T H E A C C U R A C Y O F P A R A M E T E R E S T I M A T E S AND C O V E R A G E PROBABILITY O F P O P U L A T I O N V A L U E S IN REGRESSION M O D E L S U P O N D I F F E R E N T T R E A T M E N T S O F S Y S T E M A T I C A L L Y MISSING D A T A by L U C A S O N Y A N G O A. O T H U O N B. Ed. (Hons.), The University of Nairobi, 1980 M . A., The University of British Columbia, 1993 A THESIS SUBMITTED IN P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F PHILOSOPHY in T H E F A C U L T Y O F G R A D U A T E STUDIES Department of Educational Psychology & Special Education We acceDt this thesis as conforming to the required standard T H E UNIVERSITY O F BRITISH C O L U M B I A Sep 18,1998 © Lucas Onyango A. Othuon, 1998 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department o , £jL.cdZ«LJ (jLt . V SjL The University of British Columbia Vancouver, Canada Date <X% U 7 f r DE-6 (2/88) 11 ABSTRACT Several methods are available for the treatment of missing data. Most of the methods are based on the assumption that data are missing completely at random (MCAR). However, data sets that are M C A R are rare in psycho-educational research. This gives rise to the need for investigating the performance of missing data treatments (MDTs) with non-randomly or systematically missing data, an area that has not received much attention by researchers in the past. In the current simulation study, the performance of four MDTs, namely, mean substitution (MS), pairwise deletion (PW), expectation-maximization method (EM), and regression imputation (RS), was investigated in a linear multiple regression context. Four investigations were conducted involving four predictors under low and high multiple R 2 , and nine predictors under low and high multiple R 2 . In addition, each investigation was conducted under three different sample size conditions (94, 153, and 265). The design factors were missing pattern (2 levels), percent missing (3 levels) and non-normality (4 levels). This design gave rise to 72 treatment conditions. The sampling was replicated one thousand times in each condition. MDTs were evaluated based on accuracy of parameter estimates. In addition, the bias in parameter estimates, and coverage probability of regression coefficients, were computed. The effect of missing pattern, percent missing, and non-normality on absolute error for R2estimate was of practical significance. In the estimation of R 2 , E M was the most accurate under the low R 2 condition, and PW was the most accurate under the high R 2 condition. No M D T was consistently least biased under low R 2 condition. However, with nine predictors under the high R 2 condition, PW was generally the least biased, with a tendency to overestimate population R 2 . The mean absolute error (MAE) tended to increase with increasing non-normality and increasing percent missing. Also, the M A E in R2estimate tended to be smaller under monotonic pattern than under non-monotonic pattern. MDTs were most differentiated at the highest level of percent missing (20%), and under non-monotonic missing pattern. Ill In the estimation of regression coefficients, RS generally outperformed the other M D T s with respect to accuracy of regression coefficients as measured by M A E . However, E M was competitive under the four predictors, low R 2 condition. M D T s were most differentiated only in the estimation of Pi, the coefficient of the variable with no missing values. M D T s were undifferentiated in their performance in the estimation for b2,.. .,bp, p = 4 or 9, although the M A E remained fairly the same across all the regression coefficients. The M A E increased with increasing non-normality and percent missing, but decreased with increasing sample size. The M A E was generally greater under non-monotonic pattern than under monotonic pattern. Wi th four predictors, the least bias was under RS regardless of the magnitude of population R 2 . Under nine predictors, the least bias was under P W regardless of population R 2 . The results for coverage probabilities were generally similar to those under estimation of regression coefficients, with coverage probabilities closest to nominal alpha under R S . A s expected, coverage probabilities decreased with increasing non-normality for each M D T , with values being closest to nominal value for normal data. M D T s were most differentiated with respect to coverage probabilities under non-monotonic pattern than under monotonic pattern. Important implications of the results to researchers are numerous. First, the choice of M D T was found to depend on the magnitude of population R 2 , number of predictors, as well as on the parameter estimate of interest. With the estimation of R 2 as the goal of analysis, use of E M is recommended i f the anticipated R 2 is low (about .2). However, i f the anticipated R 2 is high (about .6), use of P W is recommended. Wi th the estimation of regression coefficients as the goal of analysis, the choice of M D T was found to be most crucial for the variable with no missing data. The RS method is most recommended with respect to estimation accuracy of regression coefficients, although greater bias was recorded under RS than under P W or M S when the number of predictors was large (i.e., nine predictors). Second, the choice of M D T seems to be of little concern i f the proportion of missing data is 10 percent, and also i f the missing pattern is monotonic rather than non-monotonic. Third, the proportion of missing data seems to have less iv impact on the accuracy of parameter estimates under monotonic missing pattern than under non-monotonic missing pattern. Fourth, it is recommended for researchers that in the control of Type I error rates under low R 2 condition, the E M method should be used as it produced coverage probability of regression coefficients closest to nominal value at .05 level. However, in the control of Type I error rates under high R 2 condition, the R S method is recommended. Considering that simulated data were used in the present study, it is suggested that future research should attempt to validate the findings of the present study using real field data. Also , a future investigator could modify the number of predictors as well as the confidence interval in the calculation of coverage probabilities to extend generalization of results. TABLE OF CONTENTS A B S T R A C T i i LIST OF T A B L E S viii LIST OF FIGURES x A C K N O W L E D G M E N T S xii CHAPTER 1 1 INTRODUCTION 1 Research Issues 2 Current Study 4 CHAPTER 2 8 L I T E R A T U R E REVIEW 8 Types of Missing Data 8 Unit Non-response 8 Item Non-response 9 Methods of Treating Missing Data 10 Deletion Methods 11 Listwise Deletion 11 Pairwise Deletion 11 Imputation Methods 12 Mean Substitution 12 Regression Imputation 12 Expectation-maximization (EM) Method 13 Previous Research on MDTs 18 Proportion of Missing Data 18 Violation of Normality 19 Missing Pattern 20 Studies with Randomly Missing Data 25 Studies with Non-randomly Missing Data 30 Studies in Structural Equations Modeling 34 Summary 39 CHAPTER 3 42 M E T H O D 42 Advantages of Simulation Methodology 42 Issues in Data Generation 43 Factors Influencing M D T Performance 43 Proportion of Missing Data 45 Sample Size 45 Non-normality 46 Number of Predictors 47 Size of Multiple R 2 47 Pattern of Missing Data 48 Study Design 51 Procedure 51 Simulation Program 54 VI Evaluation Criteria 58 Analysis 61 Summary 63 CHAPTER 4 65 RESULTS 65 Study 1: Four Predictors under Low R 2 Condition 65 Effects on Absolute Error for R2estimate 65 Relative Performance of MDTs in the Estimation of Population R 2 65 Effects on Absolute Error for Regression Coefficients . 72 Relative Performance of MDTs in the Estimation of Regression Coefficients....72 Coverage Probability for Pi 79 Summary 84 Study 2: Four Predictors under High R 2 Condition 86 Effects on Absolute Error for R2estimate.. 86 Relative Performance of MDTs in the Estimation of Population R 2 87 Effects on Absolute Error for Regression Coefficients 95 Relative Performance of MDTs in the Estimation of Regression Coefficients....96 Coverage Probability for Pi 101 Summary 106 Study 3: Nine Predictors under Low R 2 Condition 108 Effects on Absolute Error for R2estimate 108 Relative Performance of MDTs in the Estimation of Population R 2 108 Effects on Absolute Error for Regression Coefficients 115 Relative Performance of MDTs in the Estimation of Regression Coefficients..115 Coverage Probability for Pi 122 Summary 128 Study 4: Nine Predictors under High R 2 Condition 130 Effects on Absolute Error for R2estimaie 130 Relative Performance of MDTs in the Estimation of Population R 2 131 Effects on Absolute Error for Regression Coefficients 136 Relative Performance of MDTs in the Estimation of Regression Coefficients.. 