per-formed better in terms of producing more balanced covariate resultsfor smaller sample sizes.• Gap in the study: The authors did not find any significant differ-ence between the overall PS approach and subgroup-specific approach.Therefore, better approach was not achieved. Another problem is that,the authors did not perform any simulation study to confirm their find-ings. Therefore, PS matching and PS stratification approach was notexplored more extensively in terms of simulation studies for varyingscenarios.Fourth study• To address effect modifier Green et al. [59] provided several approachesbesides the use of overall PS and subgroup-specific PS approaches.• Approaches considered for handling effect modifier: In thisstudy, the authors provided several approaches such as (i) estimatingoverall PS same as equation (1.18) and during PS matching sometimesthey did matching on all covariates. (ii) estimating overall PS same asequation (1.18) and then did exact matching on the subgroup variable.(iii) Subgroup-specific PS approach same as equations (1.19 and 1.20).(iv) Useful interaction during PS estimation were recoreded as follows,for example, consider that, based on literature review we got the idea ofpossible interaction between subgroup variable g and covariates C2,C3and Ck, So, the PS will be calculated as follows,logit(Pr(A = 1)) = α0 + α1C1 + ...+ αkCk + βsg +β1C2 × g + β3C3 × g + βkCk × g (1.21)and then matching on all covariates using that PS. (v) Considereduseful interaction during PS estimation same as equation (1.21)andperformed exact matching on subgroup variable.• Simulation and data analysis: The authors did an empirical dataanalysis in their study.• Findings of the study: They found that, in their data analysiscompared to overall PS subgroup-specific approach performed betterby producing lower standardized mean difference. The use of usefulinteraction approach performed worst in their study.• Gap in the study: The authors evaluated their various proposed ap-proaches under PS matching (full matching) and all of their findingswere based on an empirical data analysis only. Their proposed vari-ous combination could be applicable in other PS methods so that, amore general idea could be explored which approach performed betteragainst other PS methods.Fifth study• To address effect modifier Eeren et al. [60] provided uni-variate PSand generalized approach.• Approaches considered for handling effect modifier: In thisstudy, the authors provided propensity score covariate adjustmentmethod to address effect modifier. Two PS covariate adjustment meth-ods they provided were,Univariate PS method: They estimated the PS by ignoring subgroupvariable as follows,logit(Pr(A = 1|C)) = α0 + α1C1 + ...+ αkCk (1.22)and then used that PS in the outcome model with treatment, subgroupand subgroup treatment interaction.Generalized PS method: The authors at first created a combinationvariable based on the levels of treatment and subgroup variable andthen used that multi category combination variable as a treatmentvariable in PS estimation using treatment variable A, subgroup vari-able G, covariates C. Then the joint conditional probability of treat-ment assignment A and subgroup variable G given the baseline covari-ates will be,Pr(A = a,G = g | C = c) = α0 + α1C1 + ...+ αkCk (1.23)The generalized PS can be estimated using multinomial regressionmodel. After getting the PS the outcome model would consist of esti-mated generalized PSs and dummy variables denoting the group mem-bership (G).• Simulation and data analysis: The authors undertook empiricaland simulation studies. In the simulation they considered two sce-narios, (i) the subgroup variable was independent of treatment as-signment. (ii) The subgroup variable was related to the treatmentassignment. The subgroup was related to the outcome in both scenar-ios.• Findings of the study: They found that, compared to univariate PS, the generalized PS approach produced lower bias and, lower meansquared error.• Gap in the study: The authors did not considered the use of overallPS or subgroup-specific or useful interaction approach as a covariateadjustment in univariate PS approach. The authors also did not eval-uate performance of other PS methods within same simulation study.Sixth study• To address effect modifier Wang et al.[61, 63] performed some ap-proaches as mentioned by Green et al. [59] with slight modificationfor multiple subgroup variables.• Approaches considered for handling effect modifiers: In thisstudy, the authors provided considered various approaches that werementioned by Green et al. [59]. For multiple subgroup variables theauthors considered creating a combination variable based on the levelsof subgroup variables, then did separate PS matching and finally addedthe matched data for outcome analysis.• Simulation and data analysis: The authors did ran plasmode sim-ulation in their study by varying cohort size, varying the exposure andoutcome prevalence, strengths of the relationship between baseline co-variates and exposure, and applying various degrees of confoundingwithin subgroup.• Findings of the study: They found that, each approaches had per-formed similar in terms of bias, mean squared error, coverage proba-bility and covariate balance. Depending on the performance matricesone should use different methods.• Gap in the study: The considered only PS matching methods, al-though that did a comprehensive plasmode simulation study. Com-parison with other methods might provide more insight about variouseffect modifier approaches. Moreover, the authors did not perform anysimulation study which would provide some more complex scenario toevaluate the performance of various methods.1.4 Knowledge gapWe observed that, researcher commonly used an overall PS approach andsubgroup-specific approaches [55–57, 59, 61] to handle effect modifier withinvarious propensity score methods. In some studies researcher used someeffect modifier approaches under certain propensity score method for ex-ample within propensity matching method only [59, 61]. Overall, gaps inthe previous studies existed as none of the studies reviewed all the vari-ous approaches of addressing effect modifier within various propensity scoremethods in a single study. Without those comparison the performance (bias,means squared error, coverage probability) of various methods of addressingeffect modifier cannot be explored. None of the study, evaluate the similareffect modifier approaches in the context of various propensity score meth-ods such as propensity score weighting, propensity score stratification andpropensity score as covariate adjustment. Some of the studies were basedon the empirical data analysis only [58, 59], whereas to establish differentapproaches one must evaluate their performance both in simulation andempirical data analysis.Select studies undertook Monte Carlo simulation studies [55–57, 60],but their Monte Carlo simulations were limited to few scenarios. On theother hand, to discuss various propensity score methods using simulationstudy, Setoguchi et al. [30] provided extensive simulation scenarios. Noneof the studies compared the various methods to handle effect modifier byconsidering the various scenarios discussed by [30]. One study by Wang et al.[61] considered extensive plasmode simulation study. But they consideredvarious approaches to dealing with effect modifier within propensity scorematching methods only, other propensity score methods were not explored.Moreover, we found some studies that did simulation study [55–57, 60,61], reported a few performance measures of their simulation study, whereas[64] described several other extensive measures such as bias, empirical stan-dard error, mean squared error and coverage probability of simulation mea-sure. Moreover, none of the study compared various combination of han-dling effect modifier methods within all the propensity score methods con-text using both Monte Carlo and plasmode simulation together which wouldprovide more sensitivity of the findings.1.5 Objectives of the thesisTo overcome these gaps, in the current thesis we focused on four primaryobjectives. First, we will review various methodological techniques to han-dle effect modifiers within all propensity score methods suggested in theliterature. Second, besides various suggested approaches, we will comparecombination s of effect modifier approaches amongst different propensityscore methods that were not previously explored using Monte Carlo simu-lation studies under multiple scenarios described by Setoguchi et al. [30].Required modifications for the adding of effect modifiers in each scenarioto compare various propensity score methods of addressing effect modifierwill be explored. Third, we will apply all of the methods to empirical datato understand how these approaches work in practical scenarios. Lastly, wewill also perform a plasmode simulation study to compare all the approachesof addressing effect modifiers within propensity score methods. This studywill provide a guideline for future researchers with a more thorough under-standing to address effect modifiers under the propensity score paradigm.Chapter 2Propensity Score Methods toDeal with Effect ModifierThis Chapter provides a comprehensive description of all the methods dis-cussed in literature and our proposed methods to handle effect modifiers inthe context of propensity score analysis. We have provided details of eachof the methods that will build the basis of our simulation study. In all ofthe approaches, we have considered the outcome Y as a continuous variable,treatment indicator A as binary (for treatment and control the A=1 andA=0 respectively), covariates Ci both continuous and categorical, and theeffect modifier is gender G (M=male, F=Female). We have described themethods that use PS matching, PS weighting, PS stratification, and PS asa covariate adjustment separately.Overview of all the methods to handle effect modifier underpropensity score methodsWe found that researchers adapted various approaches to handle effectmodifier under propensity score methods based on the extensive literaturesearch. In those approaches, researchers followed multiple strategies in dif-ferent propensity score analysis stages, such as the estimation of the propen-sity score, then in the propensity score implementation, and finally in theoutcome analysis. In this Chapter, we will describe all the existing ap-proaches and also our proposed approaches. In the next section, we willbriefly describe the different strategies considered in the propensity scoreanalysis to understand the whole idea easily. In the Table 2.1 we pro-vided brief summary for each methods of handling effect modifiers underthe propensity score paradigm.23Table 2.1: Propensity score effect modifier methodsPropensityscoremethodsEffect modifiermethodsSub-setting onsubgroup variablePropensity scoreestimationPropensity scoreimplementationOutcome analysis ReferenceMatching M1 No G as a covariate Full matching Separate on each level of G Rassen et al. [55],Wang et al. [61]approach E (NNmatching)M2 (Jointapproach)No G as a covariate Full matching Treatment, G, Treatmentand G interactionGreen et al. [59]approach 2 ,Wang et al. [61]approach B (NNmatching)M3 (Joint model,separatematching)No G as a covariate Full matching +exact matching onsubgroup variableTreatment, G, Treatmentand G interactionGreen et al. [59]approach 3, Wanget al. [61]approach A (NNmatching)M4 (Stratifiedapproach)Yes G will not beincludedFull matching Treatment, G, Treatmentand G interactionGreen et al. [59]approach 1, Wanget al. [61]approach C (NNmatching)M5 (Interactionapproach)No Useful interactionwith GFull matching Treatment, G, Treatmentand G interactionGreen et al. [59]approach 4M6 (Interactionmodel, separatematching)No Useful interactionwith GFull matching +exact matching onsubgroup variableTreatment, G, Treatmentand G interactionGreen et al. [59]approach 5Weighting W1 No G as a covariate Weighting Treatment, G, Treatmentand G interactionKreif et al. [56]W2 No Useful interactionwith GWeighting Treatment, G, Treatmentand G interactionNew proposedmethodW3 Yes G will not beincludedWeighting Treatment, G, Treatmentand G interactionKreif et al.[56]W4 No G as a covariate Stabilized Weight Treatment, G, Treatmentand G interactionHernan et al. [23]W5 No Useful interactionwith GStabilized Weight Treatment, G, Treatmentand G interactionNew proposedmethodW6 No G as a covariate Weighting Separate on each level of G New proposedmethodStratificationS1 No G as a covariate Stratified on PS Treatment, G, Treatmentand G interactionGirman et al. [58]S2 No Useful interactionwith GStratified on PS Treatment, G, Treatmentand G interactionNew proposedmethodS3 Yes G will not beincludedStratified on PS Separate on each level of G Girman et al. [58]Covariateadjust-mentC1 No G will not beincludedUse PS as acovariatePS, Treatment, G,Treatment and GinteractionEeren et al. [60]univariate PSC2 Yes G will not beincludedUse PS as acovariateSeparate on each level of G New proposedmethodC3 No Useful interactionwith GUse PS as acovariatePS, Treatment, G,Treatment and GinteractionNew proposedmethodC4 No G will not beincludedUse PS as acovariatePS for different categoriesof treatment, groupmembership dummyvariableEeren et al. [60]generalized PSC5 No G as a covariate Use PS as acovariatePS, Treatment, G,Treatment and GinteractionNew proposedmethodC6 No G as a covariate Use PS as acovariateSeparate on each level of G New proposedmethodNote:G: subgroup variable, PS: Propensity score2.1 Propensity score matching approachMethods proposed in the literatureMethod M1Description: Estimate the propensity score (PS) by including the sub-group variable as a covariate with other covariates. Then use this PS tomatch within subgroup variable. After that fit the separate outcome modelfor each level of the effect modifier variable.Details: This approach was proposed by Rassen et al. [55] where theauthors considered the subgroup variable as covariates along with otherconfounders in the PS estimation model.After getting the PS, perform PSmatching (i.e. full matching) to match within subgroup. Finally, a separateoutcome model will be fitted at each level of subgroup variable separately.Given below the steps of this method discussed by Rassen et al. [55]:Steps:-Step 1: PS score will be calculated based on the available covariates andsubgroup of interest. The subgroup variable will be added as covariate inPS score estimation. This PS is based on the entire cohort is called “cohortPS” as follows,logit(Pr(A = 1 | C,G)) = β0 + β1C1 + β2C2 + ...+ βkCk + βgG (2.1)–Step 2: Do PS matching based on the cohort PS from step 1. Here, weconsidered PS full matching.-Step 3: Outcome regression will be performed using the matched datafrom step 1 as follows,E(Y | G = M,C) = β0 + β1C1 + β2C2 + ...+ βkCk + βtA (2.2)E(Y | G = F,C) = β0 + β1C1 + β2C2 + ...+ βkCk + βtA (2.3)One caution that was mentioned by Rassen [55] was that, to apply theoverall PS to draw inference within subgroup level, sample size of subgrouplevel should be large enough (n > 1000) and correct specification of thecohort PS is needed. In some other papers, such as Kreif et al. [56] andRadice et al. [57] also applied the similar method proposed by Rassen etal. [55] regarding matching on subgroup-specific PS to compare the per-formance of PS matching with using overall PS (cohort PS) matching ap-proach. Wang et al. [61] applied this to observe relative performance ofvarious matching strategy within subgroup levels using overall PS. One ofthe recent population-based study [65], where the authors adjusted the sub-group effect of age, gender and recent antibiotic use history on the studyof association measure between sodium glucose cotransporter two inhibitorsand risk of genital mycotic and urinary trac infection. They mentioned that,separate subgroup analysis may restrict the findings of the study and therewas chance to have smaller sample in any subgroup level. Therefore, theyused the overall PS score estimated from full cohort in their study.Method M2 (Joint approach)Description: Estimate PS by adding the subgroup variable as covariateand do PS full matching [59].Details: At first in the estimation of PS the subgroup variable will beadded as a covariate besides other covariates following equation (2.1). Afterthat, in the matching stage, matching will be done using PS full matching.Given below the steps to conduct this method by following [59]:Steps:-Step 1: Estimate PS score based on the available covariates and sub-group of interest. The subgroup variable will be added as covariate in PSscore estimation similar to equation (2.1).-Step 2: Using PS score from step 1 to perform PS full matching.-Step 3: After getting the matched data from step 3 fit outcome modelby following model:E(Y | A,G) = β0 + β1A+ β2G+ β4A×G (2.4)This approach is easy to implement, but without considering the sub-group variable in the matching stage, it ignores the interest in differentialeffects by subgroup [59].Method M3 (Joint model, separate matching)Description: Estimating PS by adding the subgroup variable as covari-ate and besides full matching perform exact matching on subgroup variableof interest [59].Details: Similar to the method M2, under this method, the overallPS will be calculated by considering the subgroup variable as a covariatefollowing equation (2.1). After that, in the PS matching stage, besides PSmatching, exact matching on the subgroup variable will be conducted. If wehave more than one effect modifier, then by considering various combinationsof the levels of all effect modifiers we need form separate variable and performexact matching on those variables [61]. Given below the steps to conductthis method by following [59]:Steps:-Step 1: Estimate PS score based on the available covariates and sub-group of interest. The subgroup variable will be added as covariate in PSscore estimation similar to equation (2.1).-Step 2: Using PS score from step 1 perform PS score full matching andalso exact matching on the subgroup (gender) variable.-Step 3: After getting the matched data from step 3 fit outcome modelfollowing equation (2.4).If any interest lies on checking covariate balance and the estimation oftreatment effect in terms of levels of subgroup variable with any specialconsideration, then this method is applicable [59].Method M4 (Stratified approach)Description: Estimating the PS score separately for each level of effectmodifier and then perform matching within levels of effect modifier sepa-rately Rassen et al. [55] . Then by combining the matched data fit outcomemodel including the treatment, subgroup and subgroup interaction with thetreatment variable.Details: This approach was proposed by Rassen et al. [55] and alsoimplemented in [56, 57, 59, 61]. Green et al. [59] called this method as“stratified approach.” To implement this method, at first the dataset needsto be separated based on levels of subgroup that is, for our case, one formale and another one for female. Then do matching and using matcheddata perform the weighted outcome regression by considering the treatment,gender and gender interaction with the treatment variable. For more thanone effect modifier, we need to create a combination variable consideringthe various combinations of levels of effect modifiers and they by stratifyingthem we can perform PS matching same as single effect modifier Wang etal. [61].Steps:- Step 1: Entire dataset will be separated into male and female group(i.e., levels of subgroup variable).- Step 2: Estimate PS score for each of the dataset from step 1 usingsame covariates. For a set of covariates Ci’s as a measured confoundersestimation of PS for male and female separately will be,logit(Pr(A = 1 | G = M,C)) = α0 + α1C1 + ...+ αkCk (2.5)andlogit(Pr(A = 1 | G = F,C)) = α0 + α1C1 + ...+ αkCk (2.6)- Step 3: Using PS score from step 2 perform PS score full matchingseparately for male and female.- Step 4: After getting the matched data from step 3 fit the weightedoutcome model following equation (2.4).One of the limitations of the previous method M1 was that, sometimesdifferent covariates act differently for different levels of effect modifier, there-fore, getting correct specification of PS model for entire cohort is difficult[66]. Besides this, to apply the method M1 bigger sample size is requiredfor subgroups Rassen et al. [55]. In this case, separate or stratified PS ap-proach at various levels of effect modifier would capture the variability atvarious level of subgroup variable and the correct specification of PS wouldbe possible. One major problem of this method M4 is that, if there aremany subgroup variables in any study then smaller sample size in differentlevels of subgroup may raise and affect the convergence in the PS estimation[61, 63].Method M5 (Interaction approach)Description: Estimating PS by considering useful interaction betweensubgroup variable and other suitable covariate(s) and do PS matching with-out consideration of subgroup variable of interest [59].Details: At first the PS will be calculated by considering the subgroupvariable and its interaction terms with other required covariates. Here, thecovariates that will be used to form interaction with subgroup variable,will be decided based on the literature review and expert knowledge. Forexample, consider that, based on literature review we got the idea of possibleinteraction between subgroup variable g and covariates C2,C3 and Ck, So,the PS will be calculated as follows,logit(Pr(A = 1)) = α0+α1C1+...+αkCk+βsg+β1C2×g+β3C3×g+βkCk×g(2.7)Then at the PS matching stage, matching will be done without anyspecial consideration of the subgroup variable. Given below the steps toconduct this method by following Green et al. [59]:Steps:-Step 1: Estimating a PS score based on the available covariates, sub-group. Also include interaction with subgroup variable and some of theother covariates. Suppose that, in our case, considering the variables thatwould be added as interaction with the gender variable and C2,C3 and Ckby getting idea from literature. So, the PS will be estimated as,logit(Pr(A = 1)) = α0+α1C1+...+αkCk+βgG+β1C2×G+β3C3×G+βkCk×G(2.8)-Step 2: Using PS score from step 1 to perform PS score full matchingand matching without consideration of gender variable.-Step 3: After getting the matched data from step 3 fit outcome modelfollowing equation (2.4).After estimating propensity score, like the method M3, this method alsoignores the subgroup variable during matching which ignores the interest indifferential effects by subgroup.Method M6 (Interaction model, separate matching)Description: Estimating PS by considering useful interaction betweensubgroup variable and other suitable covariates and besides full matchingperform exact matching on subgroup variable of interest [59].Details: At first the PS will be calculated by considering the subgroupvariable and its interaction terms with other required covariates similar tothe equation (2.7).After that, in the PS matching stage, besides PS match-ing, perform exact matching on the subgroup variable. Given below thesteps to conduct this method by following (Green and Stuart 2014):Steps:-Step 1: Estimating a PS score based on the available covariates, sub-group. Also include interaction with subgroup variable and some of theother covariates similar to the equation (2.8).-Step 2: Using PS score from step 1 perform PS score full matching andalso exact matching on the subgroup variable.-Step 3: After getting the matched data from step 3 fit outcome modelfollowing equation (2.4).This method has all the beneficial characteristics like method M5 ad-ditionally this method addressed the issue of particular consideration ofsubgroup variable during matching. If interest lies on checking the covari-ate balance on various levels of subgroup variable this approach would be afeasible approach.2.2 Propensity score weighting approach2.2.1 Methods proposed in the literatureMethod W1Description: Estimate the PS by including the subgroup variable as acovariate with other covariates. Use the PS to calculate the Inverse proba-bility treatment weight (IPTW) and the outcome model will be performedusing weighted least squares [56, 57].Details: Estimate the PS by considering the covariates and subgroupvariable in the PS model. After estimating PS following equation (2.1)calculate the weight wi defined as the inverse of the estimated probabilityof the observed treatment as follows,wi =Aipsi+1−Ai1− psi (2.9)where, psi is the estimated PS score for the individual i and Ai is theindicator of treatment. This weight should be added in the original dataset.After that, the outcome model from the weighted data will be calculated as,E(Y | A,G) = β0 + β1A+ β2G+ β3A×G (2.10)Given below the steps of conducting IPTW to address subgroup effectfollowing Kreif et al. [56]:Steps:-Step 1: PS score will be calculated based on the available covariates andsubgroup of interest. The subgroup variable will be added as covariate inPS score estimation. So, the PS will be estimated following equation (2.1).-Step 2: After getting the PS score from step 1, this PS score will beused to calculate the weight wi for each individual following equation (2.9).-Step 3: From the weighted data calculate the outcome model followingequation (2.10).In the IPTW approach, individual with a high probability of the observedtreatment will receive low weight. In case of PS matching method manyobservations could be missing for further analysis because of utilization ofmatched samples only. In that case, IPTW would provide opportunity to re-serve individuals by creating pseudo population formed by using the weight.Kreif et al. [56] and Radice et al. [57], utilized this method in their study. Intheir simulation studies, they tried to explore the performance of propensityscore weighting with propensity score genetic matching for any misspecifi-cation in the PS estimation model by ignoring subgroup-interaction termfrom the model. They mentioned that, in the presence of subgroup variableproper model specification in the PS estimation is challenging because ofdifferential treatment assignment mechanism. They performed both overallPS method and subgroup-specific method in their study.Method W3Description: Estimating the PS score separately for each level of effectmodifier and then apply PS weighting approach separately for each level ofsubgroup variable [56, 57].Details: Separate the entire dataset based on the levels of subgroupvariable. Then estimate the PS for each dataset, calculate weight and per-form weighted outcome regression for each dataset separately [56, 57]. Theweight will be calculated similar to the equation (2.9) for each data setseparately. Kreif et al. [56] and Radice et al. [57] mentioned that, if thetreatment assignment differ by subgroup, then the utilization of PS scorebased on overall PS score would not achieve covariate balance in subgrouplevel, separate subgroup level analysis would be preferred then. Given belowthe steps of the method as described in Kreif et al. [56]:Steps:-Step 1: Entire dataset will be separated one for male and another onefor female.-Step 2: Estimate PS score for male and female separately using samecovariates using equation (2.5) and (2.6).-Step 3: Using PS score from step 2 calculate weights for male and femaledatasets as follows by considering g as gender and M=male, F=Female,wi,g=M =Ai,g=Mpsi,g=M+1−Ai,g=M1− psi,g=M (2.11)andwi,g=F =Ai,g=Fpsi,g=F+1−Ai,g=F1− psi,g=F (2.12)where psi is the propensity score for individual i.-Step 4: Using the weights in step 3 perform outcome model followingequation (2.10).Method W4Description: In the context of marginal structural model (MSM), theeffect modifier will be added in the numerator to calculate the stabilizedweight and in the marginal mean model the effect modifier will be added asan interaction term with treatment variable [23].Details: This method addressed the issue of effect modifier in case ofmarginal structural model fitting and was described in Hernan et al. [23].Under this method, when any subgroup variable present in the study, thenthat variable should be added as a covariate along with other covariates inthe estimation of PS to get the denominator of the stabilized weight andto get numerator only subgroup variable will be added as covariate in theestimation of PS. After that, using the stabilized weight, marginal meanmodel will be calculated where the subgroup variable will be added as aninteraction term with the treatment variable. Given below the steps of thisapproach:Steps:-Step 1: Estimating a PS score based on the available covariates andsubgroup of interest. The subgroup variable will be added as covariate inPS score estimation to get the denominator of the stabilized weight and toget numerator only subgroup variable will be added in the PS estimation asfollows,For denominator, PS will be,P (A = 1 | C,G) = α0 + α1C1 + ...+ αkCk + βgG (2.13)and for numerator, PS will be,P (A = 1 | G) = α0 + βgG (2.14)-Step 2: Calculate the stabilized weight using the denominator and nu-merator from step 1 as,SW (G) =P (A = 1 | G)P (A = 1 | C,G) (2.15)-Step 3: In the outcome model using stabilized weight the main exposurevariable, subgroup variable G (gender) and interaction between exposureand subgroup variable will be included,E[Y | G] = β0 + β1A+ β3G+ β3A×G (2.16)Under this MSM approach, SW (G) calculated like method W1, with anexception that, the covariate “gender” variable was added in the estimationof numerator weights in the method W4.2.2.2 New methods we proposedMethod W2Description: Estimating PS by considering useful interaction betweensubgroup variable and other suitable covariates. Use the PS to calculate theInverse probability treatment weight (IPTW) and the outcome model willbe performed using weighted least squares like method W1.Details: Estimate the PS by considering the covariates and subgroupvariable as well as useful interaction with subgroup variable with subgroupand other covariates in the PS model. After estimating the PS calculate theweight wi. This weight should be added in the original dataset. After that,the outcome model from the weighted data will be calculated. Given belowthe steps of this method:Steps:-Step 1: PS score will be calculated based on the available covariates andsubgroup of interest. The subgroup variable will be added as covariate inPS score estimation. So, the PS will be estimated following equation (2.7).-Step 2: After getting the PS score from step 1, this PS score will beused to calculate the weight wi for each individual following equation (2.9).-Step 3: From the weighted data calculate the outcome model followingequation (2.10).Method W5Description: Similar to the method W4, in the context of marginalstructural model (MSM), the effect modifier will be added in the numeratorto calculate the stabilized weight. For the denominator the propensity scorewill be estimated by considering covariates and some useful interactionsof covariates and effect modifier. In the marginal mean model the effectmodifier will be added as an interaction term with treatment variable.Details: This method similar to the method W4 address the issue of ef-fect modifier in case of marginal structural model fitting. Under this method,variable should be added as covariate along with other covariates in the es-timation of PS to get the denominator of the stabilized weight and to getnumerator only subgroup variable will be added as covariate in the estima-tion of PS. After that, using the stabilized weight, marginal mean model willbe calculated where the subgroup variable will be added as an interactionterm with the treatment variable. Given below the steps of this approach:Steps:-Step 1: Estimating a PS score based on the available covariates andsubgroup of interest. The subgroup variable will be added as covariate inPS score estimation to get the denominator of the stabilized weight and toget numerator only subgroup variable will be added in the PS estimation asfollows,For denominator, PS will be,P (A = 1 | C,G) = α0+α1C1+...+αkCk+βgG+β1C2×G+β3C3×G+βkCk×G(2.17)and for numerator, PS will be,P (A = 1 | G) = α0 + βgG (2.18)-Step 2: Calculate the stabilized weight using the denominator and nu-merator from step 1 as,SW (G) =P (A = 1 | G)P (A = 1 | C,G) (2.19)-Step 3: In the outcome model using stabilized weight the main exposurevariable, subgroup variable G (gender) and interaction between exposureand subgroup variable will be included following equation (2.16)Method W6Description: Estimate the PS by including the subgroup variable as acovariate with other covariates. Use the PS to calculate the Inverse proba-bility treatment weight (IPTW) and the outcome model will be performedseparately for each level of subgroup.Details: Estimate the PS by considering the covariates and subgroupvariable in the PS model by following the equation (2.1) and calculate theweight wi following the equation (2.9). The fitted the outcome model sepa-rately for each level of subgroup.Steps:-Step 1: Estimate the PS by following the equation (2.1).-Step 2: Estimate the wi using PS from step 1 by following the equation(2.9).-Step 3: Outcome regression will be performed using the matched datafrom step 1 as follows,E(Y | G = M,C) = β0 + β1C1 + β2C2 + ...+ βkCk + βtA (2.20)E(Y | G = F,C) = β0 + β1C1 + β2C2 + ...+ βkCk + βtA (2.21)2.3 Propensity score stratification approach2.3.1 Methods proposed in the literatureMethod S1Description: Estimate the PS by including the subgroup variable ascovariates with other confounders. Using the PS perform PS stratification.Details: Estimate the PS by considering the covariates and subgroupvariable in the PS model. Stratify the dataset based on the quintiles of theoverall PS. Fit the outcome model for each stratum and pooled the stratum-wise PS using Mantel-Haenszel method to get the combined treatment effectGirman et al. [58]. Girman et al. [58] applied this method in their studyto do subgroup analysis by considering the overall cohort PS with strati-fication approach. They stratified the dataset based on the decile of PS.Then calculated the stratum-wise treatment effect and estimate the pooledtreatment effect. Each time of the stratum-wise treatment effect estimationoutcome model will be same as (2.4).Steps:-Step 1: Estimating a PS score based on the available covariates andsubgroup of interest. The subgroup variable will be added as covariate inPS score estimation following equation (2.1).-Step 2: Using PS score from step 1 sort the subjects and stratify thedataset into quantiles based on overall PS.-Step 3: Calculate stratum-specific treatment effect and pooled themusing Mantel-Haenszel method.When outcome variable is continuous, wecan estimate the stratum-wise treatment effect using equation (2.4) andthen use the weighted average method to get the pooled treatment effect asfollows:ATE =∑sNsNATEs (2.22)Method S2Description: Estimating the PS score separately for each level of ef-fect modifier and then apply PS stratification and estimate the treatmentseparately for each dataset Girman et al. [58].Details: The dataset will be divided for male and female separately.Then estimate the PS for each dataset. Stratify the individuals in eachdataset based on the decile of PS. Calculate the stratum-specific treatmenteffect for male and female data separately. Finally, apply Mantel-Haenszelestimator to get the pooled treatment effect for male and female. Givenbelow the steps of the method:Steps:-Step 1: Entire dataset will be separated into male and female group.