Science, Irving K. Barber Faculty of (Okanagan)
Computer Science, Mathematics, Physics and Statistics, Department of (Okanagan)
DSpace
UBCO
Epp, Joyce
Master of Science - MSc
University of British Columbia
Detecting breast cancer in its early stages improves patient outcomes, as the disease is often more treatable at early stages. In British Columbia, BC Cancer Breast Screening provides screening mammography as the primary means of early detection. As with any screening test, false positive results do occur. However, false positive results from screening mammography have been shown to be associated with the risk of a future breast cancer. Moreover, the mammographic features that indicate a mammogram is abnormal have been shown to stratify the increase in risk. Models to predict breast cancer risk currently have a modest ability to distinguish between who will be diagnosed and who will not, with concordances in the 0.55 to 0.71 range. Using risk factor, screening, and diagnosis data from BC Cancer Breast Screening, we build several models that incorporate mammographic features with the aim of creating a risk prediction model that provides a better risk prediction than current models. We fit a Cox proportional hazards model, a time varying Cox model, an accelerated failure time model and a Poisson regression model. We found that mammographic features are associated with an increased risk of breast cancer and increased model concordance by about 0.006. However, while the internal calibration was satisfactory, as it was close to 1, the discrimination of the models was unsatisfactory, with concordances around 0.63. Compared to several models in the literature, our model performance was similar, with concordances near 0.63 and internal calibration close to 1 for both ours and the published models examined. We conclude that breast cancer risk prediction is not substantially improved by the incorporation of mammographic features in a categorical format.
Development of a prognosticationmodel for breast cancerin British ColumbiabyJoyce EppB.Sc., The University of British Columbia, 2015A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe College of Graduate Studies(Computer Science)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)July 2020c© Joyce Epp, 2020The following individuals certify that they have read, and recommendto the College of Graduate Studies for acceptance, a thesis/dissertation en-titled:Development of a prognostication model for breast cancerin British Columbiasubmitted by Joyce Epp in partial fulfilment of the requirements of thedegree of Master of ScienceDr. Jeff Andrews, I. K. Barber School of Arts & SciencesSupervisorDr. Rasika Rajapakshe, I. K. Barber School of Arts & SciencesCo-supervisorDr. John Braun, I. K. Barber School of Arts & SciencesSupervisory Committee MemberDr. Ramon Lawrence, I. K. Barber School of Arts & SciencesSupervisory Committee MemberDr. Abbas Milani, School of EngineeringUniversity ExamineriiAbstractDetecting breast cancer in its early stages improves patient outcomes, asthe disease is often more treatable at early stages. In British Columbia, BCCancer Breast Screening provides screening mammography as the primarymeans of early detection. As with any screening test, false positive resultsdo occur. However, false positive results from screening mammography havebeen shown to be associated with the risk of a future breast cancer. More-over, the mammographic features that indicate a mammogram is abnormalhave been shown to stratify the increase in risk. Models to predict breastcancer risk currently have a modest ability to distinguish between who willbe diagnosed and who will not, with concordances in the 0.55 to 0.71 range.Using risk factor, screening, and diagnosis data from BC Cancer BreastScreening, we build several models that incorporate mammographic featureswith the aim of creating a risk prediction model that provides a better riskprediction than current models. We fit a Cox proportional hazards model,a time varying Cox model, an accelerated failure time model and a Poissonregression model. We found that mammographic features are associatedwith an increased risk of breast cancer and increased model concordanceby about 0.006. However, while the internal calibration was satisfactory, asit was close to 1, the discrimination of the models was unsatisfactory, withconcordances around 0.63. Compared to several models in the literature, ourmodel performance was similar, with concordances near 0.63 and internalcalibration close to 1 for both ours and the published models examined. Weconclude that breast cancer risk prediction is not substantially improved bythe incorporation of mammographic features in a categorical format.iiiLay SummarySometimes when women get a screening mammogram there are featureson the mammogram that indicate an abnormality, but after further testingno breast cancer is found. Though no cancer was found at the time, thefeatures on the mammogram that indicated something was abnormal mayincrease the woman’s risk of a future breast cancer down the road.Here we use statistical models and mammographic features to predictthe risk of a woman developing breast cancer using data from BC CancerBreast Screening. We confirmed that mammographic features do increasethe risk of breast cancer, but not enough to be useful for individual riskprediction.ivPrefaceThis thesis is covered by UBC-BCCA REB Certificate of Ethics Approvalnumber H18-00172.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . 11.1 Objective and thesis outline . . . . . . . . . . . . . . . . . . . 2Chapter 2: Background . . . . . . . . . . . . . . . . . . . . . . . 42.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.1 Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.2 Risk models, concordance and calibration . . . . . . . 42.1.3 Risk distribution . . . . . . . . . . . . . . . . . . . . . 52.1.4 Hazard function . . . . . . . . . . . . . . . . . . . . . 52.1.5 Surveyimpute procedure in SAS . . . . . . . . . . . . 52.1.6 Schoenfeld residuals . . . . . . . . . . . . . . . . . . . 62.1.7 dfbetas . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.8 Martingale residuals . . . . . . . . . . . . . . . . . . . 62.1.9 Akaike Information Criterion . . . . . . . . . . . . . . 72.1.10 Lexis split . . . . . . . . . . . . . . . . . . . . . . . . . 7viTABLE OF CONTENTS2.2 Breast cancer risk prediction models . . . . . . . . . . . . . . 7Chapter 3: Methods . . . . . . . . . . . . . . . . . . . . . . . . . 103.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Variable characteristics . . . . . . . . . . . . . . . . . . . . . . 153.4 Cox proportional hazards model . . . . . . . . . . . . . . . . 243.5 Time-dependent Cox regression model . . . . . . . . . . . . . 313.6 Accelerated Failure Time model . . . . . . . . . . . . . . . . . 363.7 Poisson regression model . . . . . . . . . . . . . . . . . . . . . 39Chapter 4: Application . . . . . . . . . . . . . . . . . . . . . . . 444.1 Concordance comparison to other models . . . . . . . . . . . 444.1.1 Cox proportional hazards model concordance compar-ison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.1.2 Cox proportional hazards model comparison of abso-lute risk predictions . . . . . . . . . . . . . . . . . . . 464.1.3 Time-dependent Cox model concordance comparison . 484.1.4 AFT model concordance comparison . . . . . . . . . . 504.2 Calibration comparison to other models . . . . . . . . . . . . 514.2.1 Cox proportional hazards model calibration comparison 514.2.2 AFT model calibration comparison . . . . . . . . . . . 554.3 Birth cohort and risk . . . . . . . . . . . . . . . . . . . . . . . 56Chapter 5: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 61Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62viiList of TablesTable 3.1 Exclusions made to get to the final cohort . . . . . . . 13Table 3.2 Table describing the final risk factors in the data . . . 16Table 3.3 Table showing the breakdown of false positives andother screens on the index screens and all screens . . . 17Table 3.4 Table showing the values of the risk factors in the data 23Table 3.5 Cox proportional hazards model . . . . . . . . . . . . . 26Table 3.6 2 and 5-year absolute risk predicted from Cox propor-tional hazards model . . . . . . . . . . . . . . . . . . . 30Table 3.7 Comparison of 2-year risk predicted from our Cox modelto [ECP+17] . . . . . . . . . . . . . . . . . . . . . . . . 31Table 3.8 Cox time varying regression model . . . . . . . . . . . 35Table 3.9 AIC of AFT models . . . . . . . . . . . . . . . . . . . 37Table 3.10 Weibull AFT model . . . . . . . . . . . . . . . . . . . 40Table 3.11 Time predictions for different percentiles . . . . . . . . 41Table 3.12 Poisson model . . . . . . . . . . . . . . . . . . . . . . . 42Table 4.1 Training and testing concordance values for publishedmodels, published model on our data, and the samemodel with mammographic features added . . . . . . . 44Table 4.2 Risk distribution for published models with and with-out mammographic features added. . . . . . . . . . . . 49Table 4.3 AIC of our time varying Cox model and the time vary-ing Cox models using the published predictors. . . . . 50Table 4.4 AIC of our AFT model and the AFT models using thepublished predictors. . . . . . . . . . . . . . . . . . . . 50Table 4.5 Comparisons of Cox model calibration on test set . . . 53Table 4.6 Comparisons of AFT model calibration on test set . . 56Table 4.7 Proportional hazards model adjusted for birth cohortand number of births . . . . . . . . . . . . . . . . . . . 58viiiLIST OF TABLESTable 4.8 Cox proportional hazards model stratified by birth co-hort. Only showing the coefficients for ethnicity as therest remained stable. . . . . . . . . . . . . . . . . . . . 59ixList of FiguresFigure 3.1 Example follow-up of two women undergoing screen-ing. Normal screens are black and abnormal screensare red. . . . . . . . . . . . . . . . . . . . . . . . . . . 12Figure 3.2 Total number of screens in the cohort. . . . . . . . . . 14Figure 3.3 Frequency of mammographic features for the indexscreens where there was an abnormality indicated. . . 18Figure 3.4 Average number of children per half decade of birthstratified by self-reported education level. . . . . . . . 20Figure 3.5 Average density for each age group at index screenstratified by ethnicity. . . . . . . . . . . . . . . . . . . 21Figure 3.6 Average density for each age group at index screenstratified by select mammographic features. . . . . . . 22Figure 3.7 Checking the proportional hazards assumptions forthe variables and variable categories that showed asignificant p-value in the Schoenfeld residuals. . . . . 28Figure 3.8 The discrimination from the Cox model is quite low,as seen in the overlap of the curves for those who hada diagnosis and those who did not. . . . . . . . . . . . 29Figure 3.9 Number of screens remaining to use in the model thatchanged from the previous screen. . . . . . . . . . . . 33Figure 3.10 Remaining features on screens with a feature identi-fied after formatting the data prior to fitting the Coxregression model. . . . . . . . . . . . . . . . . . . . . 34Figure 3.11 Plot of the log-log transformation of the Kaplan-Meiersurvival vs the log transformation of time. Used toassess whether the Weibull distribution is appropriatefor the data. . . . . . . . . . . . . . . . . . . . . . . . 37Figure 3.12 The deviance residuals of select variables in the AFTmodel. The distributions are similar. . . . . . . . . . 38Figure 3.13 Diagnostics of Poisson model. . . . . . . . . . . . . . 43xLIST OF FIGURESFigure 4.1 (a) [TCSB+08], (b) [TCSB+08] with features, (c)[TCZK05]no BD, (d) [TCZK05] no BD, with features, (e) [TCZK05]with BD, (f) [TCZK05] with BD and features. Com-parison of the absolute risk predicted by our modeland the published models. The blue line is the 45◦line. The red dashed line is the line of best fit. . . . 47Figure 4.2 Calibration of our Cox model for training set (left)and testing set (right) for 2, 4 and 5 year survival.The blue line is the linear model line. The red lineis a flexible calibration line fitted to the data. Theblack line is a 45◦ line, the ideal calibration. . . . . . 52Figure 4.3 Calibration of the comparison Cox models. (a) [TCSB+08],(b) [TCSB+08] with features, (c)[TCZK05] no BD,(d) [TCZK05] no BD, with features, (e) [TCZK05]with BD, (f) [TCZK05] with BD and features. Theblue line is a linear model line. The red line is a flex-ible calibration line fitted to the data. The black lineis a 45◦ line, the ideal calibration. . . . . . . . . . . . 54Figure 4.4 Calibration of our AFT model for training set (left)and testing set (right) for 2, 4 and 5 year survival.The blue line is the linear model line. The red lineis a flexible calibration line fitted to the data. Theblack line is a 45◦ line, the ideal calibration. . . . . . 55Figure 4.5 Calibration of the comparison AFT models. (a) [TCSB+08],(b) [TCSB+08] with features, (c)[TCZK05] no BD,(d) [TCZK05] no BD, with features, (e) [TCZK05]with BD, (f) [TCZK05] with BD and features. Theblue line is a linear model line. The red line is a flex-ible calibration line fitted to the data. The black lineis a 45◦ line, the ideal calibration. . . . . . . . . . . . 57xiAcknowledgementsI would like to thank my supervisors Dr. Rasika Rajapakshe and Dr.Jeff Andrews for their enthusiasm, guidance and advice, and for makingthis project possible.I would also like to thank my committee members Dr. Ramon Lawrenceand Dr. John Braun and examiner Dr. Abbas Milani for their insights andtime taken to review this work and Dr. Mikael Hartman and Dr. Hua Shenfor helpful discussions from near and far.xiiTo my familyxiiiChapter 1IntroductionBreast cancer is the most common cancer in Canadian women, account-ing for a quarter of new cancer diagnoses each year [BWD+20]. In 2020,it is estimated that over 27,000 women in Canada will be diagnosed withthe disease and 5,100 will die from it [BWD+20]. Fortunately, patient out-comes are improved by detecting a breast cancer in its early stages, whilethe disease is often more treatable [SBSTL15, KGR+95].The primary method for the early detection of breast cancer in Canadais screening mammography [oCS12]. Canada’s first organized screeningmammography program was BC Cancer Breast Screening, which startedin 1988 and currently provides screening mammograms to women across theprovince of British Columbia through 36 fixed and 3 mobile screening sites[oC11, Scr18]. Symptom-free women aged 40-74 without a prior breast can-cer diagnosis are eligible to self-refer to screening mammography subject tothe guidelines of the program. These guidelines stratify the frequency andeligibility with which a woman is recommended to receive screening [Scr18].In general, women assessed to be in a low risk group are recommended toattend screening less often than a woman deemed to be at a high risk ofdeveloping breast cancer. The risk factors currently used by BC CancerBreast Screening to determine screening eligibility are age, family historyand presence of a mutation in the breast cancer genes BRCA1 and BRCA2[Scr18].Due to the nature of screening tests, which balance the trade-off be-tween correctly identifying a high number of those who have a diseaseand recalling an appropriate number for additional tests, an unavoidablebyproduct of screening mammography is the phenomenon of false positives[Gor13, GSW+17]. False positives are defined as an abnormality detectedby the radiologist that does not result in a cancer diagnosis after furtherassessments. Since this leads women to sometimes undergo invasive pro-cedures as well as experience the emotional stress of a potentially loomingdiagnosis, false positives have traditionally been seen as a harm of screeningmammography [EBM+98, VdSKDDVR11, MSG02].However, it has been shown that false