137 Coverage Probability for Pi 143 Summary 148 CHAPTER 5 150 DISCUSSION 150 Effects on Absolute Error of Parameter Estimates 150 Relative Performance of MDTs 151 MDTs and Coverage Probabilities 155 Implications of the Study 156 Strengths and Limitations of the Present Study , 157 REFERENCES 160 APPENDIX A: BIAS IN P A R A M E T E R ESTIMATES 169 vii A P P E N D I X B : G R A P H S F O R B I A S I N P A R A M E T E R E S T I M A T E S 178 A P P E N D I X C : A R T I C L E S U S E D I N T H E R E V I E W 187 LIST OF TABLES T A B L E 1: Number of studies using multiple regression in four journals 44 2: Number of data points deleted to create missing patterns in the 4-predictor model..50 3: Number of data points deleted to create missing patterns in the 9-predictor model..50 4: Design factors and levels for the study 51 STUDY 1 5: Effects on absolute error for R2estimate fulfilling the r) 2 > .03 criterion 66 6: Means and standard deviations of absolute error for R estimate 68 7: Effects on absolute error for regression coefficients fulfilling the r\2 > .03 criterion.72 8: Means and standard deviations of absolute error for bi 75 9: Mean coverage probability (%) for p i 80 STUDY 2 10: Effects on absolute error for R2estimate fulfilling the r\2 > .03 criterion 88 11: Means and standard deviations of absolute error for R2estimate 91 12: Effects on absolute error for regression coefficients fulfilling ther| 2> .03 criterion 95 13: Means and standard deviations of absolute error for bi 97 14: Mean coverage probability (%) for Bi 102 STUDY 3 15: Effects on absolute error for R2estimate fulfilling the T) 2 > .03 criterion 108 16: Means and standard deviations of absolute error for R2estimate H I 17: Effects on absolute error for regression coefficients fulfilling the T] 2 > .03 criterion 115 18: Means and standard deviations of absolute error for bj 118 19: Mean coverage probability (%)for P i 124 STUDY 4 20: Effects on absolute error for R2estimate fulfilling the rj 2 > .03 criterion 130 21: Means and standard deviations of absolute error for R2estimate 132 22: Effects on absolute error for regression coefficients fulfilling the TV2 > .03 criterion 136 23: Means and standard deviations of absolute error for bi 139 24: Mean coverage probability (%) for P i 144 DISCUSSION 25: Best MDTs in the estimation of parameters 152 26: Least biased MDTs in the estimation of parameters 152 APPENDIX A: Bias in parameter estimates 169 Table A . l : Bias in R2estimate: 4 predictors and low R 2 condition 170 Table A.2: Bias in b i : 4 predictors and low R 2 condition 171 Table A.3: Bias in R2estimate: 4 predictors and high R 2 condition 172 Table A.4: Bias in bi: 4 predictors and high R 2 condition 173 Table A.5: Bias in R2estimate: 9 predictors and low R 2 condition 174 Table A.6: Bias in bj: 9 predictors and low R 2 condition 175 Table A.7: Bias in R2 e sumate: 9 predictors and high R 2 condition 176 Table A.8: Bias in bi: 9 predictors and high R 2 condition LIST OF FIGURES F I G U R E 1: Monotonic pattern of missing data 49 2: Non-monotonic pattern of missing data 49 3: Flowchart for E M method used in the study 57 S T U D Y 1 4: Mean absolute error for R2estimate across levels of non-normality ( N = 94) 69 5: Mean absolute error for R 2 e sumate across levels of non-normality ( N = 153) 70 6: Mean absolute error for R2estimate across levels of non-normality ( N = 265) 71 7: Mean absolute error for bi across levels of non-normality ( N = 94) 76 8: Mean absolute error for bi across levels of non-normality ( N = 153) 77 9: Mean absolute error for bi across levels of non-normality ( N = 265) 78 10: Mean coverage probability for Pi (N = 94) 81 11: Mean coverage probability for P] ( N = 153) 82 12: Mean coverage probability for Pi ( N = 265) 83 S T U D Y 2 13: Interaction effect of pattern and non-normality on absolute error for R2esumate under E M ( N = 153) 89 14: Interaction effect of pattern and non-normality on absolute error for R2esumate under R S ( N = 153) 89 15: Interaction effect of pattern and non-normality on absolute error for R2estimate under E M ( N = 265) 90 16: Interaction effect of pattern and non-normality on absolute error for R2estimate under RS ( N = 265) 90 17: Mean absolute error for R2estimate across levels of non-normality ( N = 94) 92 18 Mean absolute error for R2esumate across levels of non-normality ( N = 153) 93 19: Mean absolute error for R2esumate across levels of non-normality ( N = 265) 94 20: Mean absolute error for bi across levels of non-normality ( N = 94) 98 21: Mean absolute error for bi across levels of non-normality ( N = 153) 99 22: Mean absolute error for bi across levels of non-normality ( N = 265) 100 23: Mean coverage probability for pi (N = 94) 103 24: Mean coverage probability for Pi ( N = 153) 104 25: Mean coverage probability for Pi ( N = 265) 105 S T U D Y 3 26: Mean absolute error for R2estimate across levels of non-normality ( N = 94) 112 27 Mean absolute error for R2estimate across levels of non-normality ( N = 153) 113 28: Mean absolute error for R2estimate across levels of non-normality ( N = 265) 114 29: Mean absolute error for bi across levels of non-normality ( N = 94) 119 30: Mean absolute error for bi across levels of non-normality ( N = 153) 120 31: Mean absolute error for bi across levels of non-normality ( N = 265) 121 32: Mean coverage probability for Pi ( N = 94) 125 33: Mean coverage probability for p i (N = 153) 126 34: Mean coverage probability for pi ( N = 265) 127 XI S T U D Y 4 35: Mean absolute error for R2estimate across levels of non-normality ( N = 94) 133 36: Mean absolute error for R2estimaie across levels of non-normality ( N = 53) 134 37: Mean absolute error for R2estimate across levels of non-normality ( N = 265) 135 38: Mean absolute error for bi across levels of non-normality ( N = 94) 140 39: Mean absolute error for bi across levels of non-normality ( N = 153) 141 40: Mean absolute error for bi across levels of non-normality ( N = 265) 142 41: Mean coverage probability for (3i (N = 94) 145 42: Mean coverage probability for Pi (N = 153) 146 43: Mean coverage probability for Pi (N = 265) 147 A P P E N D I X B : G R A P H S F O R B I A S I N P A R A M E T E R E S T I M A T E 178 Figure B . l : Bias in R2estimate under four predictors and low R 2 condition 179 Figure B.2 : Bias in bi under four predictors and low R 2 condition 180 Figure B .3 : Bias in R2esumate under four predictors and high R 2 condition 181 Figure B.4: Bias in bi under four predictors and high R 2 condition 182 Figure B .5 : Bias in R2estimate under nine predictors and low R 2 condition 183 Figure B.6: Bias in bi under nine predictors and low R 2 condition 184 Figure B .7 : Bias in R2esumate under nine predictors and high R 2 condition 185 Figure B.8: Bias in bi under nine and high R 2 condition 186 Xll ACKNOWLEDGEMENTS I would like to extend my sincere gratitude to members of my research committee, Dr. Nand Kishor (Research Supervisor), Dr. James Steiger, Dr. Don Al l i son , and Dr. Bob Conry, for their support in the development of this thesis. Without their commitment, nothing much could have been achieved. M y acknowledgements go to Maseno University College for according me study leave to pursue the program that has culminated in the writing of this dissertation, and to the Department of Educational Psychology and Special Education, University of British Columbia, for the academic and financial support during my stay in Canada. I must thank all my friends in the Department of Educational Psychology and Special Education, University of Brit ish Columbia, for their encouragement whenever the going was tough. I would like to sincerely thank my wife, Akeyo, for taking good care of our lovely children, A k i n y i , Ochieng' , and Omondi, during the four years I was away. Nobody else could have done it better! A n d lastly, I dedicate this dissertation to my parents, Ayado and Wamwayi ; although not formally schooled by default, they had a vision beyond measure. 