-Step 2: Estimate PS score for each of the dataset from step 1 usingsame covariates following equations (2.5) and (2.6).-Step 3: Stratify the PS into decile based one the PS from step 1 formale and female separately.-Step 4: Calculate stratum-specific treatment effect and then applyMantel-Haesnzel estimator to get pooled treatment for male and femaleseparately. We have our outcome variable as continuous, so we estimatedthe stratum wise treatment effect using the outcome model and then com-bined to get the pooled treatment effect for male and female separately usingequation (2.22).2.3.2 New method we proposedMethod S3Description: Estimate the PS by including the subgroup variable as co-variates with other confounders and using the useful interactions of subgroupvariable with other covariates. Using the PS perform PS stratification.Details: Estimate the PS by considering the covariates and subgroupvariable as well as including the useful interaction with other covariates withsubgroup variable. Then stratify the dataset based on the quintiles of theoverall PS. Fit the outcome model for each stratum and pooled the stratum-wise PS using Mantel-Haenszel method to get the combined treatment effectGirman et al. [58]. Finally calculate the stratum-wise treatment effectand estimate the pooled treatment effect. Each time of the stratum-wisetreatment effect estimation outcome model will be same as (2.4).Steps:-Step 1: Estimating a PS score based on the available covariates, sub-group of interest and the useful interactions of subgroup variable with othercovariates by following equation (2.7).-Step 2: Using PS score from step 1 sort the subjects and stratify thedataset into quantiles based on overall PS.-Step 3: Calculate stratum-specific treatment effect and pooled themusing Mantel-Haenszel method.When outcome variable is continuous, wecan estimate the stratum wise treatment effect using equation (2.4) andthen use the weighted average method to get the pooled treatment effect asfollows:ATE =∑sNsNATEs (2.23)2.4 Propensity score covariate adjustmentapproach2.4.1 Methods proposed in the literatureMethod C1Description: Estimate the PS score using the covariates and use the PSscore as a covariate in the outcome model along with treatment, subgroupand subgroup treatment variable interaction proposed by Eeren et al. [60].Details: The PS model will be estimated using all available covariates.But in the estimation of PS, subgroup variable will not added as a covariate.The outcome model will include the estimated PS as covariate, the treatmentvariable, subgroup variable and interaction between treatment and subgroupvariable. Given below the steps of this method following Eeren et al. [60] :Steps:-Step 1: Estimating a PS score based on the available covariates. Thesubgroup variable will not be added as covariate in PS score estimation. So,the PS will be calculated as,logit(Pr(A = 1|C)) = α0 + α1C1 + ...+ αkCk (2.24)-Step 2: Fit outcome model by considering the following covariates: PSscore from step 1, main exposure variable, subgroup variable and interactionbetween exposure and subgroup variable using the following model:E(Y | A,G) = β0 + β1ps+ β2A+ β3G+ β4A×G (2.25)In the outcome model we can add other covariates.Method C4Description: Using a combination variable using treatment and sub-group variable based on the levels of both variables. Then considering thatcombination variable as exposure variable, a generalized PS is estimatedusing baseline covariates and not including any interaction terms betweenthe covariates. After getting the PS the outcome model would consist ofestimated generalized PS(s) for each category of the combined treatmentvariable and dummy variables denoting the group membership Eeren et al.[60].Details: This is also a PS score covariate adjustment approach proposedby Eeren et al.[60] to deal with subgroup variable. Generally, generalizedpropensity score is applicable when the treatment variable has more thantwo categories and a multinomial logistic regression method been appliedto calculate the generalized PS. To deal with subgroup this generalized PShas been adopted here. This method requires a combination variable usingtreatment and subgroup variable based on the levels of both variables. Thenconsidering that combination variable as exposure variable, a generalizedPS is estimated using baseline covariates and not including any interactionterms between the covariates. For our treatment variable A, subgroup vari-able G, covariates C. Then the joint conditional probability of treatmentassignment A and subgroup variable G given the baseline covariates will be,Pr(A = a,G = g | C = c) = α0 + α1C1 + ...+ αkCk (2.26)If the treatment variable A has 2 categories and group variable has 2categories, then PS will be estimated for 4 groups. The generalized PS canbe estimated using multinomial regression model. After getting the PS theoutcome model would consist of 3 estimated generalized PSs (consideringthe 1st category (0,0) as baseline group) and 3 dummy variables denotingthe group membership (G) as follows:E(Y ) = β0 + β1ps1 + β2ps2 + β3ps3 + β4G1 + β5G2 + β6G3 (2.27)Given below the details steps of the method C4 following Eeren et al.[60]:Steps:-Step 1: Create a combination variable using the main exposure variableand subgroup variable of interest. E.g. let us consider we have an exposurevariable and a subgroup variable having levels (0 and 1). So, a combinationvariable will be created by considering all possible combinations [(0,0), (0,1),(1,0) and (1,1)] of the levels of main exposure variable and subgroup variable.(0,0) category will be considered as reference group.-Step 2: Estimate PS score by considering the new combination vari-able as dependent variable using multinomial logistic regression model byfollowing (2.26).-Step 3: Fit outcome model by considering the following covariates: 3generalized PS score from step 2, 3 dummy variables indicating the member-ship in each main exposure variable. Here we reported the outcome modelseparately for each level of subgroup variable. using the following equation(2.27). In the outcome model we can add other covariates.2.4.2 New methods we proposedMethod C2Description: Estimate the PS score using the covariates separately foreach level of subgroup variable and use the PS score as a covariate alongwith treatment in the outcome model separately.Details: At first divide the dataset into the levels of subgroup variableand then PS model will be estimated using all available covariates separatelyfor each level of subgroup. The outcome model will include the estimatedPS as covariate along with the treatment variable. Given below the steps ofthis method:Steps:-Step 1: Entire dataset will be separated into male and female group(i.e., levels of subgroup variable).-Step 2: Estimate PS score for each of the dataset from step 1 usingsame covariates by following (2.5) and (2.6)-Step 3: Outcome regression will be performed using the PS as a covari-ate for each data from step 1 as follows,E(Y | G = M) = β0 + β1ps+ β2A (2.28)E(Y | G = F ) = β0 + β1ps+ β2A (2.29)In the outcome model we can add other covariates.Method C3Description: Estimate the PS score using the covariates and usefulinteractions between subgroup and other covariates. Then use the PS scoreas a covariate in the outcome model along with treatment, subgroup andsubgroup treatment variable interaction.Details: The PS model will be estimated using all available covariatesand useful subgroup covariates interactions. The outcome model will includethe estimated PS as covariate, the treatment variable, subgroup variable andinteraction between treatment and subgroup variable. Given below the stepsof this method:Steps:-Step 1: Estimating a PS score based on the available covariates, sub-group of interest and the useful interactions of subgroup variable with othercovariates by following equation (2.7).-Step 2: Fit outcome model by considering the following covariates: PSscore from step 1, main exposure variable, subgroup variable and interactionbetween exposure and subgroup variable using the following model:E(Y | A,G) = β0 + β1ps+ β2A+ β3G+ β4A×G (2.30)In the outcome model we can add other covariates.Method C5Description: Estimate the PS score using the covariates and also byconsidering the subgroup variable as a covariate. Then use the PS scoreas a covariate in the outcome model along with treatment, subgroup andsubgroup treatment variable interaction.Details: The PS model will be estimated using all available covariatesand subgroup variable as a covariate. The outcome model will include theestimated PS as covariate, the treatment variable, subgroup variable andinteraction between treatment and subgroup variable. Given below the stepsof this method:Steps:-Step 1: Estimating a PS score based on the available covariates andsubgroup of interest by following equation (2.1).-Step 2: Fit outcome model by considering the following covariates: PSscore from step 1, main exposure variable, subgroup variable and interactionbetween exposure and subgroup variable using the following model:E(Y | A,G) = β0 + β1ps+ β2A+ β3G+ β4A×G (2.31)In the outcome model we can add other covariates.Method C6Description: Estimate the PS score using the covariates and also byconsidering the subgroup variable as a covariate. Then fit the separateoutcome model for each level of subgroup variable by considering the PS asa covariate with treatment variable.Details: The PS model will be estimated using all available covariatesand subgroup variable as a covariate. The outcome model will be fittedseparately for each level of subgroup variable (i.e. separately for male andfemale). In the outcome model will include the treatment variable and PSas a covariate. Given below the steps of this method:Steps:-Step 1: Estimating a PS score based on the available covariates andsubgroup of interest by following equation (2.1).-Step 2: Fit outcome model by considering the following covariates: PSscore from step 1, main exposure variable and subgroup variable separatelyfor each subgroup level.In the outcome model we can add other covariates.2.5 Features of the various methods to handleeffect modifier in Propensity score contextPropensity score matching methods to handle effect modifierTable 2.2: Propensity score matching methods to handle effect modifierPS EffectmodifiermethodsAdvantage DisadvantageM1 Can handle multiple subgroupvariables.(i) overall PS may not capture the truedifferences in PS across subgroups. (ii)The patients included from the overallPS and subgroup are different.M2 (Jointapproach)Multiple subgroup variables canbe handled.Exploring a particular subgroupvariable during matching is notpossibleM3 (Jointmodel,separatematching)Subgroup variable level-wisecovariate balance is possible toachieve.smaller sample size might createproblem in exact matching formultiple subgroupsM4(Stratifiedapproach)(i) Better covariate balance canbe achieved in each subgroupand subgroup wise treatmenteffect is easy to calculate(i) For multiple subgroup variables,smaller samples in separate subgrouplevel increase the potentiality ofconvergence problem in PS estimation,so, subgroup-level effect might notachieve (ii) There is a chance of largervariance in treatment effect estimation.M5(Interactionapproach)Joint effect of subgroup variablecan be explored and reducemodel subgroup-treatmentmisspecification.A matching is done for full cohort so ifany there is any interest lies onparticular subgroup variable then thismethod cannot be applicable.M6(Interactionmodel,separatematching)Besides joint effect, particularinterest on a subgroup variablecan be analyzed.smaller sample size might createproblem in exact matching formultiple subgroups.Propensity score weighting methods to handle effect modifierTable 2.3: Propensity score weighting methods to handle effect modifierPS EffectmodifiermethodsAdvantage DisadvantageW1 Can handle multiple subgroupvariables by using weight based onoverall cohort PS to inferencewithin subgroups.Overall PS may not capture thetrue differences in PS acrosssubgroup level.W2* Joint effect of subgroup variablecan be explored and reduce modelsubgroup-treatmentmisspecificationIncorrect specification of modelinteraction will lead to bias in thetreatment effect estimates.W3 Better covariate balance can beachieved in each subgroup andsubgroup wise treatment effect iseasy to calculate(i) For multiple subgroupvariables, smaller samples inseparate subgroup level increasethe potentiality of convergenceproblem in PS estimation (ii)There is a chance of larger variancein treatment effect estimationW4 If there are time-varying covariatesin the data, then with this methodeffect modifier can be also behandled.(i) Overall PS may not capture thetrue differences in PS acrosssubgroup level. (ii) The patientsincluded from the overall PS andsubgroup are different.W5* Joint effect of subgroup variablecan be explored and reduce modelsubgroup-treatmentmisspecificationIncorrect specification of modelinteraction will lead to bias in thetreatment effect estimates.W6* Can handle multiple subgroupvariables by using weight based onoverall cohort PS to inferencewithin subgroups.(i) Overall PS may not capture thetrue differences in PS acrosssubgroup level.Note:*: New proposed methodsPropensity score stratification methods to handle effect modi-fierTable 2.4: Propensity score stratification methods to handle effect modifierPS EffectmodifiermethodsAdvantage DisadvantageS1 Can handle multiple subgroupvariables by using weight based onoverall cohort PS to inferencewithin subgroups.Overall PS may not capture thetrue differences in PS acrosssubgroup level.S2* Joint effect of subgroup variablecan be explored and reduce modelsubgroup-treatmentmisspecificationIncorrect specification of modelinteraction will lead to bias in thetreatment effect estimates atvarious stratum formed on thebasis of PS.S3 (i) Better covariate balance can beachieved in each subgroup. (ii)Subgroup wise treatment effectcan be easy to get with requiredadjustment.(i) For multiple subgroupvariables, smaller samples inseparate subgroup level increasethe potentiality of convergenceproblem in PS estimation.(ii)There is a chance of larger variancein treatment effect estimation.Note:*: New proposed methodsPropensity score covariate adjustment methods to handle effectmodifierTable 2.5: Propensity score covariate adjustment methods to handle effectmodifierPS EffectmodifiermethodsAdvantage DisadvantageC1 Adding the subgroup variable withunivariate PS in outcome modelreduces the bias of treatment effectestimation.(i) In univariate PS estimationsubgroup variable was ignored, butEeren et al.[60] mentioned that noprognostic variable should be ignoredin PS estimation. (ii) This methodreduces the bias but adding subgroupvariable affected the interpretabilityof the main treatment effect. (iii) Fortreatment variable having more than2 categories this method is notapplicable.C2* For subgroup label modelspecification will be formed(i) For multiple subgroup variables,smaller samples in separate subgrouplevel increase the potentiality ofconvergence problem in PSestimation. (ii) There is a chance oflarger variance in treatment effectestimationC3* Joint effect of subgroup variable canbe explored and reduce modelsubgroup-treatment misspecificationIncorrect specification of modelinteraction will lead to bias in thetreatment effect estimates at variousstratum formed on the basis of PS.C4 (i) Under this method subgroupvariable was not ignored so that itreduces the bias, (ii) When treatmentvariable has more than two categoriesthen this method is applicableIn mutlicategory treatment variable,the assumption of equality of jointdistribution of potential outcome andcovariates among different categoriesof treatment variable may be violated.C5* Adding the subgroup variable in PSestimation the bias in PS modelspecification and also adding PS inthe outcome model reduces bias intreatment effect estimation.Overall PS may not capture the truedifferences in PS across subgrouplevel.C6* Adding the subgroup variable in PSestimation the bias in PS modelspecification and also adding PS inthe outcome model reduces bias intreatment effect estimation.For multiple subgroup variablescorrect specification of outcome modelincluding PS as a covariate might bedifficultNote:*: New proposed methodsChapter 3Motivating Example RightHeart Catheterization StudyThis chapter provides the result from a data analysis after applying all ofthe PS effect modifier methods that we have mentioned in Chapter 2. In ourstudy, we have used a publicly available dataset based on a study conductedby Connors et al. [7]. The authors explored the association of the use ofRight Heart Catheterization with the subsequent survival, length of stay,intensity of care and cost of care, during the first day in the intensive careunit (ICU). Before analysis of the dataset, given below a description of thestudy performed by Connors et al. [7].3.1 Right Heart Catheterization studyRight Heart Catheterization (RHC) is a recommended therapy to treat crit-ically heart patients, by the cardiologist. Over the decades, the use of RHCtherapy has been increased in serving cardiac events. However, the RHC isa surgical procedure and women patients with actute myocardial infractionas well as elderly had higher risk of death when they undergone with RHCtreatment. Due to complexity in the RHC surgical procedure,the effective-ness of this therapy was not evaluated using randomized controlled trial.From ethical point of view, performing randomization would not be possibleto evaluate its effectiveness. Moreover, physician did not provide sufficientpatients for the randomization. For these reasons, to evaluate the effective-ness of the RHC the researchers needed to depend on observational study.Connors et al. [7], performed the study to explore the association betweenRHC therapy and subsequent health outcomes such as,survival, length ofstay, intensity of care and cost of care.The study was a prospective cohort study conducted on the severely illhospitalized adult patients at five medical centers in the U.S during 1989to 1994. The study included two phases: Phase I (a prospective observa-tional study to describe the decisions and process of care for severely ill45patients) and Phase II (a clustered randomized trial to explore the effect ofinterventions). In the phase I there were 4,301 patients and 4,804 patientswere enrolled in the phase II. The authors, in their study combined the twophase’s datasets for their analysis. In total there were 5,735 patients, 2,184treated patients and 3,551 control patients. For each subject the observedtreatment status, equal to 1 if RHC was applied within 24 hours of admis-sion, and 0 otherwise. Clinical outcome was an indicator for death within30 or 60 or 180 days after study entry. There were 68% of the RHC patientsthat had clinical outcome compared to 63% of the No RHC patients.The study was an observational study and confounding bias was one ofthe major issues in the analysis. In the data analysis the authors adoptedthe propensity score analysis approach to perform the measure of the effectof RHC on subsequent death within 30 days.3.1.1 Analysis of the Right Heart Catheterization (RHC)dataset• Crude and adjusted association:Let us check the crude association between RHC exposure and death.To get this we performed the following regression model:logit(Death = “Y es”) = β0 + β1RHC (3.1)Outcome death is binary (Yes or No) and main treatment variable ofinterest RHC (Yes or No). Therefor, to get the crude association betweentreatment and outcome we fitted a logistic regression and reported the oddsratio (OR) with 95% confidence interval (95% CI).Covariates: Other covariates that we included in our report to get theadjusted effect of RHC treatment were, Age (categorized into 0-50, 50-60,60-70, 70-80, 80-110), Race (White, Black, other), Disease category (ABF,CHF, MOSF, Other), Cancer (None. Localized, Metastatic), DNR sta-tus (Yes, No), APACHE III score ignoring Coma, Probability of survival 2months, DASI score two weeks prior, Temperature, heart rate, Blood pres-sure, Respiratory rate, WBC count, PaO2 by FIO2, PaCO2, pH, Creatinine,Albumin, GComa Score and comorbidity diseases such as, cardiologic dis-ease, congestiveHF, dementia, gastrointestinal bleeding, immuno suppres-sion, Hepatic failure, Chronic myolema, Psychiatric history, renal disease.Table 3.1: Crude association between treatment Right Heart Catheterization(RHC) and subsequent deathRHC Odds ratio 95% CI p-valueNo RefYes 1.25 (1.12,1.40) 0.001From crude association between right heart catheterization therapy anddeath (Table 3.1) we can see that, those patients whom taken right heartcatheterization therapy had 25% (OR = 1.25, 95% CI = 1.12-1.40) higherrisk of death compared to whom did not taken that therapy. Then weexplored the adjusted association by the following adjusted model,logit(Death = “Y es”) = β0 + β1RHC + β2age + β3race + β4Disease.category+β5Cancer + β6dnr.status + β7apache.III.score + β8Pr.2mo.survival+β9myocardial.infraction + β10cardiologic + β11congestive.HF + β12dementia+β13chronic.PD + β14gI.bleeding + β15immunosupp + β16hepaticF+β17myeloma + β18psychiatric.history + β19renal.disease + β20DASI.2wk.prior+β21temperature + β22heart.rate + β23Blood.pressure + β24Respiratory.rate+β25WBC.count + β26PaO2.by.FIO2 + β27PaCO2 + β28pH+β29Creatinine + β30Albumin + β31GComa.ScoreSimilar to the crude association to measure adjusted association betweenright heart catheterization therapy and death we fitted a logistic regressionand reported the odds ratio (OR) with 95% confidence interval (95% CI)(Table 3.2). We found from the adjusted analysis that, patietns whom hadundergone right heart cathetrization therapy had 7% (OR = 1.07, 95% CI= 0.87-1.43) higher risk of death compared to them whom did not take thetherapy.Table 3.2: Unadjusted and adjusted effect of Right heart catheterizationtreatment on deathVariable Unadjusted odds ratio(95% CI)Adjusted odds ratio (95%CI)RHCNo Ref RefYes 1.25 (1.12,1.40) 1.07 (0.87,1.43)Age< 50 Ref Ref50-60 1.62 (1.37,1.91) 1.84 (1.26,2.68)60-70 2.27 (1.94,2.65) 1.82 (1.28,2.60)70-80 2.51 (2.14,2.94) 1.65 (1.13,2.43)> 80 3.74 (3.02,4.63) 2.01 (1.22,3.27RaceWhite 1.04 (0.83,1.31) 0.62 (0.37,1.03)Black 0.99 (0.76,1.27) 0.73 (0.41,1.29)Other Ref RefDisease categoryABF Ref RefCHF 1.05 (0.86,1.29) 1.63 (1.08,2.46)MOSF 1.67 (1.46,1.91) 1.17 (0.81,1.69)Other 1.7 (1.46,1.97) 1.68 (1.15,2.46)CancerNone Ref RefLocalized 2.31 (1.96,2.71) 0.92 (0.63,1.35)Metastic 6.48 (4.58,9.19) 2.59 (1.31,5.09)DNR statusNo Ref RefYes 4.41 (3.46,5.60) 1.6 (0.96,2.69)APACHE.III.score 1.02 (1.01,1.02) 0.99 (0.98,1)Pr.mo.survival 0.009 (0.006,0.01) 0.01 (0.0001,0.05)No of comorbidity 1.32 (1.26,1.39) 1.22 (1.09,1.36)ADLs.2wk.prior 1.15 (1.08,1.22) 1.08 (1,1.16)DASI.2wk.prior 0.93 (0.92,0.94) 0.97 (0.95,0.98)Temperature 0.89 (0.86,0.92) 0.95 (0.87,1.03)Heart rate 0.99 (0.98,1.01) 1 (0.99,1)Blood pressure 0.99 (0.99,1.01) 1 (0.99,1.01)Respiratory rate 1 (1,1.01) 1.01 (1.00, 1.02)WBC count 1 (0.99,1.01) 1.01 (1.00,1.03)Note:To get adjusted treatment effect we have considered, the Age, Race (White, Black,other), Disease category (ABF, CHF, MOSF, Other), Cancer (None. Localized,Metastatic), DNR status (Yes, No), APACHE III score ignoring Coma, Probabilityof survival 2 months, DASI score two weeks prior, Temperature, heart rate, Bloodpressure, Respiratory rate, WBC count, PaO2 by FIO2, PaCO2, pH, Creatinine,Albumin, GComa Score and comorbidity diseases such as, cardiologic disease, con-gestiveHF, dementia, gastrointestinal bleeding, immuno-suppression, Hepatic failure,Chronic myolema, Psychiatric history, renal disease.3.2 Presence of effect modifier in the RightHeart Catheterization (RHC) studyConnors et al. [7] performed subgroup analysis in their study to exploresubgroup-wise treatment effect. They have considered age, gender, race,having sepsis status and post-operative care history of patients. They havegot hazard ratio of death by 30 days higher in elderly people, women, whitecolor, having shock or sepsis, patients receiving post-operative care. To de-tect the subgroup in their analysis, they have calculated the hazard ratiofor each of these variables for their corresponding levels, but their findingswere not statistically significant. King et al. [67] pointed out that, womenwho were undergone Catherterization therapy had higher risk of death com-pared to men similar to the claim made by Connors et al. [7]. From thedataset of the Right Heart Catheterization study we also found the similarassociation between RHC exposure and subsequent death of the critically illpatients who were women. Following table shows association between RHCand death for each level of gender variable:Table 3.3: Crude and adjusted association between treatment Right HeartCatheterization and subsequent death for male and female in the RightHeart Catheterization studyLevel of thesubgroup variableRHC Unadjusted effect(95% CI)Adjusted effect(95% CI)Male No Ref RefYes 1.21 (1.04,1.41) 1.23 (1.02,1.48)Female No Ref RefYes 1.30 (1.09,1.54) 1.62 (1.31,1.99)Note:RHC: Right Heart Catheterization therapy, Adjusted variables were Age, Race(White, Black, other), Disease category (ABF, CHF, MOSF, Other), Cancer(None. Localized, Metastatic), DNR status (Yes, No), APACHE III scoreignoring Coma, Probability of survival 2 months, DASI score two weeks prior,Temperature, heart rate, Blood pressure, Respiratory rate, WBC count, PaO2by FIO2, PaCO2, pH, Creatinine, Albumin, GComa Score and comorbiditydiseases such as, cardiologic disease, congestiveHF, dementia, gastrointestinalbleeding, immuno-suppression, Hepatic failure, Chronic myolema, Psychiatrichistory, renal disease.So we can see that, compared to male patients female had higher risk(adjusted OR = 1.62, 95% CI = 1.31 - 1.99) of death due to catheterization(Table 3.3). Therefore. in our study, we have considered the gender of apatient as an effect modifier.3.3 Analysis of effect modifier in the Right HeartCatheterization (RHC) studyWe have identified various methods to handle the effect modifier in the con-text of PS analysis based on extensive literature review that was discussedin the Chapter 2. We will present the results after applying each of thoseapproaches in the Right Heart Catheterization study:Results from Propensity score effect modifier matching meth-odsMethod M1For the method M1 we have considered the subgroup variable (gender)in the estimation of PS estimation along with other covariates using logisticregression as follows:logit(Pr(RHC = “Y es” | C,Gender)) = β0 + β′C + βgGender (3.2)where C is all the covariates mentioned in the section 3.1.1.Then we performed PS full matching. Then we evaluated the balancediagnostic to evaluate the covariate balance after the method M1 as givenin the figure 3.1 for male and female separately. We have observed that,we have achieved better covariate balance after applying the method M1.After that, to get the crude and adjusted effect of treatment RHC, we per-formed separate outcome model using logistic regression for each level ofeffect modifier variable for method M1 as follows,For crude analysis,logit(Pr(Death = “Y es” | Gender = Male)) = β0 + βtRHC (3.3)logit(Pr(Death = “Y es” | Gender = Female)) = β0 + βtRHC (3.4)and for adjusted analysis,logit(Pr(Death = “Y es” | Gender = Male, C)) = β0 + β′C + βtRHC (3.5)logit(Pr(Death = “Y es” | Gender = Female, C)) = β0 + β′C + βtRHC(3.6)Figure 3.1: Covariate balance plot after performing PS effect modifiermethod M1 based on the levels of subgroup variableMethod M2For method M2 we have considered the subgroup variable (gender) inthe estimation of PS estimation along with other covariates similar to the(3.2). Then we performed PS full matching. Then we evaluated the balancediagnostic to evaluate the covariate balance like method M1 as given inFigure 3.2. We also achieved covariate balance after performing methodM2. After that, to get the crude and adjusted effect of RHC we fitted thefollowing outcome model using logistic regression:For crude association,logit(Pr(Death = “Y es” | RHC,Gender)) = β0 + βtRHC + βgGender+βtgRHC×Gender (3.7)and for adjusted association,logit(Pr(Death = “Y es” | RHC, C,Gender)) = β0 + β′C + βtRHC + βgGender+βtgRHC×Gender(3.8)where C is all the covariates mentioned in the section 3.1.1.Figure 3.2: Covariate balance plot after performing PS effect modifiermethod M2 based on the levels of subgroup variableMethod M3For method M3, like method M2 and M1, we have considered the sub-group variable (gender) in the estimation of PS estimation along with othercovariates and estimated the propensity score by following equation (3.2).Then we performed PS full matching, besides full matching, we did exactmatching the effect modifier variable (gender). Then we evaluated the bal-ance diagnostic to evaluate the covariate balance as given in Figure 3.3 andwe found that we have got covariate balance from the method M3. Afterthat, to get the crude and adjusted effect of treatment RHC, we performedoutcome model by including subgroup variable interaction with treatmentRHC variable following equations (3.7) and (3.8).Figure 3.3: Covariate balance plot after performing PS effect modifiermethod M3 based on the levels of subgroup variableMethod M4For method M4, we separated the dataset into two groups based on thelevel of the subgroup variable. Then we calculated the separate PS scoreconsidering the covariates for each dataset as follows,logit(Pr(RHC = “Y es” | C,Gender = Male)) = β0 + β′C (3.9)logit(Pr(RHC = “Y es” | C,Gender = Female)) = β0 + β′C (3.10)where C is all the covariates mentioned in the section 3.1.1.For each dataset we performed the PS full matching. Then we performedthe balance diagnostic (figure 3.4 and 3.5) separately for male and female.After ensuring covariate balance within male and female we then aggregatedthe matched data from male and female. We then fitted the outcome modelboth for crude and adjusted model from the matched data using weights inthe logistic regression by considering an interaction term with the treatmentand effect modifier following equations (3.7) and (3.8).Figure 3.4: Covariate balance plot after performing PS effect modifiermethod M4 for MaleFigure 3.5: Covariate balance plot after performing PS effect modifiermethod M4 for FemaleMethod M5For method M5, we needed to calculate the PS score by including co-variates and interactions with the suitable covariates and effect modifier. Inour analysis, we considered the interaction of age, race, education, incomeand blood pressure (bp) with the effect modifier gender. Then we added allthe covariates and interactions in the estimation of PS as follows:logit(Pr(RHC = “Y es” | C,Gender)) = β0 + β′C + β1gage×Gender+β2grace×Gender + β3geducation×Gender+β4gincome×Gender + β5gbp×Gender(3.11)Then we performed full matching and balance evaluation (Figure 3.6)and after getting covariate balance we fitted the outcome model similar tomethod M4 using logistic regression by following equations (3.7) and (3.8).Figure 3.6: Covariate balance plot after performing PS effect modifiermethod M5 based on the levels of subgroup variableMethod M6For method M6, we calculated the PS score as mentioned in the equation(3.11). Then we performed full matching and exact matching on effectmodifier variable. Other balance checking (Figure 3.7) and outcome modelwas same as method M5 using logistic regression by following equations (3.7)and (3.8).Figure 3.7: Covariate balance plot after performing PS effect modifiermethod M6 based on the levels of subgroup variableDistribution of standardized mean difference for all propensityscore matching approach at each of the subgroup levelGiven below the distribution of standardized mean difference after per-forming the propensity score matching methods to handle effect modifierfigure (3.8 and 3.9)Figure 3.8: Distribution of standardized mean difference for all PS matchingapproach for MaleFigure 3.9: Distribution of standardized mean difference for all PS matchingapproach for FemaleWe can see that, for all propensity score effect modifier methods medianstandardized mean difference was same and also equal to the unmatcheddata. This happened because the in the right heart catheterization study theauthors used data from hospitals where all patients had similar backgroundcharacteristics were admitted. That’s why we have achieved similar balancefor all the matching methods of propensity score effect modifier.Unadjusted and adjusted treatment effect at each level of ef-fect modifier after applying the propensity score effect modifiermatching methodsGiven below the results from the analysis of treatment effect after con-ducting propensity score matching methods to handle effect modifier at itseach level:Table 3.4: Crude and adjusted treatment effect from propensity score effectmodifier matching methods M1-M6PS matchingmethodsLevel of thegenderUnadjusted effect(95% CI)Adjusted effect(95% CI)M1 Male 1.14 (0.98,1.32) 1.18 (0.99,1.41)Female 1.47 (1.24,1.75) 1.86 (1.51,2.30)M2 Male 1.14 (0.98,1.32) 1.22 (1.03,1.45)Female 1.47 (1.24,1.75) 2.01 (1.63,2.46)M3 Male 1.21 (1.04,1.40) 1.29 (1.09,1.53)Female 1.44 (1.21,1.71) 1.91 (1.56,2.34)M4 Male 0.97 (0.83,1.13) 1.06 (0.89,1.27)Female 1.46 (1.23,1.73) 1.83 (1.50,2.23)M5 Male 1.11 (0.95,1.28) 1.26 (1.07,1.50)Female 1.47 (1.23,1.75) 1.95 (1.59,2.40)M6 Male 1.11 (0.96,1.29) 1.24 (1.04,1.47)Female 1.52 (1.28,1.80) 1.85 (1.50,2.26)Note:Adjusted variables were Age ,Race, Disease category, Cancer, Do Not Re-suscitate status, APACHE III score ignoring Coma, Probability of survival2 months,DASI.2wk.prior, Temperature, heart rate, Blood pressure, Respira-tory rate, WBC count, PaO2 by FIO2, PaCO2, pH, Creatinine, Albumin,GComa Score and comorbidity diseases such as, cardiologic disease, conges-tiveHF, dementia, gastrointestinal bleeding, immuno-suppression, Hepatic fail-ure, Chronic myolema, Psychiatric history, renal disease.Results from Propensity score effect modifier weighting meth-odsMethod W1For method W1 we have considered the subgroup variable (gender) inthe estimation of PS estimation along with other covariates and estimatedby following equation (3.2) using logistic regression. Then we calculated theinverse probability treatment weight as described in the methods section fortreated and untreated patients as follows,wi =RHCipsi+1− RHCi1− psi (3.12)where, psi is the estimated propensity score for the individual i. This weightshould be added in the original dataset.We calculated the standardized mean difference for each covariates sepa-rately for subgroup variable label figure (3.10 and 3.11). Then from weighteddata, we estimated the treatment effect by considering an interaction termwith the treatment and effect modifier in the outcome model using logisticregression by following equations (3.7) and (3.8).Method W2For method W2, we have estimated the propensity score similar to themethod M5. Then calculated the inverse probability treatment weight fol-lowing equation (3.12) and finally outcome model using logistic regressionby following equations (3.7) and (3.8).Method W3To perform method W3, we separated the dataset into male and femalegroup. Then we calculated the PS following equations (3.9) and (3.10). Theninverse probability treatment weight as described in the methods section fortreated and untreated