positives are associated with the11.1. Objective and thesis outlinerisk of a future breast cancer [HHS+15, vECRTV12, vECKV14, CRR+13,WvECL+17]. Further, recent work has shown that the mammographic fea-tures on a false positive mammogram are a promising risk factor to includein determining screening eligibility [CTRP+16]. Mammographic featuresare the characteristics on a mammogram that a radiologist uses to deter-mine if a mammogram is abnormal, and thus, whether a woman should gofor additional procedures, such as a diagnostic mammogram, ultrasound orbiopsy. The features BC Cancer Breast Screening radiologists, and radiolo-gists worldwide, endeavour to detect are masses, calcification, architecturaldistortion and asymmetry. The presence of these features and the locationthey appear in on the mammogram are recorded in the Breast Screeningregistry.These features stratify the increased risk from a false positive mammo-gram. For example, of the four features, calcifications have been shown tohave the highest risk of a future diagnosis and the lowest increase in riskhas been found in asymmetry and distortion [CTRP+16]. Not only that,but women whose mammographic features change from one screen to thenext show an even higher risk of developing a future cancer [CTRP+16].The increased risks that these mammographic features provide are greaterthan or equal to that attributed to family history, which is currently usedto determine screening frequency [BJS+17, Scr18]. It has been suggestedthat incorporation of mammographic features into risk prediction modelswill improve the models that screening guidelines are based on by providingbetter risk stratification, and, in turn, improve the screening guidelines byincreasing personalization [CTRP+16, WRB+13].Models that use risk factors to predict breast cancer risk in the literaturecurrently do not incorporate this mammographic feature information. Falsepositives and previous biopsies have been used, among other well-known riskfactors, such as age, family history, mammographic breast density, and var-ious lifestyle and reproductive factors. Further, the ability of these modelsto distinguish between who will develop breast cancer, and who will not, isfair at best. With the added risk that the false positives contribute, andthe stratification from the mammographic features, it is possible that theirinclusion could improve the predictive ability.1.1 Objective and thesis outlineThe objective of this thesis is to build a risk prediction model for BritishColumbian women attending screening mammography which incorporates21.1. Objective and thesis outlinethe mammographic feature information and predicts risk of breast cancer.The remainder of this thesis is laid out as follows. Chapter 2 will reviewthe literature on models to predict breast cancer risk. In Chapter 3, wedetail the methodology for the Cox proportional hazards, time-varying Cox,accelerated failure time, and Poisson models explored. In Chapter 4, wemake a comparison of the results to the literature and explore the clinicalrelevance. Finally, Chapter 5 finishes with a summary and suggestions forfuture work.3Chapter 2BackgroundIn this chapter, we define some key concepts and terms and provide areview of breast cancer risk prediction models in the literature.2.1 Terminology2.1.1 RiskRisk is the probability, or how likely it is, that an event will occur. Thisrisk can be described as absolute, which is the number of events divided bythe number of people in that group; or relative, which is the absolute risk ofone group compared to the absolute risk of another [gro19]. In the contextof breast cancer, absolute risk is the probability of developing breast cancerin a given period of time, while relative risk is how much more or less likelyone group is to develop breast cancer compared with another.2.1.2 Risk models, concordance and calibrationRisk models for breast cancer are statistical models that assess the prob-ability that an individual will be diagnosed with breast cancer in a givenperiod of time [AALY+18]. These are known as prognostication models,which are a type of prediction model [SMvdW+13]. The main measuresof validation reported in the literature are discrimination and calibration[CGBB+17].Discrimination is the ability of a model to separate individuals who willhave the disease from those who will not. This is usually reported as theconcordance value [HJLM96, HJLC+84]. A concordance of 1 indicates per-fect discrimination, or women without a diagnosis had consistently lowerrisk scores than women who are diagnosed, and a value of 0.5 indicatesthat the model is no better than randomly guessing at distinguishing be-tween who will get a diagnosis and who will not. It can be calculated as theproportion of concordant pairs, where a concordant pair occurs when the in-dividual with the diagnosis has a higher risk than an individual without the42.1. Terminologydiagnosis. Graphically, a concordance of 1 would mean we see two distinct,non-overlapping curves on a risk distribution plot, with the higher risk curvefor those who had a diagnosis. This would allow us to specify a thresholdto divide individuals into diagnosis and no diagnosis groups. A concordanceof 0.5 would mean the diagnosis and no diagnosis curves perfectly overlapeach other and there is no distinguishing between them.Calibration is a ‘goodness-of-fit’ measure which gives the ratio of theobserved number of cases to predicted number of cases [vH00, Sch80]. Awell-calibrated model will have a ratio close to 1. For a stronger look atcalibration, individuals may be grouped into quantiles and the observed toexpected ratio examined for each quantile [S+19].2.1.3 Risk distributionA risk distribution describes the frequency of absolute risk within thepopulation [S+19]. It is often shown with distinguishing between those witha diagnosis and those without a diagnosis to examine whether those witha diagnosis have a higher risk than those without. It may also be used todecide on risk cut-offs for preventive measures, where individuals above aspecified risk threshold are eligible to receive the preventive measure [HJ15].2.1.4 Hazard functionA hazard function, or the hazard at time t, can be defined ash(t) =f(t)S(t)(2.1)where f(t) is the probability density function and S(t) is the survival func-tion. h(t) can be interpreted as the probability of experiencing the event inthe next small interval of time, divided by the probability of surviving upto that time [TG00].2.1.5 Surveyimpute procedure in SASThe surveyimpute procedure in SAS can use hot-deck imputation toimpute missing values [SAS19, Inc15]. Observations that do not have missingvalues are considered donor units (d) and observations with missing valuesare considered recipient units (m). For each recipient unit, a donor unit isselected at random and the missing values for that recipient unit are filledin with the values from the donor. The selection method chosen for thisanalysis was simple random sampling with replacement.52.1. Terminology2.1.6 Schoenfeld residualsSchoenfeld residuals are calculated usingri = zi − E(Zi) (2.2)where zi is the observed value of the variable and E(Zi) is the expectedvalue [Sch82]. The scaled Schoenfeld residuals can be calculated withRi = ri · d · var(β) (2.3)where ri are the Schoenfeld residuals, d is the number of events, and var(β)is the covariance matrix.The expected value of the scaled Schoenfeld residuals can be approxi-mated by adding the estimated coefficient to the coefficient as a function oftime [TG00]. As such,E(Ri) ≈ β + β(t) (2.4)Therefore, if the proportional hazards assumption is correct, then a plot ofβ(t) vs t should show an approximately linear line at zero.2.1.7 dfbetasA model diagnostic that can be examined to asses the influence of eachobservation on parameter estimation is the difference in an estimated coeffi-cient when all of the data are used and when one observation is deleted fromthe data [W+86]. The model is fitted with removing one observation at atime and the difference between the coefficient estimation with and withoutthat one observation is computed. A simple plot of the change in coefficientagainst the observations will reveal if any observations have a large influenceon parameter estimation and should be removed.2.1.8 Martingale residualsMartingale residuals can be used to assess the functional form of a vari-able [TGF90]. They are calculated withmi = δi −H0(ti) exp(z′iβ) (2.5)where δi is the censoring indicator, H0(ti) is the integrated baseline hazardfunction, and exp(z′iβ) is the linear predictor from the proportional hazardsmodel. Since the Martingale residuals are asymmetric, deviance residualsare an attempt to make the Martingale residuals symmetric around zero.62.2. Breast cancer risk prediction models2.1.9 Akaike Information CriterionThe Akaike Information Criterion (AIC) is given byAIC = −2l(β) + 2k (2.6)where l(β) is the value of the log-likelihood of the model and k is the numberof parameters in the model [SIK86]. The better the model fits the data, thesmaller the value of −2l(β). The more complex the model, the larger 2k is,which penalizes the model for complexity. Therefore, a better model shouldhave a smaller AIC because it fits the data well with a small number ofparameters.2.1.10 Lexis splitA Lexis split allows for following individuals on multiple time scales[Kei90, Car07, PC+11]. These time scales may be age, calendar time, ortime since exposure, to name a few. These times scales are cut into groups,or time bands. During the follow-up of an individual, they pass throughdifferent time bands. Each time band they pass through generates a newrow in the data for that observation. The hazard is constant in each timeband.2.2 Breast cancer risk prediction modelsModels to predict the risk of breast cancer are ubiquitous in the breastcancer literature [LPB+19, EF15, ATW+12, CGBB+17, AALY+18]. Pos-sibly the most well known is the logistic regression Gail model publishedin 1989, which used white women’s age at menarche, number of previousbiopsies, age at first live childbirth and number of first-degree female familymembers who had a breast cancer diagnosis [GBB+89]. The concordance ofthe model was not reported in the original paper.Since then, it has been extensively externally validated and updated. Forinstance, [TCZK05] externally validated the Gail model for mixed ethnici-ties and updated it using a Cox regression model using the Gail risk factorswith the addition of breast density. This resulted in a concordance of 0.68,a promising update. Interestingly, in this same study, they fit a Cox modelwith age, ethnicity and breast density and this had the same concordanceof 0.67 as when they validated the Gail model. [Hoe13] validated the Gailmodel on BC women attending screening mammography, and found that72.2. Breast cancer risk prediction modelswhile the calibration was good, the concordance was only 0.61. Among oth-ers, BMI, alcohol use, diet and exercise have been added to the Gail modelto improve the discriminatory performance and modify the model as newrisk factors become available. It has since been validated and/or updatedon various populations, such as Asian and African, in which it providesa low discrimination for and overestimates risk, respectively [CGBB+17].Most, if not all, of the modified Gail models used previous breast biopsiesas a predictor, however they did not use false positives or the associatedmammographic features.Another well-known model is the Rosner-Colditz model, which was a log-incidence model built from data on white women aged 30-64. This modelbegan with primarily hormonal factors such as age at menarche, menopausalstatus, age, and age at first and subsequent births. With a concordance valueno higher than 0.68, it has been updated to include estrogen use, family his-tory, weight, BMI, alcohol consumption and benign breast disease. Mostrecently, hormone biomarker measures have been added to this model andthe Gail model, such as testosterone and estrogen levels, which show im-provement to these models [CGK+19, ZRT+18]. It appears that most, ifnot all, of the updates to the model have used white women as well, thusthis line of models may not be applicable to the multi-ethnic population ofwomen attending screening mammography in BC. While benign breast dis-ease is used in nearly all the updated Rosner-Colditz models, mammographicfeatures and false positives have not been added to this model.There is also a group of models that include detailed family historyand presence of genetically inherited gene mutations. Since the Gail modelperforms best in women without a family history, [CRT93] built the Clausmodel for individuals with a family history before the discovery of the ge-netic mutation on the BRCA1 and BRCA2 genes. The Tyrer-Cuzick modelis another genetically based model which uses a host of predictors, suchas hormonal factors, biopsies, family history of breast and ovarian cancer,and BMI [CGBB+17]. Though they did not report any performance mea-sures in the original paper, it has been validated on high risk populationsand performs better than the Gail and Claus models, although it tends tooverestimate risk in certain sub-populations [CGBB+17].There are also a number of independently derived models, such as theThai risk prediction model, which is a logistic regression model for Thaiwomen which attained a concordance of 0.65 and calibration of 1.0 using only4 variables; age, contraceptive use, BMI and menopausal status [ATW+14].