1 CHAPTER 1 INTRODUCTION Researchers employing multivariate data analysis techniques (e.g., multiple regression, discriminant analysis, and canonical correlation) commonly encounter the problem of dealing with missing data on one or more variables (Gleason & Staelin, 1975). In particular, the problem of missing data is most acute in field experiments (Lepkowski, Landis, & Stehouer, 1987) and surveys (Kim & Curry, 1977). With missing data, it may be necessary to apply a suitable treatment during the statistical analysis. The basic aim of treating missing data is to improve the accuracy of the parameter estimates of interest. This may reduce the likelihood of making a wrong statistical inference. Missing data may be planned by a researcher or unplanned. The present study is primarily concerned with unplanned missing data. Common causes of unplanned missing data include omission and attrition. Omissions occur when some respondents fail to answer some items on a questionnaire. Such may occur in the middle of a survey or at the end. Omissions that occur in the middle of a survey instrument may be due to a respondent simply not seeing a question, or forgetting to go back to skipped questions. Attrition occurs when subjects drop out prematurely. Examples of studies where attrition is common include panel surveys or cohort studies with designs that involve repeated measurements. The resulting data are unbalanced, sometimes with unequal numbers of measures for each respondent. The easiest method for handling missing data is to discard all cases with missing values. However, Raymond and Roberts (1987) noted a serious limitation of this approach is that it encourages researchers to overlook available data. Further, as the number of variables increases, increasing amounts of data are ignored, even if the total number of missing values remains constant. Several techniques, referred to as missing data treatments (MDTs), have been proposed for handling missing data. One popular method is the pairwise deletion or piecemeal method. With this method, all available data are used to compute means and variances, while all available 2 pairs of values are employed in the computation of covariances. Another common procedure is mean substitution in which missing values are replaced by the mean of the variable. In addition, numerous regression imputation procedures have been proposed. These methods make use of information present in the covariates to estimate the missing values for a variable of interest. The more complex procedures include the iterated linear regression methods, one form of which is known as expectation-maximization method (EM). Listwise deletion, pairwise deletion, and mean substitution are already included in various statistical computer softwares. In the present study, I investigated the performance of commonly used MDTs that are popular in computer software, and also E M which is currently much talked about in electronic forums, SEMNET and AERA-D*. Important research issues discussed below guided the investigation. Research Issues There are a number of issues pertaining to current knowledge about MDTs that require attention. Firstly, whereas MDTs have been around for over six decades, the number of studies on some MDTs has remained inadequate. In particular, Roth (1994) noted that although the number of studies that have assessed the bias of simple MDTs (e.g., mean substitution and pairwise deletion) is substantial, research on more complex MDTs like E M is limited. The small number of studies on MDTs did confine comparisons of the E M method to listwise and pairwise deletion methods, and paid little or no attention to other treatments. Few studies have systematically incorporated the various patterns of missing data, and most researchers have made use of randomly deleted data. Although most MDTs were developed for use with randomly missing data, such data sets are rare in psycho-educational research where missingness in one variable is likely to be related to measurements in another variable within the same data set. Only two studies that made use of non-randomly missing data were found in the * SEMNET is an electronic forum for academic exchanges on topics related to structural equations modeling. AERA-D is an electronic forum for exchanges on methodological issues among members of the American Educational Research Association. 3 literature. Azen, Van Guilder, and Hil l (1989) compared the performance of three approaches (complete cases, pairwise deletion, and E M algorithm) in estimating regression parameters and missing values under varying design conditions that included pattern of missing data. However, they did not investigate how the MDTs might affect the test of hypotheses about the regression coefficients. In addition, their simulation consisted of only 50 replications under design conditions whose selection criteria were arbitrary. Kromrey and Hines (1994) compared the performance of MDTs by investigating the effects of systematically missing data on the parameter estimates of a multiple linear regression model. As in Azen et al., they did not investigate the relative performance of MDTs in hypothesis testing. Also, they used only a two-predictor regression model. Two predictor linear regression models are rarely found in applied research. There is need to compare different MDTs within or across data sets with different non-random patterns of missing data and under simulation conditions that are more representative of real data. The common procedure in most M D T studies is that samples with known statistical properties are randomly selected from a population, and portions of the samples are then randomly deleted and treated with various MDTs. For each MDT, the bias or accuracy of parameter estimates is assessed with respect to the population parameters. Many of the probability statements yielded by methods of statistical inference are based on the assumption of normality. In particular, the OLS estimation of regression parameters is one of the statistical procedures most commonly understood to be affected by distribution assumptions (Mooney & Duval, 1993). To make inferential statements using OLS, we need to assume that the random residual in the model is normally distributed (Draper & Smith, 1981, p.23). This assumption is required because the sampling distribution of the OLS estimator is based on the random residual of the model. If we assume normality when non-normality actually holds, our hypothesis tests could lead to false rejection of the null hypothesis, that is, our confidence intervals and hypothesis tests could have a greater than nominal probability of Type I error. Alternatively, such hypothesis tests could lead to Type I error rates that are smaller than 4 the nominal value, and this can be problematic i f they come with a corresponding reduction in statistical power. Whereas researchers have investigated the effectiveness of various M D T s under different sample sizes, proportion of missing data, size of correlation, and number of variables, very few studies have explicitly investigated the relative performance of M D T s under violation of multivariate normality. A s observed by Fleishman (1978), many of the psychological variables found in practice are skewed and/or kurtotic to various degrees. In a study based on over 500 score distributions which were reasonably representative samples of psychometric, achievement, and ability measures, Miccer i (1989) found that only 6.8% of the distributions exhibited both tail weight and symmetry approximating that expected of normal distributions. In addition, 100% of the distributions were significantly non-normal at an alpha level of .01. This shows how prevalent non-normal distributions are in psycho-educational research and the importance of investigating how M D T s might function under non-normality of distributions. Current Study Previous researchers have investigated the accuracy of parameter estimates after different treatments of missing data. However, most of these studies have tended to focus more on normally distributed data sets in which data were missing completely at random ( M C A R ) . Little is known about the bias and accuracy of parameter estimates when different M D T s are used to treat systematically missing data. Also , little is known about the performance of M D T s under violation of normality. In addition, researchers have tended to ignore investigating how hypothesis testing might be affected under various design conditions. In the present study, systematically missing data were used, and non-normality was incorporated as a factor in the design. The present study was based on multiple linear regression models in which data were missing only in the predictors (X's). The related problem of missing values in the outcome variable (Y) is less interesting in the sense that i f the predictors are complete, then the missing 5 cases contribute no information to the regression equation of Y on the set of predictors under conditions of M C A R (cf. Little, 1992). Multiple regression models have been used in a majority of previous studies. Raymond and Roberts (1987), Beale and Little (1975), and Little (1992) compared MDTs in a linear regression context. With respect to non-randomly missing data, Azen, Guilder, and Hil l (1989) and Kromrey and Hines (1994) also investigated the effectiveness of MDTs in a multiple linear regression context. Further, as noted by Little (1992), the focus on regression models is justified by the fact that many studies in social and behavioral sciences often make use of these models. Considering that regression imputation and the E M method have not received much attention, I investigated the accuracy and bias of parameter estimates, as well as the coverage