patients for each dataset as follows,wi,Gender=Male =RHCi,Gender=Malepsi,Gender=Male+1− RHCi,Gender=Male1− psi,Gender=Male (3.13)andwi,Gender=Female =RHCi,Gender=Femalepsi,Gender=Female+1− RHCi,Gender=Female1− psi,Gender=Female (3.14)We calculated the standardized mean difference for each covariates sepa-rately for subgroup variable label (figure 3.10 and 3.11). Then from weighteddata, we estimated the treatment effect by the outcome model using logisticregression by following equations (3.7) and (3.8).Method W4For method W4, we have estimated the PS by considering subgroup vari-able (gender) as a covariate. Then using the PS we calculated the stabilizedweight for treatment as mentioned in the method section for the methodW4 as follows,SW (Gender) =P (RHC = “Y es” | Gender)P (RHC = “Y es” | C,Gender) (3.15)where the denominator was calculated as follows,P (RHC = “Y es” | C,Gender) = β0 + β′C + βgGender (3.16)and the numerator was calculated as follows,P (RHC = “Y es” | Gender) = β0 + βgGender (3.17)From the weighted data, we evaluated the standardized mean difference(figure 3.10 and 3.11) and estimated the adjusted and unadjusted treatmenteffect using logistic regression by following equations (3.7) and (3.8).Method W5For method W5, we have estimated the PS similar to the method W2considering useful interactions with subgroup variable using logistic regres-sion by following equation (3.11). Then calculated the stabilized weight fortreatment similar to the equation (3.15) and finally outcome model usinglogistic regression by following equations (3.7) and (3.8).Method W6For method W6, we have estimated the PS similar to the method W1following equation (3.12). Then calculated the stabilized weight for treat-ment similar to the equation (3.15). We separated the weighted data intomale and female group and then evaluated the standardized mean difference(figure 3.10 and 3.11) and estimated the adjusted and unadjusted treatmenteffect using logistic regression by following equations (3.7) and (3.8).We have presented the distribution of standardized mean differences af-ter propensity score weighting approach for each level of subgroup variable(male and female). We observed that, almost all the propensity score weight-ing methods perfomed better by producing lower standardized mean differ-ence, Among them the method W3 produced the lowest standardized meandifference which was estimation of propensity score separately for male andfemale and performing PS weighting separately.Distribution of standardized mean difference for propensityscore weighting approach at each of the subgroup levelFigure 3.10: Distribution of standardized mean difference for PS weightingapproach for MaleUnadjusted and adjusted treatment effect at each level of ef-fect modifier after applying the propensity score effect modifierweighting methodsGiven below the results from the analysis of treatment effect after con-ducting propensity score weighting methods to handle effect modifier at itseach level:Figure 3.11: Distribution of standardized mean difference for all PS match-ing approach for FemaleTable 3.5: Crude and adjusted treatment effect from propensity score effectmodifier weighting methods W1-W6PS weightingmethodLevel of thegenderUnadjusted effect(95% CI)Adjusted effect(95% CI)W1 Male 1.20 (1.08,1.34) 1.16 (1.03,1.30)Female 1.36 (1.21,1.53) 1.79 (1.56,2.05)W2 Male 1.17 (1.05,1.30) 1.15 (1.02,1.29)Female 1.39 (1.23,1.56) 1.75 (1.52,2.00)W3 Male 1.08 (0.98,1.20) 1.12 (0.99,1.26)Female 1.48 (1.32,1.66) 1.73 (1.51,1.98)W4 Male 1.20 (1.03,1.40) 1.16 (0.97,1.37)Female 1.36 (1.14,1.62) 1.78 (1.46,2.18)W5 Male 1.17 (1.01,1.36) 1.15 (0.97,1.37)Female 1.39 (1.16,1.65) 1.42 (1.42,2.12)W6 Male 1.20 (1.08,1.34) 1.13 (1.00,1.28)Female 1.36 (1.21,1.53) 1.38 (0.94,2.03)Note:Adjusted variables were Age , Sex, Race, Disease category, Cancer, Do NotResuscitate status, APACHE III score ignoring Coma, Probability of survival2 months,DASI.2wk.prior, Temperature, heart rate, Blood pressure, Respira-tory rate, WBC count, PaO2 by FIO2, PaCO2, pH, Creatinine, Albumin,GComa Score and comorbidity diseases such as, cardiologic disease, conges-tiveHF, dementia, gastrointestinal bleeding, immuno-suppression, Hepatic fail-ure, Chronic myolema, Psychiatric history, renal disease.Results from Propensity score effect modifier stratification meth-odsMethod S1For method S1, we have considered the subgroup variable (gender) inthe estimation of PS estimation along with other covariates similar to theequation (3.2). Then we sorted the PS and divided the data into five groupsbased on the PS. Within each stratum we evaluated the distribution of PSfor treated and untreated patients figure (3.12 and 3.13). After that, withineach stratum we calculated the unadjusted and adjusted treatment effectusing logistic regression by following equations (3.7) and (3.8) and thenpooled them to get the weighted average effect as follows,ATE =∑sNsNATEs (3.18)Figure 3.12: Distribution of standardized mean difference for PS stratifica-tion approach method S1 for MaleMethod S2For method S2, we estimated the PS by incorporating the useful interac-tions with gender and other covariates using logistic regression by followingequation (3.11). Then we sorted the PS and divided the data into five groupsbased on the PS. Within each stratum we evaluated the distribution of PSfor treated and untreated patients Figure (3.14 and 3.15). After that, withinFigure 3.13: Distribution of standardized mean difference for PS stratifica-tion approach method S1 for Femaleeach stratum we calculated the unadjusted and adjusted treatment effect us-ing logistic regression by following equations (3.7) and (3.8) and then pooledthem to get the weighted average effect following equation (3.18)Figure 3.14: Distribution of standardized mean difference for PS stratifica-tion approach method S2 for MaleFigure 3.15: Distribution of standardized mean difference for PS stratifica-tion approach method S2 for FemaleMethod S3For method S3, we separated the dataset into male and female group.Then estimate PS for each dataset separately by following equations (3.9)and (3.10). Then stratified each dataset based on PS and calculated thestratum specific and average unadjusted and adjusted treatment effect usinglogistic regression by following equations (3.7) and (3.8) and then pooledthem to get the weighted average effect following equation (3.18) separately.We presented the distribution of the PS in each stratum for treated anduntreated patient from both datasets (3.16 and 3.17).Figure 3.16: Distribution of standardized mean difference for PS stratifica-tion approach method S3 for MaleFigure 3.17: Distribution of standardized mean difference for PS stratifica-tion approach method S3 for FemaleUnadjusted and adjusted treatment effect at each level of ef-fect modifier after applying the propensity score effect modifierstratification methodsGiven below the results from the analysis of treatment effect after con-ducting propensity score stratification methods to handle effect modifier atits each level:Table 3.6: Crude and adjusted treatment effect after performing propensityscore effect modifier methods S1-S3PS stratificationmethodLevel of thegenderUnadjusted effect(95% CI)Adjusted effect(95% CI)S1 Male 1.29 (0.93,1.77) 1.25 (0.89,1.74)Female 1.35 (0.95,1.92) 1.39 (0.97,1.99)S2 Male 1.28 (0.92,1.76) 1.27 (0.90,1.7)Female 1.43 (1.00,2.05) 1.43 (0.99,2.07)S3 Male 1.4 (0.95,2.08) 1.13 (0.81,1.57)Female 1.23 (0.50,2.98) 1 (0.92,1.09)Note:Adjusted variables were Age , Sex, Race, Disease category, Cancer, Do NotResuscitate status, APACHE III score ignoring Coma, Probability of survival2 months,DASI.2wk.prior, Temperature, heart rate, Blood pressure, Respira-tory rate, WBC count, PaO2 by FIO2, PaCO2, pH, Creatinine, Albumin,GComa Score and comorbidity diseases such as, cardiologic disease, conges-tiveHF, dementia, gastrointestinal bleeding, immuno-suppression, Hepatic fail-ure, Chronic myolema, Psychiatric history, renal disease.Results from Propensity score covariate adjustment methodsMethod C1For method C1, we have calculated the PS considering covariates onlyas follows,logit(Pr(RHC = “Y es” | C)) = β0 + β′C (3.19)and use the PS as a covariate along with interaction term between treat-ment and subgroup variable in the outcome model using logistic regressionas follows,For crude association,logit(Pr(Death = “Y es” | ps,RHC,Gender)) = β0 + β1ps + βtRHC+βgGender + βtgRHC×Gender (3.20)and for adjusted association,logit(Pr(Death = “Y es” | ps, RHC,C,Gender)) = β0 + β′C + β1ps+βtRHC + βgGender + βtgRHC×Gender (3.21)where ps is the estimated propensity score.Method C2For method C2, at first we have divided the dataset into male and femalegroups. Then calculated the PS separately for each group by followingequation (3.19). Then used that PS to fit separate outcome model followingequations (3.20) and (3.21).Method C3For method C3, we have calculated the PS considering covariates andsubgroup variable as a covariate and also considering the useful interactionsbetween them following equation (3.11). Then use the PS as a covariatealong with interaction term between treatment and subgroup variable inthe outcome model following equations (3.20) and (3.21).Method C4For method C4, we have a formed a combination variable by consideringlevels of treatment and subgroup variable. Then used that combinationvariable as a treatment variable we fitted multinomial logistic regression tocalculate the PS. Then fitted the the outcome model considering 3 estimatedgeneralized PSs (considering the 1st category (0,0) as baseline group) and 3dummy variables denoting the group membership (G) as follows:logit(Pr(Death = “Y es” | ps,G)) = β0 + β1ps1 + β2ps2 + β3ps3+β4G1 + β5G2 + β6G3 (3.22)where ps is the estimated propensity score.Other covariates were considered in case of adjusted model.Method C5For method C5, we have calculated the PS considering covariates andsubgroup variable as a covariate similar to the equation (3.2). Then use thePS as a covariate along with interaction term between treatment and sub-group variable in the outcome model following equations (3.20) and (3.21).Method C6For method C6, we have calculated the PS considering covariates andsubgroup variable as a covariate similar to the equation (3.2). Then use thePS as a covariate to fit separate outcome model for each level of subgroupvariable following equations (3.20) and (3.21).Unadjusted and adjusted treatment effect at each level of ef-fect modifier after applying the propensity score effect modifiercovariate adjustment methodsGiven below the results from the analysis of treatment effect after con-ducting propensity score covariate adjustment methods to handle effect mod-ifier at its each level:Table 3.7: Crude and adjusted treatment effect after performing propensityeffect modifier methods C1-C6PS covariateadjustmentmethodsLevel of thegenderUnadjusted effect(95% CI)Adjusted effect(95% CI)C1 Male 1.26 (1.07,1.49) 1.25 (1.04,1.51)Female 1.35 (1.12,1.62) 1.64 (1.33,2.03)C2 Male 1.12 (0.94,1.33) 1.17 (0.96,1.44)Female 1.52 (1.24,1.86) 1.69 (1.35,2.13)C3 Male 1.26 (1.07,1.48) 1.25 (1.03,1.50)Female 1.35 (1.12,1.62) 1.64 (1.33,2.02)C4 Male 1.09 (0.91,1.31) 1.12 (0.91,1.37)Female 1.16 (0.98,1.37) 1.24 (1.02,1.49)C5 Male 1.26 (1.07,1.48) 1.25 (1.04,1.51)Female 1.35 (1.12,1.62) 1.64 (1.33,2.02)C6 Male 1.12 (0.94,1.33) 1.17 (0.96,1.44)Female 1.52 (1.24,1.86) 1.69 (1.35,2.13)Note:Adjusted variables were Age , Sex, Race, Disease category, Cancer, Do NotResuscitate status, APACHE III score ignoring Coma, Probability of survival2 months,DASI.2wk.prior, Temperature, heart rate, Blood pressure, Respira-tory rate, WBC count, PaO2 by FIO2, PaCO2, pH, Creatinine, Albumin,GComa Score and comorbidity diseases such as, cardiologic disease, conges-tiveHF, dementia, gastrointestinal bleeding, immuno suppression, Hepatic fail-ure, Chronic myolema, Psychiatric history, renal disease.Chapter 4Two Data GenerationMechanisms for SimulationIn this chapter we have described the simulation methods that we followedto implement all of the methods that described in the Chapter 2. We haveprovided a Monte Carlo simulation and a plasmode simulation based on thedataset from Right Heart Catheterization study conducted by Connors etal. [7].4.1 Monte Carlo simulation4.1.1 Monte Carlo simulation without consideringinteraction between subgroup variable and a covariate4.1.2 Simulation settingsTo perform Monte-Carlo simulation we have followed the simulation strat-egy described in Setoguchi et al. [30] with modification to generate effectmodifier variable. We have considered a subgroup variable in our study,say, gender having levels male and female. We considered the n = 1000subjects for the simulation. For each the n=1000 subjects, we generated10 covariates Wi. Among those 10 covariates four (W1 − W4) were con-founders, three (W5 −W7) were predictors of exposure variable only, and,three (W8−W10) were the predictors of outcome only. In our study we con-sidered the exposure variable, say, A as binary and a continuous outcomeY . In 4.1 we have provided a DAG for this simulation. Among 10 covariatessix (W1,W3,W5,W6,W8,W9) were binary and four (W2,W4,W7,W10) werecontinuous nature.4.1.3 Data generationCovariates of simulation studyAt first, we considered a subgroup variable with two levels of size equalto the half of the dataset at each level. Then for each level we, generated72Figure 4.1: DAG for the simulation studyeach of the covariates Wi(i = 1, 2, 3, 4, 7, 10) from a normal distribution withmean = 0 and standard deviation = 1. Other covariates, Wi(i = 5, 6, 8, 9)were generated by considering the correlation with other covariates. Thecorrelation between W1 and W5, W2 and W6, W3 and W8 and W4 andW9, were set to 0.2, 0.9, 0.2, and 0.9, respectively. Then the covariates,W1,W3,W5,W6,W8, and W9 were dichotomized, by setting the value 1 ifeach value of the covariates greater than its mean and 0 otherwise. Bydichotomizing these covariates would weaken the values of these covariates.Exposure of simulation studyThe exposure variable was generated using propensity score for variousscenarios. For each scenario, the logistic regression was used to calculatepropensity score p(A|Wi). Then to generate the binary exposure variable,a random number from a uniform(0, 1) distribution and then by settingA = 1 if the random number was less than the previously calculated PSscore and A = 0 otherwise. The details of the scenario is given in the nextsection.4.1.4 Scenarios of simulation study without consideringinteraction between subgroup variable and a covariateWe have considered several scenarios by differing the nature of the associa-tion between the exposure A and the covariates Wi. Based on the differentassumption of the structure of the association between exposure and covari-ates we have considered total seven scenarios (A - G) as follows:• Scenario A: model with additivityThis scenario was the simplest one, where we assumed the associationbetween exposure and covariates was linear and additive effect of the covari-ates as follows:P (A = 1 |Wi) = (1 + exp(−(0 + b1w1 + b2w2 + b3w3 + b4w4 +b5w5 + b6w6 + b7w7)))−1 (4.1)• Scenario B: model with mild non-linearityIn this scenario, a mild non-linearity was considered by assuming theassociation between exposure and covariates linear. And mild non-linearityby adding a quadratic term of a covariate as follows:P (A = 1 |Wi) = (1 + exp(−(0 + b1w1 + b2w2 + b3w3 + b4w4+b5w5 + b6w6 + b7w7 + b2w22)))−1 (4.2)• Scenario C: model with moderate non-linearityBesides linear association between exposure and covariates, we assumedthe moderate non-linearity using the two more variables with quadratic termas follows:P (A = 1 |Wi) = (1 + exp(−(0 + b1w1 + b2w2 + b3w3 + b4w4 + b5w5+b6w6 + b7w7 + b2w22 + b4w24 + b7w27)))−1 (4.3)• Scenario D: model with mild non-additivityUnder this scenario we have added four interaction terms in the modelby assuming mild non-additivity as follows:P (A = 1 |Wi) = (1 + exp(−(0 + b1w1 + b2w2 + b3w3 + b4w4 + b5w5 +b6w6 + b7w7 + b1 × 0.5× w1w3 + b2 × 0.7× w2w4 +b4 × 0.5× w4w5 + b5 × 0.5× w5w6)))−1(4.4)• Scenario E: model with mild non-additivity and non-linearityTo model by assuming non-additivity and non-linearity we have includeda quadratic term and four interaction terms in the model as follows:P (A = 1 |Wi) = (1 + exp(−(0 + b1w1 + b2w2 + b3w3 + b4w4 + b5w5 +b6w6 + b7w7 + b2w22 + b1 × 0.5× w1w3 + b2 × 0.7× w2w4 +b4 × 0.5w4w5 + b5 × 0.5× w5w6)))−1(4.5)• Scenario F: model with moderate non-additivityIn this scenario, we have added ten interaction terms and along with aquadratic term in the model as follows:P (A = 1 |Wi) = (1 + exp(−(0 + b1w1 + b2w2 + b3w3 + b4w4 + b5w5 + b6w6+b7w7 + b1 × 0.5× w1w3 + b2 × 0.7× w2w4 + b3 × 0.5× w3w5+b4 × 0.7× w4w6 + b5 × 0.5× w5w7 + b1 × 0.5× w1w6+b2 × 0.7× w2w3 + b3 × 0.5× w3w4 + b4 × 0.5× w4w5+b5 × 0.5× w5w6)))−1(4.6)• Scenario G: moderate non-additivity and non-linearityUnder this scenario more complex association between exposure and co-variates have been assummed, by adding three quadratic terms and teninteraction terms in the model as follows:P (A = 1 |Wi) = (1 + exp(−(0 + b1w1 + b2w2 + b3w3 + b4w4 + b5w5 + b6w6 +b7w7 + b2w2w2 + b4w4w4 + b7w7w7 + b1 × 0.5× w1w3 +b2 × 0.7× w2w4 + b3 × 0.5× w3w5 + b4 × 0.7× w4w6 +b5 × 0.5× w5w7 + b1 × 0.5× w1w6 +b2 × 0.7× w2w3 + b3 × 0.5× w3w4 +b4 × 0.5× w4w5 + b5 × 0.5× w5w6)))−1(4.7)Outcome model of simulation studyTo generate outcome Y for each scenario we have considered the func-tion of covariates Wi and treatment A. For our study, we were interested toaddress the effect modifier, therefore, we changed the outcome generationsuch that, we generated the outcome for each level of the subgroup variableseparately, then we combined each dataset to get the combined dataset foranalysis. For each level of subgroup variable, during the outcome Yi gen-eration we varied the coefficient of the treatment variable A. The outcomemodel for each level of subgroup variable has been generated as follows:Y = a0 + a1w1 + a2w2 + a3w3 + a4w4 + a5w8 + a6w9 + a7w10 + γ1AandY = a0 + a1w1 + a2w2 + a3w3 + a4w4 + a5w8 + a6w9 + a7w10 + γ2ARegression coefficient values were considered as: b0 = 0; b1 = 0.8; b2 =−0.25; b3 = 0.6; b4 = −0.4; b5 = −0.8; b6 = −0.5; b7 = 0.7; a0 = −3.85;a1 = 0.3; a2 = −0.36; a3 = −0.73; a4 = −0.2; a5 = 0.71; a6 = −0.19;a7 = 0.26; γ1 = 0.7; γ2 = 1.3For each of the scenario we have performed 1000 simulation by varyingthe random seed from 1 to 1000. For each iteration we considered the datasethaving size n=1000, We have applied all the methods discussed in Chapter2. In the propensity score matching approach, we have followed the PSmatching methods (full matching) to do matching after estimating the PSfor each scenario as mentioned in the original paper.4.1.5 Monte Carlo simulation: considering an interactionbetween subgroup variable and a covariateIn the above-mentioned simulation scenarios there were no interaction be-tween covariate and subgroup variable. In reality there is a possibility tohave joint effect between subgroup and covariates, which is important toconsider. To mimic this situation we have considered a covariate randomlychosen to form interaction with subgroup variable in the simulation scenar-ios. That’s why we have modified the above-mentioned simulation scenariosby adding considering a joint effect between subgroup variable and a covari-ate. We considered the covariate w5 to form a joint effect with subgroupvariable and modified the above-mentioned simulation scenarios as follows:4.1.6 Scenarios of simulation study with consideringinteraction between subgroup variable and a covariate• Scenario A1: model with additivityP (A = 1 |Wi) = (1 + exp(−(0 + b1w1 + b2w2 + b3w3 + b4w4 +b5w5 + b6w6 + b7w7 + γ3 × w5G)))−1 (4.8)• Scenario B1: model with mild non-linearityP (A = 1 |Wi) = (1 + exp(−(0 + b1w1 + b2w2 + b3w3 + b4w4+b5w5 + b6w6 + b7w7 + b2w22 + γ3 × w5G)))−1 (4.9)• Scenario C1: model with moderate non-linearityP (A = 1 |Wi) = (1 + exp(−(0 + b1w1 + b2w2 + b3w3 + b4w4 + b5w5+b6w6 + b7w7 + b2w22 + b4w24 + b7w27 + γ3 × w5G)))−1(4.10)• Scenario D1: model with mild non-additivityP (A = 1 |Wi) = (1 + exp(−(0 + b1w1 + b2w2 + b3w3 + b4w4 + b5w5 +b6w6 + b7w7 + b1 × 0.5× w1w3 + b2 × 0.7× w2w4 +b4 × 0.5× w4w5 + b5 × 0.5× w5w6 + γ3 × w5G)))−1(4.11)• Scenario E1: model with mild non-additivity and non-linearityP (A = 1 |Wi) = (1 + exp(−(0 + b1w1 + b2w2 + b3w3 + b4w4 + b5w5 +b6w6 + b7w7 + b2w22 + b1 × 0.5× w1w3 + b2 × 0.7× w2w4 +b4 × 0.5w4w5 + b5 × 0.5× w5w6 + γ3 × w5G)))−1(4.12)• Scenario F1: model with moderate non-additivityP (A = 1 |Wi) = (1 + exp(−(0 + b1w1 + b2w2 + b3w3 + b4w4 + b5w5 + b6w6 +b7w7 + b1 × 0.5× w1w3 + b2 × 0.7× w2w4 +b3 × 0.5× w3w5 + b4 × 0.7× w4w6 +b5 × 0.5× w5w7 + b1 × 0.5× w1w6 +b2 × 0.7× w2w3 + b3 × 0.5× w3w4 +b4 × 0.5× w4w5 + b5 × 0.5× w5w6 + γ3 × w5G)))−1(4.13)• Scenario G1: moderate non-additivity and non-linearityP (A = 1 |Wi) = (1 + exp(−(0 + b1w1 + b2w2 + b3w3 + b4w4 + b5w5 + b6w6 +b7w7 + b2w2w2 + b4w4w4 + b7w7w7 + b1 × 0.5× w1w3 +b2 × 0.7× w2w4 + b3 × 0.5× w3w5 + b4 × 0.7× w4w6 +b5 × 0.5× w5w7 + b1 × 0.5× w1w6 +b2 × 0.7× w2w3 + b3 × 0.5× w3w4 +b4 × 0.5× w4w5 + b5 × 0.5× w5w6 + γ3 × w5G)))−1(4.14)In all scenarios we considered γ3 = −3 and γ3 = 3 when the subgroupvariable G = 0 and G = 1 respectively by keeping other parameter valuesand outcome model same as we did for Monte Carlo simulation where didnot consider any interaction between a covariate and subgroup variable.4.2 Plasmode simulationMonte Carlo simulation is a way to investigate the performance of vari-ous methods of addressing effect modifier under PS context. On the otherhand, real world databases often have hundreds, or even thousands of mea-sured covariates with complex covariance structures. Beyond confoundingbias in the real-world data there are several other situation exits, such as,patient follow-up time and censoring patterns may be associated with ex-posure and outcome due to the underlying baseline characteristics. ButMonte Carlo simulation approaches for evaluating the performance of thestatistical methods, did not capture the correlation structure between co-variates and exposure of real-world data. Moreover, sometimes the MonteCarlo simulation of data-generation process may produce spurious correla-tions among the variables that were not present in the actual dataset. Onthe other hand, due to the development in electronic health care databasesand inability to perform randomized controlled trial, many epidemiologicalstudies have been carried out based on the dataset available in the healthcare databases. Any simulation that would capture the real scenario ofthe association between various covariates under study would be helpful tounderstand the performance of different statistical methods. In this case,another simulation strategy known as “Plasmode” simulation can be appli-cable. Basically, a plasmode is a simulated dataset that is created from areal-world dataset such a way that, the simulated data have the reflection ofnature of the complex relationship within covariates similar to the real-worlddata. This is a type of simulation process where subjects were sampled withreplacement from the observed data by keeping the subjects’ covariates as itis and simulates exposure, outcome or both. Besides this, under plasmodesimulation framework, during simulation the researcher can assign a desiredvalue for the treatment effect in the data generating processThe plasmode simulation uses the boostrap sampling technique to dosampling with replacement from the real world data. In each bootstrapsampling, a bootstrap sample of size similar to the original real world datais collected from a complete set of covariates. Within each sample, noneof the covariates are modified or changed. Through this process in eachbootstrap sample, the relationship among covariates, exposure, remain sameas the original real world data. Therefore, like Monte Carlo simulation thedistribution of covariates do not need to be fixed arbitrarily. Given belowthe steps of performing plasmode simulation:• Base cohort construction• Covariates selection for simulation• Estimate outcome• Predict outcome• Resampling and perform simulation• Analyze the resampled dataPlasmode simulation of Right heart Catheterization dataTo compare the various methods of handling effect modifier under PS con-text, we performed a plasmode simulation in addition to Monte Carlo sim-ulation. At Monte Carlo simulation we have decided the distribution ofcovariates, exposure, outcome arbitrarily and in different scenarios we eval-uated the effect modifier methods. To compare those effect modifier methodsin more rigorous way, we performed a plasmode simulation in our study. Thedetails of our plasmode simulation is given below:Steps of plasmode simulation• Step 1: Base cohort selectionTo perform plasmode simulation, we have selected a publicly avail-able dataset from a study conducted by Connors et al. [7], wherethe researchers wanted to explore the association between exposure toRight Heart Catheterization (RHC) therapy, also known as Swan-Ganzcatheter and subsequent death among the patient in the intensive careunit. This study was a prospective cohort study based on the severelyill hospitalized adult patients at 5 medical centers in the U.S during1989 to 1994.As we mentioned earlier, the study included two phases: Phase I (aprospective observational study to describe the decisions and processof care for severely ill patients) and Phase II (a clustered randomizedtrial to explore the effect of interventions). In the phase I there were4301 patients and 4804 patients were enrolled in the phase II. Connorset al. [7] in their study combined the two phases datasets for theiranalysis. In total there were 5735 patients, 2184 treated patients and3551 control patients. For each subject the observed treatment status,equal to 1 if RHC was applied within 24 hours of admission, and 0otherwise. Clinical outcome was an indicator for death within 30 or60 or 180 days after study entry. There were 68% of the RHC patientsthat had clinical outcome compared to 63% of the No RHC patients.The study was an observational study and confounding bias was one ofthe major issues in the analysis. For this reason, the authors adoptedthe propensity score analysis approach to perform the measure of theeffect of RHC on subsequent death within 30 days.For our study, we considered the patient’s gender status as an effectmodifier. Among the 5,735 patients, 1500 were female and 3300 weremale. Therefore, similar to the Monte Carlo simulation study, we havedivided the whole dataset into two datasets for each level of the effectmodifier with respective size. These two datasets were used as thebase cohorts for plasmode simulation.• Step 2: Covariates selection and defining outcomeFor each set dataset in our study, our outcome was death, expo-sure was RHC catheterization and subgroup variable was gender. Wehave selected the following covariates to model the exposure and out-come model: Age , Race (White, Black, other), Disease category(ABF,CHF,MOSF,Other), Cancer (None. Localized, Metastatic), DoNot Resuscitate status (Yes, No), APACHE III score ignoring Coma,Probability of survival 2 months,DASI.2wk.prior, Temperature, heartrate, Blood pressure, Respiratory rate, WBC count, PaO2 by FIO2,PaCO2, pH, Creatinine, Albumin, GComa Score and comorbidity dis-eases such as, cardiologic disease, congestiveHF, dementia, gastroin-testinal bleeding, immuno suppression, Hepatic failure, Chronic my-olema, Psychiatric history, renal disease.• Step 3: Predicting outcomeTo simulate outcome according to our desired value of treatment vari-able RHC, we at first fitted the following logistic regresssion separatelyfor male and female,logit(Pr(Death = “Y es” | Gender = Male, C)) = β0 +β′C+βtRHC (4.15)logit(Pr(Death = “Y es” | Gender = Female, C)) = β0 + β′C + βtRHC(4.16)where C is the covariates mentioned in step 2.• Step 4: Resampling and performing simulationBy setting up the outcome model in the previous step, we have gener-ated total 1000 datasets from each cohort where each had size n=4000.In each of the simulated dataset we had two variables, patient id andoutcome.• Step 5: Analyze the resampled dataAfter performing resampling from the base cohort for male and fe-male separately each of the two simulated datasets had been mergedwith each other to get the complete data from both cohort. For eachcompleted dataset, we then performed each of the PS effect modifiermethods. Finally, we averaged the treatment effect and other per-formance measure discussed in the next section obtained from eachsimulated plasmode dataset.After that, we have used the regression coefficients from these mod-els to build the outcome model for each of the simulated dataset bykeeping the patients original covariance structure. We have insertedour desired treatment effect in male and female group separately andusing the effect we have predicted the outcome.logit(Pr(Death = “Y es” | Gender = Male, C)) = β0 + β′C + αtRHC(4.17)logit(Pr(Death = “Y es” | Gender = Female, C)) = β0 +β′C+αtRHC(4.18)where C is the covariates mentioned in step 2 and for male and femalethe values of αt are log(1.23) and log(1.62) respectively.We performed the plasmode simulation both for unadjusted and ad-justed cases. In addition to that, we also performed plasmode sim-ulation for a reduced sample size (n=1000) situation to evaluate anyvariation in performance of simulation.4.3 Performance measures of simulation studyTo assess the performance of the various methods in the context of propen-sity score to address effect modifier, we used the following performance mea-sures both for Monte Carlo and Plasmode simulation as described by [64].• Bias:∑Ni=1(θˆi − θ)N, i = 1, 2, ..., n (4.19)• Absolute Bias:∑Ni=1 |(θˆi − θ)|N, i = 1, 2, ..., n (4.20)• Percent Bias [68]:∑Ni=1(θˆi − θ)θ× 100, i = 1, 2, ..., n (4.21)• Empirical Standard error (EmpSE):√∑Ni=1(θˆi − θ¯)2N − 1 , i = 1, 2, ..., n (4.22)whereθ¯ =∑Ni=1 θˆiN(4.23)• Mean squared error (MSE):∑Ni=1(θˆi − θ)2N, i = 1, 2, ..., n (4.24)Other than bias, mean squared error, coverage, we are not very interestedabout rest as we do not have a specific hypothesis to test, but we reportedfor sake of completeness.