[Hoe13] built a stratified Cox model with BC women, which attained a con-cordance of 0.65 and 0.63 for in situ and invasive, respectively, and among82.2. Breast cancer risk prediction modelsthe typical variables of age, ethnicity, family history, biopsy and estrogenuse, incorporated a variable indicating whether the baseline mammogramwas abnormal. [UTT+03] built a model for long term risk based on Japanesewomen using only age, BMI, family history, and age at menarche, but didnot report concordance, while [WGB+14] built a model for Chinese womento fit the low-incidence population and achieved a concordance of 0.64.While it has been found that false positives are associated with the riskof breast cancer, so far, only one published model has explicitly used them.False positives were added by [BWBB+06] which gave a concordance of 0.63when fitting a logistic regression model with 11,638 diagnoses and 2 millionscreens.9Chapter 3Methods3.1 DataThe data used in this study were from the BC Cancer Breast Screen-ing registry. It contained risk factor, screening, and diagnosis informa-tion collected during BC Cancer Breast Screening’s routine operations from6,455,264 screening mammograms for 1,099,298 women who attended screen-ing mammography from 1988 to March 31, 2019.Ethics approval was obtained from the UBC-BCCA Research EthicsBoard with certificate number H18-00172. Access to the data was grantedafter a formal BC Cancer Data Access Request was approved by BC Canceron July 22, 2018. Data were provided to us de-identified by BC CancerBreast Screening with a randomly assigned numeric subject identifier.Risk factors that are assumed to be unchanging when a woman first at-tends screening at BC Cancer Breast Screening are collected with a volun-tary background survey at the first screening mammography appointment.These factors include ethnicity, highest education level achieved, number offull-term pregnancies, the age at the first full-term pregnancy, if applicable,and the age at menarche [LZW+12, DQ20, RCW94, MCL+70, oHFiBC+12].Risk factors that can change over time are voluntarily surveyed at eachscreening visit. These include the woman’s family history of breast cancer,current estrogen use, whether she has had a breast biopsy, menopausal statusand if she has had a hysterectomy and ovary removal [PDD+97, KKLL18,CG01, DYFC15, oHFiBC+12, SPF+97]. Since the risk factor information inthis dataset is obtained from voluntary surveys, it is subject to errors andmissingness. The risk factors collected by BC Cancer Breast Screening havebeen shown to affect breast cancer risk and have been used in previous riskprediction studies.Once screening mammograms are completed, they are assessed by cre-dentialed and trained radiologists [CP12]. The radiologists ascertain andrecord the mammographic breast density, the abnormal status, and if ab-normal, which mammographic features were observed along with their loca-tion. As such, the screening mammogram information available in the Breast103.2. ParticipantsScreening registry relies completely on the radiologist’s interpretation of themammogram.Screen-detected and non-screen detected cancers were also available inthe data. Screen-detected cancers are those which are diagnosed as a resultof abnormal findings from a screening mammogram. Non-screen detectedcancers include: interval cancers, which are cancers discovered within therecommended screening interval from the last normal screen; false negatives,which are cancers discovered after additional assessments from an abnormalscreen did not find cancer; and cancers that occur when a woman does notscreen according to the recommended guidelines.Each woman can have at most one diagnosis. Once a woman is diagnosedwith ductal carcinoma in situ (DCIS) or invasive breast cancer then sheis no longer eligible to attend screening mammography. Recurrences andtreatment are not recorded in the BC Cancer Breast Screening Registry,but are in the BC Cancer Registry.3.2 ParticipantsThe individuals in the study can be divided into two groups; those whodid not have a false positive and those who had at least one false positive.These two groups are important because normal screens for those who didnot have a false positive serve as the reference category for which risks fromfalse positive screens with mammographic features can be compared. Asin [CTRP+16], for women who did not have a false positive, the start offollow-up was defined to begin from the first screening mammogram, but forwomen who did have a false positive, their follow-up began from the firstfalse positive (Figure 3.1). This adjustment to the index screen was done tobetter understand the risk following a false positive screening mammogram.In the following analysis, the methods require choosing a time origin, whichwill be the index screening mammogram. From this index screen, womenwere followed for 5 years, until diagnosis, until their last screen if there wasno diagnosis, or until study end, whichever came first. The official study endwas Dec 31, 2016 as the data were considered complete (as of August 2019)by BC Cancer until this date, thus an additional censoring of all screens anddiagnoses was done at this date.Although all women who underwent screening at BC Cancer BreastScreening until March 31, 2019 were included in the data we received, wehad to exclude some individuals prior to analysis (Table 3.1). First, 680women were excluded due to missing diagnosis information or birth date.113.2. Participantsl ll ll llFPNo FP1 n n+1 n+2Screen numberFalse Positive (FP) statusFigure 3.1: Example follow-up of two women undergoing screening. Normalscreens are black and abnormal screens are red.123.2. ParticipantsTable 3.1: Exclusions made to get to the final cohortReason removed Women remaining Women removedall women 1,099,298incorrect diagnosis date 1,099,288 10missing birth date 1,099,286 2missing diagnosis details 1,098,618 668age outside 40-74 1,054,660 43,958index screen after Dec 30, 2016 962,318 92,342true positive at index screen 957,276 5,042false negative at index screen 954,760 2,516one screen 813,280 141,480Total 813,280 287,736As the screening program regularly screens women in the age range of 40-74, it is not helpful to predict risk for women outside these ages. Therefore,43,958 women younger than 40 or older than 74 at the time of the indexscreen were excluded. 92,342 women who had their index screen after thestudy end date were also excluded.We further excluded 5,042 women with a true positive at the index screenand 2,516 women with a false negative at the index screen. True positiveswere excluded because the cancer had developed prior to the start of follow-up and were only detected at the index screen because it was already present.False negatives were excluded because these ideally should have been truepositives and would have been excluded for the same reasons as true posi-tives. Lastly, 141,480 women who only had one screen were excluded, sinceno useful information could be learned from them.This left us with a cohort of 813,280 women, 11,166 diagnoses and2,347,901 screening mammograms, an average of 2.9 ± 1.3 screens per per-son within the 5 years of follow-up (Figure 3.2). Of the 2,347,901 screeningmammograms, 296,774 (12.6%) were false positives and 259,794 (87.5%) ofthese false positives were the woman’s first false positive (which occurredat the index screen as defined earlier for women with a false positive). Theremaining 12.5% of the false positives were additional false positives, e.g.second or third, that occurred on a subsequent screen at some point dur-ing follow-up. The median follow-up for women with and without a falsepositive was 4.97 and 5 years, respectively.133.2. Participants0500001000001500002000002500001 2 3 4 5 6 7Number of screensNumber of womenFigure 3.2: Total number of screens in the cohort.143.3. Variable characteristics3.3 Variable characteristicsData cleaning was undertaken prior to model fitting and the results aresummarized in Table 3.2. First, there were several different ways missingentries were recorded in the data, such as ‘Refuse to Answer’, ‘Unknown’,‘NA’ , ‘N’, and sometimes a special numeric value such as 0 or 99. Thesewere recoded to a single value, ‘NA’, for consistency.Second, at the start of the screening program, breast density was recordedas a binary variable, representing < 50% and ≥ 50% density. However, in2005, density started being recorded as a quaternary variable, representing0-24%, 25-49%, 50-74% and ≥ 75% density, but the changeover was gradualand only 24% of the mammograms in the dataset are recorded in the newway. While less informative, to reduce the number of missing values, allbreast density values were recoded as a binary factor.Education and ethnicity were also recoded to contain fewer categoriesin an effort to make the categories more meaningful. European and Britishethnicities were combined into one ‘Caucasian’ ethnicity. Education levelswere grouped into more coarse groups defining no high school education (‘NoHS’), some or graduated high school (‘Some/HS’), and some or graduatedpost-secondary (‘Some/Co’).Family history was initially coded as a 19 level categorical variable repre-senting combinations of family members ever diagnosed with breast cancer,such as ‘mother and sister‘ or ‘sister and father’. Family history was con-verted into a binary variable to indicate ‘Yes’ or ‘No’. In light of the findingsfrom [Hoe13], who found that the type of relationship is important and notjust the presence of family history, two additional binary variables werecreated from the original family history variable indicating whether theirmother or sister had a history of breast cancer. Women who indicated thattheir family history was unknown were given the baseline value of ‘No’ toboth these variables.The mammographic features also underwent some modifications. Origi-nally, each of the four features was in its own data field; these were combinedinto one field. If a feature was seen at least once and was the only featureobserved on the mammogram, it was labeled with ‘feature only’. For ex-ample, a mammogram with only calcification would be coded as ‘calc only’.If only two features were seen on the mammogram at least once each, thiscombination received its own category, but 3 or more distinct features weregrouped into one category due to the low number of occurrences. Whilegrouping the features into one variable and creating a category for eachcombination makes for many categories, it is important clinically to see the153.3. Variable characteristicsTable 3.2: Table describing the final risk factors in the dataVariable Description Valuesage age at index screen numericethnic ethnic/cultural heritage of client First NationsAfricanEast AsianSouth AsianCaucasianOthereducation highest level of education achieved No HSSome/CoSome/HSnum full term number of full-term pregnancies integerdeliveriesage at 1st age at first full-term pregnancy integerdeliveryage at menarche age menstruation began integerever mom mother had breast cancer 0 (No), 1 (Yes)ever sis sister had breast cancer 0 (No), 1 (Yes)Ever Estrogen currently using estrogen 0 (No), 1 (Yes)Ever Biopsy ever had a breast biopsy 0 (No), 1 (Yes)menstr menopausal status 0 (No), 1 (Yes)hyster had hysterectomy 0 (No), 1 (Yes)ovary both ovaries removed 0 (No), 1 (Yes)result abnormal status 0 (No), 1 (Yes)density breast density 1 (< 50%)2 (≥ 50%)feat mammographic features 3 or morearch asymarch calcarch onlyasym calcasym onlycalc onlymass archmass calcmass onlymissingno feat163.3. Variable characteristicsinfluence of the solo features and combinations. To handle missing features,an additional category ‘missing’ was created to represent screens that thedata indicated were abnormal, but were missing the corresponding mam-mographic feature(s) present. This occurred for 107,525 screens, which isroughly a third of all the false positives (Table 3.3). Prior to 2003, only thepresence and location of an abnormality was recorded, but not the type ofmammographic feature, resulting in the missing features.Table 3.3: Table showing the breakdown of false positives and other screenson the index screens and all screensCount %Index screennormal screen 553,486 68false positive 259,794 32false positive with features recorded 164,519 20false positive without features recorded 95,275 12Total 813,280 100All screensnon-false positive screen 2,051,127 87false positive 296,774 13false positive with features recorded 189,249 8false positive without features recorded 107,525 5Total 2,347,901 100After data cleaning, some exploratory analysis was performed to under-stand the data. We found that of the 813,280 index screens, 32% (259,794)were false positives (Table 3.3). Of these false positive screens, 37% (95,275)did not have mammographic features recorded (Figure 3.3). Therefore, ofthe 813,280 index screens, 20% (164,519) were false positives with featuresrecorded and 12% (95,275) were false positives without any mammographicfeatures recorded.From Table 3.3, we note that roughly one third of the women who at-tended screening mammography had a false positive at some point duringtheir screening journey. This is a similar finding to [EBM+98], who foundthat for women attending screening with a median of 4 screening mammo-grams, one third of the women screened had a false positive.The distribution in Figure 3.3 of the index screens with mammographicfeatures present reveals that asymmetry (‘asym only’) is recorded most often(32%) and the combination of architectural distortion with calcifications173.3. Variable characteristics0.00.10.20.3arch_calc3_or_moremass_archmass_calcasym_calcmass_asymarch_asymarch_onlycalc_onlymass_onlyasym_onlymissingFeatures at index screenRelative FrequencyFigure 3.3: Frequency of mammographic features for the index screenswhere there was an abnormality indicated.183.3. Variable characteristics(‘arch calc’) occurs least often (0.2%). Examining the distribution of themammographic features using all the features from all screens during follow-up revealed a nearly identical frequency distribution to the distribution offeatures on the index screen.When examining education, it was found that 50% of the women hadsome college or completed a post-secondary degree, 36% had graduated highschool or completed some high school, and 6% had no high school education.On further inspection, the average age of those women who reported theyhad no high school was 58.5 at the index screen, compared to the averageage of 53.3 for those who had a high school education and 49.7 for thosewho had post-secondary education. Education was also found to be mildlyrelated to the number of children. Figure 3.4 shows that as the educationlevel increases the number of children decreases on average. We can alsoobserve the increase in children for older women. When compared withbirth year, it was found that women born in the 1920s and 1930s had themost children on average. It appears that both education and number ofchildren are associated with age at index screen and birth cohort.With respect to ethnicity, 47% reported one of the ethnicities that wereused to create the Caucasian group, 10% reported East Asian, 3% SouthAsian and only 1.2% and 0.2% First Nations and African, respectively. Theremaining 38% of the women did not have an ethnicity recorded, with 4.1%responding ‘other’, 0.3% ‘refused to answer’, and 33.2% stated their ethnicitywas ‘unknown’. These were grouped into the category ‘Other’, making this‘Other’ group comprised of 11.0% ‘other’, 0.7% ‘refused to answer’, and88.3% ‘unknown’. This category may be thought of as all individuals whodid not place themselves into one of the predefined ethnicities.We confirmed that the breast density decreases with age and is stratifiedby ethnicity (Figure 3.5). Breast density is a measure of how much fibroglan-dular tissue is seen on the mammogram, represented as a percentage of totalbreast area. On a mammogram, breast tissue appears white and fat appearsdark [MRMTB14]. Of particular note is that the East Asian women havethe highest breast density, while the First Nations women have the lowest.When examining associations between age, mammographic breast den-sity and the type of feature, we found that calcifications were associatedwith dense breast. Mass and calcification (‘mass calc’) falls in between cal-cifications (‘calc only’) alone and mass (‘mass only’) alone. Figure 3.6 showsselect features.Table 3.4 provides a summary of the risk factor values for the cohortused in this analysis. For mammographic breast density and features, thevalues used represent the values recorded at the index screens. A summary193.3. Variable characteristicsllllllllllllllllllllllllllllllllllll1.52.02.53.03.515−19 20−24 25−29 30−34 35−39 40−44 45−49 50−54 55−59 60−64 65−69 70−74Year of birth (1900s)Average number of childreneducationlllNo HSSome/HSSome/CoFigure 3.4: Average number of children per half decade of birth stratifiedby self-reported education level.203.3. Variable characteristicslllllllllllllllllllllllllllllllllllllllll1.01.21.41.61.840−44 45−49 50−54 55−59 60−64 65−69 70−74Age CategoryAverage breast density ethnicllllllEast AsianOtherSouth AsianAfricanCaucasianFirst NationsFigure 3.5: Average density for each age group at index screen stratifiedby ethnicity.213.3. Variable characteristicsllllllllllllllllllllllllllllllllll1.21.41.640−44 45−49 50−54 55−59 60−64 65−69 70−74Age CategoryAverage breast densityfeatlllllcalc_onlyasym_calcmass_calcmass_onlyno_featFigure 3.6: Average density for each age group at index screen stratifiedby select mammographic features.223.3. Variable characteristicsTable 3.4: Table showing the values of the risk factors in the dataVariable Missing (%) Average or count (%)age 0 51.8ethnic 305,596 (37.5)First Nations 9,911 (1.2)African 1,794 (0.2)East Asian 85,512 (10.5)Caucasian 382,942 (47.0)South Asian 27,525 (3.4)education 60,897 (7.5)No HS 49,542 (6.1)Some/Co 407,344 (50.1)Some/HS 295,497 (36.3)num full term deliveries 2,617 (0.3) 1.9ever mom 00 753,566 (92.7)1 59,714 (7.3)ever sis 00 779,616 (95.9)1 33,664 (4.1)Ever Estrogen 6588 (0.8)0 702,591 (86.4)1 104,101 (12.8)Ever Biopsy 5010 (0.6)0 730,691 (89.8)1 77,579 (9.5)result 00 553,486 (68.1)1 259,794 (31.9)density 143,807 (17.7)1 433,171 (53.3)2 236,302 (29.1)feat 95,275 (11.7)3 or more 578 (0.1)arch asym 5,350 (0.7)arch calc 452 (0.1)arch only 9,941 (1.2)asym calc 2,685 (0.3)233.4. Cox proportional hazards modelVariable Missing (%) Average or count (%)featasym only 82,510 (10.1)calc only 23,823 (2.9)mass arch 582 (0.1)mass asym 3,135 (0.4)mass calc 1,547 (0.2)mass only 33,916 (4.2)no feat 553,486 (68.1)of the subsequent screens will be provided when these screens are used inanalysis.3.4 Cox proportional hazards modelTo model the risk of a future breast cancer we first use a Cox proportionalhazards model (Equation 3.1) [Cox72]. The first point of this model is todetermine which risk factors are important to breast cancer risk and by whatmagnitudes, an insight of clinical relevance. The second point is to obtaina risk estimate for each observation, which can be used to predict breastcancer risk at an individual level [Col15].The overall framework of the model is as follows. Predictors, in our case,risk factors, are measured at an initial point. Here, that initial point is theindex screening mammogram. Individuals are followed for a period of time,in our case, 5 years, and the outcome, diagnosis or no diagnosis, at the endof the 5 years and when it happened, is determined.In this way, this semi-parametric model allows for incorporating the timeto diagnosis while taking into account the censored observations. It has abaseline hazard function, h0(t), which makes no assumptions on the formof its distribution. The baseline hazard is the value of the hazard function,hi(t), when all the predictor values are set to zero. The predictors are anexponentiated linear combination, where xj is the jth covariate in the modeland βj the jth coefficient. The hazard function, hi(t), is the hazard at timet. This is the outcome of interest, or response [All10]. The assumptions ofthe model are proportional hazards, independent observations and that thepredictors are multiplicative on the baseline hazard.hi(t) = exp(β1x1i + β2x2i + . . .