probability for population regression coefficients under expectation-maximization method (EM), mean substitution (MS) method, pairwise deletion (PW) method, and simple regression imputation (RS) method. Shchigolev (1960/1965) noted that the two most useful measures in the evaluation of errors are the mean absolute error of estimation (MAE) and the mean square error of estimation (MSE). Note that the M A E is not the same as the mean absolute deviation of the parameter estimate, and the M S E is not the same as the standard deviation of the parameter estimate. The difference arises from the fact that for mean absolute deviation and standard deviation, the deviations are from the mean. However, for M A E and MSE, the deviations are from the true population value. Both the M A E and the M S E provide information on the spread (closeness) of the parameter estimates around the true population value, the only advantage of M S E over M A E being that M S E is more mathematically tractable, an advantage that was not of necessity in the present study. Therefore, the M A E was used to evaluate the accuracy of parameter estimates in the present study. The M A E was obtained by first computing the absolute error of estimation, defined as the absolute deviation of each parameter estimate from the model value, then calculating the mean of the absolute values. In addition, as suggested by Judge, Hi l l , Griffiths, Lutkepohl, and Lee (1988), an optimal performance was obtained when both M A E and bias of parameter estimates were smallest. Therefore, as in Anderson, Stone-Rumero, and Tisak (1996), 6 the bias of parameter estimates, defined as the difference between the true parameter (that derived from the population model) and the mean of the estimators was also computed. The following research questions guided the conduct of the present study: 1. What are the effects of missing pattern, percent missing, and non-normality on absolute error of parameter estimates of a multiple regression model (MRM) when missing data are treated with selected MDTs? 2. What is the relative performance of MDTs with respect to accuracy and bias of parameter estimates under various conditions? 3. What is the relative performance of MDTs with respect to coverage probabilities of regression coefficients at the conventional alpha level of .05? In real world settings, missing data are the rule rather than the exception (Kolb & Dayton, 1995). The common practice of deleting cases with missing observations is of questionable validity since, typically, there is little evidence supporting that the data are missing randomly. Unfortunately, researchers have continued to give greater attention to evaluating MDTs under M C A R condition, simply because commonly used MDTs are based on the M C A R assumption. Whether data sets violate the M C A R assumption or not, researchers will continue using MDTs. It is therefore important to evaluate MDTs using non-MCAR data, as this is what researchers commonly encounter in the field. Consequently, the researchers would be able to make informed judgment concerning which M D T to apply for various types of data. This may improve the accuracy of decision-making in research, with the possibility of influencing practice in psycho-educational settings. The E M method of treating missing data is based on normal theory. However, most field data are non-normal (Micceri, 1989). By employing simulation methodology, it would be possible to provide evidence of conditions under which normal theory based E M method breaks 7 down. Such findings would be of importance to practitioners in understanding when not to apply the E M method. 8 CHAPTER 2 LITERATURE REVIEW This chapter begins with a review of types of missing data. This is followed by a description of methods of treating missing data, and an analysis of previous studies involving MDTs that were identified through manual and electronic searches. The databases searched included ERIC: Educational Resources, Education Index, and Psychlnfo. The search began by using the title words "missing data" and "incomplete data." This led to a number of books and articles with the following descriptors: statistical data, statistical regression, multiple regression, models, maximum likelihood, mathematical modeling, multivariate analysis, and algorithms. These descriptors were used for a more in-depth search. Types of Missing Data According to Cochran (1983), missing data may be of two types, namely, unit non-response or item non-response. This section describes these two types of missing data. Unit Non-response Sometimes, none of the variables of interest is measured for a unit or subunit in a survey. For example, the mailed questionnaire may not be returned, or the interviewer may not find anyone at home. Cochran (1983) called this type of non-response "unit non-response." Unit non-response occurs if a unit is selected for the sample and is eligible for the survey, but no response is obtained for the unit or the obtained response is unusable. Causes of unit non-response include language problems, total refusal to respond, or the responses given are classified as unusable. The problem of how to handle unit non-response goes back to the 1930s, the same period when many of the standard techniques and results in probability sampling were developed (Cochran, 1983). Quota sampling is an early example of a group of methods whose aim was to cut down or eliminate non-response when data are being collected in the field (King, 1983). This 9 method is based on full probability sampling of a target population. The method was developed to cut down on field costs by taking stratified samples for public opinion surveys. It enables one to obtain a desired number of completed cases in a relatively short period of field time. Sudman (1967) noted that the method involves drawing sampling locations down to the block level with exactly the same technique of multistage selection with probabilities proportional to size as in full probability sampling. Miss ing data are encountered in randomized experiments as well , and the development of methods for making estimates of the effects of the treatment from experiments with missing data also began in the 1930s. According to Cochran (1983), the technique most widely used in field experiments applies the standard least squares model for the analysis of the complete results of field experiments to the missing data that are present. This is an approach that makes use of "fil l ing in" or "imputing" values for all the missing observations by formulae so that the standard analysis of the completed data gives the same estimates as the least squares analysis of the missing data. This approach does not require that the units (e.g., respondents) for whom the data are missing be a random subsample of the units in the experiment (Cochran, 1983). Item Non-response In a survey, it sometimes occurs that for certain questions, either no answer is given or the answer is judged to contain a gross error and is deleted during editing. Usually, such questions are sensitive. Examples of such questions may concern income, or when the respondent does not have the information. This type of missing data is called an item non-response (Cochran, 1983). Cochran (1983) provided a more detailed explanation of circumstances that may lead to item non-response. He noted that item non-response occurs i f questions that should be answered are not answered, or i f the answers are classified as unusable. Sometimes, item non-response occurs because response to an instrument or question is broken off after being partly completed, 10 but the partial response is not classified as a unit non-response. Item non-response may occur because the respondent does not have the information needed for one or more questions, because the respondent refuses to answer specific questions, or because the interviewer or respondent skips the question. Item non-response may occur for blocks of questions: an interviewer may miss a branch point in an interview, or a respondent may refuse to answer all questions on a specific subject, say, income or the final questions in an interview. Such data sets tend to display non-random patterns because the cause of missingness is related to the measurement process and instrument design. According to Cochran (1983), the earliest paper on item non-response was that of Wi lks in 1932. In that paper, a random sample from a bivariate normal distribution was considered. Wi lks (1932) considered the following situation under M C A R condition. Suppose in n ) 2 observations, Xx and X2 were both measured. Also , suppose that in nio observations, Xx is measured but X2 is missing. Further, suppose that in n02 observations, X2 is measured but Xx is missing. Wi lks found maximum likelihood estimates of the parameters px, j l 2 , O",, r J 2 , and A A p , and gave large sample formulas for the variances and covariance of px and p2 . However, he noted that his estimates were not particularly simple to compute. Therefore, he proposed mean substitution as a simpler method for treating missing data. In this case, i f the no2 units in which Xx was missing were a random sample, the mean of all the n 1 2 + n i 0 measured values of Xx is used as an estimate of px . Methods of Treating Missing Data Existing techniques of handling unplanned missing data may be classified into two broad categories: deletion methods and imputation (filling in) methods. 