• Model standard error (modelSE):√∑Ni=1 se(θˆi)2N, i = 1, 2, ..., n (4.25)• Power:∑Ni=1 1(pi ≤ α)N, i = 1, 2, 3, ..., n (4.26)• Coverage:P (θˆlower ≤ θ ≤ θˆupper), i = 1, 2, 3, ..., n (4.27)• Confidence interval length:∑Ni=1 se(θˆi)2 × zαN, i = 1, 2, 3, ..., n (4.28)• Bias-adjusted coverage:P (θˆlower ≤ θ¯ ≤ θˆupper), i = 1, 2, 3, ..., n (4.29)Chapter 5Simulation ResultsThis chapter provides the Monte Carlo simulation results for each of thescenarios described in Chapter 4. We also provided the plasmode simula-tion results based on the Right Heart Catheterization (RHC) dataset fromthe study by [7] that we used for data analysis in Chapter 3. To measuresimulation performance, we have considered the various performance mea-sure described in chapter 4 section 4.3. In the current thesis, we exploredthe different approaches to address effect modifiers under the literature’spropensity score methods. Besides the existing approach, we also proposedsome new approaches. The following section will describe the simulationresults from the Monte Carlo simulation and plasmode simulation.5.1 Monte Carlo simulation without consideringinteraction between subgroup variable and acovariate5.1.1 Monte Carlo simulation result for scenario A: modelwith additive effectsFrom Monte Carlo simulation for the scenario A where the additive effectwas considered during propensity score estimation we found that,Performance of overall PS approach: In this simplest scenario,overall PS matching methods (methods M1, M2, M3) to match within sub-groups approach produced higher bias, higher mean squared error and lowercoverage probability than other PS matching methods. The PS matchingmethod (method M1) did not achieve the desired 95% coverage probabil-ity. The outcome was estimated separately based on the subgroup vari-able levels, which has improved in the method M2, where the outcome wasbuilt by considering treatment, subgroup variable, and interaction betweentreatment and subgroup variable. On the other hand, under this approach,different propensity score methods such as the PS weighting and covariateadjustment methods had lower bias and mean squared error and achieved84the desired 95% coverage probability. The PS stratification method underthis approach had a higher bias than any other PS method.Performance of subgroup-specific PS approach: The subgroup-specific propensity score matching (method M4) had lower bias, mean squarederror and achieved the desired 95% coverage probability compared to otherpropensity score matching methods. Under this approach other PS methodsPS weighting (method W3) and covariate adjustment method (method C2)produced lower bias, mean squared error and higher coverage probabilitycompared to PS matching method. The PS stratification method (methodS3) had higher bias than PS matching but mean squared error was lowerunder the same approach.Performance of use of useful interaction PS approach: In thisscenario, the use of useful interaction approach the PS matching methods(methods M5, M6) produced higher bias and mean squared error than thesubgroup-specific approach similar to the Green et al. [59], although thecoverage probability was achieved as desired. Under the use of useful in-teraction approach, the propensity score weighting method (method W2),propensity score stratification method (method S3) produced higher bias,lower mean squared error and higher coverage probability. For propensityscore adjustment method (method C3) had lower bias having higher meansquared error and achieved the desired coverage probability.Performance of univariate and generalized PS approach: Wefound the covariate adjustment method generalized propensity score method(method C4) produced similar bias to the univariate PS menthod C1 whereasEeren [60] found the generalized PS produced lower bias.Results from the Monte Carlo simulation result for scenario A:additive effectsTable 5.1: Results from the Monte Carlo simulation result for scenario A:additive effectsPS effect modifiermethodsGender Absolute Bias Percent Bias SD MSE MDSE Power CP CIL BACM1 M 0.0063 90% 0.091 0.091 0.064 1 0.84 0.25 0.84F 0.001 8% 0.093 0.093 0.064 1 0.82 0.25 0.82M2 M 0.0063 90% 0.091 0.091 0.102 1 0.96 0.39 0.97F 0.001 8% 0.093 0.093 0.103 1 0.97 0.39 0.97M3 M 0.0066 94% 0.088 0.088 0.103 1 0.98 0.39 0.98F 0.0035 27% 0.093 0.093 0.104 1 0.97 0.4 0.97M4 M 0.0048 69% 0.08 0.08 0.104 1 0.99 0.4 0.99F 0.0047 36% 0.076 0.076 0.105 1 0.99 0.4 0.99M5 M 0.0087 124% 0.083 0.083 0.102 1 0.98 0.39 0.99F 0.001 8% 0.081 0.081 0.103 1 0.99 0.4 0.99M6 M 0.0081 116% 0.081 0.081 0.103 1 0.99 0.4 0.99F 0.0021 16% 0.087 0.087 0.105 1 0.98 0.4 0.98W1 M 0.0007 10% 0.061 0.061 0.079 1 0.98 0.31 0.98F 0.001 8% 0.059 0.059 0.079 1 0.99 0.31 0.99W2 M 0.0185 264% 0.024 0.03 0.161 0.99 1 0.63 1F 0.0174 134% 0.024 0.03 0.161 1 1 0.63 1W3 M 0.0005 7% 0.033 0.033 0.08 1 1 0.31 1F 0.0009 7% 0.033 0.033 0.08 1 1 0.31 1W4 M 0.0007 10% 0.061 0.061 0.079 1 0.98 0.31 0.98F 0.001 8% 0.059 0.059 0.079 1 0.99 0.31 0.99W5 M 0.0007 10% 0.061 0.061 0.079 1 0.98 0.31 0.98F 0.001 8% 0.059 0.059 0.079 1 0.99 0.31 0.99W6 M 0.0007 10% 0.061 0.061 0.065 1 0.95 0.25 0.95F 0.001 8% 0.059 0.059 0.065 1 0.97 0.25 0.97S1 M 0.0177 253% 0.055 0.058 0.158 0.99 1 0.62 1F 0.0171 132% 0.056 0.058 0.158 1 1 0.62 1S2 M 0.0185 264% 0.039 0.043 0.158 1 1 0.62 1F 0.0169 130% 0.039 0.043 0.158 1 1 0.62 1S3 M 0.0185 264% 0.024 0.03 0.161 0.99 1 0.63 1F 0.0174 134% 0.024 0.03 0.161 1 1 0.63 1C1 M 0.001 14% 0.045 0.045 0.066 1 1 0.26 1F 0.0001 1% 0.045 0.045 0.066 1 1 0.26 1C2 M 0.001 14% 0.002 0.002 0.071 1 1 0.28 1F 0.001 8% 0.002 0.002 0.071 1 1 0.28 1C3 M 0.0001 1% 0.046 0.046 0.067 1 1 0.26 1F 0.0001 1% 0.046 0.046 0.067 1 1 0.26 1C4 M 0.0004 6% 0.01 0.01 0.061 1 1 0.24 1F 0.0001 1% 0.01 0.01 0.061 1 1 0.24 1C5 M 0.001 14% 0.045 0.045 0.066 1 1 0.26 1F 0.0001 1% 0.045 0.045 0.066 1 1 0.26 1C6 M 0.0002 3% 0.052 0.052 0.07 1 0.99 0.27 0.99F 0.0002 2% 0.052 0.052 0.07 1 0.99 0.28 0.99Note:M:Male, F:Female,SD: standard deviation, MSE: Mean squared error,MDSE: Model standard error, CP: Coverage probability, CIL:Confidenceinterval length estimate,BAC:Bias adjusted coverageBias comparison for simulation scenario A: additive effectsFigure 5.1: Bias for each of the approaches to handle effect modifier forsimulation scenario A : additive effects for subgroup label MaleFigure 5.2: Bias for each of the approaches to handle effect modifier forsimulation scenario A : additive effects for subgroup label femaleMean squared error comparison for simulation scenario A :additive effectsFigure 5.3: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario A : additive effects for subgroup label MaleFigure 5.4: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario A : additive effects for subgroup label femaleCoverage probability for simulation scenario A : additive effectFigure 5.5: Coverage probability for each of the approaches to handle effectmodifier for subgroup label Male for simulation scenario A : additive effectsFigure 5.6: Coverage probability for each of the approaches to handle effectmodifier for subgroup label Female for simulation scenario A : additive effects5.1.2 Monte Carlo simulation result for scenario B : mildnon-linearityFrom this Monte Carlo simulation scenario we found that,Performance of overall PS approach: In this scenario, under theuse of overall PS methods the PS matching methods M1, M2, M3 pro-duced lower bias than the subgroup-specific PS matching approach whereasin the previous scenario the overall PS method produced higher bias thansubgroup-specific PS approach. But the mean squared error was higher forthe methods M1, M2 and M3 compared to the subgroup-specific approachmethod M4 and did not achieve the desired 95% coverage probability. Sim-ilar to the additive effects scenario under this approach the PS weightingmethod W1 and covariate adjustment method C5 produced lower bias, meansquared error and higher coverage probability than the PS matching methodM1. On the other hand, the stratification method had higher bias than allother PS methods in this scenario.Performance of subgroup-specific PS approach: Interestingly inthis scenario, the subgroup-specific propensity score matching method M4had higher bias than the PS matching methods M1, M2 and M3 whereoverall PS matching approach has been used. In this scenario within thesubgroup-specific approach PS weighting method W3 and covariate adjust-ment method C2 had lower bias than PS matching method and PS stratifi-cation method S3 had higher bias.Performance of use of useful interaction PS approach: Under theuse of useful interaction approach the PS matching methods (methods M5,M6) produced lower bias in the male subgroup but higher bias in the femalesubgroup than the other PS matching methods. Within this approach otherthan PS stratification method the PS weighting and covariate adjustmentmethod performed well by producing lower bias, mean squared error andsimilar coverage probability as desired 95%.Performance of univariate and generalized PS approach: In thisscenario we found the generalized PS method produced similar bias of theunivariate PS method but the generalized PS had lower mean squared errorthan univariate PS having similar coverage probability as desired 95%.Results from the Monte Carlo simulation result for scenario B :mild non-linearityTable 5.2: Results from the Monte Carlo simulation result for scenario B :mild non-linearityPS effect modifiermethodsGender Absolute Bias Percent Bias SD MSE MDSE Power CP CIL BACM1 M 0.0127 181% 0.088 0.089 0.065 1 0.84 0.25 0.85F 0.0117 90% 0.09 0.091 0.065 1 0.86 0.25 0.86M2 M 0.0127 181% 0.088 0.089 0.1 1 0.97 0.39 0.97F 0.0117 90% 0.09 0.091 0.101 1 0.97 0.39 0.97M3 M 0.0127 181% 0.087 0.088 0.102 1 0.98 0.39 0.98F 0.0154 118% 0.087 0.088 0.103 1 0.98 0.39 0.97M4 M 0.0142 203% 0.073 0.074 0.103 1 1 0.4 1F 0.0176 135% 0.075 0.077 0.104 1 1 0.4 0.99M5 M 0.0117 167% 0.08 0.081 0.1 1 0.99 0.39 0.99F 0.0184 142% 0.084 0.086 0.103 1 0.99 0.4 0.99M6 M 0.0104 149% 0.081 0.082 0.103 1 0.99 0.39 0.99F 0.02 154% 0.081 0.083 0.104 1 0.99 0.4 0.99W1 M 0.0011 16% 0.06 0.06 0.078 1 0.99 0.3 0.98F 0.0019 15% 0.057 0.057 0.078 1 0.98 0.3 0.98W2 M 0.002 29% 0.044 0.044 0.078 1 1 0.3 1F 0.001 8% 0.042 0.042 0.078 1 1 0.3 1W3 M 0.0016 23% 0.028 0.028 0.079 1 1 0.31 1F 0.0015 12% 0.029 0.029 0.079 1 1 0.31 1W4 M 0.0011 16% 0.06 0.06 0.078 1 0.99 0.3 0.98F 0.0019 15% 0.057 0.057 0.078 1 0.98 0.3 0.98W5 M 0.002 29% 0.044 0.044 0.078 1 1 0.3 1F 0.001 8% 0.042 0.042 0.078 1 1 0.3 1W6 M 0.0011 16% 0.06 0.06 0.065 1 0.96 0.25 0.96F 0.0019 15% 0.057 0.057 0.065 1 0.96 0.25 0.96S1 M 0.0174 249% 0.054 0.057 0.158 0.99 1 0.62 1F 0.0184 142% 0.055 0.058 0.157 1 1 0.62 1S2 M 0.0184 263% 0.037 0.041 0.158 0.99 1 0.62 1F 0.0169 130% 0.038 0.042 0.158 1 1 0.62 1S3 M 0.0197 281% 0.024 0.031 0.162 1 1 0.63 1F 0.0174 134% 0.023 0.029 0.162 1 1 0.63 1C1 M 0.0007 10% 0.045 0.045 0.067 1 1 0.26 1F 0.0007 5% 0.045 0.045 0.066 1 1 0.26 1C2 M 0.0001 1% 0.002 0.002 0.071 1 1 0.28 1F 0.0001 1% 0.002 0.002 0.07 1 1 0.28 1C3 M 0.0008 11% 0.045 0.045 0.067 1 1 0.26 1F 0.0007 5% 0.045 0.045 0.067 1 1 0.26 1C4 M 0.0005 7% 0.012 0.012 0.062 1 1 0.24 1F 0.0001 1% 0.012 0.012 0.062 1 1 0.24 1C5 M 0.0007 10% 0.045 0.045 0.067 1 1 0.26 1F 0.0007 5% 0.045 0.045 0.066 1 1 0.26 1C6 M 0.0006 9% 0.05 0.05 0.07 1 0.99 0.27 0.99F 0.0007 5% 0.05 0.05 0.07 1 1 0.27 1Note:M:Male, F:Female,SD: standard deviation,MSE: Mean squared error,MDSE: Model standard error,CP: Coverage probability,CIL:Confidenceinterval length estimate,BAC:Bias adjusted coverageBias comparison for simulation scenario B : mild non-linearityFigure 5.7: Bias for each of the approaches to handle effect modifier forsimulation scenario B : mild non-linearity for subgroup label MaleFigure 5.8: Bias for each of the approaches to handle effect modifier forsimulation scenario B : mild non-linearity for subgroup label femaleMean squared error comparison for simulation scenario B : mildnon-linearityFigure 5.9: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario B : mild non-linearity for subgroup labelMaleFigure 5.10: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario B : mild non-linearity for subgroup labelfemaleCoverage probability for simulation scenario B : mildnon-linearityFigure 5.11: Coverage probability for each of the approaches to handle ef-fect modifier for subgroup label Male for simulation scenario B : mild non-linearityFigure 5.12: Coverage probability for each of the approaches to handle effectmodifier for subgroup label Female for simulation scenario B : mild non-linearity5.1.3 Monte Carlo simulation result for scenario C :moderate non-linearityFrom this Monte Carlo simulation scenario we found that,• Performance of overall PS approach: Under the use of overallPS methods the PS matching methods produced higher bias, highermean squared error than the subgroup-specific PS matching approachand did not achieve the desired 95% coverage probability. Under thisapproach compared to PS matching methods the PS weighting, strat-ification and covariate adjustment had lower bias, lower mean squarederror and achieved desired 95% coverage probability.• Performance of subgroup-specific PS approach: In this sce-nario, the subgroup-specific propensity score matching method M4 hadlower bias, lower mean squared error than the PS matching methodsM1, M2 and M3 where overall PS matching approach has been used.In this scenario within the subgroup-specific approach PS weightingmethod W3 and covariate adjustment method C2 and stratificationmethod S3 had lower bias, lower mean squared error and than the cor-responding PS matching method M4 as well as desired 95% coverageprobability.• Performance of use of useful interaction PS approach: Underthe use of useful interaction approach the PS matching method M5had lower bias in male subgroup and higher bias in female subgroupcompared to overall and subgroup-specific PS approach. Other PSmethods such as PS weighting, stratification and covariate adjustmentproduced lower bias, lower mean squared error and than PS matchingmethods as well as had desired 95% coverage probability.• Performance of univariate and generalized PS approach: Thegeneralized PS method produced similar bias of the univariate PSmethod but the generalized PS had lower mean squared error thanunivariate PS having similar coverage probability as desired in thisscenario.Results from the Monte Carlo simulation result for scenario C :moderate non-linearityTable 5.3: Results from the Monte Carlo simulation result for scenario C :moderate non-linearityPS effect modifiermethodsGender Absolute Bias Percent Bias SD MSE MDSE Power CP CIL BACM1 M 0.047 671% 0.088 0.1 0.065 1 0.79 0.25 0.84F 0.051 392% 0.089 0.102 0.065 1 0.78 0.25 0.85M2 M 0.047 671% 0.088 0.1 0.098 1 0.96 0.38 0.96F 0.051 392% 0.089 0.102 0.1 1 0.96 0.38 0.96M3 M 0.0493 704% 0.086 0.099 0.1 1 0.97 0.38 0.97F 0.0493 379% 0.085 0.098 0.1 1 0.96 0.39 0.98M4 M 0.0446 637% 0.072 0.085 0.101 1 0.99 0.39 0.99F 0.0486 374% 0.07 0.086 0.101 1 1 0.39 1M5 M 0.0458 654% 0.076 0.089 0.099 1 0.99 0.38 0.99F 0.0533 410% 0.078 0.094 0.099 1 0.98 0.38 0.99M6 M 0.047 671% 0.074 0.088 0.099 1 0.98 0.38 0.98F 0.0502 386% 0.074 0.089 0.101 1 0.98 0.39 0.99W1 M 0.0043 61% 0.055 0.056 0.073 1 0.99 0.29 0.99F 0.0048 37% 0.056 0.056 0.074 1 0.99 0.29 0.99W2 M 0.0042 60% 0.039 0.04 0.074 1 1 0.29 1F 0.0051 39% 0.039 0.04 0.074 1 1 0.29 1W3 M 0.0051 73% 0.022 0.022 0.074 1 1 0.29 1F 0.0046 35% 0.022 0.023 0.075 1 1 0.29 1W4 M 0.0043 61% 0.055 0.056 0.073 1 0.99 0.29 0.99F 0.0048 37% 0.056 0.056 0.074 1 0.99 0.29 0.99W5 M 0.0042 60% 0.039 0.04 0.074 1 1 0.29 1F 0.0051 39% 0.039 0.04 0.074 1 1 0.29 1W6 M 0.0043 61% 0.055 0.056 0.065 1 0.98 0.25 0.98F 0.0048 37% 0.056 0.056 0.065 1 0.98 0.25 0.98S1 M 0.0088 126% 0.053 0.053 0.15 0.97 1 0.59 1F 0.0082 63% 0.054 0.054 0.151 1 1 0.59 1S2 M 0.0094 134% 0.035 0.036 0.151 0.98 1 0.59 1F 0.0082 63% 0.036 0.037 0.151 1 1 0.59 1S3 M 0.0098 140% 0.018 0.021 0.153 0.99 1 0.6 1F 0.0088 68% 0.019 0.021 0.153 1 1 0.6 1C1 M 0.0007 10% 0.045 0.045 0.065 1 1 0.25 0.99F 0.0012 9% 0.045 0.045 0.065 1 0.99 0.25 0.99C2 M 0.001 14% 0.002 0.002 0.068 1 1 0.26 1F 0.0009 7% 0.002 0.002 0.068 1 1 0.26 1C3 M 0.0008 11% 0.046 0.046 0.065 1 0.99 0.26 0.99F 0.0012 9% 0.046 0.046 0.065 1 0.99 0.25 0.99C4 M 0.0011 16% 0.017 0.017 0.059 1 1 0.23 1F 0.0005 4% 0.017 0.017 0.059 1 1 0.23 1C5 M 0.0007 10% 0.045 0.045 0.065 1 1 0.25 0.99F 0.0012 9% 0.045 0.045 0.065 1 0.99 0.25 0.99C6 M 0.0006 9% 0.05 0.05 0.067 1 0.99 0.26 0.99F 0.0013 10% 0.05 0.05 0.067 1 0.99 0.26 0.99Note:M:Male, F:Female,SD: standard deviation,MSE: Mean squared error,MDSE: Model standard error,CP: Coverage probability,CIL:Confidenceinterval length estimate,BAC:Bias adjusted coverageBias comparison for simulation scenario C : moderatenon-linearityFigure 5.13: Bias for each of the approaches to handle effect modifier forsimulation scenario C : moderate non-linearity for subgroup label MaleFigure 5.14: Bias for each of the approaches to handle effect modifier forsimulation scenario C : moderate non-linearity for subgroup label femaleMean squared error comparison for simulation scenario C :moderate non-linearityFigure 5.15: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario C : moderate non-linearity for subgrouplabel MaleFigure 5.16: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario C : moderate non-linearity for subgrouplabel femaleCoverage probability for simulation scenario C : moderatenon-linearityFigure 5.17: Coverage probability for each of the approaches to handle effectmodifier for subgroup label Male for simulation scenario C : moderate non-linearityFigure 5.18: Coverage probability for each of the approaches to handle effectmodifier for subgroup label Female for simulation scenario C : moderate non-linearity5.1.4 Monte Carlo simulation result for scenario D : mildnon-additivityFrom this Monte Carlo simulation scenario we found that,• Performance of overall PS approach: Under the use of overallPS methods, the PS matching methods M1, M2, M3 produced samebias as the subgroup-specific PS matching approach in the male sub-group but in the female subgroup those overall PS methods had higherbias, higher mean squared error and did not achieve the desired 95%coverage probability than subgroup-specific approach. Under the sameapproach the PS weigthing and covariate adjustment method producedlower bias, lower mean squared error than PS matching methods andachieved desired 95% coverage probability. The other PS method PSstratification produced higher bias than all other PS methods in malesubgroup whereas in the female group the bias was lower than PSmatching method.• Performance of subgroup-specific PS approach: In this sce-nario, the subgroup-specific propensity score matching method M4performed better in terms of producing lower bias, lower mean squarederror in the female subgroup. The other PS methods, the PS weight-ing and covariate adjustment method produced lower bias, lower meansquared error than PS and had 95% coverage probability matchingmethod M4. The PS stratification method S3 had higher bias in themale subgroup whereas the bias was lower in the female subgroup formethod S3 than PS matching method M4.• Performance of use of useful interaction PS approach: Un-der the use of useful interaction approach the PS matching methodM5 had lower bias, lower mean squared error than other PS match-ing methods whereas the other interaction approach method M6 hadhigher bias than other PS matching methods. Under this approachthe other PS weighting and covariate adjustment methods had lowerbias than PS matching methods. On the other hand, the PS stratifi-cation methods had higher bias in the male subgroup which was lowerin female subgroup.• Performance of univariate and generalized PS approach: Inthis scenario the generalized PS method produced similar bias of theunivariate PS method but the generalized PS had lower mean squarederror than univariate PS with desired coverage probability.Results from the Monte Carlo simulation result for scenario D :mild non-additivityTable 5.4: Results from the Monte Carlo simulation result for scenario D :mild non-additivityPS effect modifiermethodsGender Absolute Bias Percent Bias SD MSE MDSE Power CP CIL BACM1 M 0.0146 209% 0.099 0.1 0.064 1 0.78 0.25 0.79F 0.023 177% 0.1 0.103 0.064 1 0.78 0.25 0.81M2 M 0.0146 209% 0.099 0.1 0.109 1 0.96 0.42 0.95F 0.023 177% 0.1 0.103 0.11 1 0.96 0.42 0.97M3 M 0.015 214% 0.099 0.1 0.11 1 0.96 0.42 0.96F 0.0235 177% 0.099 0.102 0.112 1 0.96 0.43 0.96M4 M 0.0151 216% 0.083 0.085 0.111 1 0.99 0.43 0.99F 0.0214 177% 0.085 0.087 0.113 1 0.99 0.43 0.99M5 M 0.012 171% 0.095 0.095 0.111 1 0.98 0.42 0.98F 0.0188 177% 0.093 0.095 0.11 1 0.98 0.42 0.98M6 M 0.0164 234% 0.088 0.09 0.112 1 0.99 0.43 0.99F 0.0237 177% 0.092 0.095 0.112 1 0.99 0.43 0.98W1 M 0.0023 33% 0.069 0.069 0.084 1 0.98 0.32 0.98F 0.0021 177% 0.066 0.066 0.084 1 0.98 0.32 0.98W2 M 0.0026 37% 0.054 0.054 0.084 1 1 0.33 1F 0.0023 177% 0.052 0.052 0.085 1 1 0.33 1W3 M 0.0027 39% 0.041 0.041 0.085 1 1 0.33 1F 0.0018 177% 0.044 0.044 0.086 1 1 0.33 1W4 M 0.0023 33% 0.069 0.069 0.084 1 0.98 0.32 0.98F 0.0021 177% 0.066 0.066 0.084 1 0.98 0.32 0.98W5 M 0.0026 37% 0.054 0.054 0.084 1 1 0.33 1F 0.0023 177% 0.052 0.052 0.085 1 1 0.33 1W6 M 0.0023 33% 0.069 0.069 0.065 1 0.93 0.25 0.93F 0.0021 177% 0.066 0.066 0.065 1 0.95 0.25 0.95S1 M 0.0179 256% 0.058 0.06 0.163 0.98 1 0.64 1F 0.018 177% 0.06 0.062 0.163 1 1 0.64 1S2 M 0.0185 264% 0.041 0.045 0.163 0.99 1 0.64 1F 0.0175 177% 0.042 0.046 0.163 1 1 0.64 1S3 M 0.0188 269% 0.028 0.034 0.167 0.99 1 0.65 1F 0.0188 177% 0.029 0.035 0.167 1 1 0.65 1C1 M 0.0002 3% 0.046 0.046 0.068 1 1 0.26 1F 0.0002 177% 0.046 0.046 0.068 1 1 0.26 1C2 M 0.0002 3% 0.002 0.002 0.072 1 1 0.28 1F 0.0001 177% 0.002 0.002 0.072 1 1 0.28 1C3 M 0.0004 6% 0.047 0.047 0.068 1 1 0.27 1F 0.0002 177% 0.046 0.046 0.068 1 1 0.27 1C4 M 0.001 14% 0.011 0.011 0.062 1 1 0.24 1F 0.0005 177% 0.012 0.012 0.062 1 1 0.24 1C5 M 0.0002 3% 0.046 0.046 0.068 1 1 0.26 1F 0.0002 177% 0.046 0.046 0.068 1 1 0.26 1C6 M 0.0001 1% 0.053 0.053 0.072 1 0.99 0.28 0.99F 0.0004 177% 0.053 0.053 0.072 1 1 0.28 1Note:M:Male, F:Female,SD: standard deviation,MSE: Mean squared error,MDSE: Model standard error,CP: Coverage probability,CIL:Confidenceinterval length estimate,BAC:Bias adjusted coverageBias comparison for simulation scenario D : mild non-additivityFigure 5.19: Bias for each of the approaches to handle effect modifier forsimulation scenario D : mild non-additivity for subgroup label MaleFigure 5.20: Bias for each of the approaches to handle effect modifier forsimulation scenario D : mild non-additivity for subgroup label femaleMean squared error comparison for simulation scenario D : mildnon-additivityFigure 5.21: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario D : mild non-additivity for subgroup labelMaleFigure 5.22: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario D : mild non-additivity for subgroup labelfemaleCoverage probability for simulation scenario D : mildnon-additivityFigure 5.23: Coverage probability for each of the approaches to handle ef-fect modifier for subgroup label Male for simulation scenario D : mild non-additivityFigure 5.24: Coverage probability for each of the approaches to handle effectmodifier for subgroup label Female for simulation scenario D : mild non-additivity5.1.5 Monte Carlo simulation result for scenario E : mildnon-additivity and non-linearityFrom this Monte Carlo simulation scenario we found that,• Performance of overall PS approach: Under the use of overallPS methods, the PS matching methods M1, M2, M3 produced samebias as the subgroup-specific PS matching approach in the male sub-group but in the female subgroup those overall PS methods had higherbias, higher mean squared error than subgroup-specific approach andmethod M1 did not achieve desired 95% coverage probability. Underthe same approach the PS weighting and covariate adjustment methodproduced lower bias, lower mean squared error than PS matchingmethods and had desired coverage probability. The other PS methodPS stratification produced higher bias than all other PS methods inmale subgroup whereas in the female group the bias was lower thanPS matching method.• Performance of subgroup-specific PS approach: In this sce-nario, the subgroup-specific propensity score matching method M4performed better in terms of producing lower bias, lower mean squarederror in the female subgroup. The other PS methods, the PS weight-ing and covariate adjustment method produced lower bias, lower meansquared error than PS matching method M4 and had desired coverageprobability. The PS stratification method S3 had higher bias in themale subgroup whereas the bias was lower in the female subgroup formethod S3 than PS matching method M4.• Performance of use of useful interaction PS approach: Un-der the use of useful interaction approach the PS matching methodM5 had lower bias, lower mean squared error than other PS match-ing methods whereas the other interaction approach method M6 hadhigher bias than other PS matching methods. Under this approachthe other PS weighting and covariate adjustment methods had lowerbias than PS matching methods. On the other hand, the PS stratifi-cation methods had higher bias in the male subgroup which was lowerin female subgroup.• Performance of univariate and generalized PS approach: Inthis scenario the generalized PS method produced similar bias of theunivariate PS method but the generalized PS had lower mean squarederror than univariate PS with desired coverage probability.Results from the Monte Carlo simulation result for scenario E :mild non-additivity and non-linearityTable 5.5: Results from the Monte Carlo simulation result for scenario E :mild non-additivity and non-linearityPS effect modifiermethodsGender Absolute Bias Percent Bias SD MSE MDSE Power CP CIL BACM1 M 0.0367 524% 0.097 0.104 0.065 1 0.77 0.25 0.81F 0.0416 320% 0.101 0.109 0.065 1 0.77 0.25 0.79M2 M 0.0367 524% 0.097 0.104 0.108 1 0.96 0.42 0.97F 0.0416 320% 0.101 0.109 0.11 1 0.95 0.42 0.96M3 M 0.0371 530% 0.095 0.102 0.109 1 0.97 0.42 0.98F 0.0414 318% 0.097 0.105 0.11 1 0.96 0.42 0.97M4 M 0.0364 520% 0.078 0.086 0.11 1 0.99 0.42 0.99F 0.0384 295% 0.084 0.092 0.111 1 0.99 0.43 0.99M5 M 0.0326 466% 0.089 0.094 0.108 1 0.97 0.42 0.98F 0.0421 324% 0.091 0.1 0.11 1 0.98 0.42 0.98M6 M 0.034 486% 0.087 0.093 0.11 1 0.99 0.42 0.99F 0.0412 317% 0.088 0.097 0.11 1 0.98 0.42 0.98W1 M 0.0003 4% 0.068 0.068 0.082 1 0.99 0.32 0.99F 0.0005 4% 0.064 0.064 0.082 1 0.98 0.32 0.98W2 M 0.0007 10% 0.051 0.051 0.082 1 1 0.32 1F 0.0009 7% 0.049 0.049 0.083 1 1 0.32 1W3 M 0.0005 7% 0.038 0.038 0.083 1 1 0.32 1F 0.0005 4% 0.038 0.038 0.084 1 1 0.32 1W4 M 0.0003 4% 0.068 0.068 0.082 1 0.99 0.32 0.99F 0.0005 4% 0.064 0.064 0.082 1 0.98 0.32 0.98W5 M 0.0007 10% 0.051 0.051 0.082 1 1 0.32 1F 0.0009 7% 0.049 0.049 0.083 1 1 0.32 1W6 M 0.0003 4% 0.068 0.068 0.065 1 0.93 0.25 0.93F 0.0005 4% 0.064 0.064 0.065 1 0.95 0.25 0.95S1 M 0.0211 301% 0.059 0.062 0.163 0.98 1 0.64 1F 0.0204 157% 0.058 0.062 0.162 1 1 0.63 1S2 M 0.0216 309% 0.04 0.046 0.163 0.98 1 0.64 1F 0.0196 151% 0.042 0.046 0.163 1 1 0.64 1S3 M 0.0223 319% 0.028 0.036 0.167 0.99 1 0.65 1F 0.0208 160% 0.029 0.035 0.167 1 1 0.65 1C1 M 0.0001 1% 0.046 0.046 0.068 1 1 0.27 1F 0.0001 1% 0.046 0.046 0.068 1 1 0.26 1C2 M 0.001 14% 0.002 0.002 0.072 1 1 0.28 1F 0.0001 1% 0.002 0.002 0.072 1 1 0.28 1C3 M 0.0003 4% 0.047 0.047 0.068 1 1 0.27 1F 0.001 8% 0.047 0.047 0.068 1 1 0.27 1C4 M 0.001 14% 0.014 0.014 0.062 1 1 0.24 1F 0.0004 3% 0.014 0.014 0.062 1 1 0.24 1C5 M 0.0001 1% 0.046 0.046 0.068 1 1 0.27 1F 0.0001 1% 0.046 0.046 0.068 1 1 0.26 1C6 M 0.0001 1% 0.053 0.053 0.072 1 0.99 0.28 0.99F 0.0001 1% 0.053 0.053 0.072 1 0.99 0.28 0.99Note:M:Male, F:Female,SD: standard deviation,MSE: Mean squared error,MDSE: Model standard error,CP: Coverage probability,CIL:Confidenceinterval length estimate,BAC:Bias adjusted coverageBias comparison for simulation scenario E : mild non-additivityand non-linearityFigure 5.25: Bias for each of the approaches to handle effect modifier forsimulation scenario E : mild non-additivity and non-linearity for subgrouplabel MaleFigure 5.26: Bias for each of the approaches to handle effect modifier forsimulation scenario E : mild non-additivity and non-linearity for subgrouplabel femaleMean squared error comparison for simulation scenario E : mildnon-additivity and non-linearityFigure 5.27: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario E : mild non-additivity and non-linearityfor subgroup label MaleFigure 5.28: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario E : mild non-additivity and non-linearityfor subgroup label femaleCoverage probability for simulation scenario E : mildnon-additivity and non-linearityFigure 5.29: Coverage probability for each of the approaches to handle ef-fect modifier for subgroup label Male for simulation scenario E : mild non-additivity and non-linearityFigure 5.30: Coverage probability for each of the approaches to handle effectmodifier for subgroup label Female for simulation scenario E : mild non-additivity and non-linearity5.1.6 Monte Carlo simulation result for scenario F:moderate non-additivityFrom this Monte Carlo simulation scenario we found that,• Performance of overall PS approach: In this scenario, under theuse of overall PS methods, the matching methods M1, M2 had lowerbias than other PS matching methods in the male subgroup but themean squared error was higher compared to other PS matching meth-ods and method M1 did not have desired 95% coverage probability.Performance of subgroup-specific PS approach: In the femalesubgroup these methods M1 and M2 had higher bias than subgroup-specific PS matching method M4 but the bias was lower compared touseful interaction approach. In this scenario, all the other PS methodssuch as PS weighting, stratification and covariate adjustment methodhad lower bias, lower mean squared error than PS matching methodsand had achieved desired coverage probability under all approaches ofaddressing effect modifier.• Performance of univariate and generalized PS approach: Thegeneralized PS method produced similar bias of the univariate PSmethod but the generalized PS had lower mean squared error thanunivariate PS with desired coverage probability in this scenario.Results from the Monte Carlo simulation result for scenario F:moderate non-additivityTable 5.6: Results from the Monte Carlo simulation result for scenario F:moderate non-additivityPS effect modifiermethodsGender Absolute Bias Percent Bias SD MSE MDSE Power CP CIL BACM1 M 0.0442 631% 0.09 0.1 0.065 1 0.79 0.25 0.83F 0.0532 409% 0.092 0.106 0.065 1 0.78 0.25 0.84M2 M 0.0442 631% 0.09 0.1 0.104 1 0.96 0.4 0.97F 0.0532 409% 0.092 0.106 0.105 1 0.96 0.4 0.97M3 M 0.0488 697% 0.088 0.1 0.105 1 0.96 0.4 0.98F 0.0524 403% 0.088 0.103 0.106 1 0.97 0.41 0.97M4 M 0.0485 693% 0.076 0.09 0.106 1 0.99 0.41 0.99F 0.0485 373% 0.075 0.089 0.107 1 0.99 0.41 0.99M5 M 0.046 657% 0.082 0.094 0.105 1 0.98 0.4 0.98F 0.055 423% 0.081 0.098 0.106 1 0.98 0.41 0.99M6 M 0.047 671% 0.08 0.092 0.106 1 0.98 0.41 0.99F 0.0554 426% 0.079 0.096 0.106 1 0.98 0.41 0.99W1 M 0.007 100% 0.065 0.065 0.081 1 0.99 0.31 0.99F 0.0103 79% 0.062 0.063 0.081 1 0.99 0.31 0.99W2 M 0.007 100% 0.05 0.05 0.081 1 1 0.31 1F 0.0106 82% 0.048 0.049 0.082 1 1 0.32 1W3 M 0.0083 119% 0.035 0.036 0.082 1 1 0.32 1F 0.0091 70% 0.035 0.036 0.082 1 1 0.32 1W4 M 0.007 100% 0.065 0.065 0.081 1 0.99 0.31 0.99F 0.0103 79% 0.062 0.063 0.081 1 0.99 0.31 0.99W5 M 0.007 100% 0.05 0.05 0.081 1 1 0.31 1F 0.0106 82% 0.048 0.049 0.082 1 1 0.32 1W6 M 0.007 100% 0.065 0.065 0.065 1 0.95 0.25 0.96F 0.0103 79% 0.062 0.063 0.065 1 0.95 0.26 0.96S1 M 0.0179 256% 0.056 0.059 0.16 0.97 1 0.62 1F 0.0157 121% 0.056 0.058 0.159 1 1 0.62 1S2 M 0.0185 264% 0.037 0.042 0.16 0.97 1 0.63 1F 0.015 115% 0.037 0.04 0.159 1 1 0.62 1S3 M 0.0182 260% 0.025 0.031 0.162 0.98 1 0.63 1F 0.0161 124% 0.024 0.029 0.162 1 1 0.63 1C1 M 0.001 14% 0.045 0.045 0.067 1 1 0.26 1F 0.0005 4% 0.045 0.045 0.067 1 1 0.26 0.99C2 M 0.0003 4% 0.002 0.002 0.071 1 1 0.28 1F 0.0003 2% 0.002 0.002 0.071 1 1 0.28 1C3 M 0.0003 4% 0.047 0.047 0.068 1 1 0.26 1F 0.0006 5% 0.046 0.046 0.067 1 0.99 0.26 0.99C4 M 0.0012 17% 0.014 0.014 0.062 1 1 0.24 1F 0.0007 5% 0.014 0.014 0.062 1 1 0.24 1C5 M 0.001 14% 0.045 0.045 0.067 1 1 0.26 1F 0.0006 5% 0.045 0.045 0.067 1 0.99 0.26 0.99C6 M 0.0003 4% 0.052 0.052 0.071 1 0.99 0.28 0.99F 0.0008 6% 0.052 0.052 0.071 1 0.99 0.28 0.99Note:M:Male, F:Female,SD: standard deviation,MSE: Mean squared error,MDSE: Model standard error,CP: Coverage probability,CIL:Confidenceinterval length estimate,BAC:Bias adjusted coverageBias comparison for simulation scenario F: moderatenon-additivityFigure 5.31: Bias for each of the approaches to handle effect modifier forsimulation scenario F: moderate non-additivity for subgroup label MaleFigure 5.32: Bias for each of the approaches to handle effect modifier forsimulation scenario F: moderate non-additivity for subgroup label femaleMean squared error comparison for simulation scenario F:moderate non-additivityFigure 5.33: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario F: moderate non-additivity for subgrouplabel MaleFigure 5.34: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario F: moderate non-additivity for subgrouplabel femaleCoverage probability for simulation scenario F: moderatenon-additivityFigure 5.35: Coverage probability for each of the approaches to handle effectmodifier for subgroup label Male for simulation scenario F: moderate non-additivityFigure 5.36: Coverage probability for each of the approaches to handle effectmodifier for subgroup label Female for simulation scenario F: moderate non-additivity5.1.7 Monte Carlo simulation result for scenario G:moderate non-additivity and non-linearityFrom Monte Carlo simulation for the scenario where moderate non-additivityand non-linearity were considered during propensity score estimation wefound that,• Performance of overall PS approach and subgroup-specific PSapproach: Under the use of overall PS approach, the PS matchingmethods M1, M2 and M3 had lower bias and higher mean squarederror than subgroup-specific and use of useful interaction approach inmale subgroup. On the other hand, in the female subgroup the over-all PS approach produced higher bias compared to subgroup-specificapproach within PS matching methods.Performance of use of useful interaction PS approach: Underthe useful interaction approach the bias was higher than other PSmatching approach, although mean squared error was lower.• Performance of univariate and generalized PS approach: In-terestingly in this scenario the generalized PS methods produced higherbias but lower mean squared error than univariate PS methods.Results from the Monte Carlo simulation result for scenario G:moderate non-additivity and non-linearityTable 5.7: Results from the Monte Carlo simulation result for scenario G:moderate non-additivity and non-linearityPS effect modifiermethodsGender Absolute Bias Percent Bias SD MSE MDSE Power CP CIL BACM1 M 0.0601 859% 0.087 0.106 0.065 1 0.78 0.26 0.87F 0.0674 518% 0.088 0.111 0.065 1 0.74 0.26 0.85M2 M 0.0601 859% 0.087 0.106 0.1 1 0.94 0.39 0.97F 0.0674 518% 0.088 0.111 0.099 1 0.94 0.39 0.97M3 M 0.0629 899% 0.086 0.107 0.101 1 0.94 0.39 0.97F 0.0656 505% 0.084 0.106 0.101 1 0.95 0.39 0.98M4 M 0.0641 916% 0.07 0.095 0.103 1 0.98 0.4 1F 0.0642 494% 0.069 0.094 0.103 1 0.99 0.4 1M5 M 0.0615 879% 0.079 0.1 0.101 1 0.97 0.39 0.99F 0.0674 518% 0.08 0.105 0.101 1 0.96 0.39 0.99M6 M 0.0634 906% 0.074 0.097 0.102 1 0.97 0.39 0.99F 0.067 515% 0.073 0.099 0.102 1 0.98 0.39 0.99W1 M 0.0245 350% 0.062 0.066 0.079 1 0.99 0.31 0.99F 0.0231 178% 0.062 0.066 0.079 1 0.99 0.31 0.99W2 M 0.024 343% 0.047 0.052 0.079 1 1 0.31 1F 0.024 185% 0.047 0.052 0.079 1 1 0.31 1W3 M 0.0244 349% 0.029 0.038 0.08 1 1 0.31 1F 0.0242 186% 0.03 0.038 0.08 1 1 0.31 1W4 M 0.0245 350% 0.062 0.066 0.079 1 0.99 0.31 0.99F 0.0231 178% 0.062 0.066 0.079 1 0.99 0.31 0.99W5 M 0.024 343% 0.047 0.052 0.079 1 1 0.31 1F 0.024 185% 0.047 0.052 0.079 1 1 0.31 1W6 M 0.0245 350% 0.062 0.066 0.065 1 0.95 0.26 0.96F 0.0103 79% 0.062 0.063 0.065 1 0.95 0.26 0.96S1 M 0.0022 31% 0.055 0.055 0.154 0.94 1 0.6 1F 0.0052 40% 0.056 0.056 0.154 1 1 0.6 1S2 M 0.0031 44% 0.038 0.038 0.155 0.93 1 0.61 1F 0.0043 33% 0.037 0.037 0.155 1 1 0.61 1S3 M 0.0038 54% 0.021 0.021 0.156 0.95 1 0.61 1F 0.0039 30% 0.021 0.022 0.156 1 1 0.61 1C1 M 0.0023 33% 0.046 0.046 0.067 1 0.99 0.26 0.99F 0.0005 4% 0.046 0.046 0.066 1 0.99 0.26 0.99C2 M 0.0014 20% 0.002 0.003 0.069 1 1 0.27 1F 0.0014 11% 0.002 0.003 0.069 1 1 0.27 1C3 M 0.0025 36% 0.048 0.048 0.067 1 0.99 0.26 0.99F 0.0005 4% 0.047 0.047 0.067 1 0.99 0.26 0.99C4 M 0.0034 49% 0.019 0.019 0.061 1 1 0.24 1F 0.0029 22% 0.019 0.019 0.061 1 1 0.24 1C5 M 0.0023 33% 0.046 0.046 0.067 1 0.99 0.26 0.99F 0.0005 4% 0.046 0.046 0.066 1 0.99 0.26 0.99C6 M 0.0024 34% 0.052 0.052 0.069 1 0.99 0.27 0.99F 0.0002 2% 0.052 0.052 0.069 1 0.99 0.27 0.99Note:M:Male, F:Female,SD: standard deviation,MSE: Mean squared error,MDSE: Model standard error,CP: Coverage probability,CIL:Confidenceinterval length estimate,BAC:Bias adjusted coverageBias comparison for simulation scenario G: moderatenon-additivity and non-linearityFigure 5.37: Bias for each of the approaches to handle effect modifier for sim-ulation scenario G: moderate non-additivity and non-linearity for subgrouplabel MaleFigure 5.38: Bias for each of the approaches to handle effect modifier for sim-ulation scenario G: moderate non-additivity and non-linearity for subgrouplabel femaleMean squared error comparison for simulation scenario G:moderate non-additivity and non-linearityFigure 5.39: Mean squared error for each of the approaches to handle ef-fect modifier for simulation scenario G: moderate non-additivity and non-linearity for subgroup label MaleFigure 5.40: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario A : additive effects for subgroup label femaleCoverage probability for simulation scenario G: moderatenon-additivity and non-linearityFigure 5.41: Coverage probability for each of the approaches to handle effectmodifier for subgroup label Male for simulation scenario G: moderate non-additivity and non-linearityFigure 5.42: Coverage probability for each of the approaches to handle effectmodifier for subgroup label Female for simulation scenario G: moderate non-additivity and non-linearity5.2 Monte Carlo simulation by considering aninteraction between subgroup variable and acovariateIn the previous section we have described the Monte Carlo simulation resultswhere we did not consider the presence of joint effect or interaction betweena covariate and the subgroup variable. In our study we have considered an-other type of Monte Carlo simulation where in each simulation scenarios wehave considered an interaction between a covariate and subgroup variable.In the next section we have described our findings from this Monte Carlosimulation results.5.2.1 Monte Carlo simulation result for scenario A1 :additive effects with an interaction between subgroupvariable and a covariate• Performance of overall PS approach: For use of overall PS ap-proach, we found as our expectation the PS matching methods (meth-ods M1, M2, M3) produced higher bias than the matching methodsM5 and M6 where useful interaction was considered during propen-sity score estimation. This is because in this simulation in the datageneration we have considered an interaction between a covariate andthe subgroup variable, In this same scenario, the previous Monte Carlosimulation the PS matching methods M1, M2, M3 produced lower biasthan the corresponding methods M1, M2 and M3 under this simula-tion strategy. Under the overall PS approach compare to PS matchingmethods M1, M2 the PS weighting methods had lower bias but thecorresponding PS stratification and covariate adjustment methods hadhigher bias.• Performance of subgroup-specific PS approach: Under the subgroup-specific approach among the PS matching methods the PS matchingmethod M4 produced the lowest bias compared to other PS match-ing methods in the male subgroup but in the female subgroup thebias from this method M4 was higher than the useful interaction ap-proaches M5 and M6. Compared to PS weighting method W3 the PSmatching method M4 had lower bias in the male subgroup but hadhigher bias in the female subgroup having higher mean squared errorboth groups. The covariate adjustment method C2 had lower biasthan the PS matching method M4 but the stratification method S3had higher bias and higher mean squared error.