+ βpxpi)h0(t) (3.1)243.4. Cox proportional hazards modelWe can write Equation 3.1 more concisely as in Equation 3.2.hi(t) = expp∑j=1βjxjih0(t) (3.2)The hazard associated with a covariate is provided by exp(βj). If it takesa value >1, then there is an increase in hazard, if it is <1 then there is areduction in hazard. For continuous variables, exp(βj) is the increase inhazard with one unit increase in the predictor, holding the other predictorsconstant (Equation 3.3).exp {β(x+ 1)}exp(βx)= eβ (3.3)For categorical variables, a reference category must be chosen to takethe baseline value of γ1 = 0 and the hazard estimates are made with respectto that reference category. So for j = 1, . . . ,m groups, where j = 1 is thereference category, the hazard function for an individual in the jth group ishj(t) = exp(γj)h0(t) (3.4)Rearranging, we find that the relative hazard for someone in group jrelative to group 1, is eγj .hj(t)h0(t)= eγj (3.5)Prior to model fitting, missing values for all variables were imputed withSAS’s surveyimpute procedure [SAS19, Inc15]. This function imputes miss-ing values by sampling from other values that are present for that variable.Next, the data was split into a randomly selected training and testing setwhere 70% was used for training and 30% for testing. To reduce the numberof variables in the model, variable selection was performed using backwardsselection through the selectCox() function in R, which uses the reductionin AIC as a stopping rule [R C19, MIG12, SIK86]. The resulting model isshown in Table 3.5.From the model in Table 3.5, we find that the features that have thelargest increase in risk compared to the reference category of no features(‘no feat’) are when 3 or more (‘3 or more’), architectural distortion andcalcifications (‘arch calc’), mass and architectural distortions (‘mass arch’)and calcifications alone (‘calc only’) are seen. We also note that the confi-dence intervals for the features are quite large, making it difficult to deducehow much hazard is actually attributed by the presence of these features.253.4. Cox proportional hazards modelTable 3.5: Cox proportional hazards modelcoef exp(coef) p-value 2.5 % 97.5 %age 0.04 1.03 0.00 1.03 1.04featno feat (ref) 0.00 1.00feat3 or more 0.84 2.31 0.01 1.28 4.17featarch asym -0.04 0.96 0.81 0.71 1.31featarch calc 1.17 3.23 0.00 1.88 5.57featarch only 0.03 1.03 0.74 0.85 1.26featasym calc 0.42 1.53 0.01 1.09 2.15featasym only -0.12 0.88 0.00 0.81 0.96featcalc only 0.70 2.01 0.00 1.82 2.22featmass arch 0.96 2.62 0.00 1.52 4.52featmass asym 0.43 1.53 0.01 1.11 2.11featmass calc 0.46 1.58 0.03 1.04 2.41featmass only 0.03 1.03 0.56 0.92 1.16featmissing 0.16 1.17 0.00 1.10 1.25density1 (ref) 0.00 1.00density2 0.39 1.48 0.00 1.41 1.55ethnicCaucasian (ref) 0.00 1.00ethnicFirst Nations 0.21 1.23 0.04 1.01 1.50ethnicAfrican 0.21 1.24 0.33 0.80 1.90ethnicEast Asian -0.05 0.95 0.20 0.88 1.03ethnicOther 0.01 1.01 0.58 0.96 1.06ethnicSouth Asian -0.32 0.73 0.00 0.62 0.85Ever Estrogen0 (ref) 0.00 1.00Ever Estrogen1 0.16 1.17 0.00 1.10 1.24Ever Biopsy0 (ref) 0.00 1.00Ever Biopsy1 0.35 1.42 0.00 1.34 1.51ever mom0 (ref) 0.00 1.00ever mom1 0.51 1.66 0.00 1.55 1.78ever sis0 (ref) 0.00 1.00ever sis1 0.41 1.50 0.00 1.38 1.64educationNo HS (ref) 0.00 1.00educationSome/HS 0.27 1.31 0.00 1.19 1.45educationSome/Co 0.32 1.38 0.00 1.25 1.53263.4. Cox proportional hazards modelSurprisingly, asymmetry (‘asym only’) has a negative coefficient. Sincethe upper bound of the 95% confidence interval is very close to 1.0, it islikely that this is not really significant, as found in [CTRP+16], but thelarge number of screens causes it to appear significant. Another interestingobservation is that the First Nations group when compared to Caucasianhave an increase in risk. Recall from Figure 3.5 that the First Nations hadthe lowest breast density. As such, this may suggest that a high breastdensity should be taken into account with ethnicity and other factors.It is also intriguing to find that the education status shows up as signifi-cant, while none of the breast cancer risk prediction models we are aware ofinclude it. It is possible that there is still some overfitting occurring or somebehaviour that has been unaccounted for to make it appear as a significantrisk factor. Alternatively, this education variable may be more of a measureof lifestyle than just strictly education. For example, we found that womenwho had more children also had the first child at a younger age. We alsosaw in Figure 3.4 that education is associated with number of children. Itcould be that higher educated women are having less children and at a laterage, thus increasing their risk of breast cancer, resulting in education levelas a significant risk factor [DQ20].To assess whether this model adequately describes the data, we checkfor proportional hazards, influential observations, and non-linearity. Pro-portional hazards were checked using cox.zph() in the survival package in Rwhich checks the Schoenfeld residuals [Sch82, R C19, The15]. A global p-value of 0.025 < 0.05 shows us that this assumption may not be handled cor-rectly. A graphical check in Figure 3.7 of the variables which failed the testreveal that high school education, missing features, family history and breastdensity don’t seem to have any strong patterns with time [R C19, KK19].A possible concern might be breast density, which shows a slight decreaseuntil about 2 years after the index mammogram.Influential observations were assessed using dfbetas [W+86, R C19, The15].The cut-off for influential outliers was 1.0, but no values above 1.0 were ob-served. As well, non-linearity in the only numeric variable, age, was assessedusing Martingale residuals, showing no concerning patterns [TGF90, R C19,The15].The concordance for training and test set were 0.637 (95% CI 0.631,0.643) and 0.627 (95% CI 0.621, 0.633), respectively [R C19, The15]. FromFigure 3.8, we can see that there is a serious amount of overlap between therisk distributions of those with a diagnosis and those without a diagnosis,a model producing a higher concordance value would reduce the overlap,allowing for better risk stratification. The calibration was assessed at 0.96273.4. Cox proportional hazards modelllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll−2021.1 1.6 2.1 2.6 3.2 3.8 4.2 4.7Time (years)Beta(t) for density2llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll−202461.1 1.6 2.1 2.6 3.2 3.8 4.2 4.7Time (years)Beta(t) for featmissingllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll0481.1 1.6 2.1 2.6 3.2 3.8 4.2 4.7Time (years)Beta(t) for ever_mom1llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll−1001.1 1.6 2.1 2.6 3.2 3.8 4.2 4.7Time (years)Beta(t) for educationSome/HSFigure 3.7: Checking the proportional hazards assumptions for the variablesand variable categories that showed a significant p-value in the Schoenfeldresiduals.283.4. Cox proportional hazards modeland 1.04 for 5 and 2-year risk, respectively, a value of 1.0 would indicateperfect agreement between observed outcomes and predictions [SVC+10,R C19, Har19]. The overall 5-year mean absolute risk was 1.61%, while themean absolute risk for those with no diagnosis and those with a diagnosiswas 1.6% and 2.0%, respectively (p < 0.0001) [R C19, KW52].0.00.20.40.60.80.0 2.5 5.0 7.5 10.0Absolute 5−year risk (%)densityNo diagnosis DiagnosisFigure 3.8: The discrimination from the Cox model is quite low, as seen inthe overlap of the curves for those who had a diagnosis and those who didnot.Table 3.6 shows the absolute risk predictions on the test set for 2 and 5year absolute and stratified risk. Stratified risk was calculated by dividingthe mean risk in each group by the mean risk in the low risk group and isused to observe how much more risk an individual above low risk has relativeto an individual in the low risk group. We have broken the risk predictions293.4. Cox proportional hazards modelinto risk groups corresponding to the NICE guidelines [fCEN+]. The NICEguidelines specify 10-year risk divided into general, moderate and high at<3%, 3-8% and >8%. To get 5 and 2-year risk cut-offs we divided these by2 and 5, respectively. An additional low risk group was created since mostof the risk fell in the general group. We observe that compared to the lowestrisk group, the highest risk group has about 8 times more risk for 2 and 5years. We also notice that a small proportion are in the high risk group,with most individuals classified as general risk. We can interpret the meanrisk as saying that on average an individual in the moderate risk group hasa 2.2% and 0.82% probability of developing breast cancer in the next 5 and2 years, respectively.Table 3.6: 2 and 5-year absolute risk predicted from Cox proportional haz-ards modelRisk group Risk range % at risk Mean risk Stratified risk5-Yearlow 0-0.75 3.0 0.67 1.0 (reference)general 0.75-1.5 55 1.1 1.6moderate 1.5-4 40 2.2 3.2high >4 2.5 5.3 7.82-Yearlow 0-0.3 34 0.24 1.0 (reference)general 0.3-0.6 50 0.41 1.7moderate 0.6-1.6 14 0.82 3.4high >1.6 0.43 2.0 8.6We compare this risk distribution to [ECP+17], who used a logistic re-gression model with age, family history, BMI, estrogen use, and numberof masses and microcalcifications interpreted by fully automated computer-aided detection and found a concordance of 0.71 (0.69-0.73). From Table3.7, we note that the increase in concordance allowed them to find a lowrisk group, whereas our concordance lumps most of our observations intothe general risk group. They also determined a larger proportion to be at ahigh risk. While the proportions for the groups are different, the mean riskfor these groups in our model and in [ECP+17] are astoundingly similar.Since the concordance value is quite low, we move on to some alternativemodels to see if the performance may be improved.303.5. Time-dependent Cox regression modelTable 3.7: Comparison of 2-year risk predicted from our Cox model to[ECP+17]Risk group Risk range % at risk Mean risk Stratified risk[ECP+17] (c=0.71)low 0-0.15 10 0.12 1.0 (reference)general 0.15-0.6 65 0.33 2.8moderate 0.6-1.6 23 0.82 6.8high ≥1.6 2 1.95 16.2Our Cox model (c=0.64)low 0-0.15 0.37 0.14 1.0 (reference)general 0.15-0.6 85 0.34 2.4moderate 0.6-1.6 14 0.82 5.8high ≥1.6 0.43 2.00 14.33.5 Time-dependent Cox regression modelFrom clinical knowledge, we know that the mammographic features canchange over time. From the literature, we know that these changes areassociated with an increased risk of developing breast cancer [CTRP+16].Therefore, we fit a time-dependent Cox regression model to incorporate thischanging behaviour of the mammographic features [FL99, R C19, The15].Since the previous model measured hazard from the index screen, incorpo-rating the more recent features from screens closer to a diagnosis and thechanging behaviour may improve the model performance.This time-varying model allows for the value of x to change over time,denoted by x(t) (Equation 3.6). This means that if the features change fromone screen to the next, the value of x changes. Note that this is the onlychange in the model equation from Equation 3.1 in the previous section.Baseline hazard, h0(t), is defined as the hazard function for someone whohas a value of zero for every variable for all time [Col15].hi(t) = expp∑j=1βjxji(t)h0(t) (3.6)The relative hazard eβj can be interpreted as follows. For two individualss and r at a given time t whose values differ by one unit, with the same valuesof the other predictors, the relative hazard is given by Equation 3.7 below.313.5. Time-dependent Cox regression modelhr(t)hs(t)= exp[β1{xr1(t)− xs1(t)}+ . . .+ βp{xrp(t)− xsp(t)}] (3.7)In order to capture the time-varying behaviour, the false positive ab-normal screens and normal screens were selected. True positive and falsenegative screens were removed as the features on these screens are indicativeof a cancer that is already present.In order to set the data up for input into the tmerge() and coxph()functions in R, we had to take a subset of screens from the data accordingto a specific algorithm [R C19, The15]. All index screens were included. Asubsequent screen was included if the values of at least one of the densityor feature variables was different from the screen immediately preceding it.The missing features were treated as generic abnormal results, so if someonehad an abnormal mammogram but the feature was missing, a subsequentabnormal with the feature present was included as there was no way totell if the features were the same or not. Figure 3.9 shows the remainingscreens used in this portion of the analysis. An individual with one screenmeans that they did not have any changes in their screens. A woman with 2remaining screens had 1 change at some point during follow-up, and so on.Figure 3.10 shows the distribution of features on the screens used in thisanalysis which had features present. 