11 Deletion Methods Listwise Deletion. This method, also referred to as complete case analysis, is the most obvious method of handling missing data. The method involves discarding cases with any missing values. In this approach, entire data records with missing data are deleted. Advantages of listwise deletion are ease of implementation and the fact that valid inference is obtained when missingness depends on the regressors (Little, 1992). Other advantages are that it always generates consistent covariance and correlation matrices, and test statistics used with complete data can be used without modification (Kim & Curry, 1977). Although listwise deletion is commonly used (Gilley & Leone, 1991), a serious limitation of the approach is that it encourages researchers to overlook available data. Also, it sacrifices huge amounts of data (Reinecke & Schimdt, 1996; Malhotra, 1987; Stumpf, 1978; Little, 1992), with the possible result of a decrease in power (Gilley & Leone, 1991) and an introduction of bias in parameter estimation (Donner, 1982; Little & Rubin, 1987). If the number of predictors is large, then even a sparse pattern of missing data in the predictors can result in a substantial number of missing cases (Little, 1992). Researchers should therefore avoid the use of listwise deletion as a treatment method for missing data as it involves the complete loss of individual cases. If the M C A R (missing completely at random) assumption is not fulfilled, the values of the covariance matrix based on listwise deletion procedure are underestimated (Reinecke & Schimdt, 1996). Pairwise Deletion. In this method, all possible cases with valid data are used to calculate means and variances, while all available pairs of values are used in the computation of covariances (Little & Rubin, 1987). As noted by Reinecke and Schimdt (1996), this method provides more benefit than the listwise deletion because it uses all available information. Like in listwise deletion, pairwise deletion assumes that missing data are M C A R . When the sample size is small and the number of cases with missing information is high, or when the predictor variables 12 are highly correlated, the resulting covariance matrix of the predictors generated by pairwise deletion of missing data may not be positive-definite. A symmetric matrix A and its associated quadratic form are positive definite if x'Ax > 0 for all nonnull x. If a covariance matrix is not positive-definite, regression coefficients based on it are indeterminate (Kim & Curry, 1977; Little, 1992), because all required computations are not possible. In contrast to listwise deletion, the values of the covariance matrix based on pairwise deletion method are overestimated. Comparisons as to which of the matrices give more consistent and efficient results are lacking in the literature (Allison, 1987; Brown, 1994). Imputation Methods Mean Substitution. One of the simplest techniques used in the imputation of missing data is the mean substitution method, also known as unconditional mean substitution. This method, proposed by Wilks (1932), simply substitutes missing values by the mean of the non-missing values for that variable. Although mean substitution preserves data and is easy to use, it also tends to attenuate variance estimates (Roth, 1994). Computed variance estimates decrease as more means are added to calculations. For example, a researcher may have 40 subjects, but 5 have missing data. Mean substitution might suggest that we add 5 means to the 35 scores. This would increase the N in the calculation of variance, but would not increase the deviations around the mean added by the 5 additional cases. Subsequently, covariance estimates are also attenuated (Little, 1988; Malhotra, 1987). Researchers may also believe they have more degrees of freedom than is warranted, because substituted means are not independent from other observations in the data. Regression Imputation. Regression imputation is yet another method for treating missing data. This method, proposed for the first time by Buck (1960), begins with the estimation of the mean and covariance matrix from the sample mean and covariance matrix based on complete 13 cases. These estimates are then used to regress the variable with missing data on a variable with complete data. Using the resulting linear regression equation, missing data are imputed case by case. Put differently, this method computes predicted estimates for any missing entry by using a linear regression equation. Expectation-Maximization (EMI Method. The E M method for estimating missing data is one of the various maximum likelihood (ML) imputation approaches. M L methods for imputing missing data generally assume that the observed data are a sample drawn from a multivariate normal distribution and M C A R (DeSarbo et al., 1986). Little and Rubin (1987) described one such method for general patterns of missing data with ignorable non-response. In this method, it is assumed that the data are M A R and the objective is to maximize the likelihood function. If the likelihood is differentiable and unimodal, maximum likelihood estimates can be found. However, if this is not possible, then iterative methods like the Newton-Raphson algorithm or the method of scoring can be applied. Both these methods involve calculating the matrix of second derivatives of the loglikelihood. For complex patterns of incomplete data, the entries in this matrix tend to be complicated functions. As a result, to be practicable the methods can require careful algebraic manipulations and efficient programming. Little and Rubin (1987) noted that an alternative computing strategy for incomplete-data problems, which does not require second derivatives to be calculated or approximated, is the Expectation-Maximization (EM) method. This method is remarkably simple, both conceptually and computationally. Graham, Hofer, and Piccinin (1994) stressed the importance of the fact that different E M methods are required for different kinds of analysis. However, they noted that in many analyses involving the general linear model, the iterative E M method could be extensively useful because the covariance matrix can be used as input. The iterative E M method replaces missing values by estimated values, estimates parameters, reestimates the missing values assuming the new parameter estimates are correct, reestimates parameters, and so forth, iterating 14 until convergence (i.e., until there is no improvement in the estimated parameters). This procedure is available in B M D P A M (Dixon, 1988). A n important difference between the scoring algorithm and the E M method is that the former requires inversion of the information matrices of p. and Z at each iteration. The E M method consists of two steps: an E step (estimation step) and an M step (maximization step). According to Little and Rubin (1987), the E step is used to calculate the conditional expectations of the "missing data" given the observed data and current estimated parameters. The calculated conditional expectations are then substituted for the "missing data." The quotations around "missing data" imply that E M does not necessarily substitute the missing values. The M step of the E M algorithm performs maximum likelihood estimation of 9 just as i f there were no missing data, that is, as i f they had been substituted with estimates at the E step. The mathematical treatment of iterative E M algorithm as used in the present study, and discussed in greater detail in Little (1983) and Little and Rubin (1987), is as follows: Suppose that the hypothetical complete data y = ( y y ) is a random sample from the K-variate normal distribution, with mean fl and covariance matrix £ . This belongs to the regular exponential family where A and B are scalar functions, 9 is an r by 1 vector of parameters, and t(ys) is an r by 1 vector of complete data sufficient statistics. In this case, the sufficient statistics t(ys) is given by the sample means f(y,.0) = B(ys)expfyTt(ys)} A(9) 15 and the uncorrected sample sum of squares and cross-products matrix d , with (j,k) th element rfj*=Xw*« j,k = \,...,K. 1=1 The complete data maximum likelihood estimates are given by A A djk _ _ c*=-fi—yjVk' j,k = l,...,K. Now suppose we have incomplete data p . The E step of the E M algorithm consists in finding expected values of the sufficient statistics: yja = E(y j\p;na,I.a) djka=E(dJk\p;pa,I,a) where p,a and Z a are current estimates of p, and Z . The M step consists in forming new estimates \lh, Zh by substituting y . and djka for y .and djk, respectively. It remains to describe the calculation of y j a and dJka. This clearly depends on the pattern of missing data. Note that 1 N, .? 