• Performance of use of useful interaction PS approach: For useof useful interaction approach, in this scenario the covariate adjust-ment methods C3 had higher bias than PS weighting methods W2,W5, PS stratification method S2 and PS matching methods M5 andM6.• Performance of univariate and generalized PS approach: Un-der this simulation, the generalized PS method C4 produced lowerbias, mean squared error than the univariate PS method C1.• Under scenario the methods M1, M2, M3, M4, C1, C5, and C6 didnot achieve the desired 95% coverage probability.Results from the Monte Carlo simulation result for scenario A1:additive effectsTable 5.8: Results from the Monte Carlo simulation result for scenario A1:additive effectsPS effect modifiermethodsGender Absolute Bias Percent Bias SD MSE MDSE Power CP CIL BACM1 M 0.0747 1067% 0.081 0.11 0.064 1 0.73 0.25 0.88F 0.089 685% 0.09 0.126 0.064 1 0.65 0.25 0.84M2 M 0.0747 1067% 0.081 0.11 0.094 1 0.91 0.37 0.98F 0.089 685% 0.09 0.126 0.102 1 0.87 0.39 0.97M3 M 0.0941 1344% 0.091 0.131 0.102 1 0.88 0.39 0.98F 0.0919 707% 0.089 0.128 0.102 1 0.86 0.39 0.97M4 M 0.0039 56% 0.107 0.107 0.137 0.99 0.98 0.52 0.99F 0.0131 101% 0.174 0.174 0.166 1 0.93 0.61 0.94M5 M 0.0116 166% 0.123 0.123 0.13 0.99 0.96 0.5 0.96F 0.0082 63% 0.185 0.185 0.164 1 0.91 0.6 0.92M6 M 0.0138 197% 0.118 0.119 0.132 0.99 0.96 0.5 0.96F 0.0068 52% 0.186 0.186 0.165 1 0.9 0.6 0.9W1 M 0.0194 277% 0.059 0.062 0.077 1 0.98 0.3 0.99F 0.034 262% 0.067 0.075 0.087 1 0.97 0.34 0.98W2 M 0.0192 274% 0.143 0.144 0.135 0.98 0.92 0.5 0.93F 0.0044 34% 0.116 0.116 0.121 1 0.96 0.45 0.96W3 M 0.0177 253% 0.13 0.131 0.137 0.98 0.98 0.51 0.98F 0.0053 41% 0.105 0.105 0.124 1 0.99 0.46 0.99W4 M 0.0194 277% 0.059 0.062 0.077 1 0.98 0.3 0.99F 0.034 262% 0.067 0.075 0.087 1 0.97 0.34 0.98W5 M 0.0192 274% 0.143 0.144 0.135 0.98 0.92 0.5 0.93F 0.0044 34% 0.116 0.116 0.121 1 0.96 0.45 0.96W6 M 0.0194 277% 0.059 0.062 0.065 1 0.96 0.25 0.96F 0.034 262% 0.067 0.075 0.065 1 0.9 0.25 0.94S1 M 0.0017 24% 0.11 0.11 0.144 0.91 0.98 0.56 0.98F 0.1248 960% 0.102 0.161 0.134 1 0.91 0.52 0.99S2 M 0.0239 341% 0.072 0.076 0.123 1 1 0.48 1F 0.0678 522% 0.065 0.094 0.117 1 0.99 0.46 1S3 M 0.1125 1607% 0.103 0.152 0.158 0.97 0.98 0.61 1F 0.0788 606% 0.074 0.108 0.158 1 0.99 0.61 1C1 M 0.0807 1153% 0.061 0.101 0.07 1 0.84 0.28 0.97F 0.0904 695% 0.059 0.108 0.072 1 0.8 0.28 0.98C2 M 0.0001 1% 0.002 0.002 0.093 1 1 0.36 1F 0.0001 1% 0.004 0.004 0.087 1 1 0.34 1C3 M 0.0689 984% 0.057 0.089 0.083 1 0.94 0.33 1F 0.062 477% 0.051 0.08 0.079 1 0.96 0.31 1C4 M 0.0019 27% 0.014 0.014 0.086 1 1 0.34 1F 0.0013 10% 0.009 0.01 0.076 1 1 0.3 1C5 M 0.0969 1384% 0.058 0.113 0.071 1 0.78 0.28 0.98F 0.0967 744% 0.059 0.113 0.072 1 0.78 0.28 0.98C6 M 0.1016 1451% 0.064 0.12 0.076 1 0.78 0.3 0.98F 0.0985 758% 0.06 0.115 0.073 1 0.78 0.29 0.98Note:M:Male, F:Female,SD: standard deviation,MSE: Mean squared error,MDSE: Model standard error,CP: Coverage probability,CIL:Confidenceinterval length estimate,BAC:Bias adjusted coverageBias comparison for simulation scenario A1 : additive effectsFigure 5.43: Bias for each of the approaches to handle effect modifier forsimulation scenario A1 : additive effects for subgroup label MaleFigure 5.44: Bias for each of the approaches to handle effect modifier forsimulation scenario A1 : additive effects for subgroup label femaleMean squared error comparison for simulation scenario A1 :additive effectsFigure 5.45: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario A1 : additive effects for subgroup label MaleFigure 5.46: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario A1 : additive effects for subgroup labelfemaleCoverage probability for simulation scenario A1 : additive effectFigure 5.47: Coverage probability for each of the approaches to handle effectmodifier for subgroup label Male for simulation scenario A1 : additive effectsFigure 5.48: Coverage probability for each of the approaches to handle effectmodifier for subgroup label Female for simulation scenario A1 : additiveeffects5.2.2 Monte Carlo simulation result for scenario B1 : mildnon-linearity with an interaction between subgroupvariable and a covariate• Ooverall various PS approaches produced lower bias similar to the pre-vious simulation scenarios as depicted in the Figures (5.49 and 5.50).• Performance of overall PS approach: In this scenario under theuse of overall PS approach the PS matching method M1 producedlowest bias than other PS matching methods whereas under sameapproach the PS matching methods M2 and M3 produced higherbias than other PS matching methods. The corresponding weight-ing method produced lower bias than PS matching methods M2 andM3. On the other hand, the covariate adjustment methods C5 and C6had higher bias than PS matching methods M2 and M3 and the PSstratification method S1 had lower bias in the male subgroup but infemale subgroup the bias was higher than all the PS methods.• Performance of subgroup-specific PS approach: The subgroup-specific PS approach had lower bias than overall PS approach but lowerthan the PS methods M5 and M6 that considered the useful interactionapproach. Under subgroup-specific approach the PS weighting methodW3 and the PS stratification method S3 had higher bias than the PSmatching method M4 whereas the corresponding covariate adjustmentmethod C2 had lower bias than other PS methods.• Performance of use of useful interaction PS approach: Underuse of useful interaction approach the covariate adjustment method C3had higher bias, higher mean squared error than other PS methods.• Performance of univariate and generalized PS approach: Inthis scenario we also found that, the generalized PS method had lowerbias, lower mean squared error than the univariate PS method.• Under this scenario the covariate adjustment C6 did not achieve thedesired 95% coverage probability.Results from the Monte Carlo simulation result for scenario B1 :mild non-linearityTable 5.9: Results from the Monte Carlo simulation result for scenario B1 :mild non-linearityPS effect modifiermethodsGender Absolute Bias Percent Bias SD MSE MDSE Power CP CIL BACM1 M 0.0006 9% 0.05 0.05 0.07 1 0.99 0.27 0.99F 0.0007 5% 0.05 0.05 0.07 1 1 0.27 1M2 M 0.0771 1101% 0.076 0.109 0.091 1 0.91 0.36 0.98F 0.0571 439% 0.089 0.105 0.102 1 0.93 0.4 0.97M3 M 0.0869 1241% 0.08 0.118 0.096 1 0.91 0.37 0.97F 0.0607 467% 0.088 0.107 0.102 1 0.92 0.4 0.98M4 M 0.0111 159% 0.104 0.105 0.136 0.99 0.99 0.52 0.99F 0.0307 236% 0.167 0.169 0.17 1 0.97 0.62 0.96M5 M 0.0234 334% 0.13 0.132 0.134 0.98 0.96 0.51 0.96F 0.0157 121% 0.173 0.174 0.164 1 0.93 0.6 0.93M6 M 0.0155 221% 0.12 0.121 0.133 0.99 0.97 0.51 0.96F 0.0155 119% 0.174 0.175 0.166 1 0.93 0.61 0.93W1 M 0.0195 279% 0.062 0.065 0.078 1 0.98 0.31 0.98F 0.0222 171% 0.064 0.067 0.085 1 0.98 0.33 0.99W2 M 0.0133 190% 0.147 0.148 0.14 0.98 0.93 0.51 0.93F 0.0093 72% 0.108 0.108 0.119 1 0.97 0.45 0.97W3 M 0.02 286% 0.134 0.136 0.141 0.98 0.97 0.52 0.98F 0.0192 148% 0.097 0.099 0.124 1 1 0.46 0.99W4 M 0.0195 279% 0.062 0.065 0.078 1 0.98 0.31 0.98F 0.0222 171% 0.064 0.067 0.085 1 0.98 0.33 0.99W5 M 0.0133 190% 0.147 0.148 0.14 0.98 0.93 0.51 0.93F 0.0093 72% 0.108 0.108 0.119 1 0.97 0.45 0.97W6 M 0.0195 279% 0.062 0.065 0.064 1 0.94 0.25 0.96F 0.0222 171% 0.064 0.067 0.065 1 0.94 0.26 0.95S1 M 0.0162 231% 0.101 0.102 0.136 0.93 0.99 0.53 0.99F 0.1348 1037% 0.098 0.166 0.131 1 0.89 0.51 0.99S2 M 0.0087 124% 0.071 0.072 0.122 1 1 0.48 1F 0.0794 611% 0.064 0.102 0.115 1 0.99 0.45 1S3 M 0.1113 1590% 0.112 0.158 0.17 0.95 0.98 0.66 1F 0.0676 520% 0.064 0.093 0.17 1 1 0.66 1C1 M 0.0699 999% 0.059 0.092 0.071 1 0.86 0.28 0.98F 0.0701 539% 0.054 0.089 0.07 1 0.9 0.27 0.98C2 M 0.0002 3% 0.002 0.002 0.094 1 1 0.37 1F 0.0002 2% 0.004 0.004 0.084 1 1 0.33 1C3 M 0.0675 964% 0.058 0.089 0.082 1 0.95 0.32 0.99F 0.0555 427% 0.048 0.074 0.078 1 0.97 0.3 1C4 M 0.0013 19% 0.014 0.014 0.085 1 1 0.33 1F 0.0005 4% 0.008 0.008 0.073 1 1 0.29 1C5 M 0.0717 1024% 0.056 0.091 0.071 1 0.88 0.28 0.98F 0.0694 534% 0.053 0.087 0.07 1 0.9 0.28 0.99C6 M 0.0828 1183% 0.061 0.103 0.075 1 0.86 0.3 0.98F 0.0755 581% 0.054 0.093 0.071 1 0.87 0.28 0.98Note:M:Male, F:Female,SD: standard deviation,MSE: Mean squared error,MDSE: Model standard error,CP: Coverage probability,CIL:Confidenceinterval length estimate,BAC:Bias adjusted coverageBias comparison for simulation scenario B1 : mild non-linearityFigure 5.49: Bias for each of the approaches to handle effect modifier forsimulation scenario B1 : mild non-linearity for subgroup label MaleFigure 5.50: Bias for each of the approaches to handle effect modifier forsimulation scenario B1 : mild non-linearity for subgroup label femaleMean squared error comparison for simulation scenario B1 :mild non-linearityFigure 5.51: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario B1 : mild non-linearity for subgroup labelMaleFigure 5.52: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario B1 : mild non-linearity for subgroup labelfemaleCoverage probability for simulation scenario B1 : mildnon-linearityFigure 5.53: Coverage probability for each of the approaches to handle effectmodifier for subgroup label Male for simulation scenario B1 : mild non-linearityFigure 5.54: Coverage probability for each of the approaches to handle effectmodifier for subgroup label Female for simulation scenario B1 : mild non-linearity5.2.3 Monte Carlo simulation result for scenario C1 :moderate non-linearity with an interaction betweensubgroup variable and a covariate• Performance of overall PS approach: In this scenario under theuse of overall PS approach the PS matching methods M1, M2, M3produced higher bias and higher mean squared error than the otherPS matching methods ( Figures 5.55 and 5.56). In the same approachthe PS weighting method W1 had lower bias, lower mean squarederror. The PS stratification method S1 and PS covariate adjustmentmethods C5 and C6 produced higher bias than PS matching weightingmethod,• Performance of subgroup-specific PS approach: Under the subgroup-specific PS approach the PS matching method M4 had lower bias thanthe overall use of PS matching methods M1, M2 and M3. But the PSmatching method M4 had lower mean squared error than overall use ofPS approach in the male subgroup but had higher mean squared errorin the female subgroup. The corresponding PS weighting method W2and covariate adjustment method C2 had lower bias than PS matchingmethod M4. On the other hand, the corresponding PS stratificationmethod S3 had higher bias than PS matching method.• Performance of use of useful interaction PS approach: In thisscenario under the use of useful interaction approach, the PS match-ing methods M5 and M6 had lower bias than the overall PS approachbut the bias was higher compared to subgroup specific PS matchingmethod M4. The corresponding PS weighting method W2 had lowerbias than PS matching methods. Under the same approach the cor-responding covariate adjustment method C3 had higher bias than PSmatching and weighting methods. The PS stratification method S2had lower bias in the male group compared to PS matching methodsbut the bias was higher in female subgroup.• Performance of univariate and generalized PS approach: Inthis scenario the generalized PS method had lower bias, lower meansquared error than the univariate PS method.• Under this scenario methods M1, M2, M3, S2, C1, C5, and C6 did notachieve desired 95% coverage probability.Results from the Monte Carlo simulation result for scenario C1 :moderate non-linearityTable 5.10: Results from the Monte Carlo simulation result for scenario C1: moderate non-linearityPS effect modifiermethodsGender Absolute Bias Percent Bias SD MSE MDSE Power CP CIL BACM1 M 0.1064 1520% 0.083 0.135 0.064 1 0.59 0.25 0.86F 0.0647 498% 0.083 0.105 0.066 1 0.76 0.26 0.88M2 M 0.1064 1520% 0.083 0.135 0.091 1 0.82 0.35 0.96F 0.0647 498% 0.083 0.105 0.102 1 0.93 0.4 0.98M3 M 0.1358 1940% 0.091 0.163 0.1 1 0.74 0.38 0.97F 0.066 508% 0.083 0.106 0.103 1 0.93 0.4 0.98M4 M 0.0317 453% 0.099 0.103 0.129 0.99 0.99 0.49 0.99F 0.0544 418% 0.135 0.145 0.158 1 0.98 0.59 0.97M5 M 0.0532 760% 0.118 0.13 0.124 0.98 0.95 0.47 0.96F 0.0344 265% 0.143 0.147 0.156 1 0.95 0.58 0.95M6 M 0.0509 727% 0.107 0.118 0.124 0.99 0.97 0.47 0.98F 0.0353 272% 0.144 0.149 0.158 1 0.96 0.59 0.95W1 M 0.0185 264% 0.057 0.06 0.074 1 0.99 0.29 0.99F 0.0067 52% 0.064 0.064 0.086 1 0.99 0.34 0.99W2 M 0.0096 137% 0.117 0.118 0.117 0.98 0.97 0.44 0.96F 0.0235 181% 0.107 0.109 0.12 1 0.98 0.45 0.97W3 M 0.0061 87% 0.121 0.121 0.126 0.98 0.98 0.46 0.98F 0.0485 373% 0.098 0.109 0.125 1 1 0.46 0.99W4 M 0.0185 264% 0.057 0.06 0.074 1 0.99 0.29 0.99F 0.0067 52% 0.064 0.064 0.086 1 0.99 0.34 0.99W5 M 0.0096 137% 0.117 0.118 0.117 0.98 0.97 0.44 0.96F 0.0235 181% 0.107 0.109 0.12 1 0.98 0.45 0.97W6 M 0.0185 264% 0.057 0.06 0.064 1 0.97 0.25 0.97F 0.0067 52% 0.064 0.064 0.066 1 0.96 0.26 0.97S1 M 0.0885 1264% 0.13 0.157 0.15 0.69 0.92 0.57 0.97F 0.1979 1522% 0.111 0.227 0.14 1 0.75 0.54 0.98S2 M 0.0153 219% 0.071 0.073 0.116 1 1 0.46 1F 0.0914 703% 0.066 0.113 0.115 1 0.98 0.45 1S3 M 0.0722 1031% 0.072 0.102 0.135 1 0.99 0.53 1F 0.0598 460% 0.062 0.086 0.135 1 1 0.53 1C1 M 0.0905 1293% 0.059 0.108 0.068 1 0.76 0.27 0.97F 0.0991 762% 0.06 0.116 0.071 1 0.74 0.28 0.98C2 M 0.0004 6% 0.002 0.002 0.087 1 1 0.34 1F 0.0001 1% 0.004 0.004 0.081 1 1 0.32 1C3 M 0.0738 1054% 0.057 0.093 0.078 1 0.91 0.31 0.99F 0.0695 535% 0.053 0.087 0.078 1 0.94 0.3 1C4 M 0.0032 46% 0.015 0.016 0.082 1 1 0.32 1F 0.0008 6% 0.011 0.011 0.073 1 1 0.28 1C5 M 0.1069 1527% 0.059 0.122 0.069 1 0.67 0.27 0.98F 0.1038 798% 0.059 0.119 0.071 1 0.71 0.28 0.98C6 M 0.1107 1581% 0.065 0.128 0.073 1 0.69 0.29 0.97F 0.1052 809% 0.06 0.121 0.071 1 0.7 0.28 0.97Note:M:Male, F:Female,SD: standard deviation,MSE: Mean squared error,MDSE: Model standard error,CP: Coverage probability,CIL:Confidenceinterval length estimate,BAC:Bias adjusted coverageBias comparison for simulation scenario C1 : moderatenon-linearityFigure 5.55: Bias for each of the approaches to handle effect modifier forsimulation scenario C1 : moderate non-linearity for subgroup label MaleFigure 5.56: Bias for each of the approaches to handle effect modifier forsimulation scenario C1 : moderate non-linearity for subgroup label femaleMean squared error comparison for simulation scenario C1 :moderate non-linearityFigure 5.57: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario C1 : moderate non-linearity for subgrouplabel MaleFigure 5.58: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario C1 : moderate non-linearity for subgrouplabel femaleCoverage probability for simulation scenario C1 : moderatenon-linearityFigure 5.59: Coverage probability for each of the approaches to handle effectmodifier for subgroup label Male for simulation scenario C1 : moderate non-linearityFigure 5.60: Coverage probability for each of the approaches to handle effectmodifier for subgroup label Female for simulation scenario C1 : moderatenon-linearity5.2.4 Monte Carlo simulation result for scenario D1 : mildnon-additivity with an interaction between subgroupvariable and a covariate• Performance of overall PS approach: In this scenario under theuse of overall PS approach the PS matching methods M1, M2, M3produced higher bias and higher mean squared error than the otherPS matching methods ( Figures 5.61 and 5.62). In the same approachthe PS weighting method W1 had lower bias, lower mean squarederror. The PS stratification method S1 had lower bias in the malesubgroup but had higher bias in the female subgroup. Under the samescenario the corresponding covariate adjustment methods C5 and C6had higher bias than other PS methods.• Performance of subgroup-specific PS approach: Under the subgroup-specific PS approach the PS matching method M4 had lower bias thanall other PS matching methods. But the PS matching method M4 hadlower mean squared error than overall use of PS approach in the malesubgroup but had higher mean squared error in the female subgroup.The corresponding PS weighting method W2 had higher bias than thePS matching method M4. The corresponding covariate adjustmentmethod C2 had lowest bias whereas the PS stratification method S3had higher bias, higher mean squared error than PS matching method.• Performance of use of useful interaction PS approach: In thisscenario under the use of useful interaction approach, the PS match-ing methods M5 and M6 had lower bias than the overall PS approachbut the bias was higher compared to subgroup specific PS matchingmethod M4. The corresponding PS weighting method W2 had lowerbias in the male subgroup but the bias was higher was higher in femalesubgroup than the corresponding PS matching methods. The strat-ification method S2 had lower bias in the male subgroup but it washigher in the female subgroup. The corresponding covariate adjust-ment method C3 had higher bias than other PS methods under thisapproach.• Performance of univariate and generalized PS approach: Thegeneralized PS method had lower bias, lower mean squared error thanthe univariate PS method.• Under this scenario the methods M1, M2, M3, S1, C1, C5, and C6 didnot achieve the desired 95% coverage probability.Results from the Monte Carlo simulation result for scenario D1 :mild non-additivityTable 5.11: Results from the Monte Carlo simulation result for scenario D1: mild non-additivityPS effect modifiermethodsGender Absolute Bias Percent Bias SD MSE MDSE Power CP CIL BACM1 M 0.0885 1264% 0.087 0.124 0.064 1 0.69 0.25 0.86F 0.0814 626% 0.092 0.123 0.065 1 0.68 0.25 0.84M2 M 0.0885 1264% 0.087 0.124 0.095 1 0.88 0.37 0.96F 0.0814 626% 0.092 0.123 0.107 1 0.88 0.41 0.97M3 M 0.1166 1666% 0.098 0.152 0.108 0.99 0.83 0.41 0.97F 0.0845 650% 0.091 0.124 0.107 1 0.88 0.41 0.97M4 M 0.018 257% 0.113 0.114 0.142 0.98 0.98 0.53 0.98F 0.0026 20% 0.185 0.185 0.175 0.99 0.94 0.64 0.94M5 M 0.0424 606% 0.149 0.155 0.144 0.96 0.95 0.54 0.94F 0.0152 117% 0.18 0.18 0.167 1 0.92 0.61 0.93M6 M 0.0245 350% 0.128 0.13 0.139 0.98 0.96 0.53 0.96F 0.0103 79% 0.188 0.188 0.171 1 0.91 0.62 0.92W1 M 0.0248 354% 0.061 0.066 0.077 1 0.98 0.3 0.98F 0.0244 188% 0.075 0.079 0.093 1 0.97 0.36 0.98W2 M 0.0183 261% 0.132 0.133 0.131 0.98 0.95 0.49 0.95F 0.0214 165% 0.11 0.112 0.117 1 0.95 0.44 0.97W3 M 0.0312 446% 0.119 0.123 0.132 0.98 0.98 0.49 0.98F 0.006 46% 0.1 0.101 0.123 1 0.99 0.46 0.99W4 M 0.0248 354% 0.061 0.066 0.077 1 0.98 0.3 0.98F 0.0244 188% 0.075 0.079 0.093 1 0.97 0.36 0.98W5 M 0.0183 261% 0.132 0.133 0.131 0.98 0.95 0.49 0.95F 0.0214 165% 0.11 0.112 0.117 1 0.95 0.44 0.97W6 M 0.0248 354% 0.061 0.066 0.065 1 0.95 0.25 0.96F 0.0244 188% 0.075 0.079 0.065 1 0.89 0.26 0.91S1 M 0.0271 387% 0.095 0.099 0.134 0.94 0.98 0.52 0.99F 0.1626 1251% 0.086 0.184 0.123 1 0.82 0.48 0.99S2 M 0.0093 133% 0.074 0.075 0.124 1 1 0.48 1F 0.0882 678% 0.066 0.11 0.117 1 0.98 0.46 1S3 M 0.1124 1606% 0.095 0.147 0.155 0.98 0.99 0.6 1F 0.077 592% 0.075 0.107 0.155 1 0.99 0.6 1C1 M 0.0908 1297% 0.06 0.109 0.071 1 0.79 0.28 0.98F 0.1003 772% 0.059 0.116 0.073 1 0.78 0.28 0.98C2 M 0.0004 6% 0.002 0.002 0.094 1 1 0.37 1F 0.0012 9% 0.004 0.005 0.086 1 1 0.34 1C3 M 0.0891 1273% 0.06 0.107 0.083 1 0.89 0.33 0.99F 0.0794 611% 0.052 0.095 0.079 1 0.91 0.31 1C4 M 0.0028 40% 0.019 0.019 0.086 1 1 0.34 1F 0.0034 26% 0.008 0.009 0.076 1 1 0.3 1C5 M 0.1075 1536% 0.058 0.122 0.072 1 0.73 0.28 0.98F 0.1054 811% 0.058 0.12 0.073 1 0.75 0.28 0.98C6 M 0.1178 1683% 0.066 0.135 0.077 1 0.7 0.3 0.98F 0.1099 845% 0.06 0.125 0.073 1 0.72 0.29 0.98Note:M:Male, F:Female,SD: standard deviation,MSE: Mean squared error,MDSE: Model standard error,CP: Coverage probability,CIL:Confidenceinterval length estimate,BAC:Bias adjusted coverageBias comparison for simulation scenario D1 : mild non-additivityFigure 5.61: Bias for each of the approaches to handle effect modifier forsimulation scenario D1 : mild non-additivity for subgroup label MaleFigure 5.62: Bias for each of the approaches to handle effect modifier forsimulation scenario D1 : mild non-additivity for subgroup label femaleMean squared error comparison for simulation scenario D1 :mild non-additivityFigure 5.63: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario D1 : mild non-additivity for subgroup labelMaleFigure 5.64: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario D1 : mild non-additivity for subgroup labelfemaleCoverage probability for simulation scenario D1 : mildnon-additivityFigure 5.65: Coverage probability for each of the approaches to handle effectmodifier for subgroup label Male for simulation scenario D1 : mild non-additivityFigure 5.66: Coverage probability for each of the approaches to handle effectmodifier for subgroup label Female for simulation scenario D1 : mild non-additivity5.2.5 Monte Carlo simulation result for scenario E1 : mildnon-additivity and non-linearity with an interactionbetween subgroup variable and a covariate• Performance of overall PS approach: In this scenario under theuse of overall PS approach the PS matching methods M1, M2, M3produced higher bias and higher mean squared error than the otherPS matching methods ( Figures 5.67 and 5.68). In the same approachthe PS weighting method W1 had lower bias, lower mean squared error.The PS stratification method S1 had lower bias in the male subgroupbut had higher bias in the female subgroup. Under the same scenariothe corresponding covariate adjustment methods C5 and C6 had higherbias than other PS methods except the PS matching method M3.• Performance of subgroup-specific PS approach: Under the subgroup-specific PS approach the PS matching method M4 had lower bias thanall other PS matching methods. But the PS matching method M4 hadlower mean squared error than other PS matching methods in the malesubgroup but had higher mean squared error than overall PS approachin the female subgroup. The corresponding PS weighting method W3in the female subgroup and covariate adjustment method C2 had lowerbias than the PS matching method M4. The PS weighting method W3in the male subgroup had higher bias than the PS matching methodM4. The corresponding PS stratification method S3 had higher bias,higher mean squared error than PS matching, weighting and covariateadjustment method.• Performance of use of useful interaction PS approach: In thisscenario under the use of useful interaction approach, the PS matchingmethods M5 and M6 had lower bias than the overall PS approachbut the bias was higher compared to subgroup specific PS matchingmethod M4. The corresponding PS weighting method W2 had lowerbias than the corresponding PS matching methods. The stratificationmethod S2 had lower bias in the male subgroup but it was higher inthe female subgroup. The corresponding covariate adjustment methodC3 had higher bias than other PS methods under this approach.• Performance of univariate and generalized PS approach: Thegeneralized PS method had lower bias, lower mean squared error thanthe univariate PS method.• Under this scenario the methods M1,M2,M3,M4,M5,S1,C1,C3,C5, andC6 did not achieve desired 95% coverage probability.Results from the Monte Carlo simulation result for scenario E1 :mild non-additivity and non-linearityTable 5.12: Results from the Monte Carlo simulation result for scenario E1: mild non-additivity and non-linearityPS effect modifiermethodsGender Absolute Bias Percent Bias SD MSE MDSE Power CP CIL BACM1 M 0.0977 1396% 0.084 0.129 0.064 1 0.63 0.25 0.86F 0.0532 409% 0.092 0.106 0.066 1 0.78 0.26 0.84M2 M 0.0977 1396% 0.084 0.129 0.093 1 0.86 0.36 0.97F 0.0532 409% 0.092 0.106 0.107 1 0.93 0.41 0.97M3 M 0.1179 1684% 0.088 0.147 0.101 1 0.83 0.39 0.97F 0.0543 418% 0.09 0.105 0.106 1 0.93 0.41 0.97M4 M 0.0343 490% 0.107 0.112 0.143 0.99 0.99 0.54 0.99F 0.0453 348% 0.175 0.18 0.178 0.99 0.96 0.65 0.95M5 M 0.0755 1079% 0.155 0.172 0.147 0.94 0.93 0.55 0.93F 0.0138 106% 0.172 0.172 0.171 0.99 0.94 0.63 0.93M6 M 0.0582 831% 0.127 0.14 0.138 0.98 0.95 0.53 0.96F 0.0167 128% 0.181 0.182 0.176 0.99 0.93 0.64 0.92W1 M 0.0261 373% 0.063 0.068 0.078 1 0.98 0.3 0.98F 0.0128 98% 0.072 0.073 0.092 1 0.98 0.35 0.98W2 M 0.015 214% 0.136 0.137 0.131 0.98 0.94 0.49 0.94F 0.0072 55% 0.103 0.104 0.118 1 0.98 0.44 0.98W3 M 0.0376 537% 0.122 0.128 0.132 0.99 0.97 0.49 0.98F 0.0217 167% 0.096 0.098 0.125 1 1 0.46 1W4 M 0.0261 373% 0.063 0.068 0.078 1 0.98 0.3 0.98F 0.0128 98% 0.072 0.073 0.092 1 0.98 0.35 0.98W5 M 0.015 214% 0.136 0.137 0.131 0.98 0.94 0.49 0.94F 0.0072 55% 0.103 0.104 0.118 1 0.98 0.44 0.98W6 M 0.0261 373% 0.063 0.068 0.064 1 0.94 0.25 0.96F 0.0128 98% 0.072 0.073 0.066 1 0.93 0.26 0.93S1 M 0.0438 626% 0.089 0.1 0.127 0.94 0.98 0.49 0.99F 0.1688 1298% 0.084 0.189 0.121 1 0.79 0.47 1S2 M 0.0066 94% 0.074 0.074 0.122 1 1 0.48 1F 0.0991 762% 0.063 0.118 0.115 1 0.97 0.45 1S3 M 0.1126 1609% 0.104 0.153 0.166 0.96 0.98 0.65 1F 0.0633 487% 0.062 0.089 0.166 1 1 0.65 1C1 M 0.0856 1223% 0.059 0.104 0.071 1 0.82 0.28 0.98F 0.0846 651% 0.055 0.101 0.071 1 0.84 0.28 0.99C2 M 0.0004 6% 0.002 0.002 0.094 1 1 0.37 1F 0.0008 6% 0.004 0.004 0.083 1 1 0.32 1C3 M 0.0887 1267% 0.06 0.107 0.082 1 0.88 0.32 0.99F 0.0729 561% 0.049 0.088 0.078 1 0.93 0.3 1C4 M 0.0018 26% 0.017 0.017 0.085 1 1 0.33 1F 0.0022 17% 0.008 0.008 0.073 1 1 0.29 1C5 M 0.0887 1267% 0.056 0.105 0.071 1 0.82 0.28 0.98F 0.0836 643% 0.052 0.099 0.071 1 0.86 0.28 0.99C6 M 0.1048 1497% 0.063 0.122 0.077 1 0.76 0.3 0.98F 0.0921 708% 0.055 0.107 0.071 1 0.8 0.28 0.98Note:M:Male, F:Female,SD: standard deviation,MSE: Mean squared error,MDSE: Model standard error,CP: Coverage probability,CIL:Confidenceinterval length estimate,BAC:Bias adjusted coverageBias comparison for simulation scenario E1 : mild non-additivityand non-linearityFigure 5.67: Bias for each of the approaches to handle effect modifier forsimulation scenario E1 : mild non-additivity and non-linearity for subgrouplabel MaleFigure 5.68: Bias for each of the approaches to handle effect modifier forsimulation scenario E1 : mild non-additivity and non-linearity for subgrouplabel femaleMean squared error comparison for simulation scenario E1 : mildnon-additivity and non-linearityFigure 5.69: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario E1 : mild non-additivity and non-linearityfor subgroup label MaleFigure 5.70: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario E1 : mild non-additivity and non-linearityfor subgroup label femaleCoverage probability for simulation scenario E1 : mildnon-additivity and non-linearityFigure 5.71: Coverage probability for each of the approaches to handle effectmodifier for subgroup label Male for simulation scenario E1 : mild non-additivity and non-linearityFigure 5.72: Coverage probability for each of the approaches to handle effectmodifier for subgroup label Female for simulation scenario E1 : mild non-additivity and non-linearity5.2.6 Monte Carlo simulation result for scenario F1:moderate non-additivity with an interaction betweensubgroup variable and a covariate• Performance of overall PS approach: In this scenario under theuse of overall PS approach the PS matching methods M1, M2, M3produced higher bias and higher mean squared error than the otherPS matching methods ( Figures 5.73 and 5.74). In the same approachthe PS weighting method W1 had lower bias, lower mean squared error.The PS stratification method S1 had lower bias in the male subgroupbut had higher bias in the female subgroup. Under the same scenariothe corresponding covariate adjustment methods C5 and C6 had higherbias than other PS methods except the PS matching method M3 inthe male subgroup.• Performance of subgroup-specific PS approach: Under the subgroup-specific PS approach the PS matching method M4 had lower bias thanall other PS matching methods in the male subgroup but producedhigher bias in the female subgroup. But the PS matching methodM4 had lower mean squared error than other PS matching methodsin the male subgroup but had higher mean squared error than overallPS approach in the female subgroup. The corresponding PS weight-ing method W3 and covariate adjustment method C2 had lower biasthan the PS matching method M4. The corresponding PS stratifica-tion method S3 had higher bias, higher mean squared error than PSmatching, weighting and covariate adjustment method.• Performance of use of useful interaction PS approach: In thisscenario under the use of useful interaction approach, the PS match-ing methods M5 and M6 had lower bias than the overall PS approachand subgroup-specific approach but the bias was higher compared tosubgroup specific PS matching method M4 in the male subgroup. Thecorresponding PS weighting method W2 had lower bias than the cor-responding PS matching methods. The stratification method S2 hadlower bias in the male subgroup but it was higher in the female sub-group. The corresponding covariate adjustment method C3 had higherbias than other PS methods under this approach.• Performance of univariate and generalized PS approach: Thegeneralized PS method had lower bias, lower mean squared error thanthe univariate PS method.• Under this scenario the methods M1, M2, M3, S1, C1, C5, and C6 didnot achieve the desired 95% coverage probability.Results from the Monte Carlo simulation result for scenario F1:moderate non-additivityTable 5.13: Results from the Monte Carlo simulation result for scenario F1:moderate non-additivityPS effect modifiermethodsGender Absolute Bias Percent Bias SD MSE MDSE Power CP CIL BACM1 M 0.1081 1544% 0.081 0.135 0.064 1 0.59 0.25 0.87F 0.0571 439% 0.091 0.108 0.066 1 0.76 0.26 0.84M2 M 0.1081 1544% 0.081 0.135 0.093 1 0.83 0.36 0.98F 0.0571 439% 0.091 0.108 0.109 1 0.93 0.42 0.98M3 M 0.1316 1880% 0.084 0.156 0.099 1 0.78 0.38 0.98F 0.059 454% 0.092 0.109 0.109 1 0.93 0.42 0.97M4 M 0.0467 667% 0.106 0.115 0.144 0.98 1 0.54 0.99F 0.0657 505% 0.178 0.19 0.181 0.99 0.96 0.67 0.94M5 M 0.0948 1354% 0.15 0.177 0.146 0.93 0.91 0.55 0.93F 0.0274 211% 0.182 0.184 0.177 1 0.94 0.65 0.94M6 M 0.0692 989% 0.121 0.139 0.138 0.98 0.95 0.52 0.97F 0.0317 244% 0.188 0.191 0.182 0.99 0.94 0.67 0.93W1 M 0.0219 313% 0.059 0.063 0.075 1 0.98 0.29 0.98F 0.0036 28% 0.076 0.076 0.095 1 0.98 0.37 0.98W2 M 0.0107 153% 0.132 0.132 0.132 0.98 0.95 0.49 0.95F 0.0095 73% 0.126 0.126 0.133 1 0.96 0.49 0.96W3 M 0.0175 250% 0.112 0.113 0.131 0.99 0.98 0.49 0.99F 0.0492 378% 0.124 0.133 0.14 1 0.99 0.52 0.98W4 M 0.0219 313% 0.059 0.063 0.075 1 0.98 0.29 0.98F 0.0036 28% 0.076 0.076 0.095 1 0.98 0.37 0.98W5 M 0.0107 153% 0.132 0.132 0.132 0.98 0.95 0.49 0.95F 0.0095 73% 0.126 0.126 0.133 1 0.96 0.49 0.96W6 M 0.0219 313% 0.059 0.063 0.064 1 0.96 0.25 0.97F 0.0036 28% 0.076 0.076 0.066 1 0.92 0.26 0.92S1 M 0.0621 887% 0.091 0.11 0.129 0.9 0.97 0.5 1F 0.1893 1456% 0.089 0.209 0.124 1 0.71 0.48 0.99S2 M 0.0174 249% 0.074 0.076 0.123 0.99 1 0.48 1F 0.1082 832% 0.063 0.125 0.118 1 0.97 0.46 1S3 M 0.0971 1387% 0.089 0.131 0.153 0.98 0.99 0.6 1F 0.0539 415% 0.071 0.089 0.153 1 1 0.6 1C1 M 0.0909 1299% 0.058 0.108 0.07 1 0.78 0.27 0.98F 0.096 738% 0.056 0.111 0.072 1 0.8 0.28 0.99C2 M 0.0004 6% 0.002 0.002 0.093 1 1 0.36 1F 0.0002 2% 0.005 0.005 0.085 1 1 0.33 1C3 M 0.0901 1287% 0.058 0.107 0.081 1 0.88 0.32 1F 0.078 600% 0.05 0.093 0.08 1 0.93 0.31 1C4 M 0.0006 9% 0.014 0.014 0.085 1 1 0.33 1F 0.0013 10% 0.009 0.009 0.075 1 1 0.29 1C5 M 0.0995 1421% 0.055 0.114 0.07 1 0.76 0.27 0.98F 0.0981 755% 0.055 0.112 0.073 1 0.79 0.28 0.99C6 M 0.1102 1574% 0.062 0.126 0.076 1 0.74 0.3 0.98F 0.1037 798% 0.057 0.118 0.072 1 0.75 0.28 0.99Note:M:Male, F:Female,SD: standard deviation,MSE: Mean squared error,MDSE: Model standard error,CP: Coverage probability,CIL:Confidenceinterval length estimate,BAC:Bias adjusted coverageBias comparison for simulation scenario F1: moderatenon-additivityFigure 5.73: Bias for each of the approaches to handle effect modifier forsimulation scenario F1: moderate non-additivity for subgroup label MaleFigure 5.74: Bias for each of the approaches to handle effect modifier forsimulation scenario F1: moderate non-additivity for subgroup label femaleMean squared error comparison for simulation scenario F1:moderate non-additivityFigure 5.75: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario F1: moderate non-additivity for subgrouplabel MaleFigure 5.76: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario F1: moderate non-additivity for subgrouplabel femaleCoverage probability for simulation scenario F1: moderatenon-additivityFigure 5.77: Coverage probability for each of the approaches to handle effectmodifier for subgroup label Male for simulation scenario F1: moderate non-additivityFigure 5.78: Coverage probability for each of the approaches to handle effectmodifier for subgroup label Female for simulation scenario F1: moderatenon-additivity5.2.7 Monte Carlo simulation result for scenario G1:moderate non-additivity and non-linearity with aninteraction between subgroup variable and a covariate• Performance of overall PS approach: In this scenario under theuse of overall PS approach the PS matching methods M1, M2, M3 pro-duced higher bias and higher mean squared error than the other PSmatching methods ( Figures 5.79 and 5.80). These PS matching meth-ods produced lower bias than subgroup-specific approach in the femalesubgroup. In the same approach the PS weighting method W1 hadlower bias, lower mean squared error. The PS stratification methodS1 had lower bias in the male subgroup but had higher bias in the fe-male subgroup. Under the same scenario the corresponding covariateadjustment methods C5 and C6 had lower bias in the male subgroupand higher bias in the female subgroup than other PS methods exceptcorresponding PS stratification method S1.• Performance of subgroup-specific PS approach: Under the subgroup-specific PS approach the PS matching method M4 had lower bias thanall other PS matching methods in the male subgroup but producedhigher bias in the female subgroup. But the PS matching methodM4 had lower mean squared error than other PS matching methodsin the male subgroup but had higher mean squared error than overallPS approach in the female subgroup. The corresponding PS weight-ing method W3 and covariate adjustment method C2 had lower biasthan the PS matching method M4. The corresponding PS stratifica-tion method S3 had higher bias, higher mean squared error than PSmatching, weighting and covariate adjustment method.• Performance of use of useful interaction PS approach: In thisscenario under the use of useful interaction approach, the PS match-ing methods M5 and M6 had lower bias than the overall PS approachand subgroup-specific approach but the bias was higher compared tosubgroup specific PS matching method M4 in the male subgroup. Thecorresponding PS weighting method W2 had lower bias than the cor-responding PS matching methods. The stratification method S2 hadhigher bias than PS matching method. The corresponding covariateadjustment method C3 had lower bias than corresponding PS match-ing method M5 in the male subgroup but the bias was higher in thefemale subgroup.• Performance of univariate and generalized PS approach: Thegeneralized PS method had lower bias, lower mean squared error thanthe univariate PS method.