78% of the remaining 914,111 screensdid not have features, leaving 101,441 screens with features. The relativefrequency is nearly identical to when looking at the entire dataset, despiteall the modifications made to get the data into the proper format. Thiswas checked because it was unknown what the effect would be when movingfrom the original dataset with all screens to the dataset with only screensthat were different from the previous screen.Using the same methods as in the previous section, we fit a time-varyingCox model [FL99, R C19, The15]. We assessed goodness of fit using dfbetasand Martingale residuals, but there is no longer the assumption of propor-tional hazards, so we did not need to check that this time [Col15, TCA17,TGF90, BP88]. The dfbetas and Martingale residual plots are similar tothe Cox proportional hazards model, so are not shown here [R C19, The15].The resulting model is in Table 3.8. The concordance was 0.643 (95% CI0.637, 0.649) for the training set.We again see that 3 or more features, architectural distortion with cal-cification and mass with architectural distortion have large coefficients, in-dicative of an increase in risk. The 95% confidence intervals for the expo-nentiated coefficients are again quite large, making the estimates subject to323.5. Time-dependent Cox regression model0.00.20.41 2 3 4 5 6Number of screensRelative FrequencyFigure 3.9: Number of screens remaining to use in the model that changedfrom the previous screen.333.5. Time-dependent Cox regression model0.00.10.20.3arch_calc3_or_moremass_archmass_calcasym_calcmass_asymarch_asymarch_onlycalc_onlymass_onlyasym_onlymissingFeatures during the 5 years of follow−up after necessary time−dependent modificationsRelative FrequencyFigure 3.10: Remaining features on screens with a feature identified afterformatting the data prior to fitting the Cox regression model.343.5. Time-dependent Cox regression modelTable 3.8: Cox time varying regression modelcoef exp(coef) p-value 2.5 % 97.5 %age 0.03 1.04 0.00 1.03 1.04featno feat (ref) 0.00 1.00feat3 or more 1.45 4.25 0.00 2.21 8.21featarch asym 0.56 1.75 0.00 1.21 2.54featarch calc 1.70 5.46 0.00 2.94 10.11featarch only 0.53 1.70 0.00 1.29 2.24featasym calc 0.73 2.08 0.00 1.35 3.20featasym only 0.39 1.48 0.00 1.33 1.65featcalc only 1.05 2.84 0.00 2.51 3.22featmass arch 1.52 4.56 0.00 2.45 8.48featmass asym 0.69 1.99 0.00 1.31 3.02featmass calc 1.05 2.86 0.00 1.83 4.47featmass only 0.51 1.66 0.00 1.43 1.92featmissing 0.56 1.74 0.00 1.60 1.89density1 (ref) 0.00 1.00density2 0.42 1.52 0.00 1.45 1.59ethnicCaucasian (ref) 0.00 1.00ethnicFirst Nations 0.21 1.23 0.04 1.01 1.50ethnicAfrican 0.22 1.25 0.31 0.81 1.92ethnicEast Asian -0.04 0.96 0.34 0.89 1.04ethnicOther 0.03 1.03 0.29 0.98 1.08ethnicSouth Asian -0.32 0.73 0.00 0.62 0.84Ever Estrogen0 (ref) 0.00 1.00Ever Estrogen1 0.16 1.17 0.00 1.10 1.24Ever Biopsy0 (ref) 0.00 1.00Ever Biopsy1 0.35 1.42 0.00 1.33 1.51ever mom0 (ref) 0.00 1.00ever mom1 0.51 1.66 0.00 1.55 1.78ever sis0 (ref) 0.00 1.00ever sis1 0.41 1.50 0.00 1.38 1.64educationNo HS (ref) 0.00 1.00educationSome/HS 0.27 1.30 0.00 1.18 1.44educationSome/Co 0.31 1.36 0.00 1.23 1.50353.6. Accelerated Failure Time modelmuch variability. While slightly better than the previous model, since theconcordance is again quite low, namely too low to be used in individualizedrisk prediction, we abandon the Cox models and move on to other modelsto see if there is a better option available.3.6 Accelerated Failure Time modelAccelerated failure time (AFT) models are a parametric alternative tothe Cox regression model [Wei92, KP11, Col15, Moo16]. While they requirestronger assumptions to be met, they can be used to predict the length oftime to an event. The model can be described as in Equation 3.8, whereηi = α1x1i + . . .+ αpxpi is the linear combination of predictors, h0(t) is thebaseline hazard function, and t is time. The baseline hazard can be thoughtof as the hazard of diagnosis at time t for someone who has the baselinevalue of zero for all predictors.hi(t) = exp(−ηi)h0( texp(ηi)) (3.8)A somewhat simpler way to view this model is the log-linear form as inEquation 3.9, where σ is the scale parameter, µ is the intercept, and  isthe error which has a certain distribution depending on the type of modelchosen. Interpreting the coefficients, we get that a positive α increasessurvival time, while negative values decrease survival time. Note that thisis an exactly opposite interpretation to the previous Cox models, where apositive coefficient was indicative of an increase in risk and negative valuesdecrease the risk.log Ti = µ+ α1x1i + . . .+ αpxpi + σi (3.9)Using the same training and testing set and variable selection methodas before, we fit an accelerated failure time model for several different typesof AFT models, to determine which will satisfactorily fit the data [R C19,The15]. According to Table 3.9, the Weibull, lognormal or loglogistic arebest suited as they have the lowest AIC (Akaike Information Criterion)[SIK86]. AIC was calculated with R’s AIC() function and it is merely a wayto compare models where the smaller the AIC, the better the fit [R C19].We first examine how well the data follow a Weibull distribution. Wecan do this graphically, by plotting the log of survival time and the log-log of the estimated survival (Figure 3.11) [Moo16]. The Kaplan-Meierestimate of survival is computed using the survfit() function in the survival363.6. Accelerated Failure Time modelTable 3.9: AIC of AFT modelsModel AICLognormal 102,521Loglogistic 102,604Weibull 102,612Exponential 103,952Gaussian 105,298ll l l l lll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll−12−10−8−6−4−4 −2 0log of survival timelog−log of estimated survivalFigure 3.11: Plot of the log-log transformation of the Kaplan-Meier sur-vival vs the log transformation of time. Used to assess whether the Weibulldistribution is appropriate for the data.373.6. Accelerated Failure Time modelpackage in R [R C19, The15]. The Kaplan-Meier survival estimates survivalempirically without taking into account any predictors [KM58]. The log-logtransformation of the survival estimate is plotted vs the log transformationof the time variable and a regression line through the points is fitted (Figure3.11). The closer the points are to the straight line, the better. There issome concern in the bottom left of the graph where the points deviate fromthe line, but as this only concerns a few points, we proceed with this model.We compared the plotted deviance residuals as a model diagnostic andfound the distributions were reasonably comparable (Figure 3.12) [R C19,The15]. We also examined the dfbeta residuals for influential points, andlllllllllllllllllllllllll−1.0−0.50.00.51.01 2densityresid.deviancelllllllllllllllllllllllllllllllllllllllllllllllllllllllll−1.0−0.50.00.51.0CA AP BL EA OT SAethnicresid.deviancelllllllllllllllllllllllll−1.0−0.50.00.51.00 1Ever_Estrogenresid.deviancelllllllllllllllllllllll−1.0−0.50.00.51.00 1Ever_Biopsyresid.deviancellllllllllllllllllllll−1.0−0.50.00.51.00 1ever_momresid.deviancellllllllllllllllllllllllllllllllllll−1.0−0.50.00.51.0No HS Some/HS Some/Coeducationresid.devianceFigure 3.12: The deviance residuals of select variables in the AFT model.The distributions are similar.383.7. Poisson regression modelfound several influential points [W+86, R C19, The15].Table 3.10 shows the coefficients. We see that the survival time for in-dividuals with architectural distortion and calcification (‘arch calc’) have adecrease in survival time by a factor of exp(−0.64), the most devastating im-pact of all the features. This is followed by the mass-architectural distortioncombination and calcifications alone.Table 3.11 shows some of the risk percentiles predicted by the model ontest data. We decided to hold several variables constant and vary the mam-mographic features to examine the time to event prediction for the variousfeatures. Holding the other variables constant at age 40, Caucasian, motherwho had breast cancer, high school education, has never used estrogen orhad a biopsy, and a low breast density, we see how different features alterthe time prediction. The median time to a breast cancer diagnosis for allfeatures is quite high, and indicates that these women will not likely bediagnosed with breast cancer in their lifetime. However, women in the 1stpercentile, or 1% of women, with architectural distortion and calcificationon their mammogram, are predicted to have a breast cancer diagnosis in 2years (95% CI: 1.3, 2.7). This agrees with the findings from the proportionalhazards model, as there is a large majority of women who are not likely tobe diagnosed with breast cancer, but a small proportion are at risk of diag-nosis within a few years. The training concordance of this model was 0.637(0.631, 0.643), which is again too low to be used clinically.3.7 Poisson regression modelWe now turn to Poisson regression. Poisson regression is used when theresponse variable is a count of a relatively rare event [MMVR12]. Sinceout of 813,280 individuals only 11,166 (1.4%) had a diagnosis, the eventcan be considered as relatively rare. The point of using this model is tosee if our risk factors influence the diagnosis count. The model is shown inEquation 3.10 where the βs are the coefficients and g(µ) is the link function.A positive coefficient indicates that the variable increases the number ofdiagnoses while a negative coefficient decreases the number of diagnoses.g(µi) = β0 + β1x1 + · · ·+ βkxk (3.10)Prior to fitting the model, a Lexis split was run on the data [PC+11,R C19]. The Lexis split allows us to follow individuals on different timescales. Several different splits and combinations were tried, including agegroups, calendar time and time since false positive. It was determined that393.7. Poisson regression modelTable 3.10: Weibull AFT modelcoef exp(coef) p-value 2.5 % 97.5 %(Intercept) 5.93 375.53 0.00 324.31 434.85age -0.02 0.98 0.00 0.97 0.98featno feat (ref) 0.00 1.00feat3 or more -0.42 0.65 0.05 0.43 1.00featarch asym -0.04 0.96 0.65 0.79 1.16featarch calc -0.64 0.53 0.00 0.36 0.78featarch only -0.04 0.96 0.57 0.85 1.10featasym calc -0.35 0.70 0.00 0.57 0.87featasym only 0.06 1.06 0.03 1.00 1.12featcalc only -0.45 0.64 0.00 0.60 0.68featmass arch -0.49 0.61 0.02 0.41 0.92featmass asym -0.18 0.84 0.13 0.67 1.05featmass calc -0.30 0.74 0.03 0.56 0.97featmass only -0.07 0.93 0.06 0.87 1.00featmissing -0.12 0.89 0.00 0.85 0.92density1 (ref) 0.00 1.00density2 -0.26 0.77 0.00 0.75 0.79ethnicCaucasian (ref) 0.00 1.00ethnicFirst Nations -0.14 0.87 0.03 0.76 0.99ethnicAfrican -0.03 0.97 0.84 0.72 1.31ethnicEast Asian 0.05 1.05 0.04 1.00 1.11ethnicOther 0.01 1.01 0.48 0.98 1.04ethnicSouth Asian 0.17 1.19 0.00 1.08 1.31Ever Estrogen0 (ref) 0.00 1.00Ever Estrogen1 -0.10 0.90 0.00 0.86 0.94Ever Biopsy0 (ref) 0.00 1.00Ever Biopsy1 -0.22 0.80 0.00 0.77 0.83ever mom0 (ref) 0.00 1.00ever mom1 -0.32 0.72 0.00 0.69 0.76ever sis0 (ref) 0.00 1.00ever sis1 -0.22 0.80 0.00 0.75 0.85educationNo HS (ref) 0.00 1.00educationSome/HS -0.13 0.88 0.00 0.83 0.94educationSome/Co -0.17 0.84 0.00 0.79 0.90403.7. Poisson regression modelTable 3.11: Time predictions for different percentilesfeature/percentile 1st 2.5th 10th 50th1 no feat 4.4 8.0 20.4 69.92 3 or more 2.5 4.6 11.8 40.33 arch asym 4.5 8.2 20.8 71.24 arch calc 2.0 3.7 9.5 32.45 arch only 4.3 7.9 19.9 68.26 asym calc 3.3 6.1 15.4 52.77 asym only 4.8 8.7 22.1 75.58 calc only 2.8 5.1 12.9 44.19 mass arch 2.3 4.3 10.8 37.010 mass asym 3.3 6.1 15.4 52.711 mass calc 3.2 5.9 15.1 51.612 mass only 4.3 7.8 19.9 68.113 missing 4.0 7.3 18.4 63.1splitting on time since false positive and 10-year age groups were most rel-evant to the problem and the model was adjusted for these variables. Wenote that architectural distortion with calcification and mass with architec-tural distortion have the largest impact on the diagnosis count (Table 3.12).The log link was used as this ensures that the predicted values will notbe negative, which is an impossible count option. An offset of log-time wasused to account for differing amounts of time contributed to each time band.The model was assessed for goodness of fit using deviance and influentialdiagnostics (Figure 3.13) [R C19].The plots in Figure 3.13 show that the Poisson model is not a good fit asthere is a pattern in the residuals [PS86, TC90]. Since the pattern involvesa lot of zero values, a transformation of the data would result in a largeamount of transformed zeros, which may not address the problem. Overand underdispersion were assessed using the dispersiontest() in R’s AERpackage and a non-significant p-value meant the null hypothesis of the truedispersion equalling 1 could not be rejected [DL89, R C19, KZ08]. However,the deviance divided by the degrees of freedom showed a value of 0.29, whichis not close to 1 and thus this model is underdispersed. Various combinationsof Lexis split were tried and the resulting models fitted, but none passed thediagnostic tests. Predictions made from this model would not be reliable.Future work could examine whether a generalized or zero-inflated Poissonmodel can accommodate for the problems in this model [CF92, Lam92].413.7. Poisson regression modelTable 3.12: Poisson modelcoef exp(coef) p-value 2.5 % 97.5 %(Intercept) -11.96 0.00 0.00 0.00 0.00Age40-50 0.00 1.0Age50-60 0.25 1.29 0.00 1.23 1.35Age60-70 0.61 1.84 0.00 1.74 1.94Age70+ 0.90 2.47 0.00 2.31 2.64featno feat (ref) 0.00 1.00feat3 or more 0.64 1.89 0.02 1.04 3.12featarch asym -0.02 0.98 0.89 0.76 1.25featarch calc 0.87 2.38 0.00 1.34 3.86featarch only 0.06 1.06 0.51 0.89 1.25featasym calc 0.51 1.66 0.00 1.25 2.16featasym only -0.17 0.84 0.00 0.78 0.91featcalc only 0.62 1.85 0.00 1.70 2.02featmass arch 0.85 2.33 0.00 1.37 3.67featmass asym 0.38 1.46 0.01 1.10 1.91featmass calc 0.31 1.36 0.12 0.90 1.95featmass only 0.06 1.06 0.25 0.96 1.17featmissing 0.24 1.28 0.00 1.21 1.35density1 (ref) 0.00 1.00density2 0.35 1.42 0.00 1.37 1.48ethnicCaucasian (ref) 0.00 1.00ethnicFirst Nations 0.20 1.22 0.02 1.02 1.44ethnicAfrican 0.01 1.01 0.95 0.65 1.49ethnicEast Asian -0.07 0.94 0.06 0.87 1.00ethnicOther -0.06 0.94 0.00 0.90 0.98ethnicSouth Asian -0.31 0.74 0.00 0.64 0.83Ever Estrogen0 (ref) 0.00 1.00Ever Estrogen1 0.18 1.20 0.00 1.14 1.26Ever Biopsy0 (ref) 0.00 1.00Ever Biopsy1 0.38 1.46 0.00 1.38 1.54ever mom0 (ref) 0.00 1.00ever mom1 0.45 1.57 0.00 1.48 1.67ever sis0 (ref) 0.00 1.00ever sis1 0.39 1.47 0.00 1.36 1.58educationNo HS (ref) 0.00 1.00educationSome/HS 0.22 1.25 0.00 1.15 1.36educationSome/Co 0.23 1.26 0.00 1.16 1.37423.7. Poisson regression model−15 −10 −5 0 5−4−2024Predicted valuesResidualsllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllll ll llll ll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllResiduals vs Fitted3701817664367000.00 0.02 0.04 0.060204060LeverageStd. Pearson resid.l lll ll lllllllllllllllllllllllllllllllll llll llllllll l lll lllllllllllll llllllllllllll llllllllll lllllllllllllllllll llll llll lllllll llllllllllllllllllllllllllllllllll lllll llllllllllll lll lll lllllllllllllllllllllll lllllllllllllllllllllllllll llll lllll llllllll ll llllllCook's distance0.51Residuals vs Leverage212362536291Figure 3.13: Diagnostics of Poisson model.43Chapter 4ApplicationIn this chapter, we refit some of the published models on our data andmake comparisons to our models of Chapter 3, focusing on the concordance,risk distributions and calibration [R C19, The15, MIG12, Har19]. To thatend, in this chapter we did not perform variable selection as the goal was tosee how the combinations of predictors from published models perform onour data, and not to find them.4.1 Concordance comparison to other models4.1.1 Cox proportional hazards model concordancecomparisonIn this section we compare the concordance from our models to thepublished models. We begin with the model developed by [TCSB+08], whichincluded only age, breast density and ethnicity and achieved a concordanceof 0.670. When fitting this model to our data, we find that the concordanceis only 0.609 (Table 4.1).Table 4.1: Training and testing concordance values for published models,published model on our data, and the same model with mammographicfeatures addedModel Published Our data Our data with featuresTraining concordance[TCSB+08] 0.660 (0.650, 0.670) 0.609 (0.603, 0.615) 0.618 (0.612, 0.623)[TCZK05] no BD 0.670 (0.650, 0.680) 0.620 (0.614, 0.626) 0.628 (0.622, 0.634)[TCZK05] with BD 0.680 (0.660, 0.700) 0.632 (0.626, 0.638) 0.638 (0.632, 0.645)Testing concordance[TCSB+08] NA 0.608 (0.599, 0.618) 0.614 (0.604, 0.624)[TCZK05] no BD NA 0.607 (0.597, 0.617) 0.614 (0.604, 0.624)[TCZK05] with BD NA 0.620 (0.610, 0.630) 0.625 (0.615, 0.635)Aside