1=1 and djka=^E(yijyik\P^a^a)-16 To calculate the terms in these sums, consider unit i with present items /?, and missing items mi. With independent observations, given (pa , X a ) , A E(yi\p\lia^a) = E(yij\pi,pa,Zu) = yiju and E(yuync\p>l1a>'Za) = E(yijyik\pi;pa,Tla) which is equal to A A y aa y ika+Cov( yu • ^ I P.- ; ^ . za). If y i ; / is present, that is, belongs to pi, then yija is simply the observed value y-, and C"ov(y,j:.yik)\p, \H-a , £ a ) is zero. If y}j is missing, then yija is the predicted value of y- from a linear regression of y . on the variables present in unit i, with intercept and regression coefficients calculated from pa and Z a . If ytj and yik are both missing, Coviy^,yik\pt; / i a , £ a ) is the residual covariance (or if j = k , variance) from the regression of y . and yk on the variables present in unit i, calculated from E a . Summing over units, _ l Ns A 1 V s i=l and 1 r A A ^ = —{ yija yiM+cov(yij,yik\Pi;pa,i:a)}. Hence, the complete E M algorithm is as follows: 17 (i) Calculate initial estimates fla and Ha. (ii) Complete the data matrix by calculating A f y.. if yu is present y i j a \E(yu\p,;fia,I,a) if ytj is not present where E(yij\pi\pa,'La) is obtained by the linear regression of yy- on variables present in unit i, with coefficients calculated from p,a and Z a . (iii) Form means and sums of squares cross product matrix of the completed data matrix; whenever y.. and yik are missing add to the (j, k)th element of the sums of squares cross product matrix the residual covariance (variance if j = k) of y, and yk given the variables present in unit i, calculated from E a . (iv) Calculate new estimates pb and S 6 from the means and adjusted cross-products matrix, and iterate until convergence. Little and Rubin (1987) suggested four options of obtaining initial values of the parameters. The options are: (1) to use listwise deletion; (2) to use pairwise deletion (all-value); (3) to form the sample mean and covariance matrix of the data filled in by one of the imputation methods (e.g., linear regression); and (4) to form means and variances from observed values of each variable and set all correlations equal to zero. Option 1 provides consistent estimates of the parameters if the data are M C A R and there are at least K + 1 complete observations. Option 2 makes use of all the available data but can yield an estimated covariance matrix that is not positive definite, leading to problems in the first iteration. Options 3 and 4 generally yield inconsistent estimates that are positive semidefinite and hence 18 usually workable as initial estimates. Based on the characteristics of the four options, option 3 (linear regression) was chosen for computing the initial value for use in iterative E M method. Previous Research on MDTs A number of factors were identified in the literature that may influence the performance of MDTs. These factors include pattern of missing data, sample size, proportion of missing data, magnitude of correlations among dependent and independent variables, number of predictors, and non-normality. However, the effect of these factors on absolute error in parameter estimates and control of Type I error rates when various MDTs are used in the treatment of missing data, has not received adequate attention. In the following section, important findings from previous studies on the effects of some of the factors are presented. Strengths and weaknesses in these studies, if notable, are also highlighted. Proportion of Missing Data Researchers should consider a number of factors when choosing a missing data treatment. One of the most obvious factors is the amount of missing data. Generally, the choice of M D T is not critical if the amount of missing data is small (Frane, 1976; Gilley & Leone, 1991). Monte Carlo studies suggest there is little difference in the parameter estimates and answers to research questions when less than 10% of the data are missing in random patterns (Raymond & Roberts, 1987) or systematic patterns (Malhotra, 1987). The choice of MDTs seems to become more important as the amount of missing data approaches 15 - 20% of the data set (Raymond & Roberts, 1987) and most important as missing data approaches 30-40% (Malhotra, 1987). However, when missing data approaches 30 - 40% of the data set, one might question the wisdom of conducting any analyses (Roth, 1994). Therefore, in this study, use was made of data sets in which 10 - 20% was missing. 19 Violat ion of Normality A regression model often used in research settings takes the form Y = po + PiXu+P2X2,+...+ PpXp,+el where Y is the dependent variable, P0 is an intercept, Px,...,Pp are unknown regression coefficients for Xu,...,Xpi, respectively, and £, is a random residual for the i * observation, i = 1, 2,...,n. D i l lon and Goldstein (1984) suggested that i f we are only concerned with obtaining "good" estimators of regression coefficients, then the following four assumptions are necessary: 1. The expected value of the residual vector £ i is zero, i.e., E(e) = 0. 2. There is no correlation between the i * and y* residual terms, and the residuals exhibit constant variance, i.e., E(ez') = o2!. In scalar notation, this corresponds to the following two assumptions: Zs(e,-e,-) = 0 for i ^ j and £(£,£,) = E(e2) = o2 for i = j. These two assumptions are better known as the assumptions of no serial correlation and homoscedasticity. 3. The covariance between the X ' s and the residual terms e, is zero, an assumption that is automatically fulfilled i f the X variables are nonprobabilistic, so that the nxp data matrix X consists of fixed numbers. 4. The rank of the data matrix X is p, the number of columns in X , and less than n, the number of observations, i.e., r(X) = p, where p < n. This assumption means that there are no exact linear relationships among the X variables. It is better known as the assumption of no multicollinearity. However, typically we want to go beyond mere point estimation and make statements or inferences about population regression coefficients on the basis of sample coefficients; or, more generally, we want to use the sample-based regression model to draw inferences about the population model. Therefore, we need to impose distribution assumptions on the residual terms. 20 Because the ordinary least squares (OLS) estimators are linear functions of e, (since Y, is a linear function of £,), the probability distributions of the OLS estimators will depend on the specification of the probability distribution of e,. The classical multiple linear regression model assumes that each e, is normally distributed with mean 0, common variance (7 I, and uncorrected residuals. Given normality, each element of b, the OLS estimator of p, is normally distributed with mean equal to the corresponding element of the true P and variance given by o2 times the appropriate diagonal element of the inverse matrix (X'XY1. Although it is assumed that the matrix X is fixed, yielding different values in repeated sampling, Dunteman (1984) noted that this assumption can be relaxed to provide for predictors that are randomly selected (called stochastic regressors) for each sample replication and the classical model based upon the assumption of fixed regressors will still apply assuming that X is multivariate normal. That is, the parameters are estimated in exactly the same way, and hypothesis testing of these parameters proceeds identically. Few studies have specifically evaluated the effectiveness of MDTs when the assumption of normality is violated. Non-normality may affect the performance of MDTs, especially with large proportions of missing data. In their study, Graham, Hofer, and MacKinnon (1995) found that estimates obtained for skewed data were somewhat less accurate than for normal data, and that the expectation-maximization (EM) estimates were satisfactory when skew was in the interval + 3. Missing Pattern Simonoff (1988) and Little and Rubin (1987) stated that heuristic methods of treating missing data require the assumption that data are missing completely at random (MCAR). However, most field data do not miss randomly, and very little is known about the effectiveness 21 of MDTs for such data sets (Little, 1992; Roth, 1994). Graham, Hofer, and Piccinin (1994) noted that non-randomly missing data occur when some variables are correlated with some important attribute of the individuals tested with those measures. Examples