• Under this scenario the methods M1, M2, M3, M4, M5, M6, S1, C1,C3, C5, and C6 did not achieve the desired 95% coverage probability.Results from the Monte Carlo simulation result for scenario G1:moderate non-additivity and non-linearityTable 5.14: Results from the Monte Carlo simulation result for scenario G1:moderate non-additivity and non-linearityPS effect modifiermethodsGender Absolute Bias Percent Bias SD MSE MDSE Power CP CIL BACM1 M 0.134 1914% 0.083 0.157 0.064 1 0.47 0.25 0.88F 0.071 546% 0.093 0.117 0.066 1 0.7 0.26 0.83M2 M 0.134 1914% 0.083 0.157 0.092 1 0.72 0.36 0.98F 0.071 546% 0.093 0.117 0.107 1 0.91 0.41 0.97M3 M 0.159 2271% 0.09 0.183 0.1 1 0.63 0.39 0.97F 0.071 546% 0.092 0.117 0.107 1 0.91 0.41 0.98M4 M 0.055 786% 0.099 0.114 0.133 0.99 0.98 0.5 0.99F 0.114 877% 0.154 0.191 0.177 1 0.94 0.66 0.95M5 M 0.113 1614% 0.144 0.183 0.142 0.94 0.92 0.53 0.94F 0.054 415% 0.166 0.174 0.173 1 0.95 0.64 0.94M6 M 0.081 1157% 0.11 0.137 0.127 0.99 0.93 0.48 0.97F 0.068 523% 0.172 0.185 0.179 1 0.95 0.66 0.94W1 M 0.025 357% 0.056 0.062 0.072 1 0.98 0.28 0.99F 0.015 115% 0.075 0.076 0.097 1 0.99 0.38 0.98W2 M 0.03 429% 0.11 0.114 0.115 0.99 0.97 0.44 0.97F 0.031 238% 0.13 0.134 0.134 1 0.97 0.49 0.96W3 M 0.005 71% 0.095 0.096 0.118 0.98 1 0.44 1F 0.094 723% 0.136 0.166 0.146 0.99 0.98 0.53 0.97W4 M 0.025 357% 0.056 0.062 0.072 1 0.98 0.28 0.99F 0.015 115% 0.075 0.076 0.097 1 0.99 0.38 0.98W5 M 0.03 429% 0.11 0.114 0.115 0.99 0.97 0.44 0.97F 0.031 238% 0.13 0.134 0.134 1 0.97 0.49 0.96W6 M 0.025 357% 0.056 0.062 0.064 1 0.96 0.25 0.97F 0.004 31% 0.076 0.076 0.066 1 0.92 0.26 0.92S1 M 0.109 1557% 0.104 0.15 0.135 0.72 0.91 0.52 0.98F 0.216 1662% 0.092 0.235 0.128 1 0.63 0.5 0.99S2 M 0.034 486% 0.075 0.083 0.119 0.98 0.99 0.47 0.99F 0.115 885% 0.067 0.133 0.118 1 0.96 0.46 1S3 M 0.079 1129% 0.065 0.103 0.128 1 0.99 0.5 1F 0.033 254% 0.067 0.075 0.128 1 1 0.5 1C1 M 0.099 1414% 0.057 0.114 0.068 1 0.72 0.26 0.98F 0.114 877% 0.062 0.13 0.074 1 0.68 0.29 0.98C2 M 0.001 14% 0.002 0.002 0.086 1 1 0.34 1F 0.001 8% 0.005 0.005 0.083 1 1 0.33 1C3 M 0.091 1300% 0.058 0.108 0.078 1 0.85 0.3 1F 0.089 685% 0.056 0.105 0.08 1 0.88 0.31 1C4 M 0.001 14% 0.017 0.017 0.083 1 1 0.32 1F 0.001 8% 0.01 0.01 0.075 1 1 0.29 1C5 M 0.118 1686% 0.058 0.131 0.069 1 0.62 0.27 0.98F 0.118 908% 0.061 0.133 0.074 1 0.66 0.29 0.98C6 M 0.12 1714% 0.064 0.136 0.073 1 0.63 0.29 0.98F 0.119 915% 0.062 0.134 0.073 1 0.64 0.28 0.98Note:M:Male, F:Female,SD: standard deviation,MSE: Mean squared error,MDSE: Model standard error,CP: Coverage probability,CIL:Confidenceinterval length estimate,BAC:Bias adjusted coverageBias comparison for simulation scenario G1: moderatenon-additivity and non-linearityFigure 5.79: Bias for each of the approaches to handle effect modifier forsimulation scenario G1: moderate non-additivity and non-linearity for sub-group label MaleFigure 5.80: Bias for each of the approaches to handle effect modifier forsimulation scenario G1: moderate non-additivity and non-linearity for sub-group label femaleMean squared error comparison for simulation scenario G1:moderate non-additivity and non-linearityFigure 5.81: Mean squared error for each of the approaches to handle ef-fect modifier for simulation scenario G1: moderate non-additivity and non-linearity for subgroup label MaleFigure 5.82: Mean squared error for each of the approaches to handle ef-fect modifier for simulation scenario G1: moderate non-additivity and non-linearity for subgroup label femaleCoverage probability for simulation scenario G1: moderatenon-additivity and non-linearityFigure 5.83: Coverage probability for each of the approaches to handle effectmodifier for subgroup label Male for simulation scenario G1: moderate non-additivity and non-linearityFigure 5.84: Coverage probability for each of the approaches to handle effectmodifier for subgroup label Female for simulation scenario G1: moderatenon-additivity and non-linearity5.3 Plasmode simulation result5.3.1 Plasmode simulation result: crude analysisIn our study, we performed plasmode simulation besides the Monte Carlosimulation using the data from the study conducted by (Connors et al.1996). For plasmode simulation we reported the simulation performancemeasure from both crude and adjusted simulation. From the crude plasmodesimulation study we found that,• Like Monte Carlo simulation study overall all the propensity scoreeffect modifier methods performed well by producing lower bias, meansquared error and higher coverage probability presented in the Figures(5.85 and 5.86).• In case of use of overall PS for matching within subgroup approachfor male the propensity score matching methods M1, M2 and M3 pro-duced higher bias then PS subgroup-specific matching method M4 andfor female group the bias was lower compared to method M4. Simi-larly, mean squared error also lower than the method for both maleand female group. The coverage probability was lower in method M1compared to method M4 as well as from other PS matching methods.• On the other hand, we found that, under the use of overall PS fordrawing inference within subgroup approach, compared to PS match-ing method other PS methods such as PS weighting, PS stratificationand covariate adjustment produced lower bias, mean squared error andhigher coverage probability except the covariate adjustment methodgeneralized PS that showed highest bias compared to all other meth-ods.• For PS subgrouping approach, the PS matching method (method M4)had lower bias for male subgroup compared to other PS matchingmethods but for female subgroup bias was higher than other PS match-ing methods. Under similar approach we found higher bias for malegroup in other PS methods where as for female subgroup the bias waslower in other PS methods than PS matching method.• For PS useful interaction approach we found, the bias was higher(method M5 and M6) compared to other PS matching methods. Com-pared to PS matching methods other PS methods weighting (methodsW2 and W6), stratification method (method S3) and covariate adjust-ment method (C3) produced lower bias, lower mean squared error andhigher coverage probability.• From the plasmode simulation in terms of covariate adjustment meth-ods, we observed that, compared to univariate PS method (methodC1), the generalized PS method (method C4) produced higher biasand lower mean squared error and coverage probability whereas inMonte Carlo simulation study we found lower bias in case of general-ized PS method.• Under this simulation the methods M1, W6, C2,C6 did not achieve95% coverage probability.Bias comparison for plasmode simulation (crude)Figure 5.85: Bias for each of the approaches to handle effect modifier forplasmode simulation (crude) for subgroup label MaleFigure 5.86: Bias for each of the approaches to handle effect modifier forplasmode simulation (crude) for subgroup label FemaleMean squared error for plasmode simulationFigure 5.87: Mean squared error for each of the approaches to handle effectmodifier for plasmode simulation (crude) for subgroup label MaleFigure 5.88: Mean squared error for each of the approaches to handle effectmodifier for plasmode simulation (crude) for subgroup label FemaleCoverage probability from plasmode simulationFigure 5.89: Coverage probability for each of the approaches to handle effectmodifier for plasmode simulation (crude) for subgroup label MaleFigure 5.90: Coverage probability for each of the approaches to handle effectmodifier for plasmode simulation (crude) for subgroup label Female5.3.2 Plasmode simulation result from adjusted analysisBesides plasmode simulation crude analysis we also did plasmode simulationfor adjusted case. By adjusted we are meaning that, in the outcome ateach simulated dataset we have considered the other covariates in the asmentioned in the section 3.2 from chapter 3.In our study, we performed plasmode simulation besides the Monte Carlosimulation using the data from the study conducted by (Connors et al.1996). For plasmode simulation we reported the simulation performancemeasure from both crude and adjusted simulation. From the crude plasmodesimulation study we found that,• For adjusted plasmode simulation analysis we observed all the PS effectmodifier methods showed lower bias illustrated in the Figures (5.91 and5.92)), mean squared error and higher coverage probability (except PSstratification method S3)• Under the method use of overall PS for matching within subgroupapproach for male the propensity score matching method M1 producedhigher bias than PS subgroup-specific matching methods M4 and forfemale group the PS matching methods M2 and M3 produced higherbias then subgroup-specific PS matching approach (method M4). Themean squared error and coverage probability was same in both overallPS approach and subgroup-specific PS approach.• Compared to PS overall matching methods under the same approachof effect modifier PS weighting produced lower bias whereas PS strat-ification (method S1) and covariate adjustment method (methods C5and C6) produced higher bias, lower mean squared error and highercoverage probability.• For PS sub-grouping approach, the PS matching method (methodM4) had lower bias for male subgroup compared to other PS match-ing methods but for female subgroup bias was higher than other PSmatching methods. Under similar approach we found higher bias formale group in PS weighting methods where as for female subgroup thebias was lower PS weighting method than PS matching methods. Theother PS methods, PS stratification and covariate adjustment methodsproduced higher bias than PS matching and PS weighting methods.• For PS useful interaction approach we found, the bias was lower (methodM5 and M6) for male subgroup and bias was higher for female sub-group compared to other PS matching methods. Similar to the plas-mode simulation crude analysis, compared to PS matching methodsother PS methods weighting (methods W2 and W6), PS stratificationmethod (method S3) produced lower bias, means squared error andhigher coverage probability. Under this approach the covariate adjust-ment method C3 produced higher than other PS methods in the malesubgroup but it was lower than PS matching method in the femalesubgroups.• In the plasmode simulation (adjusted) analysis, the covariate adjust-ment method (method C4) produced lower bias compared to univariatePS method (method C1) in the male subgroup whereas the bias washigher in the female subgroup. The generalized PS had lower meansquared error than univariate PS method having similar coverage prob-ability.Bias comparison for plasmode simulation (adjusted)Figure 5.91: Bias for each of the approaches to handle effect modifier forplasmode simulation (adjusted) for subgroup label MaleFigure 5.92: Bias for each of the approaches to handle effect modifier forplasmode simulation (adjusted) for subgroup label FemaleMean squared error for plasmode simulation (adjusted)Figure 5.93: Mean squared error for each of the approaches to handle effectmodifier for plasmode simulation (adjusted) for subgroup label MaleFigure 5.94: Mean squared error for each of the approaches to handle effectmodifier for plasmode simulation (adjusted) for subgroup label FemaleCoverage probability from plasmode simulation (adjusted)Figure 5.95: Coverage probability for each of the approaches to handle effectmodifier for plasmode simulation (adjusted) for subgroup label MaleFigure 5.96: Coverage probability for each of the approaches to handle effectmodifier for plasmode simulation (adjusted) for subgroup label Female5.4 Sensitivity analysis5.4.1 Propensity score weighting Methods: Overlap weightsExtreme propensity score problem in inverse probability propen-sity score method: The inverse probability weight is a widely used propen-sity score weighting method. Now, if the propensity scores have extremedifferentiability that is, some patients had propensity score near to 0 andfor others near to 1, in that case, the inverse probability weight produced bi-ased estimates, large weights for some patients, and higher variability of thetreatment effect estimates [12, 27, 33]. Some researchers proposed trimmingmethods to overcome the extreme weight problem, through which some pa-tients having extreme weights would be excluded [69, 70]. But to apply thetrimming method, we need to rely on some threshold values, which are usu-ally determined arbitrarily and resulted in the loss of a large sample fromthe study [71].Advantage of overlap weight over inverse probability weight:Overlap weight is another weighting method to overcome the problem men-tioned above of trimming method [72]. Under the overlap weighting method,each patient’s weight is the probability of that patient being assigned to theopposite group [72]. Overlap weight has all the properties of inverse proba-bility weight in terms of covariate balance and precision. Most importantly,in extreme propensity scores, we do not need to discard the patients fromtail of the distribution of propensity score using overlap weight. Overlapweight can minimize the large sample variance of the estimator [71]. Inaddition to this, the propensity score overlap weighting method can workin other settings such as multiple treatments, subgroup analysis, and whentime-varying treatments present in the study [71].The weight function that used to do overlap weighting method for thetreatment indicator Ai and estimated propensity score psi is as follows,wi = Ai × (1− psi) + (1−Ai)× psi (5.1)where psi is the estimated PS score for the individual i and Ai is the indicatorof treatment.Applying propensity score overlap weighting method to analyzeeffect modifier:To apply overlap weight to evaluate various effect modifier method’s per-formance, we have changed the weight function used for inverse probabilityweighting methods described in Chapter 2 following the equation (5.1).Results from Monte Carlo simulation without interaction afterapplying overlap weighting methodPerformance of overall PS approach: In all simulation scenarios,the overall PS approach produced higher bias and mean squared error thanthe subgroup-specific approach. In the scenarios (i) additive effects and (ii)mild non-linearity, the overall PS approach produced higher bias and meansquared error than interaction approach, whereas in other scenarios usefulinteraction approach produced higher bias and mean squared error thanoverall and subgroup-specific approach (Table 5.15).Performance of subgroup-specific PS approach: Under this ap-proach the subgroup-specific approach performed best compared to othertwo approaches in terms of lower bias and mean squared error (Table 5.15).Performance of useful interaction PS approach: The interactionapproach produced higher bias and mean squared error compared to othertwo effect modifier approaches in all scenarios except the additive effectsand mild non-linearity effects (Table 5.15).Table 5.15: Monte Carlo simulation result without interaction from overlapweighting methodSimulation scenario PS effect modifier methods Gender Absolute Bias SD MSE MDSE Power CP CIL BACAdditive effect(scenario A)W7 M 0.0002 0.05254 0.05252 0.072 1 0.99 0.28 0.99F 0.0001 0.05259 0.05256 0.072 1 0.99 0.28 0.99W8 M 0.09 0.04015 0.09854 0.072 1 0.89 0.28 1F 0.0001 0.0001 0.0001 0.073 1 1 0.28 1W9 M 0.0002 0.05254 0.05252 0.072 1 0.99 0.28 0.99F 0.0001 0.05259 0.05256 0.072 1 0.99 0.28 0.99Mild non-linearity(scenario B)W7 M 0.0006 0.05071 0.05068 0.072 1 0.99 0.28 0.99F 0.0008 0.05071 0.05069 0.072 1 0.99 0.28 0.99W8 M 0.0001 0.0001 0.0001 0.073 1 1 0.28 1F 0.0001 0.0001 0.0001 0.072 1 1 0.28 1W9 M 0.0004 0.03218 0.03217 0.072 1 1 0.28 1F 0.0002 0.03216 0.03215 0.072 1 1 0.28 1Moderate non-linearity(scenario C)W7 M 0.0005 0.05115 0.05112 0.07 1 0.99 0.27 0.99F 0.0002 0.05111 0.05109 0.07 1 0.99 0.27 0.99W8 M 0.0001 0.0001 0.0001 0.07 1 1 0.27 1F 0.0001 0.0001 0.0001 0.07 1 1 0.27 1W9 M 0.0006 0.0319 0.03189 0.07 1 1 0.27 1F 0.0003 0.03187 0.03185 0.07 1 1 0.27 1Mild non-additivity(scenario D)W7 M 0.0005 0.05419 0.05417 0.074 1 0.99 0.29 0.99F 0.0002 0.05422 0.0542 0.074 1 0.99 0.29 0.99W8 M 0.0001 0.0001 0.0001 0.075 1 1 0.29 1F 0.0001 0.0001 0.0001 0.075 1 1 0.29 1W9 M 0.0006 0.03269 0.03268 0.074 1 1 0.29 1F 0.0003 0.03273 0.03272 0.074 1 1 0.29 1Mild non-additivityand non-linearity(scenario E)W7 M 0.0004 0.05413 0.0541 0.074 1 1 0.29 1F 0.0001 0.05404 0.05401 0.074 1 0.99 0.29 0.99W8 M 0.0001 0.0001 0.0001 0.075 1 1 0.29 1F 0.0001 0.0001 0.0001 0.074 1 1 0.29 1W9 M 0.001 0.03348 0.03348 0.074 1 1 0.29 1F 0.0008 0.0334 0.03339 0.074 1 1 0.29 1Moderatenon-additivity((scenario F))W7 M 0.0011 0.05265 0.05263 0.074 1 1 0.29 1F 0.0008 0.05256 0.05254 0.073 1 0.99 0.29 0.99W8 M 0.0001 0.0001 0.0001 0.074 1 1 0.29 1F 0.0001 0.0001 0.0001 0.074 1 1 0.29 1W9 M 0.0015 0.03293 0.03295 0.074 1 1 0.29 1F 0.0012 0.03294 0.03295 0.074 1 1 0.29 1moderatenon-additivity andnon-linearity (scenarioG)W7 M 0.0006 0.05365 0.05363 0.072 1 0.99 0.28 0.99F 0.001 0.05367 0.05365 0.072 1 0.99 0.28 0.99W8 M 0.0001 0.0001 0.0001 0.073 1 1 0.28 1F 0.0001 0.0001 0.0001 0.073 1 1 0.28 1W9 M 0.0001 0.03397 0.03395 0.072 1 1 0.28 1F 0.0002 0.03392 0.0339 0.072 1 1 0.28 1Note:W7: overall PS approach, W8: subgroup-specific PS approach, W9: Useful interaction approach, M:Male, F:Female,SD: standard deviation,ESE: Empirical Standard error,MSE: Mean squared error,MDSE: Model standard error,CP: Coverage probability,CIL:Confidence interval lengthestimate,BAC:Bias adjusted coverageResults from Monte Carlo simulation with interaction after ap-plying overlap weighting methodPerformance of overall PS approach: The overall PS approach pro-duced higher bias and mean squared error than the subgroup-specific andinteraction approach in all simulation scenarios (Table 5.16).Performance of subgroup-specific PS approach: The subgroup-specific approach produced lowest bias, mean squared error than other twoapproaches (Table 5.16).Performance of useful interaction PS approach: This approachproduced higher bias and mean squared error compared to subgroup specificapproach (Table 5.16).Table 5.16: Monte Carlo simulation result with interaction from overlapweighting methodSimulation scenario PS effect modifier methods Gender Absolute Bias SD MSE MDSE Power CP CIL BACAdditive effect(scenario A1)W7 M 0.0392 0.05399 0.0667 0.074 1 0.97 0.29 0.99F 0.04 0.05511 0.06807 0.078 1 0.97 0.3 0.99W8 M 0.0001 0.0001 0.0001 0.093 1 1 0.36 1F 0.0001 0.0001 0.0001 0.088 1 1 0.34 1W9 M 0.0021 0.06725 0.06725 0.092 1 0.99 0.36 0.99F 0.0013 0.06063 0.06061 0.087 1 1 0.34 1Mild non-linearity(scenario B1)W7 M 0.0434 0.05612 0.07094 0.074 1 0.97 0.29 0.99F 0.0406 0.0525 0.06637 0.076 1 0.98 0.3 0.99W8 M 0.0001 0.0001 0.0001 0.093 1 1 0.36 1F 0.0001 0.0001 0.0001 0.086 1 1 0.34 1W9 M 0.0073 0.06858 0.06893 0.092 1 0.99 0.36 0.99F 0.0064 0.05679 0.05713 0.086 1 0.99 0.33 1Moderatenon-linearity((scenario C1))W7 M 0.0458 0.05452 0.07118 0.071 1 0.96 0.28 0.99F 0.0467 0.05531 0.07235 0.078 1 0.98 0.3 0.99W8 M 0.0001 0.0001 0.0001 0.086 1 1 0.34 1F 0.0001 0.0001 0.0001 0.086 1 1 0.33 1W9 M 0.018 0.06422 0.06665 0.085 1 0.99 0.33 0.99F 0.0171 0.05959 0.06197 0.085 1 0.99 0.33 0.99Mild non-additivity(scenario D1)W7 M 0.055 0.0563 0.0787 0.074 1 0.94 0.29 0.99F 0.0557 0.05695 0.07963 0.079 1 0.96 0.31 0.99W8 M 0.0001 0.0001 0.0001 0.094 1 1 0.37 1F 0.0001 0.0001 0.0001 0.088 1 1 0.34 1W9 M 0.014 0.06929 0.07066 0.093 1 0.99 0.36 0.99F 0.0127 0.06066 0.06194 0.087 1 0.99 0.34 0.99Mild non-additivityand non-linearity(scenario E1)W7 M 0.0604 0.05779 0.08359 0.074 1 0.93 0.29 0.99F 0.0565 0.05409 0.07821 0.078 1 0.96 0.3 1W8 M 0.0001 0.0001 0.0001 0.094 1 1 0.37 1F 0.0001 0.0001 0.0001 0.087 1 1 0.34 1W9 M 0.0249 0.07132 0.07551 0.093 1 0.98 0.36 0.99F 0.0206 0.05771 0.06124 0.086 1 0.99 0.34 1Moderatenon-additivity(scenario F1)W7 M 0.0622 0.05567 0.08342 0.073 1 0.93 0.29 0.99F 0.0617 0.05537 0.08288 0.08 1 0.95 0.31 0.99W8 M 0.0001 0.0001 0.0001 0.092 1 1 0.36 1F 0.0001 0.0001 0.0001 0.09 1 1 0.35 1W9 M 0.0302 0.06833 0.07467 0.092 1 0.98 0.36 1F 0.0269 0.05996 0.0657 0.089 1 0.99 0.35 1Moderatenon-additivity andnon-linearity(scenario G1)W7 M 0.0608 0.05493 0.08194 0.07 1 0.92 0.28 0.99F 0.0658 0.05943 0.08867 0.081 1 0.94 0.32 0.99W8 M 0.0001 0.0001 0.0001 0.085 1 1 0.33 1F 0.0001 0.0001 0.0001 0.089 1 1 0.35 1W9 M 0.0356 0.06439 0.07353 0.085 1 0.98 0.33 0.99F 0.0353 0.06333 0.07247 0.088 1 0.98 0.35 1Note:W7: overall PS approach, W8: subgroup-specific PS approach, W9: Useful interaction approach, M:Male, F:Female,SD: standard deviation,ESE: Empirical Standard error,MSE: Mean squared error,MDSE: Model standard error,CP: Coverage probability,CIL:Confidence intervallength estimate,BAC:Bias adjusted coverage5.4.2 Application of machine learning methods in theestimation of the propensity score for each of theeffect modifier methodsWe previously used binary logistic regression the propensity score for eachof the effect modifier methods. However, in the logistic regression approach,the assumptions related to variable selection, the functional form of theoutcome and exposure, distribution of covariates, interactions are required.Failing to meet these assumptions leads to biased effect estimates [26, 27].Alternatively, researchers nowadays use various machine learning methods toestimate the PS [26–29]. Some standard machine learning methods are clas-sification and regression trees (CARTs), neural networks, random forests,Generalized Boosted modeling [27, 30, 31].One of the propensity score effect modifier method was the useful inter-action approach. We had to specify the interaction terms between covariatesand subgroup variable for binary logistic regression during PS estimation.In contrast to this, with the decision tree method, we can automaticallycapture the interactions present in the model. Therefore, if we could use thedecision tree method, we can work with the overall PS, subgroup-specificapproach, and covariate adjustment approach. We do not need to applyuseful interaction approach to handle effect modifier within PS methods.To verify whether this idea works in our study, we have used the decisiontree method to estimate the propensity score for the Monte Carlo simulationstudy and then evaluated the performance of the methods of handling effectmodifier as described in Chapter 2.5.4.3 Results from the Monte Carlo simulation withoutinteraction for scenario “A” after applying thedecision tree method in the PS estimation• Performance of overall PS approach and subgroup-specificapproach: The overall PS approach produced higher bias than thesubgroup-specific in all PS methods as depicted in the Table (5.17)and Figures (5.97 and 5.98). Besides higher bias, in all PS methodswe found that, the overall PS approach had higher mean squared errorthan the subgroup-specific approach (Figures 5.99 and 5.100).• Performance of various PS methods under overall PS ap-proach: We observed that, under overall PS approach, PS match-ing methods had higher bias (Figures 5.97 and 5.98) and higher meansquared error (Figures 5.99 and 5.100) than other PS methods.• Performance of various PS methods under subgroup-specificPS approach: Under subgroup-specific PS approach we found that,the PS matching method had lower bias (Figures 5.97 and 5.98) thanother PS methods but had higher mean squared error (Figures 5.99and 5.100).Table 5.17: Monte Carlo simulation without interaction for scenario APS effect modifiermethodsGender Absolute Bias SD MSE MDSE Power CP CIL BACM1 M 0.1337 0.07632 0.15395 0.071 1 0.53 0.28 0.94F 0.1162 0.16992 0.20578 0.106 1 0.68 0.41 0.78M2 M 0.1337 0.07632 0.15395 0.071 1 0.53 0.28 0.94F 0.1162 0.16992 0.20578 0.105 1 0.67 0.41 0.77M3 M 0.0612 0.08922 0.10818 0.083 1 0.87 0.33 0.94F 0.0574 0.09315 0.1094 0.083 1 0.86 0.33 0.92M4 M 0.0337 0.12193 0.12596 0.091 1 0.84 0.36 0.86F 0.02 0.10844 0.10978 0.091 1 0.87 0.36 0.91W1 M 0.0632 0.07291 0.09645 0.072 1 0.86 0.28 0.94F 0.0619 0.07516 0.09735 0.072 1 0.84 0.28 0.94W3 M 0.0348 0.07244 0.08035 0.079 1 0.94 0.31 0.96F 0.034 0.07239 0.07995 0.078 1 0.94 0.31 0.97W4 M 0.0633 0.07295 0.09656 0.072 1 0.86 0.28 0.94F 0.0621 0.07526 0.09755 0.072 1 0.84 0.28 0.94W6 M 0.0632 0.07291 0.09645 0.064 1 0.81 0.25 0.92F 0.0619 0.07516 0.09735 0.064 1 0.8 0.25 0.92S1 M 0.0546 0.08059 0.09667 0.158 0.98 1 0.62 1F 0.0696 0.06951 0.09789 0.157 1 1 0.61 1S3 M 0.0431 0.07362 0.08526 0.165 1 1 0.64 1F 0.041 0.07571 0.08608 0.165 1 1 0.64 1C1 M 0.0607 0.06785 0.09102 0.066 1 0.86 0.26 0.95F 0.0601 0.07012 0.09234 0.066 1 0.83 0.26 0.94C2 M 0.0317 0.06836 0.07531 0.075 1 0.94 0.29 0.96F 0.0304 0.06909 0.07547 0.075 1 0.94 0.29 0.97C5 M 0.0608 0.06783 0.09104 0.066 1 0.86 0.26 0.95F 0.0602 0.07002 0.09233 0.066 1 0.84 0.26 0.94C6 M 0.0616 0.07167 0.09446 0.07 1 0.86 0.27 0.94F 0.0594 0.07388 0.09475 0.07 1 0.84 0.27 0.94Note:M:Male, F:Female,SD: standard deviation, ESE: Empirical Standard error,MSE: Mean squared error,MDSE: Model standard er-ror,CP: Coverage probability,CIL:Confidence interval length estimate,BAC:Bias adjusted coverageBias comparison for simulation scenario A: additive effectsFigure 5.97: Bias for each of the approaches to handle effect modifier forsimulation scenario A : additive effects for subgroup label MaleFigure 5.98: Bias for each of the approaches to handle effect modifier forsimulation scenario A : additive effects for subgroup label femaleMean squared error comparison for simulation scenario A :additive effectsFigure 5.99: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario A : additive effects for subgroup label MaleFigure 5.100: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario A : additive effects for subgroup label female5.4.4 Results from the Monte Carlo simulation withoutinteraction for scenario “G” after applying thedecision tree method in the PS estimation• Performance of overall PS approach and subgroup-specificapproach: Similar to the scenario “A,” under the scenario “G” wefound that, the overall PS approach produced higher bias than thesubgroup-specific in all PS methods as depicted in the Table (5.18)and Figures (5.101 and 5.102). Similarly, in all PS methods we foundthat, the overall PS approach had higher mean squared error than thesubgroup-specific approach (Figures 5.103 and 5.104).• Performance of various PS methods under overall PS ap-proach: We observed that, under overall PS approach, PS matchingmethods had higher bias (Figures 5.101 and 5.102) and higher meansquared error (Figures 5.103 and 5.104) than other PS methods.• Performance of various PS methods under subgroup-specificPS approach: Under subgroup-specific PS approach we found that,the PS matching method had lower bias (Figures 5.101 and 5.102)than other PS methods but had higher mean squared error (Figures5.103 and 5.104).Table 5.18: Monte Carlo simulation without interaction for scenario GPS effect modifiermethodsGender Absolute Bias SD MSE MDSE Power CP CIL BACM1 M 0.0944 0.08812 0.12914 0.076 1 0.73 0.3 0.91F 0.1175 0.22916 0.25741 0.111 1 0.57 0.44 0.64M2 M 0.0944 0.08812 0.12914 0.076 1 0.73 0.3 0.91F 0.1175 0.22916 0.25741 0.111 1 0.56 0.43 0.64M3 M 0.0348 0.10966 0.11501 0.088 1 0.86 0.35 0.89F 0.0283 0.10448 0.10821 0.088 1 0.9 0.35 0.9M4 M 0.012 0.12222 0.12275 0.097 1 0.88 0.38 0.88F 0.0106 0.11573 0.11616 0.097 1 0.9 0.38 0.91W1 M 0.0469 0.08937 0.1009 0.078 1 0.85 0.3 0.9F 0.047 0.08556 0.0976 0.078 1 0.87 0.3 0.93W3 M 0.0219 0.08545 0.08817 0.085 1 0.94 0.33 0.96F 0.023 0.08392 0.08697 0.085 1 0.94 0.33 0.96W4 M 0.0472 0.08949 0.10113 0.078 1 0.85 0.3 0.9F 0.0472 0.08564 0.09773 0.078 1 0.87 0.3 0.93W6 M 0.0469 0.08937 0.1009 0.065 1 0.79 0.25 0.84F 0.0619 0.07516 0.09735 0.064 1 0.8 0.25 0.92S1 M 0.0404 0.09339 0.10108 0.167 0.98 1 0.65 1F 0.0416 0.09393 0.10205 0.162 1 1 0.63 1S3 M 0.0279 0.08786 0.09215 0.173 0.97 1 0.68 1F 0.0289 0.08682 0.09147 0.173 1 1 0.68 1C1 M 0.0375 0.08353 0.09151 0.069 1 0.86 0.27 0.89F 0.0383 0.08034 0.08895 0.069 1 0.87 0.27 0.89C2 M 0.0145 0.08032 0.08157 0.078 1 0.95 0.31 0.94F 0.0149 0.07683 0.07822 0.078 1 0.95 0.31 0.95C5 M 0.0375 0.08336 0.09139 0.069 1 0.86 0.27 0.88F 0.0383 0.08036 0.08899 0.069 1 0.87 0.27 0.89C6 M 0.0382 0.08681 0.0948 0.073 1 0.86 0.29 0.9F 0.0374 0.08319 0.09117 0.073 1 0.88 0.29 0.92Note:M:Male, F:Female,SD: standard deviation, ESE: Empirical Standard error,MSE: Mean squared error,MDSE: Model standard er-ror,CP: Coverage probability,CIL:Confidence interval length estimate,BAC:Bias adjusted coverageBias comparison for simulation scenario G: moderatenon-additivity and non-linearityFigure 5.101: Bias for each of the approaches to handle effect modifier forsimulation scenario G: moderate non-additivity and non-linearity for sub-group label MaleFigure 5.102: Bias for each of the approaches to handle effect modifier forsimulation scenario G: moderate non-additivity and non-linearity for sub-group label femaleMean squared error comparison for simulation scenario G:moderate non-additivity and non-linearityFigure 5.103: Mean squared error for each of the approaches to handleeffect modifier for simulation scenario G: moderate non-additivity and non-linearity for subgroup label MaleFigure 5.104: Mean squared error for each of the approaches to handleeffect modifier for simulation scenario G: moderate non-additivity and non-linearity for subgroup label female5.4.5 Results from the Monte Carlo simulation withinteraction for scenario “A1” after applying thedecision tree method in the PS estimation:• Performance of overall PS approach and subgroup-specificapproach: The overall PS approach produced the higher bias thanthe subgroup-specific in the PS matching, weighting and covariate ad-justment methods as depicted in the Table (5.19) and Figures (5.105and 5.106). In the PS stratification method we found that, the bias waslower in overall PS approach than the subgroup-specific PS approachof effect modifier. In all PS methods except PS stratification methodwe found that, the overall PS approach had higher mean squared errorthan subgroup-specific approach (Figures 5.107 and 5.108).• Performance of various PS methods under overall PS ap-proach: We observed that, under overall PS approach, PS weightingmethods had higher bias (Figures 5.105 and 5.106) and higher meansquared error (Figures 5.107 and 5.108) than other PS methods.• Performance of various PS methods under subgroup-specificPS approach: Under subgroup-specific PS approach we found that,the PS stratification method had higher bias (Figures 5.105 and 5.106)and higher mean squared error (Figures 5.99 and 5.100) than other PSmethods.Table 5.19: Monte Carlo simulation with interaction for scenario A1PS effect modifiermethodsGender Absolute Bias SD MSE MDSE Power CP CIL BACM1 M 0.1113 0.15093 0.18747 0.119 1 0.77 0.46 0.89F 0.1155 0.13454 0.17726 0.107 1 0.74 0.42 0.88M2 M 0.1113 0.15093 0.18747 0.118 1 0.76 0.46 0.89F 0.1155 0.13454 0.17726 0.106 1 0.73 0.41 0.88M3 M 0.1158 0.12077 0.16731 0.109 1 0.78 0.43 0.93F 0.1182 0.11672 0.16605 0.103 1 0.75 0.4 0.92M4 M 0.0713 0.13985 0.15691 0.122 1 0.86 0.48 0.92F 0.0714 0.12944 0.14775 0.116 1 0.88 0.45 0.92W1 M 0.156 0.1134 0.1928 0.113 1 0.7 0.43 0.95F 0.1522 0.09803 0.18101 0.094 1 0.62 0.36 0.92W3 M 0.1239 0.11516 0.16908 0.118 1 0.8 0.45 0.94F 0.1067 0.10338 0.14855 0.103 1 0.81 0.4 0.94W4 M 0.1562 0.11355 0.1931 0.113 1 0.69 0.43 0.95F 0.1523 0.09803 0.18108 0.094 1 0.62 0.36 0.92W6 M 0.156 0.1134 0.1928 0.064 1 0.4 0.25 0.74F 0.1522 0.09803 0.18101 0.064 1 0.4 0.25 0.81S1 M 0.0017 0.11033 0.11028 0.144 0.91 0.98 0.56 0.98F 0.1248 0.10192 0.16106 0.134 1 0.91 0.52 0.99S3 M 0.1905 0.11229 0.22111 0.15 0.99 0.82 0.58 0.99F 0.1749 0.0995 0.20116 0.15 1 0.78 0.58 0.99C1 M 0.0091 0.07347 0.07399 0.071 1 0.94 0.28 0.94F 0.1012 0.07752 0.12745 0.074 1 0.72 0.29 0.93C2 M 0.0712 0.08813 0.11325 0.098 1 0.9 0.39 0.97F 0.0751 0.0862 0.11429 0.093 1 0.89 0.36 0.97C5 M 0.0823 0.08543 0.11859 0.083 1 0.82 0.32 0.94F 0.1636 0.08048 0.18233 0.078 1 0.46 0.31 0.94C6 M 0.1181 0.08907 0.14786 0.091 1 0.74 0.36 0.96F 0.1311 0.08891 0.15841 0.085 1 0.65 0.33 0.94Note:M:Male, F:Female,SD: standard deviation, ESE: Empirical Standard error,MSE: Mean squared error,MDSE: Model standard er-ror,CP: Coverage probability,CIL:Confidence interval length estimate,BAC:Bias adjusted coverageBias comparison for simulation scenario A1 : additive effectsFigure 5.105: Bias for each of the approaches to handle effect modifier forsimulation scenario A1 : additive effects for subgroup label MaleFigure 5.106: Bias for each of the approaches to handle effect modifier forsimulation scenario A1 : additive effects for subgroup label femaleMean squared error comparison for simulation scenario A1 :additive effectsFigure 5.107: Mean squared error for each of the approaches to handle effectmodifier for simulation scenario A1 : additive effects for subgroup label MaleFigure 5.108: Mean squared error for each of the approaches to handleeffect modifier for simulation scenario A1 : additive effects for subgrouplabel female5.4.6 Results from the Monte Carlo simulation withinteraction for scenario “G1” after applying thedecision tree method in the PS estimation:• Performance of overall PS approach and subgroup-specificapproach: Under the scenario “G1” we found that, the overall PSapproach produced higher bias than the subgroup-specific in all PSmethods as depicted in the Table (5.20) and Figures (5.109 and 5.110).Similarly, in all PS methods we found that, the overall PS approachhad higher mean squared error than subgroup-specific approach (Fig-ures 5.111 and 5.112).• Performance of various PS methods under overall PS ap-proach: We observed that, under overall PS approach, PS stratifi-cation methods had higher bias (Figures 5.109 and 5.110) and highermean squared error (Figures 5.111 and 5.112) than other PS methods.• Performance of various PS methods under subgroup-specificPS approach: Similar to the overall PS approach, under subgroup-specific PS approach we found that, the PS stratification method hadhigher bias (Figures 5.109 and 5.110) and higher mean squared error(Figures 5.111 and 5.112) than other PS methods.Table 5.20: Monte Carlo simulation with interaction for scenario G1PS effect modifiermethodsGender Absolute Bias SD MSE MDSE Power CP CIL BACM1 M 0.0352 0.1298 0.13443 0.112 1 0.88 0.43 0.9F 0.0987 0.13296 0.16556 0.112 1 0.8 0.43 0.91M2 M 0.0352 0.1298 0.13443 0.11 1 0.87 0.43 0.89F 0.0987 0.13296 0.16556 0.11 1 0.8 0.43 0.91M3 M 0.0524 0.11078 0.1225 0.106 1 0.91 0.42 0.94F 0.1102 0.11384 0.15843 0.106 1 0.8 0.41 0.94M4 M 0.0274 0.12309 0.12605 0.118 1 0.93 0.46 0.94F 0.0411 0.13689 0.14287 0.121 1 0.9 0.47 0.92W1 M 0.0789 0.09225 0.12132 0.093 1 0.86 0.36 0.95F 0.1748 0.10081 0.20172 0.097 1 0.53 0.38 0.93W3 M 0.0632 0.0962 0.11505 0.1 1 0.9 0.39 0.96F 0.0929 0.12163 0.153 0.114 1 0.84 0.44 0.93W4 M 0.0793 0.09214 0.1215 0.092 1 0.86 0.36 0.95F 0.175 0.10067 0.20182 0.097 1 0.53 0.38 0.93W6 M 0.0789 0.09225 0.12132 0.064 1 0.68 0.25 0.84F 0.1522 0.09803 0.18101 0.064 1 0.4 0.25 0.81S1 M 0.1091 0.10356 0.15038 0.135 0.72 0.91 0.52 0.98F 0.2162 0.09227 0.23503 0.128 1 0.63 0.5 0.99S3 M 0.1134 0.09406 0.14732 0.132 1 0.94 0.52 0.99F 0.1672 0.11351 0.20208 0.132 1 0.77 0.52 0.98C1 M 0.0311 0.07307 0.0794 0.069 1 0.9 0.27 0.94F 0.0854 0.08393 0.11972 0.078 1 0.79 0.3 0.93C2 M 0.041 0.08176 0.09144 0.095 1 0.95 0.37 0.98F 0.0583 0.09608 0.11233 0.094 1 0.9 0.37 0.94C5 M 0.0241 0.0855 0.0888 0.079 1 0.93 0.31 0.94F 0.1658 0.08313 0.18549 0.081 1 0.46 0.32 0.95C6 M 0.0607 0.08409 0.10365 0.088 1 0.91 0.34 0.96F 0.1312 0.09057 0.15937 0.084 1 0.64 0.33 0.93Note:M:Male, F:Female,SD: standard deviation, ESE: Empirical Standard error,MSE: Mean squared error,MDSE: Model standard er-ror,CP: Coverage probability,CIL:Confidence interval length estimate,BAC:Bias adjusted coverageBias comparison for simulation scenario G1: moderatenon-additivity and non-linearityFigure 5.109: Bias for each of the approaches to handle effect modifier forsimulation scenario G1: moderate non-additivity and non-linearity for sub-group label MaleFigure 5.110: Bias for each of the approaches to handle effect modifier forsimulation scenario G1: moderate non-additivity and non-linearity for sub-group label femaleMean squared error comparison for simulation scenario G1:moderate non-additivity and non-linearityFigure 5.111: Mean squared error for each of the approaches to handleeffect modifier for simulation scenario G1: moderate non-additivity and non-linearity for subgroup label MaleFigure 5.112: Mean squared error for each of the approaches to handleeffect modifier for simulation scenario G1: moderate non-additivity and non-linearity for subgroup label female5.4.7 Plasmode simulation for reduced sample sizeIn the previous plasmode simulation we had considered total n=4000 sub-jects at each plasmode simulated datasets. We did another type of plasmodesimulation study where we conisdered a redcued sample size n = 1000 whichconsists of 500 male and 500 female by keep other covariates with reducedsize from the previous plasmode simulation.Plasmode simulation result from crude analysis for reducedsample sizeFrom the plasmode simulation crude analysis for the reduced sample sizewe found that• Under use of overall PS to match within subgroup approach, we ob-served that, for male group the PS matching method M1 had lowerbias than the subgroup-specific PS method M4 whereas in the previousMonte Carlo simulation and plasmode simulation overall PS approachproduced higher bias than subgroup-specific PS approach. On theother hand, for female group the method M1 higher bias than themethod M4 similar to the plasmode and Monte Carlo simulation.