from fitting the models on different training populations, two dif-ferences may account for some of the concordance difference. First, theyhad 4-category breast density, while we have 2-category. Second, the eth-nicity categories were slightly different as they could not be recoded in the444.1. Concordance comparison to other modelssame way. They had ‘Caucasian’, ‘African American’, ‘Latina’, ‘Asian’ and‘Other’. We combined ‘East Asian’ and ‘South Asian’ into ‘Asian’ and ‘FirstNations’ with ‘Other’, but we could not create a ‘Latina’ group as there wasno ‘Latina’ category in our ethnicity variable. We fit the model using ourpreviously coded ethnicity and compared it to the results with the adjustedcoding and found that there was little difference in our model concordancewith the recoding of the ethnicity.Next, we move on to [TCZK05], which fit a Cox Gail model to their datawith and without the addition of breast density. They had a training set of81,777 women with 955 diagnoses, compared to our training set of 569,296women and 7,859 diagnoses, which indicate a cancer rate of 1.2% and 1.4%,respectively. This model uses age at menarche, number of biopsies, age, ageat first child, and number of relatives with breast cancer, with interactionsbetween family history and age at first birth and age and biopsy. In orderto make a more fair comparison, some alterations to the coding had to bemade to create similar levels and coding style as used in the paper. Ageat menarche was recoded to a categorical variable with ‘≤12’, ‘12-13’ and‘≥14’. Age at first child was converted to categorical with ‘≤20’, ‘20-24’,‘25-29’, and ‘30+’. For women with no children, we set their age at first childto ‘25-29’, as done in the original Gail model [GBB+89] and the models weare comparing to [TCZK05]. We converted family history into a categoricalvariable with three levels, ‘0’, ‘1’, ‘2+’, representing the number of firstdegree female relatives who have had a diagnosis of breast cancer. We againused our 2-category density and our coding of ethnicity.We found that the published models did not perform as well when re-fitted to our data as seen in our training concordance that is lower than theconcordance values published (Table 4.1). The published training modelshad 95% confidence intervals of (0.65, 0.67), (0.65, 0.68) and (0.66, 0.70)for the [TCSB+08] model, [TCZK05] model and [TCZK05] with mammo-graphic density, respectively, while our confidence intervals didn’t even reach0.64. Adding the mammographic features resulted in improvement in theconcordance for all models on the training set, but only by about 0.008,while adding breast density increased the concordance by about 0.01. Forcomparison, in our Cox proportional hazards model, age increases concor-dance by 0.1, family history by 0.01, breast density by 0.01, mammographicfeatures by 0.006, and the remaining predictors by less than 0.002, whenall the other predictors are in the model. When mammographic featuresare the only predictor in the Cox model, they increase concordance by 0.04,which is more than the 0.02 increase from a mother or sister with a history ofbreast cancer. We conclude that the addition of mammographic features was454.1. Concordance comparison to other modelsstatistically significant, as seen in comparing the models with and withoutfeatures using anova() in R [R C19], and shows a small increase in concor-dance. While the overall concordance is low, this increase in concordancemay be of clinical interest.Examining the testing concordance for the 6 models (3 models withmammographic features and the same 3 without mammographic features)could be more enlightening as this gives us a sense of how the model wouldpredict for future observations, since the concordance values are estimatedon data not used in the fitting of the model [R C19, The15]. In Table4.1 we see that the test concordance with the mammographic features ishigher than without the features, by about 0.006. Therefore, the additionof mammographic features does appear to improve the concordance. Wealso note that the simple model’s [TCSB+08] test concordance is in theconfidence interval of the training concordance, but the other two models’test concordance falls out of the training concordance confidence interval[TCZK05], suggesting overfitting of the latter two models.4.1.2 Cox proportional hazards model comparison ofabsolute risk predictionsWe now examine the 5-year absolute risk predictions. Predicted 5-yearabsolute risk is the probability of getting a breast cancer diagnosis within 5years. We use the model fitted on the training set to predict the absolute riskfor the test set. We compare the 5-year absolute risk predicted for the testset for these 6 models to our Cox model shown in Figure 3.8 (Figure 4.1). Inthis figure, we are plotting on the x-axis the predicted 5-year absolute riskfrom our Cox model. The 5-year predicted absolute risk from the publishedmodels is on the y-axis, where the left column y-axes in the figure containthe published models, while the right column y-axes contain the publishedmodels with the addition of mammographic features. The 5-year absoluterisk was calculated using predictSurvProb() function in the pec packagein R [R C19, MIG12]. This function calculates the survival probability atspecific time points. The resulting predictions were subtracted from 1 toget the risk, and multiplied by 100% to get the risk value as a percentage.The dotted red line represents the line of best fit, calculated using a simplelinear regression with our model risk predictions as the predictors and thepublished models risk predictions as the response. The solid blue line is the45◦ line. If the predictions from our model and a given published modelwere exactly the same, all the points in the plot would fall on this blue line.In other words, it is a visual measure of how different the predicted values464.1. Concordance comparison to other modelsFigure 4.1: (a) [TCSB+08], (b) [TCSB+08] with features, (c)[TCZK05]no BD, (d) [TCZK05] no BD, with features, (e) [TCZK05] with BD, (f)[TCZK05] with BD and features. Comparison of the absolute risk predictedby our model and the published models. The blue line is the 45◦ line. Thered dashed line is the line of best fit.474.1. Concordance comparison to other modelsare for each individual in the test set.Examining Figure 4.1 we note that the [TCZK05] model with mammo-graphic density included, graph (e), performs most similarly in terms ofpredicted absolute risk compared with the proportional hazards model ofSection 3.4, as seen by the tight grouping of predictions in (e) to the 45◦line and how closely the line of best fit matches the 45◦ line. The least sim-ilar appears to be (a), the [TCSB+08] model with only age, ethnicity anddensity. We note that this model predicts most risk to be below 5%. Whythe predicted risk only falls from 0-5% when using these predictors leadsone to question whether the risks in the 5-15% range are exaggerated or ifthese 3 variables are not enough for individual risk stratification.Upon comparing the risk distribution of these models (Table 4.2) bybreaking risks into 4 risk groups, we note that most of the individuals fall inthe general and moderate risk groups. When comparing these proportionsbetween the models with and without features, there does not seem to bemuch, if any, increase in risk stratification when features are included. De-spite using different variables, the results are similar to our model in Table3.6.Repeating the process for 2-year predicted risk shows similar behaviour,with (a) again not predicting hardly anyone to be at a high risk and model(e) predicting most similarly to our model. The risk distributions held themajority of observations in the general group, however, the low risk groupheld roughly 30% of the observations and the moderate groups held about14%. These are similar to the findings of Table 3.6.Subsetting the test data to only those with complete follow-up, i.e. thosewith a diagnosis or those with a follow-up time of 5 years, showed no dif-ferences in the absolute risk plots and risk distribution. This subset held184,007 observations with 3,307 diagnoses, compared with 243,984 observa-tions and 3,307 diagnoses in the full test set.4.1.3 Time-dependent Cox model concordance comparisonWe follow a similar process and fit these models using time varyingbreast density and features (Table 4.3). For this, we take the publishedCox models re-fitted to our data and convert it into a time varying modelfollowing the same techniques used in Section 3.5, including allowing for timevarying breast density and mammographic features. We note that there isnot much difference in concordance, as they are all in the 0.61-0.64 range.However, we do see an improvement in concordance with the addition ofmammographic features of about 0.01. As well, the simple model [TCSB+08]484.1. Concordance comparison to other modelsTable 4.2: Risk distribution for published models with and without mam-mographic features added.Risk group Risk range % at risk Mean risk StratifiedriskWithout features(a) [TCSB+08]low 0-0.75 1.0 0.6 1.0 (reference)general 0.75-1.5 51.6 1.2 1.8moderate 1.5-4 46.7 2.1 3.3high >4 0.8 4.4 6.7(c) [TCZK05]low 0-0.75 0.8 0.7 1.0 (reference)general 0.75-1.5 56.6 1.2 1.7moderate 1.5-4 41.0 2.2 3.2high >4 1.6 4.9 7.2(e) [TCZK05] with BDlow 0-0.75 4.3 0.7 1.0 (reference)general 0.75-1.5 53.4 1.1 1.6moderate 1.5-4 39.8 2.2 3.2high >4 2.6 5.3 7.8With features(b) [TCSB+08]low 0-0.75 1.4 0.7 1.0 (reference)general 0.75-1.5 53.4 1.1 1.7moderate 1.5-4 43.7 2.1 3.3high >4 1.5 5.0 7.6(d) [TCZK05]low 0-0.75 3.1 0.7 1.0 (reference)general 0.75-1.5 53.0 1.1 1.7moderate 1.5-4 41.9 2.2 3.2high >4 2.0 5.0 7.3(f) [TCZK05] with BDlow 0-0.75 1.3 0.7 1.0 (reference)general 0.75-1.5 57.0 1.1 1.7moderate 1.5-4 39.5 2.2 3.2high >4 2.2 5.2 7.7494.1. Concordance comparison to other modelsagain has a lower concordance than the other two models, but the additionof mammographic features brings the concordance into the same range asthe more complicated models, and uses only four predictors to do it.Table 4.3: AIC of our time varying Cox model and the time varying Coxmodels using the published predictors.Model AIC training c 2.5% 97.5%Our time varying Cox model 203,702 0.643 0.637 0.649(a) [TCSB+08] 204,657 0.612 0.606 0.618(b) [TCSB+08] with features 204,166 0.625 0.619 0.631(c) [TCZK05] no BD 204,474 0.620 0.614 0.626(d) [TCZK05] no BD, features 203,980 0.632 0.626 0.638(e) [TCZK05] with BD 204,152 0.634 0.628 0.640(f) [TCZK05] with BD, features 203,696 0.644 0.638 0.6504.1.4 AFT model concordance comparisonWe now take the 3 published models and use them to fit AFT modelswith and without mammographic features. For this, we are still using thepublished model set-up (predictors and interactions), but this time we areusing them in an AFT model with the Weibull baseline hazard distribu-tion, as in Section 3.6. With respect to the AIC in Table 4.4, we note thatthe models have a similar AIC, although our model is among the lowesttwo. With respect to the concordance, which was calculated using the con-cordance() function in the survival package in R [R C19, The15], we notethat the c values are again in the 0.61-0.64 range and that mammographicfeatures increase the concordance by about 0.008, on average.Table 4.4: AIC of our AFT model and the AFT models using the publishedpredictors.Model AIC training c 2.5% 97.5%Our AFT model 103,081 0.637 0.631 0.643(a) [TCSB+08] 103,800 0.609 0.603 0.615(b) [TCSB+08] with features 103,552 0.618 0.612 0.624(c) [TCZK05] no BD 103,574 0.620 0.614 0.626(d) [TCZK05] no BD, features 103,334 0.628 0.622 0.634(e) [TCZK05] with BD 103,293 0.632 0.626 0.638(f) [TCZK05] with BD, features 103,076 0.638 0.632 0.644504.2. Calibration comparison to other modelsOverall, the concordance values among the Cox proportional hazards,time varying Cox and AFT models do not go past 0.64 and are in fact quitesimilar between models. Since we have compared different models, includingdifferent predictors with some models having interactions and some without,as well as different types of models, and found similar results among all ofthem, it may not be possible to improve the concordance using these data.To be used clinically, we would prefer to have concordances above 0.7, whichwe have not seen with these data and these models.4.2 Calibration comparison to other modelsWe now examine the calibration. Calibration looks at the agreementbetween predictions and observations. In other words, if our model predictsa 20% probability of a breast cancer diagnosis for some individuals, thenwe should see 20/100 of those individuals diagnosed with breast cancer.Further, a perfectly calibrated model will have a slope of 1 and an interceptof 0 when plotting the observed outcomes and predictions [S+19].4.2.1 Cox proportional hazards model calibrationcomparisonFirst we check the calibration on the Cox model developed in Section3.4. Figure 4.2 shows the calibration on the training and test set, on theleft and right respectively for 2, 4 and 5 year survival, on the top, middleand bottom rows, respectively. Using calPlot() in R, survival probabilitieswere grouped into quantiles and the averages for the predicted and observedproportions were calculated [R C19, MIG12]. The blue line is the linearregression line of best fit. The red line is a flexible calibration line fitted tothe data. The black line is a 45◦ line which is the ideal calibration. Thecloser the other two lines are to the black line, the better the calibration[S+19].The calibration appears satisfactory for the training and test set, evenwhile a larger standard error is present for the test set. A larger standarderror on the test set is expected, as the test set observations were not usedin the fitting of the model. Calibration on the training set is not particularlyinformative even with cross-validation measures, thus we examine it only asa baseline to compare with. Since the red and blue lines overlap with theblack line, the training calibration is 1, as it should be [S+19]. The bottomleft corner of the test plots suggest that we are overestimating risk for thehigh risk individuals.514.2. Calibration comparison to other modelsllllllll ll0.99000.99250.99500.99750.992 0.994 0.996 0.998Predicted proportion (2 yr)Observed proportionllllll llll0.99250.99500.99750.992 0.994 0.996 0.998Predicted proportionObserved proportionllll llll ll0.980.990.975 0.980 0.985 0.990 0.995Predicted proportion (4 yr)Observed proportionllllll lll l0.980.990.975 0.980 0.985 0.990 0.995Predicted proportionObserved proportionllllllll ll0.970.980.990.97 0.98 0.99Predicted proportion (5 yr)Observed proportionllll ll lll l0.970.980.990.97 0.98 0.99Predicted proportionObserved proportionFigure 4.2: Calibration of our Cox model for training set (left) and testingset (right) for 2, 4 and 5 year survival. The blue line is the linear modelline. The red line is a flexible calibration line fitted to the data. The blackline is a 45◦ line, the ideal calibration.524.2. Calibration comparison to other modelsFigure 4.3 shows the calibration on the test set to the comparison mod-els for 5 year risk. We note that the calibration appears satisfactory forthe higher survival quantiles, or individuals with lower risk, but the lowersurvival quantiles, or individuals at higher risk, show that the calibration isnot ideal. We see this in graphs (c)-(f) where the black line is not insidethe shadowed area. Models (a) and (b) appear to have the best calibrationgraphically and are most similar to our training calibration, suggesting alack of overfitting.To quantify the calibration, we examine the coefficients of the blue linein Table 4.5. The ideal line would have a slope of 1.0 and an intercept of 0.We note that the models appear to be satisfactorily calibrated, though