of non-randomly missing data include situations when respondents have trouble understanding the meaning of a question and skip it, and omissions due to failure to complete the survey. Such omissions normally form clusters at the middle or end of a questionnaire, with the possibility of forming a non-monotonic pattern of missing data. Another example of non-randomly missing data pattern encountered in the analysis of social science data is the monbtonic pattern. Such a pattern is normally a result of attrition in panel survey. In such a pattern, for j = \,...,K — 1, X . is observed whenever X j + [ is observed. Further elaboration on monotonic and non-monotonic patterns is on page 48. Kromrey and Hines (1994) and Brockmeier, Kromrey, and Hines (1996) compared the effectiveness of five and eight MDTs, respectively, for non-randomly missing data within the context of two-predictor regression models in which missing data occurred on only one of the predictors. They found that existing MDTs that are intended for use with randomly missing data might be used with non-randomly missing data as well. However, they cautioned researchers to be careful in the selection of MDTs as there were substantial differences in their effectiveness across different types of data. For example, it is not clear how MDTs perform under various conditions of non-normality. Further, the researchers' results were limited to the two-predictor regression model in which systematically missing data occurred on only one of the predictors. They recommended that extensions to regression models with more predictors and to data sets with missing data on more than one predictor needed additional research. The present study was an attempt to address these issues. The pattern of missing data is closely related to what Little and Rubin (1987) called the mechanism of missing data. The key issue in understanding the mechanism of missing data is to establish whether missingness is related to the data values (Little, 1992). According to Little and 22 Rubin (1987), knowledge (or absence of knowledge) of the mechanism that led to certain values being missing is a key element in choosing an appropriate analysis and in interpreting the results. For example, when the probability that X , is missing for a case is independent of data values, then the mechanism is said to be both missing completely at random (MCAR) and missing at random (MAR). However, when the probability that X , is missing for a case depends on the value of X , X „ for that case, then the mechanism is said to be M A R and not M C A R . z p Further, when the probability that X , is missing depends on the value of X , for that case, then the mechanism is not M A R because Xx is not fully observed. Engel and Meyer (1996) noted that a major problem in any attempt to handle missing data in survey research is to cope with nonignorable non-response. Nonignorable non-response arises whenever the missing data pattern is neither missing completely at random (MCAR) nor missing at random (MAR). Little (1992) provided a more formal definition of missing data mechanism in the following manner. Let Z denote the n by (p + 1) data matrix, including observed and missing values, and let Z 0bS denote the set of observed values of Z , and let Z m i s denote the set of missing values. By introducing a missing-data indicator matrix R , with (/, / )* element R{j =1 if Xtj is observed and /?,y=0 if Xtj is missing, the notion of a missing-data mechanism is formalized in terms of a model for the conditional distribution p(R\Z,(p) of R given Z , indexed by unknown parameters (p. Data are missing at random (MAR) if the distribution depends on the data Z only through the observed values Z o h s ; that is, p(R\Z,(p) = p(R\Zobs,(p) for all Z . Data are missing completely at random (MCAR) if the distribution of R does not depend on the observed or missing values of Z ; that According to Graham, Hoffer, and Piccinin (1994), most mechanisms of missing data 23 may be classified as either "accessible" or "inaccessible." A missing data mechanism is accessible when the cause of missingness has been measured and is available for use in the analysis. If a researcher includes a measure of causes of missingness during data collection, and draws a random sample of the cases with missing data after data collection, then the researcher can use this information to account for missingness. In this case, the mechanism of missingness is accessible. Graham et al. suggested that an instrument for measuring the mechanism of missingness should include items on the reading speed of the participants, the participant's lack of motivation to complete survey, the participant's rebelliousness, and whether the participant refuses to participate because of the nature of scores on the dependent variable. However, researchers have not followed this approach. Graham et al. gave two possible ways in which inaccessible missing data mechanisms can arise. Firstly, such mechanisms can arise when the variable containing the missing data itself is the cause of missingness. For example, the mechanism would be inaccessible if the people who drop out of a drug use prevention study do so because they currently are high-level drug users. Secondly, inaccessible mechanisms also can arise if another unmeasured variable is the cause of missingness and that variable is correlated with the one containing the missing data. On the other hand, Little and Rubin (1987) referred to "accessible" missing data mechanisms as "nonignorable." However, according to Graham et al., the term "accessible" refers strictly to the mechanism and the term "ignorable" refers to a combination of the mechanism and the analysis used. Knowledge of the mechanism of missing data is important because there may be a need for the inclusion of that mechanism of missingness in the statistical model by including a distribution for response indicator variables that take the value of 1 if an item is recorded and the value of 0 otherwise (Graham et al.). Little and Rubin (1987) described three examples of planned mechanisms of missing data in survey research. These are the processes of sample selection, double sampling, and censoring. Graham et al. described two unplanned mechanisms of missing data in survey research which are 24 of interest in this study; omission and attrition. Omission is a mechanism of unplanned missing data. The occurrence of such omissions may be somewhere in the middle of the survey instrument or at the end. Omissions that occur in the middle of a survey instrument may be due to a respondent simply not seeing a question, or forgetting to go back to skipped questions. A respondent may have trouble understanding the meaning of the question and may skip it. A participant may fail to answer a particular question because s/he is afraid of possible negative consequences of answering it, or because the question evokes negative feelings he or she does not want to experience. A n omission may also be due to failure to complete the survey. Graham et al. (1994) noted that i f time is not a limiting factor, then two main reasons for an incomplete survey are lack of ability and lack of motivation. A participant may lack the ability to finish because he or she is a slow reader or because of language problems. Attrition is yet another mechanism of unplanned missing data. It occurs when a respondent is measured in an attribute for at least one wave of measurement, but is absent entirely for additional waves of measurement. Attrition may be due to a random or non-random process. A n example of a non-random cause of attrition is when a subject is i l l for the measurement session. In this case, the cause of attrition is completely independent of the measurement. Underlying values not observed in a given data set may determine whether information is missing or not. For example, in an attitudinal survey, a respondent with low cognitive skills may refuse or be unable to give answers to many questions. In this case, the pattern of missing data w i l l exhibit clustering. Such a cause of attrition is not completely independent of the measurement ( K i m & Curry, 1977). There is also the possibility of having a non-random cause of attrition related to the dependent variable itself. For example, students with disability may be more likely to drop out of a study than are students who are not disabled. The present study involved non-randomly missing data structures in which missingness may have been through omission or attrition. 