• Use of overall PS approach for PS matching method when comparedwith other PS methods (methods W1, W5, S1, C5 and C6) we observedthat, other PS methods produced lower bias, mean squared error (ex-cept PS stratification method S1 which produced higher mean squarederror) and higher coverage probability.• Under subgroup-specific PS approach, for male group compared to thePS matching method (method M4) other PS methods (method W3,S3 and C2) produced lower bias, mean squared error and higher cov-erage probability. On other hand, for female group other PS methodmethod W3 produced higher bias, higher mean squared error and cov-erage probability, but we observed lower bias under PS method C5and C6 for female group having lower mean squared error and cover-age probability.• In the use of useful interaction approach, the PS matching methods(method M5 and M6) produced higher bias than the overall PS ap-proach (methods M1, M2 and M3), almost similar mean squared errorand higher coverage probability than the overall PS approach (meth-ods M1, M2 and M3). When the same approach compared with otherPS methods we found that, other PS methods (methods W2, W6, S2,C3) had lower bias than PS matching methods (methods M5 and M6).• The generalized covariate approach (method C4) produced higher bias,mean squared error and almost similar coverage probability for bothmale and female subgroup compared to the univariate PS method(method C1).Table 5.21: Plasmode simulation result (crude) for reduced sample sizePS methods Effect modifiermethodsLevel of subgroupvariableBias SD ESE MSE MDSE Power CP CIL BACMatching M1 M 0.063 0.50 0.50 0.51 0.19 0.37 0.66 0.74 0.66F 0.11 0.56 0.56 0.57 0.20 0.44 0.69 0.77 0.69M2 M 0.063 0.51 0.51 0.51 0.35 0.10 0.93 1.35 0.93F 0.11 0.57 0.57 0.57 0.36 0.14 0.93 1.41 0.92M3 M 0.060 0.50 0.50 0.50 0.35 0.09 0.92 1.36 0.92F 0.11 0.59 0.59 0.60 0.37 0.13 0.92 1.43 0.92M4 M 0.30 0.88 0.88 0.85 0.40 0.2 0.8 1.56 1F -0.42 0.32 0.32 0.51 0.41 0 1 1.59 1M5 M 0.065 0.51 0.51 0.51 0.36 0.10 0.94 1.39 0.92F 0.12 0.58 0.58 0.59 0.37 0.15 0.93 1.43 0.92M6 M 0.076 0.53 0.53 0.54 0.36 0.10 0.94 1.40 0.93F 0.12 0.58 0.58 0.59 0.38 0.12 0.94 1.48 0.92Weighting W1 M 0.038 0.38 0.38 0.38 0.28 0.13 0.94 1.07 0.94F 0.066 0.40 0.40 0.41 0.27 0.20 0.95 1.03 0.95W2 M 0.042 0.39 0.39 0.29 0.27 0.13 0.94 1.10 0.94F 0.072 0.42 0.42 0.42 0.27 0.21 0.95 1.04 0.95W3 M -0.017 0.33 0.33 0.29 0.27 0.2 1 1.07 1F 0.954 2.03 2.03 2.05 0.55 0 1 1.81 0.6W4 M 0.039 0.38 0.38 0.38 0.28 0.13 0.94 1.08 0.94F 0.064 0.40 0.40 0.40 0.27 0.20 0.95 1.04 0.95W5 M 0.042 0.39 0.39 0.039 0.29 0.13 0.94 1.12 0.94F 0.071 0.42 0.42 0.42 0.27 0.20 0.95 1.04 0.94W6 M 0.038 0.38 0.38 0.38 0.13 0.45 0.62 0.52 0.61F 0.065 0.41 0.41 0.41 0.14 0.53 0.69 0.53 0.68Stratification S1 M 0.031 0.31 0.31 0.31 0.40 0.001 1 1.57 1F 0.502 14.12 14.12 14.12 0.42 0.003 0.99 1.63 0.98S2 M 0.037 0.32 0.32 0.32 0.40 0.003 1 1.58 0.99F 0.515 14.34 14.34 14.35 0.42 0.003 0.99 1.65 0.98S3 M 0.045 0.34 0.35 0.35 0.43 0 0.001 1.68 0.001F 0.070 0.37 0.37 0.38 0.46 0 0 1.78 0CovariateadjustmentC1 M 0.016 0.27 0.27 0.27 0.21 0.16 0.94 0.82 0.94F 0.039 0.29 0.29 0.30 0.22 0.24 0.95 0.83 0.95C2 M 0.056 0.17 0.17 0.16 0.24 0 0 0.93 0F -0.057 0.24 0.24 0.22 0.25 0 0 0.98 0C3 M 0.017 0.27 0.27 0.27 0.22 0.15 0.94 0.83 0.94F 0.040 0.30 0.30 0.30 0.22 0.25 0.95 0.84 0.94C4 M 0.101 0.30 0.30 0.31 0.22 0.20 0.95 0.86 0.95F -0.072 0.29 0.29 0.29 0.22 0.15 0.93 0.85 0.94C5 M 0.015 0.27 0.27 0.27 0.21 0.15 0.94 0.82 0.94F 0.039 0.29 0.29 0.30 0.22 0.24 0.95 0.83 0.94C6 M 0.039 0.38 0.38 0.38 0.14 0.45 0.62 0.53 0.61F 0.065 0.40 0.40 0.40 0.14 0.53 0.69 0.53 0.68Note:M:Male, F:Female,SD: standard deviation, ESE: Empirical Standard error,MSE: Mean squared error,MDSE: Model standard error,CP: Coverageprobability,CIL:Confidence interval length estimate,BAC:Bias adjusted coverage5.4.8 Plasmode simulation result from adjusted analysis forreduced sample sizeFrom the plasmode simulation adjusted analysis for the reduced sample sizeillustrated in the Table 5.22 we found that,• The use of overall PS approach the PS matching method (method M1)produced higher bias and mean squared error for male group than thesubgroup-specific PS matching method (method M4), whereas, in thefemale group we observed lower bias and mean squared error in methodM1 than method M4. The coverage probability was higher in methodM4 than the method M1.• Similar to the plasmode simulation considering the full sample wefound that, in the reduced plasmode simulation adjusted analysis un-der the overall use of PS to draw inference within subgroups the otherPS methods (methods W1, S1, C5) produced lower bias, mean squarederror and higher coverage probability than the PS matching methodM1.• We observed in the plasmode simulation with reduced sample size, thesubgroup-specific methods (methods W3, S3, C2) performed betterthan similar PS matching method (method M4) in terms of lower bias,meqn squared error and higher coverage probability.• In the useful interaction approach, compared to overall PS matchingmethod (method M1) the PS matching method M5 produced lowerbias, mean squared error and higher coverage probability. Under thisapproach, we found similar to previous simulations result that, otherPS methods (methods W2, S2, C3) performed better than methodM5 in terms of producing lower bias, mean squared error and highercoverage probability.• The covariate adjustment method generalized PS (method C4) pro-duced lower bias than univariate PS (method C1) having similar meansquared error and coverage probability.Table 5.22: Plasmode simulation result (adjusted) for reduced sample sizePS methods Effect modifiermethodsLevel of subgroupvariableBias SD ESE MSE MDSE Power CP CIL BACMatching M1 M 0.18 0.77 0.77 0.79 0.32 0.26 0.79 1.23 0.77F 0.49 1.12 1.12 1.22 0.32 0.50 0.80 1.26 0.79M2 M 0.12 0.59 0.59 0.60 0.32 0.19 0.86 1.25 0.85F 0.31 0.86 0.86 0.92 0.33 0.42 0.86 1.29 0.85M3 M 0.11 0.63 0.63 0.64 0.32 0.18 0.86 1.25 0.84F 0.31 0.90 0.90 0.95 0.33 0.42 0.86 1.29 0.85M4 M 0.10 0.48 0.48 0.44 0.34 0.2 1 1.34 1F 1.30 1.08 1.08 1.62 0.34 0.8 0.8 1.32 1M5 M 0.12 0.62 0.62 0.63 0.32 0.20 0.86 1.26 0.84F 0.33 0.89 0.89 0.96 0.33 0.43 0.85 1.29 0.85M6 M 0.10 0.60 0.60 0.62 0.33 0.17 0.87 1.26 0.86F 0.32 0.92 0.92 0.97 0.33 0.40 0.85 1.30 0.85Weighting W1 M 0.10 0.43 0.43 0.44 0.27 0.18 0.91 1.05 0.90F 0.20 0.62 0.62 0.65 0.28 0.51 0.90 1.09 0.90W2 M 0.09 0.43 0.43 0.44 0.27 0.18 0.90 1.05 0.90F 0.20 0.62 0.62 0.65 0.28 0.49 0.90 1.09 0.90W3 M 0.12 0.40 0.40 0.38 0.28 0.2 1 1.08 1F 0.27 0.24 0.24 0.35 0.30 0.6 1 1.18 1W4 M 0.10 0.43 0.43 0.43 0.27 0.18 0.91 1.05 0.89F 0.20 0.62 0.62 0.65 0.28 0.51 0.91 1.09 0.90W5 M 0.10 0.43 0.43 0.44 0.27 0.18 0.90 1.05 0.90F 0.20 0.62 0.62 0.65 0.28 0.49 0.90 1.09 0.90W6 M 0.13 0.54 0.54 0.55 0.19 0.40 0.67 0.78 0.68F 0.34 0.80 0.80 0.88 0.20 0.73 0.70 0.79 0.69Stratification S1 M 0.036 0.35 0.35 0.35 0.44 0.003 1 1.72 1F -0.16 0.40 0.40 0.43 0.44 0.006 0.99 1.73 0.99S2 M 0.036 0.35 0.35 0.35 0.44 0.002 1 1.72 0.99F -0.16 0.41 0.41 0.44 0.45 0.005 0.99 1.75 1S3 M -0.033 0.37 0.37 0.37 0.50 0 0 1.92 0F -0.056 0.49 0.49 0.49 0.51 0 0 1.98 0CovariateadjustmentC1 M 0.132 0.36 0.36 0.39 0.25 0.21 0.93 0.98 0.94F 0.054 0.46 0.46 0.47 0.26 0.45 0.94 1.01 0.94C2 M 0.087 0.46 0.46 0.46 0.31 0.015 0.043 1.20 0.03F 0.26 0.67 0.67 0.72 0.31 0 0.063 1.21 0.021C3 M 0.122 0.36 0.36 0.39 0.25 0.19 0.93 0.98 0.94F 0.044 0.46 0.46 0.47 0.26 0.44 0.94 1.01 0.94C4 M 0.048 0.36 0.36 0.36 0.26 0.14 0.93 1.01 0.93F -0.27 0.37 0.37 0.46 0.26 0.19 0.85 1.01 0.93C5 M 0.13 0.36 0.36 0.39 0.25 0.21 0.93 0.98 0.94F 0.054 0.46 0.46 0.47 0.26 0.45 0.94 1.01 0.94C6 M 0.13 0.54 0.54 0.55 0.20 0.40 0.68 0.78 0.68F 0.35 0.80 0.80 0.88 0.20 0.73 0.70 0.79 0.69Note:M:Male, F:Female,SD: standard deviation, ESE: Empirical Standard error,MSE: Mean squared error,MDSE: Model standard error,CP: Coverageprobability,CIL:Confidence interval length estimate,BAC:Bias adjusted coverage5.4.9 Plasmode simulation result when exposure prevalenceis lowIn real life, sometimes we face the situation when the prevalence of exposureis low. For example, in some pharmacoepidemiologic study, we might notneed to have the high prevalence of newly developed marketed drugs [73].In this case, the estimation of propensity score will be difficult using thelogistic regression method. In rare exposure situation, there would be achance of separation; that is, the frequency of the patients might be too lowin one group, and the logistic model might fail to achieve model convergency[73]. In the presence of an effect modifier, the low exposure prevalencemight create problems of estimation of propensity score at different levelsof the subgroup variable resulting in biased estimate and producing highervariability.To review the performance of effect modifier methods under a low ex-posure prevalence situation, we have performed a plasmode simulation byconsidering the low exposure prevalence. In the plasmode simulation, wehave considered the exposure prevalence low (i.e., p=5%) by keeping theoutcome prevalence same as the previous plasmode simulation. The lowexposure setting will provide us the idea which effect modifier methods pro-duced higher bias, mean squared error and coverage probability.From the plasmode simulation by considering exposure prevalence low(i.e. p = 0.05) illustrated in the Table 5.23 we found that,• Performance of overall PS approach: The overall PS approachproduced higher bias than the subgroup-specific and interaction ap-proach in all PS methods except covariate adjustment method. On theother hand, under this approach, compared to PS matching methods(methods M1, M3), the PS weighting methods (methods W1, W4),and PS covariate adjustment methods (methods C5, C6) producedhigher bias and higher mean squared error.• Performance of subgroup-specific PS approach: Under this ap-proach similar to the overall PS approach, compared to the PS match-ing method (method M4), the PS weighting method (method W3) andthe covariate adjustment method (method C2) produced higher biasand higher mean squared error.• Performance of useful interaction PS approach: Again underthis approach, the PS weighting methods (methods W2, W6) and thecovariate adjustment method (method C3) produced higher bias andhigher mean squared error than the corresponding PS matching meth-ods (method M5 and M6).• Performance of generalized PS and univariate PS approach:The generalized PS (method C4) produced higher bias and meansquared error than univariate PS (method C1)• Under this simulation with low exposure we found the problem ofgetting the performance of PS stratification methods. Because due tolow exposure prevalence (5%) in the stratification approach at somestrata the prevalence was too low that, the model was not converged.Table 5.23: Plasmode simulation result with low exposure prevalencePS effect modifiermethodsGender Absolute Bias SD MSE MDSE Power CP CIL BACM1 M 0.368 1.321 1.37 0.569 0.08 0.95 2.21 0.92F 0.462 1.889 1.944 0.573 0.14 0.93 2.23 0.91M2 M 0.787 4.32 4.389 2.42E5 0.02 0.03 1.4E5 0.03F 1.582 8.911 9.046 2.67E5 0 0.05 1.85E5 0.01M3 M 0.36 1.259 1.309 0.572 0.07 0.96 2.22 0.92F 0.464 1.959 2.012 0.58 0.13 0.93 2.25 0.91M4 M 0.428 1.596 1.652 0.632 0.07 0.94 2.46 0.91F 0.049 1.623 1.623 0.641 0.06 0.93 2.49 0.93M5 M 0.373 1.319 1.37 0.567 0.08 0.95 2.2 0.92F 0.469 1.908 1.964 0.575 0.14 0.93 2.24 0.91M6 M 0.389 1.109 1.175 1.31E4 0.37 0.66 1.62E3 0.65F 0.13 1.29 1.296 1.3E3 0.37 0.72 1.62E3 0.7W1 M 2.814 13.096 13.389 4.72E3 0.21 0.81 5.8E3 0.62F 5.27 21.16 21.796 0.835 0.28 0.79 3.22 0.62W2 M 0.94 3.96 4.068 1.95E11 0.09 0.91 2.42E11 0.84F 1.712 4.859 5.15 1.71E10 0.16 0.91 2.12E10 0.84W3 M 0.925 4.098 4.199 1.97E10 0.1 0.91 2.45E11 0.84F 1.724 4.6 4.911 1.90E11 0.16 0.9 2.36E11 0.85W4 M 3.514 28.473 28.675 2.61E12 0.14 0.86 4.15E11 0.67F 5.741 21.632 22.37 2.17E12 0.22 0.85 3.72E11 0.71W5 M 1.033 3.912 4.045 0.841 0.09 0.91 3.22 0.85F 2.207 8.171 8.46 0.863 0.16 0.91 3.32 0.82W6 M 1.086 5.551 5.654 1.58E11 0.09 0.91 1.96E10 0.83F 2.061 6.447 6.765 2.05E12 0.17 0.9 2.54E10 0.83C1 M 1.458 8.19 8.315 4.84E11 0.35 0.69 1.73E10 0.59F 2.396 14.368 14.559 5.39E11 0.46 0.67 1.904E11 0.53C2 M 1.494 7.931 8.067 5.10E11 0.36 0.66 1.97E10 0.59F 2.523 13.027 13.263 6.26E10 0.46 0.65 2.37E10 0.53C3 M 4.7E5 1.3E6 1.3E6 2.01E25 0.4 0.61 4.01E10 0.01F 3.393E257 2.56E258 2.57E258 2.47E10 0.42 0.57 1.6E11 0.55C4 M 1.771 7.618 7.817 0.728 0.22 0.8 2.8 0.69F 3.678 29.747 29.959 0.744 0.27 0.78 2.85 0.58C5 M 2.158 9.908 10.135 0.749 0.23 0.79 2.86 0.64F 3.986 26.414 26.7 0.763 0.3 0.76 2.92 0.57C6 M 1.515 6.808 6.971 0.352 0.51 0.5 1.37 0.41F 2.754 11.461 11.782 2.70E4 0.62 0.44 4.7E3 0.4Note:M:Male, F:Female,SD: standard deviation, ESE: Empirical Standard error,MSE: Mean squared error,MDSE: Model standard error,CP:Coverage probability,CIL:Confidence interval length estimate,BAC:Bias adjusted coverage5.4.10 Plasmode simulation result when outcomeprevalence is lowThere would be little data to estimate the exposure outcome relationship incase of a low outcome prevalence situation. In that case, using logistic re-gression produced a higher bias of the treatment effect estimate with highervariability [74, 75]. In the presence of effect modifier when the outcomeprevalence low, the bias and variability increased for each of the PS effectmodifier methods was not explored yet. Investigation of the performance ofeffect modifier methods in the low outcome prevalence setting would helpus explore whether bias and mean squared error are higher than when out-come was common and guide us which PS effect modifier methods producelower bias, mean squared error. Therefore, in addition to the low exposureprevalence, we also evaluated the performance of effect modifier methodsconsidering the outcome prevalence lower (i.e., p=5%), keeping the expo-sure prevalence similar to the one described in Chapter 5.From the plasmode simulation by considering outcome prevalence low(i.e. p = 0.05) illustrated in the Table 5.24 we found that,• Performance of overall PS approach: The overall PS approachproduced higher bias than the subgroup-specific and interaction ap-proach in all PS methods except PS stratification method. Within PSstratification method the subgroup-specific approach produced higherbias and mean squared error than overall PS approach. On the otherhand, under this approach, compared to PS matching methods (methodM1), the PS weighting methods (methods W1, W4) The PS strat-ification method (method S1) and PS covariate adjustment meth-ods (methods C5, C6) produced had lower bias bias and lower meansquared error..• Performance of subgroup-specific PS approach: Under this ap-proach, the PS matching method (method M4) produced higher biasand higher mean squared error compared to the PS weighting method(method W3)and the covariate adjustment method (method C2). ThePS stratification method (method S2) produced the higher bias andmean squared error than other PS methods.• Performance of useful interaction PS approach: Again underthis approach, the PS weighting methods (methods W2, W6) and thecovariate adjustment method (method C3) produced lower bias andlower mean squared error than the corresponding PS matching meth-ods (method M5 and M6). But the PS stratification method (methodS3) had higher bias and mean squared error than other PS methodsunder this approach.• Performance of generalized PS and univariate PS approach:The generalized PS (method C4) produced lower bias and mean squarederror than univariate PS (method C1)Table 5.24: Plasmode simulation result with low outcome prevalencePS effect modifiermethodsGender Absolute Bias SD MSE MDSE Power CP CIL BACM1 M 2.12E+125 5.74E+126 5.7E+126 5.11E4 0.65 0.52 1.09E4 0F 1.58E+61 4.3E+62 4.3E+62 4.61E4 0.57 0.72 8.68E4 0M2 M 8.0632 103.561 103.82 0.776 0.25 0.77 2.91 0.38F 6.0033 28.53 29.14 0.812 0.35 0.78 3.06 0.62M3 M 5.1918 59.019 59.21 3.32E10 0.24 0.78 4.11E10 0.52F 6.9819 35.349 36.01 2.85E10 0.38 0.76 3.5E10 0.58M4 M 15.381 314.310 314.53 0.787 0.27 0.77 2.96 0.25F 11.6378 86.108 86.84 0.858 0.36 0.76 3.19 0.48M5 M 9.548 156.721 156.93 4.06E10 0.27 0.78 5.03E10 0.35F 7.8047 46.358 46.98 3.51E10 0.37 0.77 4.35E10 0.57M6 M 10.2044 170.626 170.84 4.72E10 0.26 0.77 5.85E10 0.33F 8.2507 48.474 49.14 2.8E10 0.38 0.74 3.49E10 0.54W1 M 1.0862 8.768 8.83 0.62 0.18 0.84 2.38 0.76F 1.5927 4.619 4.88 0.643 0.29 0.85 2.48 0.8W2 M 1.069 8.131 8.19 0.62 0.18 0.85 2.38 0.77F 1.5908 4.873 5.12 0.644 0.29 0.84 2.48 0.8W3 M 0.2296 1.030 1.05 0.552 0.1 0.92 2.13 0.91F 0.357 1.980 2.01 0.553 0.17 0.92 2.14 0.9W4 M 1.0862 8.768 8.83 0.62 0.18 0.84 2.38 0.76F 1.5927 4.619 4.88 0.643 0.29 0.85 2.48 0.8W5 M 1.069 8.131 8.19 0.62 0.18 0.85 2.38 0.77F 1.5908 4.873 5.12 0.644 0.29 0.84 2.48 0.8W6 M 0.3893 1.108 1.17 1.3E4 0.37 0.66 1.6E5 0.65F 0.1296 1.29 1.29 1.3E5 0.37 0.72 1.6E5 0.7S1 M 231.1308 1.28E2 1.30E3 0.723 0.02 0.95 2.8 0.95F 382.3096 1.73E2 1.7E3 0.756 0.01 0.91 2.92 0.91S2 M 230.5418 1.3E2 1.3E3 0.729 0.01 0.95 2.82 0.95F 449.3621 1.7E3 1.8E3 0.754 0.02 0.9 2.91 0.9S3 M 8.32E+87 2.63E+89 2.63E+89 1.12E5 0.02 0.5 2.19E3 0.5F 3.18E+25 9.99E+26 9.99E+26 1.16E5 0.02 0.6 2.07E3 0.6C1 M 0.764 4.133 4.20 0.632 0.13 0.9 2.43 0.84F 1.1009 3.054 3.24 0.653 0.23 0.91 2.52 0.87C2 M 0.5628 2.88 2.93 0.666 0.01 0.03 2.56 0.01F 0.7159 3.592 3.66 0.674 0 0.04 2.61 0.01C3 M 0.3606 1.060 1.11 0.52 0.09 0.95 2.02 0.93F 0.2142 1.305 1.32 0.531 0.15 0.95 2.06 0.92C4 M 0.2949 1.018 1.05 0.543 0.09 0.93 2.12 0.92F 0.0887 0.990 0.99 0.534 0.07 0.92 2.08 0.93C5 M 0.37 1.0560 1.11 0.519 0.1 0.95 2.02 0.93F 0.225 1.3255 1.34 0.529 0.15 0.95 2.06 0.92C6 M 0.3893 1.108 1.17 1.3E4 0.37 0.66 1.6E3 0.65F 0.1296 1.2903 1.29 1.3E4 0.37 0.72 1.6E3 0.7Note:M:Male, F:Female,SD: standard deviation, ESE: Empirical Standard error,MSE: Mean squared error,MDSE: Model standard error,CP: Cov-erage probability,CIL:Confidence interval length estimate,BAC:Bias adjusted coverageChapter 6Discussion and ConclusionsStudying treatment effect modifiers is essential to explore subgroups havingdifferential beneficial effects to treatments. In the context of the propensityscore, many studies explored the treatment effect modifier. In those studies[55–59, 61] researchers used overall (full-cohort) PS to draw inference withinsubgroup or subgroup-specific PS approach to address effect modifier. Someother studies compared useful interaction approach [59], univariate and gen-eralized PS approach [60]. To apply overall PS and subgroup-specific ef-fect modifier approach within PS context some studies used PS matching,weighting, stratification and covariate adjustment method. In the currentthesis, we have reviewed all the existing and some new methods of address-ing effect modifier within all the propensity score methods. This has beenachieved through extensive Monte Carlo simulation for several scenarios con-sidering effect modifiers, as mentioned in Setoguchi et al. [30] . Along withMonte Carlo simulations, we also provided plasmode simulation based on amotivating example of data analysis from the Right Heart Catheterizationstudy conducted by Connors et al. [7].6.1 Summary of the findings of studyObjective 1: Our first objective was to review various methodologicaltechniques to handle effect modifiers within all propensity score methodssuggested in the literature. One of the earliest study conducted by Rassenet al. [55], compared the performance of overall PS and subgroup-specificPS approach of effect modifier. The authors claimed that, for large sub-group size, the overall PS approach can be used to draw inference regardingwithin subgroup variables. On the other hand, Green et al. [59] pointedout that, if substantial differentiality was present at the subgroup level in adataset, then applying overall PS would not capture the variability acrosssubgroup. Green et al. [59] found that amongst propensity score matchingmethods, the subgroup-specific propensity score approaches for effect mod-ifiers performed better than the overall PS approach. From our study, weobserved that in the simulation without interaction, under propensity score195matching, the subgroup-specific approach performed better than the over-all use of propensity score methods in most of the scenarios by producinglower bias, lower mean squared error, and achieved the desired 95% coverageprobability. However, within the simulation without interaction, the scenar-ios employing mild non-linearity, moderate non-additivity, moderate non-additivity and non-linearity, resulted in an overall PS that had lower biaswith higher mean squared error than subgroup-specific approach amongstpropensity score matching methods. Within propensity score matching,from the Monte Carlo simulation with interaction, the overall PS approachproduced higher bias and higher mean squared error than subgroup-specificapproach except in the scenario of mild non-linearity where the subgroup-specific approach had higher bias but lowered mean squared error than theoverall PS approach.Kreif et al. [56] and Radice et al. [57] compared the performance ofoverall PS and subgroup-specific approach within propensity score weightingmethod, and they also concluded that, subgroup-specific PS approach hadbetter performance than the overall PS approach. Within propensity scoreweighting method, under Monte Carlo simulation without interaction, theoverall PS approach had higher bias and higher mean squared error thanthe subgroup-specific approach in the simulation scenarios additive effect,mild non- linearity and mild non-additivity. In other scenario the overall PShad lower bias but with higher mean squared error than subgroup-specificapproach. In the Monte Carlo simulation with interaction, the overall PSapproach had higher bias than subgroup-specific approaches in all scenariosexcept the scenarios of mild non-additivity and non-linearity, moderate non-additivity where the subgroup-specific approach had higher bias and meansquared error than overall PS approach.Girman et al. [58] considered the propensity score stratification methodto compare overall PS and subgroup-specific approach and supported theprevious findings as subgroup- specific approaches produced better covari-ate balance than subgroup-specific approach. We observed that, withinstratification method, under Monte Carlo simulation without interaction,the overall PS approach had lower bias than the subgroup-specific approachin the scenarios mild non-linearity, moderate non-additivity, moderate non-additivity and non-linearity. However, in all scenarios the overall PS hadhigher mean squared error than the subgroup-specific approach. In theMonte Carlo simulation with interaction, at the male subgroup level, theoverall PS had lower bias and mean squared error than subgroup-specificapproaches, but in the female group, the overall PS had higher bias andmean squared error than the subgroup-specific PS approach.Another study by Eeren et al. [60] evaluated the univariate propensityscore method’s performance versus generalized propensity score as a co-variate adjustment method to handle effect modifier. They found that thegeneralized propensity score performed better than the univariate propen-sity score to produce lower bias and mean squared error. Our study alsosupported their findings as in most of the simulation scenarios, general-ized propensity score performed better than the univariate propensity scoremethod (Tables 6.1 and 6.2). But to apply the generalized propensity scoremethod, we need to care about the violation of the assumption of the equal-ity of joint distribution of potential outcome and covariates among differentcategories of treatment variable as mentioned by Eeren et al. [60].Objective 2: Our second objective was to compare various combina-tions of effect modifier approaches of effect modifier within propensity scoremethods that were not previously explored through Monte Carlo simula-tion study considering the scenarios mentioned in Setoguchi et al. [30]. Inthe existing literatures, researchers compared the performance of overall PSand subgroup-specific approach within PS matching methods [59, 61]. Kreifet al. [56] compared the subgroup-specific PS approach within propensityscore matching and weighting method. Girman et al. [58] compared thesubgroup-specific approach within propensity score matching and stratifica-tion method. Still some comparison among PS methods were not performed.For example, for overall PS, comparisons were undertaken among propensityscore matching, weighting, stratification and covariate adjustment methods.For subgroup-specific PS approaches, comparison among propensity scorematching, weighting, stratification and covariate adjustment method wereundertaken. For useful interaction approaches, comparisons among propen-sity score matching, weighting, stratification and co- variate adjustmentmethods were done. Therefore, in our study, we have compared all possi-ble combinations of the effect modifier approaches and the propensity scoremethods that were not previously explored.We performed the Monte Carlo simulation for the various scenarios suchas additivity and linearity, mild non-linearity, moderate non-linearity, mildnon-additivity, mild non-additivity and non-linearity, and moderate non-additivity and non-linearity as mentioned in Setoguchi et al. [30] with mod-ification to study effect modifiers. In the outcome model for each simulationscenario we considered the effect modifier which was not considered in theoriginal simulation scenario. After that, we performed another simulationwhere we did a change in each of the simulation scenarios of propensityscore estimation. We added an interaction between a covariate and sub-group variable in each of the simulation scenario to estimate PS. We didthis because, in the simulation scenarios mentioned by Setoguchi et al. [30]there was no pre-interaction between any covariate and subgroup variable,whereas, in one of the effect modifier method known as useful interactionapproach we need to consider some useful interactions between covariatesand subgroup variables in the estimation of the propensity score. The inclu-sion of interaction in the simulation scenarios would give us the idea that, ifinteraction present in the data but without considering them if we apply theoverall PS approach whether we get biased estimator or not. Moreover, withthis inclusion of interaction we would be able to evaluate the performanceof machine learning technique say, decision tree method instead of logisticregression in the PS estimation.Kreif et al. [56] evaluated the performance of overall PS approach withinPS matching and weighting methods. They found better performance withpropensity score weighting compared to matching. In our study we foundthe similar conclusion from all the simulation scenarios of Monte Carlo sim-ulation without interaction. In all simulation scenarios, un- der overall PSapproach of effect modifier, compared to propensity score matching method,propensity score weighting had lower bias and lower mean squared error.Similarly, under Monte Carlo simulation with interaction, in all scenarios,propensity score weighting method had lower bias and lower mean squarederror than propensity score matching method. Another study conductedby Girman et al. [58], compared the overall PS approach of effect modifierwithin propensity score matching and stratification method. They revealeda better performance of propensity score matching compared to propen-sity score stratification method. We also found similar results in all thesimulation scenarios of Monte Carlo simulation without interaction (Table6.1). Within Monte Carlo simulations with interaction using scenario mildnon-linearity, the propensity score matching method had lower bias andmean squared error, but in all other scenarios the propensity score matchingmethod had higher bias and mean squared error than stratification methodsat one subgroup label and in another label the propensity score matchingmethod lower bias and mean squared error (Table 6.2). Additionally, in ourstudy we have compared the performance of propensity score weighting andstratification method under overall PS approach which was not previouslydone in any study. We found that, in terms of bias, in all simulation scenar-ios both with or without in teraction, the propensity score weighting methodhad lower bias than the stratification method, but in simulation without in-teraction the weighting method had higher mean squared error than stratifi-cation method and in the simulation with interaction the weighting methodhad lower mean squared error than the stratification method (Table 6.2).Besides overall PS approach, Kreif et al. [56] compared the subgroup-specific approach within the propensity score matching and weighting ap-proach and they reported the propensity score weighting method performedbetter than propensity score matching. Our study also supported the find-ings as we observed that, for Monte Carlo simulation without in- terac-tion for scenarios under subgroup-specific approach, the propensity scoreweighting method had lower bias and mean squared error than the match-ing method (Table 6.1). On the other hand, under subgroup-specific PSapproach and Monte Carlo simulation with interaction, in simulation sce-narios: (i) in moderate non-linearity, moderate non-additivity, moderatenon-additivity and non-linearity, the propensity score weighting methodproduced lower bias, lower mean squared error than matching method; (ii)in mild non-linearity, the PS weighting method produced higher bias thanPS matching method; (iii) additive effect, mild non-additivity, mild non-additivity and non-linearity the PS weighting method resulted in lower biasbut higher mean squared error than PS matching method (Table 6.2). Inanother study by Girman et al. [58] compared the subgroup-specific PS ap-proach within propensity score matching and stratification methods. Theyachieved a better performance of propensity score matching compared topropensity score stratification methods. Our study found that under theMonte Carlo simulation without interaction for all scenarios, propensityscore matching had lower bias than stratification but the matching methodhad higher mean squared error (Table 6.1). In the Monte Carlo simulationwith interaction, all scenarios, propensity score matching methods producedlower bias and mean squared error than stratification method (Table 6.2).In comparison between propensity score weighting and stratification undersubgroup-specific effect modifier approach we found that, in Monte Carlosimulation without interaction, weighting method had lower bias an meansquared error than stratifi cation. However, in the scenarios, additive effect,mild non-additivity and non-linearity, moderate non-additivity, moderatenon-additivity and non-linearity, weighting method higher mean squarederror than stratification (Table 6.1). At Monte Carlo simulation with inter-action, in all scenarios, propensity score weighting method had lower biasand lower mean squared error than stratification method (Table 6.2).Green et al. [59] discussed using a useful interaction approach withinthe propensity score matching method. But that method was not evalu-ated within other propensity score methods. In our study, we have appliedthat useful interaction approach in all the propensity score contexts. Inthe Monte Carlo simulation, we found that when we had considered an in-teraction between a covariate and subgroup variable, the useful interactionapproach produced lower bias, lower mean squared error, and higher cover-age probability than the overall PS and subgroup-specific approach (Table6.2).In case of propensity score weighting using inverse probability weight,propensity scores have extreme differentiability. That is, some patients hadpropensity score near to 0 and for others closer to 1. In that case, the in-verse probability weight produced biased estimates, large weights for somepatients, and higher variability of the treatment effect estimates [12, 27,33]. Some researchers proposed trimming methods to overcome the extremeweight problem, through which some patients having extreme weights wouldbe excluded [69, 70]. But to apply the trimming method, we need to rely onsome threshold values, which are usually determined arbitrarily and resultedin the loss of potentially large samples from the study [71]. Overlap weightis another weighting method to overcome the problem of trimming method[72]. Under the overlap weighting method in extreme propensity scores situ-ation, we do not need to discard the patients from the tail of the distributionof propensity score using overlap weight. Overlap weight can minimize thelarge sample variance of the estimator [71]. In addition to this, the propen-sity score overlap weighting method can work in other settings, such asmultiple treatments, subgroup analysis, and when time-varying treatmentsare present in the study [71]. We have applied the propensity score over-lap weighting method as an alternative PS weighting method to evaluatethe performance of effect modifier’s approaches. Previously [71] comparingthe performance of the overlap weighting method and inverse probabilityweighting method resulted in cases of extreme weights present in the study.They found that overlap weight produced lower bias and mean square errorcompared to inverse probability weight. Li et al. [71] did not consider thepresence of treatment heterogeneity to evaluate the performance of over-lap weighting method. Moreover, in any of the propensity score weightingmethod to deal with effect modifier, the overlap weighting method was notevaluated. Therefore, in our study, we evaluated the performance of overlapweighting method and compared them with the results from inverse proba-bility weighting method.Based on the Monte Carlo simulation without interaction situation, wefound that, in all scenarios, within the propensity score overlap weightingmethod, the subgroup-specific effect modifier approach performed better interms of producing lower bias and lower mean squared error compared tooverall