theTable 4.5: Comparisons of Cox model calibration on test setmodel coef 2.5 % 97.5 %0 ideal intercept 0.00slope 1.00(a) [TCSB+08] intercept -0.01 -0.10 0.07slope 1.02 0.93 1.10(b) [TCSB+08] with features intercept 0.04 -0.05 0.14slope 0.96 0.86 1.05(c) [TCZK05] no BD intercept 0.11 -0.03 0.24slope 0.89 0.75 1.03(d) [TCZK05] no BD, features intercept 0.11 -0.03 0.25slope 0.89 0.75 1.03(e) [TCZK05] with BD intercept 0.14 0.01 0.28slope 0.85 0.72 0.99(f) [TCZK05] with BD, features intercept 0.12 -0.01 0.25slope 0.87 0.74 1.01our model intercept 0.10 -0.04 0.24slope 0.90 0.75 1.04slopes are a bit on the lower side. A calibration coefficient much less than 1is an indicator of overfitting [RACOO17]. Except for model (a), our pointestimates are lower than 1, but the confidence intervals all contain 1, exceptfor model (e), which has an upper bound of 0.99. Thus, the calibrationmeasures are reasonable. The amazing calibration in model (a) suggests alack of overfitting.534.2. Calibration comparison to other modelslllll lllll0.9700.9750.9800.9850.9900.9950.970 0.975 0.980 0.985 0.990Predicted proportion(a)lll ll lll ll0.970.980.990.970 0.975 0.980 0.985 0.990Predicted proportion(b)ll lllll l ll0.9700.9750.9800.9850.9900.9950.970 0.975 0.980 0.985 0.990Predicted proportionObserved proportion (c)llllllll ll0.970.980.990.97 0.98 0.99Predicted proportion(d)llllllllll0.9650.9700.9750.9800.9850.9900.9950.97 0.98 0.99Predicted proportion(e)lll llll l ll0.970.980.990.97 0.98 0.99Predicted proportion(f)Figure 4.3: Calibration of the comparison Cox models. (a) [TCSB+08],(b) [TCSB+08] with features, (c)[TCZK05] no BD, (d) [TCZK05] no BD,with features, (e) [TCZK05] with BD, (f) [TCZK05] with BD and features.The blue line is a linear model line. The red line is a flexible calibration linefitted to the data. The black line is a 45◦ line, the ideal calibration.544.2. Calibration comparison to other models4.2.2 AFT model calibration comparisonNext, we examine the calibration slope and plots for the AFT modelfor 2, 4 and 5 year survival on the training and testing set (Figure 4.4)[R C19, MIG12]. We note that the calibration plots are quite similar toour Cox model. As before, there is a larger standard error on the testingcalibration than the training and the predictions are better for the lowerrisk (higher survival) quantiles. Next we examine the calibration plot forlllll lll ll0.99000.99250.99500.99750.992 0.994 0.996 0.998Predicted proportion (2 yr)Observed proportionll lll lllll0.99250.99500.99750.992 0.994 0.996 0.998Predicted proportionObserved proportionllllllll ll0.980.990.975 0.980 0.985 0.990 0.995Predicted proportion (4 yr)Observed proportionllll ll lll l0.980.990.975 0.980 0.985 0.990 0.995Predicted proportionObserved proportionllllllll ll0.970.980.990.97 0.98 0.99Predicted proportion (5 yr)Observed proportionllllll llll0.970.980.990.97 0.98 0.99Predicted proportionObserved proportionFigure 4.4: Calibration of our AFT model for training set (left) and testingset (right) for 2, 4 and 5 year survival. The blue line is the linear modelline. The red line is a flexible calibration line fitted to the data. The blackline is a 45◦ line, the ideal calibration.554.3. Birth cohort and riskthe comparison models on the test set (Figure 4.5). To quantitatively ex-amine the calibration, we see in Table 4.6 that the simple model of only age,ethnicity and breast density has an amazing calibration, with a slope of 1and intercept of 0. The other models’ calibration is satisfactory, accordingto the same reasoning as used previously.Table 4.6: Comparisons of AFT model calibration on test setmodel coef 2.5 % 97.5 %0 ideal intercept 0.00slope 1.00(a) [TCSB+08] intercept 0.00 -0.08 0.08slope 1.00 0.92 1.09(b) [TCSB+08] with features intercept 0.05 -0.04 0.14slope 0.95 0.85 1.04(c) [TCZK05] no BD intercept 0.11 -0.03 0.25slope 0.89 0.74 1.03(d) [TCZK05] no BD, features intercept 0.13 0.00 0.26slope 0.87 0.74 1.00(e) [TCZK05] with BD intercept 0.12 -0.03 0.27slope 0.88 0.73 1.03(f) [TCZK05] with BD, features intercept 0.15 0.02 0.29slope 0.84 0.70 0.98our model intercept 0.11 -0.02 0.24slope 0.89 0.76 1.024.3 Birth cohort and riskRecent work has shown that predictions of breast cancer risks may beinfluenced by calendar time [SRC+15]. Using all 813,280 participants, weadjusted for birth cohort by breaking birth year into whether they wereborn prior to 1946, between 1946 and 1965, and after 1965, correspondingto the trend seen in Figure 3.4. We also adjusted for number of births bygrouping this variable into 0 children, 1 child, and 2 or more children. Wefound that First Nations have a larger relative risk compared to Caucasiandespite adjusting for birth cohort and number of children (Table 4.7). Wealso note that for this group of women it takes two or more children to seea protective effect compared to having no children.When fitting a separate model for each birth cohort, we found that the564.3. Birth cohort and risklllll ll lll0.9700.9750.9800.9850.9900.970 0.975 0.980 0.985 0.990Predicted proportion(a)lll ll lll ll0.970.980.990.970 0.975 0.980 0.985 0.990Predicted proportion(b)ll lllll lll0.9700.9750.9800.9850.9900.9950.970 0.975 0.980 0.985 0.990Predicted proportionObserved proportion (c)lll llll l ll0.970.980.990.97 0.98 0.99Predicted proportion(d)lllll lll ll0.970.980.990.97 0.98 0.99Predicted proportion(e)llllllllll0.9650.9700.9750.9800.9850.9900.9950.97 0.98 0.99Predicted proportion(f)Figure 4.5: Calibration of the comparison AFT models. (a) [TCSB+08],(b) [TCSB+08] with features, (c)[TCZK05] no BD, (d) [TCZK05] no BD,with features, (e) [TCZK05] with BD, (f) [TCZK05] with BD and features.The blue line is a linear model line. The red line is a flexible calibration linefitted to the data. The black line is a 45◦ line, the ideal calibration.574.3. Birth cohort and riskTable 4.7: Proportional hazards model adjusted for birth cohort and numberof birthscoef exp(coef) p-value 2.5 % 97.5 %age 0.03 1.03 0.00 1.03 1.03featno feat (ref) 0.00 1.00feat3 or more 0.67 1.96 0.02 1.14 3.37featarch asym 0.06 1.06 0.64 0.83 1.36featarch calc 1.01 2.75 0.00 1.68 4.50featarch only 0.07 1.07 0.40 0.91 1.27featasym calc 0.59 1.81 0.00 1.38 2.37featasym only -0.10 0.91 0.01 0.85 0.98featcalc only 0.67 1.96 0.00 1.80 2.14featmass arch 0.92 2.51 0.00 1.56 4.04featmass asym 0.41 1.51 0.00 1.15 1.98featmass calc 0.39 1.47 0.04 1.02 2.14featmass only 0.10 1.11 0.04 1.00 1.22featmissing 0.17 1.18 0.00 1.12 1.24density1 (ref) 0.00 1.00density2 0.39 1.47 0.00 1.42 1.53ethnicCaucasian (ref) 0.00 1.00ethnicFirst Nations 0.22 1.25 0.01 1.06 1.47ethnicAfrican -0.00 1.00 0.99 0.67 1.49ethnicEast Asian -0.05 0.95 0.13 0.89 1.01ethnicOther -0.00 1.00 0.88 0.95 1.04ethnicSouth Asian -0.28 0.75 0.00 0.66 0.85Ever Estrogen0 (ref) 0.00 1.00Ever Estrogen1 0.17 1.18 0.00 1.12 1.24Ever Biopsy0 (ref) 0.00 1.00Ever Biopsy1 0.34 1.40 0.00 1.33 1.47ever mom0 (ref) 0.00 1.00ever mom1 0.48 1.61 0.00 1.52 1.71ever sis0 (ref) 0.00 1.00ever sis1 0.39 1.48 0.00 1.37 1.59educationNo HS (ref) 0.00 1.00educationSome/HS 0.24 1.27 0.00 1.17 1.38educationSome/Co 0.28 1.33 0.00 1.23 1.44num deliv0 (ref) 0.00 1.00num deliv1 -0.01 0.99 0.69 0.93 1.05num deliv2+ -0.09 0.92 0.00 0.88 0.96birth yearpre1946 (ref) 0.00 1.00birth year1946-65 -0.11 0.90 0.00 0.84 0.95birth year1965+ -0.12 0.88 0.02 0.80 0.98584.3. Birth cohort and riskrelative risk of 0.74 for East Asian compared to Caucasian found by [Hoe13]was corroborated with a relative risk of 0.73 for East Asian women in onlythe cohort born prior to 1946 (Table 4.8) [R C19, The15].Table 4.8: Cox proportional hazards model stratified by birth cohort. Onlyshowing the coefficients for ethnicity as the rest remained stable.ethnicity coef exp(coef) p-value 2.5 % 97.5 %born prior to 1946First Nations 0.13 1.14 0.36 0.86 1.51African 0.20 1.22 0.51 0.68 2.21East Asian -0.32 0.73 0.00 0.64 0.83Other -0.02 0.98 0.57 0.93 1.04South Asian -0.41 0.66 0.00 0.53 0.83born between 1946 and 1965First Nations 0.19 1.21 0.08 0.97 1.51African -0.64 0.53 0.09 0.25 1.11East Asian 0.06 1.06 0.17 0.97 1.15Other 0.01 1.01 0.85 0.94 1.08South Asian -0.20 0.82 0.01 0.70 0.96born after 1965First Nations 0.69 1.99 0.01 1.16 3.42African 1.22 3.38 0.00 1.50 7.61East Asian 0.15 1.16 0.22 0.91 1.49Other -0.02 0.98 0.78 0.83 1.15South Asian -0.55 0.58 0.02 0.36 0.94This is not surprising as [SRC+15] found that the incidence rates of EastAsian and Western women are converging. They suggest that the previouslyseen protective effect of Asian ethnicity was likely due to lifestyle. Asianwomen living in BC are subject to the same change as they adapt a moreWestern lifestyle. These changes include earlier age at menarche, later ageat first birth, low number of children, rising body mass index, and shifts indiet. Further, women born prior to 1946 had access to screening at an olderage as BC Cancer Breast Screening began operations in 1988. These womenwould not have had the advantages that come with screening for the samelength of time as women born in later years.Further, as these women born prior to 1946 have become too old toparticipate in screening mammography, the influence from this cohort onpredictions will continue to diminish. In the same way, over 80% of womenattending screening born after 1975 have at least some post-secondary edu-cation. Thus, including education with ‘no high school’ as the reference maynot be relevant in a few years. A more useful predictor would be a more594.3. Birth cohort and riskcomprehensive measure of lifestyle, that is left for future work.60Chapter 5ConclusionThis research attempted to predict risk of breast cancer for BC CancerBreast Screening participants using several different models. Compared tothe literature, the performance metrics of calibration and concordance weresimilar. We did find that the mammographic features seen on a false positiveincreased the risk of breast cancer, but since the numbers of women whohave these features was not very large and the increase in risk was notsubstantial enough, they did not make a meaningful contribution to themodel discrimination.The goal of providing a risk prediction model using the mammographicfeatures was not achieved. While the calibration was fair, the concordancewas too low to be used clinically.The strengths of the research included the large number of individualsand many predictors to examine. Limitations included how the risk factorswere collected; personal risk factors were from the participants’ memory andthe mammographic information was dependent on the radiologists’ interpre-tation.The models should not be applied in clinical settings. Since the abil-ity of the models to distinguish who will be diagnosed with breast canceris unsatisfactory, the mammographic features cannot be recommended toBC Cancer Breast Screening for use in their screening eligibility guidelines.However, as there was a small increase in the concordance with the additionof mammographic features and most of the mammographic features hadsignificant hazard ratios with increased risks, further investigation into themammographic features may be recommended instead.Radiologists only point out the features that are concerning on an abnor-mal mammogram, thus the data available in the Breast Screening registrydoes not provide a complete picture of the evolution of mammographic fea-tures. Possible future research may include machine learning techniques onserial mammographic images themselves [YLS+19], which would provide amore detailed picture of the evolution of the mammographic features.61Bibliography[AALY+18] Kawthar Al-Ajmi, Artitaya Lophatananon, Martin Yuille,William Ollier, and Kenneth R Muir. Review of non-clinical risk models to aid prevention of breast cancer.Cancer Causes & Control, 29(10):967–986, 2018. → pages4, 7[All10] Paul D Allison. Survival analysis using SAS: a practicalguide. Sas Institute, 2010. → pages 24[ATW+12] Thunyarat Anothaisintawee, Yot Teerawattananon, Chol-lathip Wiratkapun, Vijj Kasamesup, and AmmarinThakkinstian. Risk prediction models of breast cancer:a systematic review of model performances. Breast cancerresearch and treatment, 133(1):1–10, 2012. → pages 7[ATW+14] Thunyarat Anothaisintawee, Yot Teerawattananon,Cholatip Wiratkapun, Jiraporn Srinakarin, PiyanootWoodtichartpreecha, Siriporn Hirunpat, Sansanee Wong-waisayawan, Panuwat Lertsithichai, Vijj Kasamesup, andAmmarin Thakkinstian. Development and validation ofa breast cancer risk prediction model for Thai women:a cross-sectional study. Asian Pac J Cancer Prev,15(16):6811–7, 2014. → pages 8[BJS+17] Hannah R Brewer, Michael E Jones, Minouk J Schoe-maker, Alan Ashworth, and Anthony J Swerdlow. Familyhistory and risk of breast cancer: an analysis accountingfor family structure. Breast cancer research and treatment,165(1):193–200, 2017. → pages 2[BP88] William E Barlow and Ross L Prentice. Residuals for rel-ative risk regression. Biometrika, 75(1):65–74, 1988. →pages 3262Bibliography[BWBB+06] William E Barlow, Emily White, Rachel Ballard-Barbash,Pamela M Vacek, Linda Titus-Ernstoff, Patricia A Carney,Jeffrey A Tice, Diana SM Buist, Berta M Geller, RobertRosenberg, et al. Prospective breast cancer risk predictionmodel for women undergoing screening mammography.Journal of the National Cancer Institute, 98(17):1204–1214, 2006. → pages 9[BWD+20] Darren R Brenner, Hannah K Weir, Alain A Demers,Larry F Ellison, Cheryl Louzado, Amanda Shaw, DonnaTurner, Ryan R Woods, and Leah M Smith. Projected es-timates of cancer in Canada in 2020. CMAJ, 192(9):E199–E205, 2020. → pages 1[Car07] Bendix Carstensen. Age–period–cohort models for theLexis diagram. Statistics in medicine, 26(15):3018–3045,2007. → pages 7[CF92] PoC Consul and Felix Famoye. Generalized Poisson re-gression model. Communications in Statistics-Theory andMethods, 21(1):89–109, 1992. → pages 41[CG01] Mark Clemons and Paul Goss. Estrogen and the riskof breast cancer. New England Journal of Medicine,344(4):276–285, 2001. → pages 10[CGBB+17] Jessica A Cintolo-Gonzalez, Danielle Braun, Amanda LBlackford, Emanuele Mazzola, Ahmet Acar, Jennifer KPlichta, Molly Griffin, and Kevin S Hughes. Breast cancerrisk models: a comprehensive overview of existing mod-els, validation, and clinical applications. Breast cancer re-search and treatment, 164(2):263–284, 2017. → pages 4, 7,8[CGK+19] Tess V Clendenen, Wenzhen Ge, Karen L Koenig, YelenaAfanasyeva, Claudia Agnoli, Louise A Brinton, FarbodDarvishian, Joanne F Dorgan, A Heather Eliassen, Roni TFalk, et al. Breast cancer risk prediction in women aged35–50 years: impact of including sex hormone concentra-tions in the Gail model. Breast Cancer Research, 21(1):42,2019. → pages 863Bibliography[Col15] David Collett. Modelling survival data in medical research.Chapman and Hall/CRC, 2015. → pages 24, 31, 32, 36[Cox72] David R Cox. Regression models and life-tables. Journalof the Royal Statistical Society: Series B (Methodological),34(2):187–202, 1972. → pages 24[CP12] Andrew J Coldman and Norman Phillips. False-positivescreening mammograms and biopsies among women par-ticipating in a Canadian provincial breast screening pro-gram. Canadian Journal of Public Health, 103(6):e420–e424, 2012. → pages 10[CRR+13] X Castells, M Roman, A Romero, J Blanch, R Zubizarreta,N Ascunce, D Salas, A Buro´n, M Sala, Cumulative FalsePositive Risk Group, et al. Breast cancer detection riskin screening mammography after a false-positive result.Cancer epidemiology, 37(1):85–90, 2013. → pages 2[CRT93] Elizabeth B Claus, Neil Risch, and W Douglas Thompson.The calculation of breast cancer risk for women with afirst degree family history of ovarian cancer. Breast cancerresearch and treatment, 28(2):115–120, 1993. → pages 8[CTRP+16] Xavier Castells, Isabel Tora´-Rocamora, Margarita Posso,Marta Roma´n, Maria Vernet-Tomas, Ana Rodr´ıguez-Arana, Laia Domingo, Carmen Vidal, Marisa Bare´, JoanaFerrer, et al. Risk of breast cancer in women with false-positive results according to mammographic features. Ra-diology, 280(2):379–386, 2016. → pages 2, 11, 27, 31[DL89] Charmaine Dean and Jerald Franklin Lawless. Testsfor detecting overdispersion in Poisson regression mod-els. Journal of the American Statistical Association,84(406):467–472, 1989. → pages 41[DQ20] Jia-Yi Dong and Li-Qiang Qin. Education level andbreast cancer incidence: a meta-analysis of cohort stud-ies. Menopause, 27(1):113–118, 2020. → pages 10, 27[DYFC15] Sara W Dyrstad, Yan Yan, Amy M Fowler, and Graham AColditz. Breast cancer risk associated with benign breast64Bibliographydisease: systematic review and meta-analysis. Breast can-cer research and treatment, 149(3):569–575, 2015. → pages10[EBM+98] Joann G Elmore, Mary B Barton, Victoria M Moceri,Sarah Polk, Philip J Arena, and Suzanne W Fletcher.Ten-year risk of false positive screening mammograms andclinical breast examinations. New England Journal ofMedicine, 338(16):1089–1096, 1998. → pages 1, 17[ECP+17] Mikael Eriksson, Kamila Czene, Yudi Pawitan, Karin Lei-fland, Hatef Darabi, and Per Hall. A clinical model foridentifying the short-term risk of breast cancer. Breastcancer research, 19(1):29, 2017. → pages viii, 30, 31[EF15] Christoph Engel and Christine Fischer. Breast cancer risksand risk prediction models. Breast care, 10(1):7–12, 2015.→ pages 7[fCEN+] National Institute for Clinical Excellence (NICE) et al.Familial breast cancer: Classification, care and managingbreast cancer and related risks in people with a familyhistory of breast cancer (cg164). → pages 30[FL99] Lloyd D Fisher and Danyu Y Lin. Time-dependent co-variates in the Cox proportional-hazards regression model.Annual review of public health, 20(1):145–157, 1999. →pages 31, 32[GBB+89] Mitchell H Gail, Louise A Brinton, David P Byar, Don-ald K Corle, Sylvan B Green, Catherine Schairer, andJohn J Mulvihill. Projecting individualized probabilities ofdeveloping breast cancer for white females who are beingexamined annually. JNCI: Journal of the National CancerInstitute, 81(24):1879–1886, 1989. → pages 7, 45[Gor13] Leon Gordis. Epidemiology 5th Edition with Student Con-sult Online Access. Elsevier Saunders, Philadelphia, PA,2013. → pages 1[gro19] BMJ Publishing group. How to calculate risk [online]. 2019[cited December 23, 2019]. → pages 465Bibliography[GSW+17] Paula Grabler, Dominique Sighoko, Lilian Wang, KristiAllgood, and David Ansell. Recall and cancer detectionrates for screening mammography: finding the sweet spot.American Journal of Roentgenology, 208(1):208–213, 2017.→ pages 1[Har19] Frank E Harrell Jr. rms: Regression Modeling Strategies,2019. R package version 5.1-3.1. → pages 29, 44[HHS+15] Louise M Henderson, Rebecca A Hubbard, Brian LSprague, Weiwei Zhu, and Karla Kerlikowske. Increasedrisk of developing breast cancer after a false-positivescreening mammogram. Cancer Epidemiology and Pre-vention Biomarkers, 24(12):1882–1889, 2015. → pages 2[HJ15] Frank E Harrell Jr. Regression modeling strategies: withapplications to linear models, logistic and ordinal regres-sion, and survival analysis. Springer, 2015. → pages 5[HJLC+84] Frank E Harrell Jr, Kerry L Lee, Robert M Califf, David BPryor, and Robert A Rosati. Regression modelling strate-gies for improved prognostic prediction. Statistics inmedicine, 3(2):143–152, 1984. → pages 4[HJLM96] Frank E Harrell Jr, Kerry L Lee, and Daniel B Mark. Mul-tivariable prognostic models: issues in developing mod-els, evaluating assumptions and adequacy, and measuringand reducing errors. Statistics in medicine, 15(4):361–387,1996. → pages 4[Hoe13] Tanja Hoegg. Statistical modelling of breast cancer riskfor British Columbian women. Master’s thesis, Universityof British Columbia, 2013. → pages 7, 8, 15, 59[Inc15] SAS Institute Inc. SAS/STAT 14.1 User’s Guide. SASInstitute Inc., Gary, NC, USA, 2015. → pages 5, 25[Kei90] Niels Keiding. Statistical inference in the Lexis dia-gram. Philosophical Transactions of the Royal Societyof London. Series A: Physical and Engineering Sciences,332(1627):487–509, 1990. → pages 766Bibliography[KGR+95] Karla Kerlikowske, Deborah Grady, Susan M Rubin,Christian Sandrock, and Virginia L Ernster. Efficacyof screening mammography: a meta-analysis. Jama,273(2):149–154, 1995. → pages 1[KK19] Alboukadel Kassambara and Marcin Kosinski. survminer:Drawing Survival Curves using ’ggplot2’, 2019. R packageversion 0.4.4. → pages 27[KKLL18] Sohyun Kim, Yeonsook Ko, Hwa Jeong Lee, and Jung-eunLim. Menopausal hormone therapy and the risk of breastcancer by histological type and race: a meta-analysis ofrandomized controlled trials and cohort studies. Breastcancer research and treatment, 170(3):667–675, 2018. →pages 10[KM58] Edward L Kaplan and Paul Meier. Nonparametric estima-tion from incomplete observations. Journal of the Ameri-can statistical association, 53(282):457–481, 1958.→ pages38[KP11] John D Kalbfleisch and Ross L Prentice. The statisticalanalysis of failure time data, volume 360. John Wiley &Sons, 2011. → pages 36[KW52] William H Kruskal and W Allen Wallis. Use of ranks inone-criterion variance analysis. Journal of the Americanstatistical Association, 47(260):583–621, 1952. → pages 29[KZ08] Christian Kleiber and Achim Zeileis. Applied Economet-rics with R. Springer-Verlag, New York, 2008. ISBN 978-0-387-77316-2. → pages 41[Lam92] Diane Lambert. Zero-inflated Poisson regression, with anapplication to defects in manufacturing. Technometrics,34(1):1–14, 1992. → pages 41[LPB+19] Javier Louro, Margarita Posso, Michele Hilton Boon,Marta Roma´n, Laia Domingo, Xavier Castells, and Mar´ıaSala. A systematic review and quality assessment of in-dividualised breast cancer risk prediction models. Britishjournal of cancer, 2019. → pages 767Bibliography[LZW+12] Lihua Liu, Juanjuan Zhang, Anna H Wu, Malcolm CPike, and Dennis Deapen. Invasive breast cancer incidencetrends by detailed race/ethnicity and age. InternationalJournal of Cancer, 130(2):395–404, 2012. → pages 10[MCL+70] Brian MacMahon, Ph Cole, TM Lin, CR Lowe, AP Mirra,B Ravnihar, EJ Salber, VG Valaoras, and S Yuasa. Ageat first birth and breast cancer risk. Bulletin of the worldhealth organization, 43(2):209, 1970. → pages 10[MIG12] Ulla B. Mogensen, Hemant Ishwaran, and Thomas A.Gerds. Evaluating random forests for survival analysis us-ing prediction error curves. Journal of Statistical Software,50(11):1–23, 2012. → pages 25, 44, 46, 51, 55[MMVR12] Raymond H Myers, Douglas C Montgomery, G GeoffreyVining, and Timothy J Robinson. Generalized LinearModels.: with Applications in Engineering and the Sci-ences, volume 791. John Wiley & Sons, 2012. → pages39[Moo16] Dirk F Moore. Applied survival analysis using R. Springer,2016. → pages 36[MRMTB14] DS AL Mousa, EA Ryan, C Mello-Thoms, and PC Bren-nan. What effect does mammographic breast density haveon lesion detection in digital mammography? Clinical ra-diology, 69(4):333–341, 2014. → pages 19[MSG02] Jenny McCann, Diane Stockton, and Sara Godward. Im-pact of false-positive mammography on subsequent screen-ing attendance and risk of cancer. Breast Cancer Research,4(5):R11, 2002. → pages 1[oC11] Public Health Agency of Canada. Organized breast can-cer screening programs in Canada: Report on programperformance in 2005 and 2006, 2011. → pages 1[oCS12] Canadian Cancer Society’s Steering Committee on Can-cer Statistics. Canadian cancer statistics 2012, 2012. →pages 168Bibliography[oHFiBC+12] Collaborative Group on Hormonal Factors in Breast Can-cer et al. Menarche, menopause, and breast cancer risk:individual participant meta-analysis, including 118 964women with breast cancer from 117 epidemiological stud-ies. The lancet oncology, 13(11):1141–1151, 2012. → pages10[PC+11] Martyn Plummer, Bendix Carstensen, et al. Lexis: An rclass for epidemiological studies with long-term follow-up.Journal of Statistical Software, 38(5):1–12, 2011. → pages7, 39[PDD+97] Paul DP Pharoah, Nicholas E Day, Stephen Duffy, Dou-glas F Easton, and Bruce AJ Ponder. Family history andthe risk of breast cancer: a systematic review and meta-analysis. International journal of cancer, 71(5):800–809,1997. → pages 10[PS86] Donald A Pierce and Daniel W Schafer. Residuals in gen-eralized linear models. Journal of the American StatisticalAssociation, 81(396):977–986, 1986. → pages 41[R C19] R Core Team. R: A Language and Environment for Statis-tical Computing. R Foundation for Statistical Computing,Vienna, Austria, 2019. → pages 25, 27, 29, 31, 32, 36, 38,39, 41, 44, 46, 50, 51, 55, 59[RACOO17] M Shafiqur Rahman, Gareth Ambler, Babak Choodari-Oskooei, and Rumana Z Omar. Review and evaluation ofperformance measures for survival prediction models in ex-ternal validation settings. BMC medical research method-ology, 17(1):60, 2017. → pages 53[RCW94] Bemard Rosner, Graham A Colditz, and Walter C Wil-lett. Reproductive risk factors in a prospective study ofbreast cancer: the nurses’ health study. American journalof epidemiology, 139(8):819–835, 1994. → pages 10[S+19] Ewout W Steyerberg et al. Clinical prediction models.Springer, 2019. → pages 5, 51[SAS19] SAS Institute Inc. SAS University Edition. SAS InstituteInc., Cary, NC, USA, 2019. → pages 5, 2569Bibliography[SBSTL15] Sepideh Saadatmand, Reini Bretveld, Sabine Siesling, andMadeleine MA Tilanus-Linthorst. Influence of tumourstage at breast cancer detection on survival in moderntimes: population based study in 173 797 patients. Bmj,351:h4901, 2015. → pages 1[Sch80] David Schoenfeld. Chi-squared goodness-of-fit tests forthe proportional hazards regression model. Biometrika,67(1):145–153, 1980. → pages 5[Sch82] David Schoenfeld. Partial residuals for the proportionalhazards regression model. Biometrika, 69(1):239–241,1982. → pages 6, 27[Scr18] BC Cancer Breast Screening. BC Cancer Breast Screening2017 program results, 2018. → pages 1, 2[SIK86] Yosiyuki Sakamoto, Makio Ishiguro, and Genshiro Kita-gawa. Akaike information criterion statistics. Dordrecht,The Netherlands: D. Reidel, 81, 1986. → pages 7, 25, 36[SMvdW+13] Ewout W Steyerberg, Karel GM Moons, Danielle Avan der Windt, Jill A Hayden, Pablo Perel, Sara Schroter,Richard D Riley, Harry Hemingway, Douglas G Altman,PROGRESS Group, et al. Prognosis research strategy(progress) 3: prognostic model research. PLoS Med,10(2):e1001381, 2013. → pages 4[SPF+97] Catherine Schairer, Ingemar Persson, Margareta Falke-born, Tord Naessen, Rebecca Troisi, and Louise A Brinton.Breast cancer risk associated with gynecologic surgery andindications for such surgery. International journal of can-cer, 70(2):150–154, 1997. → pages 10[SRC+15] Hyuna Sung, Philip S Rosenberg, Wan-Qing Chen, MikaelHartman, Wei-yen Lim, Kee Seng Chia, Oscar Wai-Kong Mang, Chun-Ju Chiang, Daehee Kang, Roger Kai-Cheong Ngan, et al. Female breast cancer incidenceamong asian and western populations: more similar thanexpected. Journal of the National Cancer Institute,107(7):djv107, 2015. → pages 56, 5970Bibliography[SVC+10] Ewout W Steyerberg, Andrew J Vickers, Nancy RCook, Thomas Gerds, Mithat Gonen, Nancy Obuchowski,Michael J Pencina, and Michael W Kattan. Assessing theperformance of prediction models: a framework for sometraditional and novel measures. Epidemiology (Cambridge,Mass.), 21(1):128, 2010. → pages 29[TC90] William Thomas and R Dennis Cook. Assessing influenceon predictions from generalized linear models. Technomet-rics, 32(1):59–65, 1990. → pages 41[TCA17] Terry Therneau, Cindy Crowson, and Elizabeth Atkinson.Using time dependent covariates and time dependent co-efficients in the Cox model. Survival Vignettes, 2017. →pages 32[TCSB+08] Jeffrey A Tice, Steven R Cummings, Rebecca Smith-Bindman, Laura Ichikawa, William E Barlow, and KarlaKerlikowske. Using clinical factors and mammographicbreast density to estimate breast cancer risk: developmentand validation of a new predictive model. Annals of inter-nal medicine, 148(5):337, 2008. → pages xi, 44, 45, 46, 47,48, 49, 50, 53, 54, 56, 57[TCZK05] Jeffrey A Tice, Steven R Cummings, Elad Ziv, and KarlaKerlikowske. Mammographic breast density and the Gailmodel for breast cancer risk prediction in a screening pop-ulation. Breast cancer research and treatment, 94(2):115–122, 2005. → pages xi, 7, 44, 45, 46, 47, 48, 49, 50, 53, 54,56, 57[TG00] Terry M Therneau and Patricia M Grambsch. Statistics forbiology and health. In Modeling survival data: extendingthe Cox model, pages 87–152. Springer-Verlag, 2000. →pages 5, 6[TGF90] Terry M Therneau, Patricia M Grambsch, and Thomas RFleming. Martingale-based residuals for survival models.Biometrika, 77(1):147–160, 1990. → pages 6, 27, 32[The15] Terry M Therneau. A Package for Survival Analysis in S,2015. version 2.38. → pages 27, 31, 32, 36, 38, 39, 44, 46,50, 5971Bibliography[UTT+03] Kimiko Ueda, Hideaki Tsukuma, Hideo Tanaka, WakikoAjiki, and Akira Oshima. Estimation of individualizedprobabilities of developing breast cancer for Japanesewomen. Breast Cancer, 10(1):54–62, 2003. → pages 9[VdSKDDVR11] AFW Van der Steeg, CMG Keyzer-Dekker, J De Vries, andJA Roukema. Effect of abnormal screening mammogramon quality of life. British Journal of Surgery, 98(4):537–542, 2011. → pages 1[vECKV14] My von Euler-Chelpin, Megumi Kuchiki, and Ilse Vejborg.Increased risk of breast cancer in women with false-positivetest: the role of misclassification. Cancer epidemiology,38(5):619–622, 2014. → pages 2[vECRTV12] My von Euler-Chelpin, Louise Madeleine Risør,Brian Larsen Thorsted, and Ilse Vejborg. Risk ofbreast cancer after false-positive test results in screeningmammography. Journal of the National Cancer Institute,104(9):682–689, 2012. → pages 2[vH00] Hans C van Houwelingen. Validation, calibration, revisionand combination of prognostic survival models. Statisticsin medicine, 19(24):3401–3415, 2000. → pages 5[W+86] Chien-Fu Jeff Wu et al. Jackknife, bootstrap and otherresampling methods in regression analysis. the Annals ofStatistics, 14(4):1261–1295, 1986. → pages 6, 27, 39[Wei92] Lee-Jen Wei. The accelerated failure time model: a usefulalternative to the Cox regression model in survival analy-sis. Statistics in medicine, 11(14-15):1871–1879, 1992. →pages 36[WGB+14] Yuan Wang, Ying Gao, Munkhzul Battsend, Kexin Chen,Wenli Lu, and Yaogang Wang. Development of a risk as-sessment tool for projecting individualized probabilities ofdeveloping breast cancer for Chinese women. Tumor Bi-ology, 35(11):10861–10869, 2014. → pages 9[WRB+13] Christina R Weisstock, Rasika Rajapakshe, ChristabelleBitgood, Steven McAvoy, Paula B Gordon, Andrew J72BibliographyColdman, Brent A Parker, and Christine Wilson. Assess-ing the breast cancer risk distribution for women under-going screening in British Columbia. Cancer PreventionResearch, 6(10):1084–1092, 2013. → pages 2[WvECL+17] Rikke Rass Winkel, My von Euler-Chelpin, ElsebethLynge, Pengfei Diao, Martin Lillholm, Michiel Kallenberg,Julie Lyng Forman, Michael Bachmann Nielsen, Wei YaoUldall, Mads Nielsen, et al. Risk stratification of womenwith false-positive test results in mammography screeningbased on mammographic morphology and density: a casecontrol study. Cancer epidemiology, 49:53–60, 2017. →pages 2[YLS+19] Adam Yala, Constance Lehman, Tal Schuster, Tally Port-noi, and Regina Barzilay. A deep learning mammography-based model for improved breast cancer risk prediction.Radiology, 292(1):60–66, 2019. → pages 61[ZRT+18] Xuehong Zhang, Megan Rice, Shelley S Tworoger,Bernard A Rosner, A Heather Eliassen, Rulla M Tamimi,Amit D Joshi, Sara Lindstrom, Jing Qian, Graham AColditz, et al. Addition of a polygenic risk score, mam-mographic density, and endogenous hormones to exist-ing breast cancer risk prediction models: A nested case–control study. PLoS medicine, 15(9):e1002644, 2018. →pages 873
Thesis/Dissertation
eng
Computer Science
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Development of a prognostication model for breast cancer in British Columbia
Text