PS and useful interaction approach. In comparison with the inverseprobability weighting method, we found that the overlapping weight pro-duced lower bias and lowered mean squared error than the inverse probabil-ity weight under the Monte Carlo sim- ulation without interaction when themodel had complex scenarios in the propensity score estimation model, suchas moderate non-linearity, mild non-linearity, mild non-additivity with non-linearity, moderate non-additivity and moderate non-additivity with non-linearity effects. Under the Monte Carlo simulation with interaction, wefound that similar to the without interaction Monte Carlo simulation, thesubgroup-specific PS approach produced lower bias and mean squared errorcompared to other PS approaches. On the other hand, in all the simulationscenarios, the inverse probability weighting method’s bias and overlappingweighting method were almost similar, whereas, in all scenarios, overlappingweight produced lower mean squared error.We previously used binary logistic regression the propensity score foreach of the effect modifier methods. However, in the logistic regressionapproach, the assumptions related to variable selection, the functional formof the outcome and exposure, distribution of co- variates and interactions arerequired. Failing to meet these assumptions leads to biased effect estimates[26, 27]. Alternatively, researchers nowadays use various machine learningmeth- ods to estimate the PS [26–29]. Some standard machine learningmethods are classification and regression trees (CARTs), neural networks,random forests and Generalized Boosted modeling [27, 30, 31].One of the propensity score effect modifier method was the useful inter-action approach. We had to specify the interaction terms between covariatesand subgroup variable for binary logistic regression during PS estimation.In contrast to this, with the decision tree method, we can automaticallycapture the interactions present in the model. Therefore, if we could use thedecision tree method, we can work with the overall PS, subgroup-specificapproach, and covariate adjustment approach. We do not need to applyuseful interaction approach to handle effect modifier within PS methods.To verify whether this idea works in our study, we have used the decisiontree method to estimate the propensity score for the Monte Carlo simulationstudy and then evaluated the performance of the methods of handling effectmodifier.We observed that, in the less complex scenario of Monte Carlo simulationwithout inter- action for additive effects, for example, the bias and meansquared error from the logistic regression were lower than the decision treemethod. On the other hand, in the more complex scenario, say, in moderatenon-additivity and non-linearity, the bias from both logistic regression anddecision was almost similar but mean squared error was higher in the decisiontree method.In the Monte Carlo simulation with interaction, we found that, in themore complex scenario, say, in case of moderate non-additivity and non-linearity effects, the decision tree method produced lower bias but highermean squared error than logistic regression method. In the more complexscenario of moderate non-additivity and non-linearity the model had morenon-linear terms and additive terms, as we found the decision tree methodperformed well by producing lower bias, so the decision tree method wouldbe a promising method in the propensity score estimation [30] in the presenceof effect modifier. Moreover, in the useful interaction approach of handlingeffect modifier within PS method, we had to specify the interactions presentin the model, but in case of decision tree we do not need to specify them.Therefore, instead of useful interaction approach, with decision tree methodwe can handle effect modifier through overall PS, subgroup-specific PS andcovariate adjustment PS approach if PS model had more complex scenariolike moderate additivity and non-linearity.Table 6.1: Summary from Monte Carlo simulation without considering aninteraction between subgroup variable and a covariateSimulationscenarioEffect modifierapproachWhich methods produced most bias Which methods had most SD Which methods had most MSE Which methods had most CPAdditive effect Overall PS Stratification and Weighting Matching Matching Weighting, StratificationSubgroup-specificPSMatching Matching Matching Matching, Weighting, StratificationUsefulinteraction PSMatching Matching Matching Matching, Weighting,Stratification,covariate adjustmentCovariateadjustment(univariate vsgeneralizedPS)Same Univariate Univariate SameMildnon-linearityOverall PS Stratification, Matching Matching Matching, covariate adjustment Weighting, Stratification,covariateadjustmentSubgroup-specificPSStratification, Matching Matching Matching Matching, Weighting,Stratification,covariate adjustmentUsefulinteraction PSStratification, Matching Matching Matching Matching, Weighting,Stratification,covariate adjustmentCovariateadjustment(univariate vsgeneralizedPS)Same univariate univariate SameModeratenon-linearityOverall PS Matching Matching Matching, Weighting Weighting, Stratification,covariateadjustmentSubgroup-specificPSMatching Matching Matching, Weighting Matching, Weighting,Stratification,covariate adjustmentUsefulinteraction PSMatching Matching Matching Matching, Weighting,Stratification,covariate adjustmentCovariateadjustment(univariate vsgeneralizedPS)Same Univariate Univariate Samemildnon-additivityOverall PS Matching, Stratification Matching, Weighting Matching, Weighting, Stratification Weighting, Stratification,covariateadjustmentSubgroup-specificPSMatching, Stratification Matching Matching, Weighting Matching, Weighting,Stratification,covariate adjustmentUsefulinteraction PSMatching, Stratification Matching, Weighting Matching, Weighting Matching, Weighting,Stratification,covariate adjustmentCovariateadjustment(univariate vsgeneralizedPS)Same Univariate Univariate SameMildnon-additivityandnon-linearityOverall PS Matching, Stratification Matching Matching, Stratification Weighting, Stratification,covariateadjustmentSubgroup-specificPSMatching, Stratification Matching Matching, Weighting, covariateadjustmentMatching, Weighting,Stratification,covariate adjustmentUsefulinteraction PSMatching, Stratification Matching, Weighting Matching, Weighting Matching, Weighting,Stratification,covariate adjustmentCovariateadjustment(univariate vsgeneralizedPS)Same Univariate Univariate SameModeratenon-additivityOverall PS Matching, Stratification Matching, Weighting Matching, Stratification Weighting, Stratification,covariateadjustmentSubgroup-specificPSMatching, Stratification Matching Matching, Weighting, covariateadjustmentMatching, Weighting,Stratification,covariate adjustmentUsefulinteraction PSMatching, Stratification Matching Matching, Weighting Matching, Weighting,Stratification,covariate adjustmentCovariateadjustment(univariate vsgeneralizedPS)Same Univariate Univariate Samemoderatenon-additivityandnon-linearityOverall PS Matching, Weighting Matching, Weighting Matching, Weighting Weighting, Stratification,covariateadjustmentSubgroup-specificPSMatching, Weighting Matching Matching, Weighting, covariateadjustmentMatching, Weighting,Stratification,covariate adjustmentUsefulinteraction PSMatching, Weighting Matching, Weighting Matching, Weighting Matching, Weighting,Stratification,covariate adjustmentCovariateadjustment(univariate vsgeneralizedPS)Same Univariate Univariate SameNote:PS: Propensity score,SD: standard deviation,MSE: Mean squared error,CP: Coverage probabilityTable 6.2: Summary from Monte Carlo simulation by considering an inter-action between subgroup variable and a covariateSimulationscenarioEffect modifierapproachWhich methods produced most bias Which methods had most SD Which methods had most MSE Which methods had most CPAdditive effect Overall PS Matching, Covariate adjustment Matching, Stratification Matching, Stratification, CovariateAdjustmentWeighting, StratificationSubgroup-specificPSWeighting, Stratification Matching, Weighting, Stratification Matching, Weighting, Stratification Matching, Weighting, StratificationUsefulinteraction PSStratification, Covariate adjustment Matching, Weighting Matching, Stratification Matching, Weighting,Stratification,covariate adjustmentCovariateadjustment(univariate vsgeneralizedPS)Univariate Univariate Univariate GeneralizedMildnon-linearityOverall PS Stratification, Covariate adjustment Weighting, Stratification, CovariateadjustmentStratification, covariate adjustment Matching,Weighting,Stratification,covariate adjustmentSubgroup-specificPSStratification Matching, Weighting, Stratification Weighting Matching, Weighting,Stratification,covariate adjustmentUsefulinteraction PSStratification, Covariate adjustment Matching, Weighting, Stratification Stratification, Weighting Matching, Weighting,Stratification,covariate adjustmentCovariateadjustment(univariate vsgeneralizedPS)Univariate univariate univariate GeneralizedModeratenon-linearityOverall PS Matching, Stratification, CovariateadjustmentMatching, Stratification Matching, Stratification, CovariateadjustmentWeighting, StratificationSubgroup-specificPSMatching, Stratification Matching, Weighting Matching, Weighting Matching, Weighting,Stratification,covariate adjustmentUsefulinteraction PSMatching Matching Matching Matching, Weighting,Stratification,covariate adjustmentCovariateadjustment(univariate vsgeneralizedPS)Univariate Univariate Univariate Generalizedmildnon-additivityOverall PS Matching, Stratification, CovariateadjustmentMatching, Stratification Matching, Covariate adjustment Weighting, StratificationSubgroup-specificPSWeighting Matching, Weighting Matching, Weighting Matching, Weighting,Stratification,covariate adjustmentUsefulinteraction PSMatching, Stratification Matching, Weighting Matching, Stratification Matching, Weighting,Stratification,covariate adjustmentCovariateadjustment(univariate vsgeneralizedPS)Univariate Univariate Univariate GeneralizedMildnon-additivityandnon-linearityOverall PS Matching, Covariate adjustment Matching, Stratification Matching, Stratification Weighting, Stratification,covariateadjustmentSubgroup-specificPSMatching Matching, Weighting Matching, Weighting Matching, Weighting,Stratification,covariate adjustmentUsefulinteraction PSMatching, Stratification Matching Matching, Stratification Matching, Weighting,Stratification,covariate adjustmentCovariateadjustment(univariate vsgeneralizedPS)Univariate Univariate Univariate GeneralizedModeratenon-additivityOverall PS Matching, Covariate adjustment Matching, Stratification Matching, Stratification Weighting, Stratification,covariateadjustmentSubgroup-specificPSMatching, Stratification Matching Matching, Weighting, covariateadjustmentMatching, Weighting,Stratification,covariate adjustmentUsefulinteraction PSMatching Matching Matching, Weighting Matching, Weighting,Stratification,covariate adjustmentCovariateadjustment(univariate vsgeneralizedPS)Univariate Univariate Univariate Generalizedmoderatenon-additivityandnon-linearityOverall PS Matching, Stratification, CovariateadjustmentMatching, Weighting Matching, Stratification Weighting, Stratification,covariateadjustmentSubgroup-specificPSMatching, Stratification Matching, Weighting Matching, Weighting Matching, Weighting,Stratification,covariate adjustmentUsefulinteraction PSMatching, Stratification Matching Matching, Weighting Matching, Weighting,Stratification,covariate adjustmentCovariateadjustment(univariate vsgeneralizedPS)Univariate Univariate Univariate GeneralizedNote:PS: Propensity score,SD: standard deviation,MSE: Mean squared error,CP: Coverage probabilityObjective 3: Our third objective was to apply all the effect modifiermethods in empirical data based on the Right Heart Catheterization studyconducted by Connors et al.[7] to explore how those methods worked inthe practical data. From the empirical data analysis, we observed that un-der propensity score matching mechanism, all the effect modifier methods,such as overall PS approach with or without exact matching on subgroupvariable, subgroup-specific PS, useful interaction approach with or withoutexact matching on subgroup variable produced similar balance in the stan-dardized mean difference for both male and female subgroup. This mighthave happened because in our empirical data originally had less variabil-ity among covariates. Therefore, with the balanced data we could not de-tect significant difference among propensity score effect modifier matchingmethods. On the other hand, under propensity score weighting methods, wefound the subgroup-specific weighting approach performed better in termsof lower standard mean difference compared to the overall propensity scoreapproach. In the propensity score stratification methods, we observed anwell-balanced propensity score between treatment and control groups withineach stratum for each effect modifier method. We evaluated the unadjustedand adjusted treatment from each of the propensity score effect modifiermethods.Objective 4: Our final objective was to perform a plasmode simulationto compare all the effect modifier approaches within propensity score meth-ods. Our plasmode simulation was based on the empirical data from theRight Heart Catheterization study conducted by Connors et al. [7]. Fromthe plasmode simulation for crude analysis, we found that, compared to theoverall PS approach, the subgroup-specific PS approach of effect modifierproduced lower bias and mean squared error within all the propensity scoremethods similar to the findings from Monte Carlo simulation.In comparison to the propensity score matching method, under the over-all and subgroup-specific approach of effect modifiers, the propensity scoreweighting method produced lower bias, lower mean squared error, and highercoverage probability similar to Monte Carlo simulation. Under overall PSapproach, the propensity score stratification method had lower bias andmean squared error than the propensity score matching method. From thecomparison between propensity score weighting method versus propensityscore stratification under overall PS approach of effect modifier we observedthat, weighting methods had higher bias and higher mean squared error.The covariate adjustment method produced higher bias and mean squarederror than stratification method, but lower than matching and weightingmethod.Under the subgroup-specific approach of effect modifier, we found that,propensity score matching, and weighting method produced higher biasand mean squared error compared to stratification and covariate adjust-ment method. The generalized PS method produced higher bias and meansquared error than the univariate PS method. In comparison with other PSmethods, the generalized produced highest bias in this plasmode simulationcrude analysis.From the plasmode simulation adjusted analysis, overall PS approach ofeffect modifier had higher bias and mean squared error than the subgroup-specific approach within all propensity score methods. Under the over-all PS approach, the propensity score matching method had higher biasand mean squared error than weighting similar to the Monte Carlo simula-tion. Compared to stratification, matching method had lower bias but meansquared error was higher. Similar to the Monte Carlo simulation, propensityscore weighting method had lower bias, but higher mean squared error thanpropensity score stratification method.Under plasmode simulation adjusted analysis, for subgroup-specific effectmodifier ap- proach, among all the propensity score methods, the propensityscore had maximum bias and propensity score matching produced the max-imum mean squared error. Within propen- sity score covariate adjustmentmethod, the generalized PS approach produced lower bias than univariatePS methods, but the mean squared error was lower than the PS matchingstratification method.Result from both the Monte Carlo simulation and PlasmodesimulationIn our study, we found that the Monte Carlo and Plasmode simulationhad consistent with each other. For example, in the Monte Carlo simulationin simple scenario say when additive effects present in the data, Propensityscore stratification method more bias than other PS methods but PS match-ing methods had higher SD and MSE than others. Similar findings we gotfrom Plasmode simulation.6.2 Sensitivity analysisOverlap weighting methodWe have applied the propensity score overlap weighting method as analternative PS weighting method to evaluate the performance of effect mod-ifier’s approaches. Previously Li et al. [71] compared the performance of theoverlap weighting method and inverse probability weighting method in caseof extreme weights present in the study. They found that overlap weightproduced lower bias, mean square error compared to inverse probabilityweight. Li et al. [71] did not consider the presence of treatment heterogene-ity to evaluate the performance of overlap weighting method. Moreover, inany of the propensity score weighting methods to deal with effect modifier,the overlap weighting method was not evaluated. Therefore, in our study,we evaluated the performance of overlap weighting method and comparedthem with the results from inverse probability weighting method. Based onthe Monte Carlo simulation without interaction situation, we found that,in all scenarios, within the propensity score overlap weighting method, thesubgroup-specific effect modifier approach performed better in terms of pro-ducing lower bias and lower mean squared error compared to overall PSand useful interaction approach. In comparison with the inverse probabilityweighting method, we found that the overlapping weight produced lower biasand lowered mean squared error than the inverse probability weight underthe Monte Carlo simulation without interaction. This was observed whenthe model had complex scenarios in the propensity score estimation model,such as moderate non-linearity, mild non-linearity, mild non-additivity withnon-linearity, moderate non-additivity and moderate non-additivity withnon-linearity effects. Under the Monte Carlo simulation with interaction, wefound that similar to the without interaction Monte Carlo simulation, thesubgroup-specific PS approach produced lower bias and mean squared errorcompared to other PS approaches. On the other hand, in all the simulationscenarios, the inverse probability weighting method’s bias and overlappingweighting method were almost similar, whereas, in all scenarios, overlappingweight produced lower mean squared error.Application of decision tree in the estimation of propensityscoreWe evaluated the performance of effect modifier methods using the de-cision tree method in the estimation of the propensity score. In the lesscomplex scenario of Monte Carlo simulation without interaction say, foradditive effects, the bias and mean squared error from the logistic regres-sion were lower than the decision tree method. On the other hand, in themore complex scenario, say, in moderate non-additivity and non-linearity,the bias from both logistic regression and decision was almost similar butmean squared error was higher in the decision tree method. In the MonteCarlo simulation with interaction, we found that, in the less complex sce-nario, say, in case of moderate non-additivity and non-linearity effects, thedecision tree method produced lower bias but higher mean squared errorthan logistic regression method. In the more complex scenario of moder-ate non-additivity and non-linearity the model had more non-linear termsand additive terms, as we found the decision tree method performed wellby producing lower bias, so the decision tree method would be a promis-ing method in the propensity score estimation [30] in the presence of effectmodifier. Moreover, in the useful interaction approach of handling effectmodifier within the PS method, we had to specify the interactions presentin the model, but in case of the decision tree, we do not need to specifythem. Therefore, instead of useful interaction approach, with decision treemethod we can handle effect modifier through overall PS, subgroup-specificPS and covariate adjustment PS approach if PS model had more complexscenario like moderate additivity and non-linearity.Application of the decision tree when a general practitionerdoes not know the model specificationFor a general practitioner it is really hard to know the model structureand taking decision about the use of a particular method. One solutionmight be the use of any machine learning method for example in our study wehave explored that, in more complex scenario compared to logistic regressionmethod, the decision tree produced lower bias. In the decision tree wedo not have to specify the different model specification such as interactionterms present among the covariates, the decision tree method automaticallycaptured those.Plasmode simulation with reduced sample size (n=1000)In the Monte Carlo simulation we have considered sample size of 1000,but in the plasmode simulation we had information on 4000 patients. There-fore, to evaluate whether in the reduced sample size affect the performanceof various effect modifier approaches we did another plasmode simulation byconsidering sample size 1000. From the analysis of plasmode simulation withreduced sample size we observed that, similar to the Monte Carlo simula-tion subgroup-specific approach produced lower bias and mean squared errorthan overall PS approach of effect modifier. Under overall PS approach, thepropensity score matching method had higher bias than weighting, stratifi-cation and covariate adjustment method. But the mean squared error washigher in stratification method compared to other PS method.Plasmode simulation considering prevalence of exposure low(p=5%)We performed the plasmode simulation by considering the low exposureprevalence. With low exposure prevalence one of the major problems inlogistic regression fitting was the separation problem which affects the esti-mates of treatment [73]. Hajage et al. [73] compared the performance of PSmatching and PS weighting method in case low exposure prevalence. Theyfound higher bias of the estimates in both the PS methods, specifically in theestimation of average treatment effect under PS weighting method. In ourstudy, we also found higher bias, mean squared error in the PS matching,weighting and covariate adjustment method of handling effect modifier. Wegot lower bias, lower mean squared under PS matching method compared tocorresponding weighting and covariate adjustment method. However, in thelow exposure prevalence, the PS stratification methods did not work well asunder the PS stratification model did not converge. This might happen dueto the fact that, with low exposure prevalence, when we divide the datasetinto 5 or 10 groups under PS stratification method, in some stratum theexposed subjects become too smaller that, stratum-wise treatment effectcould not calculated. Therefore, we need to check before analysis whetherthe exposure prevalence is low to apply effect modifier methods within PSmethods.Plasmode simulation considering prevalence of outcome low(p = 5%)Like low exposure situations, when the outcome prevalence is low, thelogistic regression had a problem with separation. However, in our study,under the low outcome prevalence situation, we found that all the PS effectmodifier methods had higher bias and mean squared error than when out-come prevalence was not low similar to the [75]. We also found the poorperformance of stratification methods in terms of producing much higherbias, mean squared error than other PS effect modifier methods. On theother hand, compared to the PS matching method, other PS effect modifiermethods had lower bias, lower mean squared error. In terms of compari-son between univariate PS and generalized PS method, we found that thegeneralized PS method produced lower bias and lower mean squared errorcompared to univariate PS.6.3 Strengths of the studyComparative analysis of all combinations of effect modifier andpropensity score methodsOur study has one of major strengths that, this is the first study wherewe have analysed all the existing and new possible combinations of effectmodifier method and propensity score analysis in extensive Monte Carlo sim-ulation, plasmode simulation, and empirical data analysis. We consideredmultiple scenarios and an extra scenario of preexisting interaction betweena covariate and a subgroup variable in the simulation study. Previously,Green et al. [59] reviewed the overall PS approach and subgroup-specific ap-proach, and useful interaction approach within the propensity score match-ing method. Another review paper by [61], evaluated the overall PS andsubgroup-specific approaches discussed by [59] within the plasmode simula-tion framework and under propensity score matching methods. Therefore,no guidelines were generated from the previous study. Using various scenar-ios of Monte Carlo simulation we have discussed the contexts where methodshad lower bias and mean squared error so that researchers would consider themodel scenario and apply the appropriate effect modifier approaches. Noneof the studies compared the performance of covariate adjustment methodsunder various simulations scenarios and not with other propensity scoremethods in a single study. In this work, we filled up that gaps of researcharea which will help the researchers in case of using covariate adjustmentmethods of effect modifier.Performing Overlap weighting method to address effect modi-fier within propensity score methodAnother strength of our study that, in case of propensity score weightingmethod to handle effect modifiers, previously researchers used inverse prob-ability weight which suffers from the extreme differentiability of propensityscore resulting in biased estimates and higher variability of the treatmenteffect estimates. To overcome this problem overlap weighting method hasbeen proposed, but performance of overlap weight method was not previ-ously explored. In our study we have used the overlap weighting methodand achieved better performance than inverse probability weight methods.This findings suggests that, in case of inverse probability weight of effectmodifier, researchers should check the distribution of propensity score andif it has extreme differentiality then researchers can apply overlap weightingmethods.Application of machine learning method (decision tree)We evaluated the performance of effect modifier approaches through esti-mating propensity score using decision tree method which was not previouslyexplored. Through decision tree method we can capture the interaction ifpresent in the model automatically. Therefore, in case of presence of interac-tions that might present in data, we do not need to apply useful interactionapproach of effect modifier. We can compare the overall PS, subgroup andcovariate approaches.Performance of effect modifier methods for reduced samplesize, low exposure and outcome prevalenceIn the plasmode simulation we also evaluated the performance of ef-fect modifier method for large sample size (n=4000) and small sample size(n=1000), so that, besides large sample situation we have explored the smallsample situation that will increase the generalizabitlity of the study findings.Under plasmode simulation we have considered two more situations, (i) ifexposure prevalence is low; (ii) if outcome prevalence was low. These twoplasmode situation can arise in real life and our study found when someeffect modifier methods did not perform well and reviewed their relativeperformance among all the methods.This analysis gives us the flexibility to overview the various propen-sity score methods’ performance under similar effect modifier approachesfor multiple scenarios in a single study. Our study’s findings will help re-searchers to select the suitable effect modifier methods during data analysiswhen they have a predefined subgroup to perform analysis depending on theresearch question.This analysis gives us the flexibility to overview the various propensityscore methods’ performance under similar effect modifier approaches formultiple scenarios in a single study. Our study’s findings will help the re-searcher to select the suitable effect modifier methods during data analysiswhen they have a predefined subgroup to perform analysis depending on theresearch question.6.4 Limitations of the studyIn our study, we have several limitations. First, we have considered a bi-nary variable gender as a subgroup variable. But in reality, more than onesubgroup variable may be present in the study. Although for some meth-ods, such as overall PS and subgroup-specific PS approaches, [61] discussedmultiple effect modifiers, that multiple effect modifiers were not included inour study.We applied propensity score full matching and NN matching during ap-plying the propen- sity score matching methods in our study. On the otherhand, Austin et al. [38] provided a comprehensive study on various matchingalgorithms that needed to be considered during matching to determine theoptimal matching algorithm that works for analysis of propensity score inthe presence of effect modifier. Although our focus was to evaluate variouseffect modifier approaches in different simulation scenarios in this study, theresearcher should consider the research objectives and set the appropriatematching algorithm.6.5 Future directionFuture studies may consider multiple effect modifiers in the same study, sothat performance of effect modifier methods can be evaluated. Issues re-lated to sample size are one of the significant issues in the application ofpropensity score methods. In our study, we have considered sample sizes ateach iteration n=1000. For propensity score analysis sample size is a biggestissue. In our study we haven’t explored the performance of various meth-ods of effect modifier for smaller sample sizes. But in that case of smallersample size within the PS stratification method, there is a chance of havingsmaller sample size within different stratum and that will create problem inthe estimation of treatment effect within different stratum. Within the PSmatching due to smaller sample size there would be a problem of not hav-ing enough control group during matching. Further research could be doneto test the performance of various effect modifier methods under a smallersample size, for example, if the sample size is less than 250.Application of the effect modifier approaches in longitudinalstudy settingsIn the longitudinal study settings, the methods that we have analysed inour study are not directly applicable. But in the longitudinal study, theremight be a chance of having time-varying covariates, in that case we can useMarginal structure model. In that case, within Marginal structural model,in the presence of effect modifier Hernan et. al. [23] mentioned to include theeffect modifier variable both in the stabilized weight calculation and in theoutcome analysis. Another one could be use of overlap weighting method.In our study we have already explored that, in more complex scenario ofpropensity score estimation, compared to inverse probability weight, theoverlap weight produced lower bias and MSE. Future research can deal withthis.Application of the effect modifier approaches if missing datapresent in the studyWhen missing data present in the study, one study [76] explored thesituation of having effect modification in the study, the authors mentionedthat, in that case, we can use multiple imputation method. During multipleimputation method, the interaction terms between the variables should becorrectly specified in the imputation model. Future research can deal withthis.Application of the effect modifier approaches in randomizedcontrolled trial studyIn case of RCT when the study is conducted for a longer period of timein that case, some selection bias may occur, some time-varying covariatesmay occur, in that case, within marginal structural model framework [23],we can apply the effect modifier approaches within PS weighting method.Future research can deal with this.Use of advanced machine learning approachesIn our study, our goal was to explore whether any tree-based methods canwork and produce better results compared to that from the logistic regressionmethod in a more complex scenario. Therefore, we did not explore othertree methods, such as random forest approach. Future research can dealwith this.Use of doubly robust methodUnder double robust method combination of exposure model and out-come regression happen. That is, within double robust method either model1 or model 2 should be correctly specified to estimate treatment effect [77].In this case, for effect modifier we can also run the doubly robust method ifwe have correctly specified model or we can apply decision tree method incase of model uncertainty. Future research can deal with this doubly robustmethod.6.6 ConclusionIn our study, we have explored the various possible combination of propen-sity score estimation and outcome analysis within various propensity scoreimplementation methods under different simulation scenarios to deal witheffect modifier. In general, in most of the simulation scenario we observedthat, the use of overall PS approach to draw inference for subgroup was nota good approach, whereas, subgroup-specific approach analysis of handlingeffect modifier performed better than overall PS approach as expected. Wealso explored that, if we could correctly specify the model having interaction,that is, if there were interaction present in the dataset between covariatesand subgroup variable and we could correctly specify them during PS estima-tion, then useful interaction approach within PS matching performed betterthan overall PS and subgroup-specific approach. In comparison among var-ious PS methods, we found that, PS matching method produced higher biasand mean squared error than the corresponding other PS methods. In mostsimulation scenarios, the generalized PS approach performed better thanunivariate PS method. In case of PS weighting method, previously no studyexplored the performance of overlap weighting method. In our study wefound that, in more complex scenario when model had more non-additiveand non-linear terms in that case, compared to inverse probability weight theoverlap weighting method performed better. In general, it is a problem toknow the model specification correctly for a general practitioner during dataanalysis. We have explored that, in more complex scenario, compared to lo-gistic regression method machine learning method say decision tree methodproduced lower bias than logistic regression method. Moreover, under theuseful interaction approach of effect modifier, we have to specify the use-ful interaction terms that were present in the data when we do the logisticregression. On the other hand, with decision tree method we do not needto specify the interactions, it automatically can capture those. Therefore,instead of useful interaction and in case of uncertainty about model specifi-cation decision tree method can be a useful approach. In the Monte Carloand Plasmode simulation the propensity score matching method producedhigher bias and mean squared than other PS methods. 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Thesis/Dissertation
eng
Population and Public Health
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Karim, Mohammad Ehsanul
Examining effect modification within propensity score methods
Text