25 Studies with Randomly Miss ing Data Limited number of studies using randomly missing data suggests that E M method produces less biased estimates than listwise deletion (Graham & Donaldson, 1993). Haitovsky (1968) conducted a simulation study with highly correlated data and found the method of pairwise deletion to be markedly superior to listwise deletion. K i m and Curry (1977) found a similar outcome for weakly correlated data as well . T i m m (1970) used simulation method to assess the ability of three methods in predicting correlation and variance-covariance matrices for observations missing at random. Instead of randomly generating numerous matrices, numbers were selected for use from the literature. With a, -1 Z = p, Kaiser's (1968) measure of average intercorrelation T = , guided the selection, p-l where Cd, is the largest eigenvalue of p and p is the number of variables. Values of OC range from zero to unity. Three matrices for (X = 0.2, 0.5, and 0.8 were chosen with p = 2, 5, and 10. Having obtained the variance-covariance matrices from the literature, Kaiser and Dickman (1962) procedure was used to generate complete data matrices of size N = 50, 100, and 200 from a multivariate normal population with p = 0 and Z = p. From these complete data matrices, two incomplete data matrices were obtained each with 1%, 10%, and 20% of the data deleted at random. Using these data with two replicates, the effect of sample size, number of variables, and average intercorrelation were examined when three M D T s were used. The three methods investigated were mean substitution, regression technique, and a modified form of Dear's (1959) technique in which the matrix Y is decomposed into its known and unknown components afterwhich a principal components solution is used to estimate the missing values. A major weakness of the study is that only two replications were used! This makes it difficult to have a reasonable degree of confidence on the results of the simulation study because of lack of 26 accuracy. Timm (1970) found no uniformly best technique for the estimation of variance-covariance matrices when observations were missing at random. However, the Dear and regression methods were generally superior to the mean substitution method for high and intermediately correlated variables, but less satisfactorily under low intercorrelation of variables. Timm (1970) observed that in the comparison of techniques for the estimation of Z , the Euclidean norm Z - Z , an additive function of eigenvalues, provides a convenient measure of "closeness" and is defined as A 2 z - z = Tr (z-z)(z-zy Because the distribution of the Euclidean norm measure is unknown, Timm (1970) formulated the following rule to compare the MDTs: "Choose that technique which minimizes E Z - Z ; ,where i denotes the various techniques used in the estimation of Z . " As a measure of the overall comparative efficiency of any pair of procedures, the ratio of the Euclidean norms was used. For example, the efficiency index for two MDTs (A and B) in estimating Z is given by < # ( Z , / Z B ) ^ J M z - z , However, Timm (1970) found that any missing data treatment given preference in the estimation of Z by employing the Euclidean criterion or the average relative efficiency index may not necessarily function best in minimizing the largest eigenvalue mean square error. Gleason and Staelin (1975) provided an approach similar to Timm's method for determining which technique produces from incomplete data a correlation matrix most similar to the correct matrix (i.e., the one calculated from complete data). This dissimilarity measure is 27 defined as follows. Let ||A|| denote the Euclidean norm of the matrix A, let R denote the correlation matrix for complete data, p the number of variables, and Ra the correlation matrix estimated from incomplete data using method a. Then ° " 1 Ra-R{ 2 \ P(P-I) represents root-mean-square deviation of predicted versus actual values for the off-diagonal correlations. The formula can be used to determine which of the MDTs yields the better estimates. To compare different MDTs for predicting missing data, Gleason and Staelin (1975) constructed the following measure of the quality of various estimates. Let xtj be the value in a complete data matrix that corresponds to a missing entry in an incomplete matrix; let ov be the variance, in the complete matrix, of the variable in which the entry lies; let x^a) be the estimate of xtj derived by method a; and let K denote the set of entries missing from the matrix. Then a measure of the dissimilarity between true and estimated values for method a is given by e „ = . (7i mpK where i, j are elements of K, K is the proportion of missing entries, m is the sample size, and p is the number of variables. This number is the root-mean-square standardized residual. Beale and Little (1975) compared six methods for handling missing data: (1) Ordinary least squares (OLS) using complete observations only (2) Buck's (1960) method (3) Iterated Buck, or corrected maximum likelihood 28 (4) OLS on observations with y present, after fitting missing values of the independent variables only, (5) Method 4, but with missing observations given fractional weights and (6) Method 5, but using a covariance matrix for all variables, found by method 3, to find the fitted values and estimate the weights. Data were generated from a multivariate normal population with one variable identified as the dependent variable, and between two and four independent variables. Samples of size 50, 100 or 200 were used. Deletion of 5, 10, 20 or 40 percent of the observed values was conducted randomly. Their criterion for judging the effectiveness of each M D T was the residual sum of squares of deviations of the observed and fitted values of the dependent variables when the deleted values were restored. This may be written as s = X(y,-*0-IvJ. where b0 and bj are the regression coefficients estimated from missing data by one of the six MDTs, and xVj and y(. are the true values of all variables without deletions. A small value of S represents a successful method. Findings showed that Buck and iterated Buck outperformed OLS using complete observations only. Finkbeiner (1979) discussed a maximum likelihood method of estimating the parameters of the multiple factor model when data were missing at random from the sample. He used a Monte Carlo study to compare the M L method with five other heuristic methods of dealing with missing data. The five methods were complete data, mean replacement, pairwise deletion, regression replacement, and principal components. Two proportions of missing data were chosen: one to have only a little data missing and the other a lot, both for samples of size 64. For each sample, random, independent normal numbers were generated by the Box and Muller (1958) 29 method with 50 replications for each of the two patterns. A l l the six M D T s were used to analyze each of the 100 samples. Sampling distributions for each method on each missingness pattern were developed. Three pieces of information were used in examining estimation effectiveness: sampling means and dispersions of parameter estimates, mean squared error of parameter estimates, and an index of common factor recovery. For the first, the least biased, minimum dispersion estimate was considered optimal. For the second, the best method produced the smallest mean squared error most consistently, regardless of bias. For the last, the average of the squared multiple correlation coefficients of congruence was used. This index is referred to as a "congruence coefficient." It varies in value from zero to one, with zero indicating no recovery of the population common factor space and 1 indicating perfect recovery. The three best methods were maximum likelihood, mean replacement and complete pair only. Tir r i and Silander (1998) proposed an imputation procedure for the treatment of missing data called stochastic complexity. They recommended the method for multivariate categorical data. The method is related to multiple imputation, which uses independent realizations of the posterior predictive distribution of the missing data under some complete-data model and prior. Intuitively, the approach involves modeling the set of data records as a matrix of incomplete discrete data. The missing data are estimated by assuming a functional form for the probability distribution of the cases (i.e., a model class). Based on the given model class assumption, an information-theoretic criterion can be derived to select between the different complete data matrices for the more likely one. Stochastic complexity represents the shortest code length needed for coding the complete data matrix relative to the model class chosen. A weakness of the method is that the exact criteria are very hard to compute for many interesting models, but it can be approximated by the Bayesian marginal likelihood computed by integrating over all the possible models in the chosen model class. 30 In their validation study, Tir r i and Silander (1998) produced synthetic missing data problems from real data sets by randomly deleting known portions of the data. Sample size was controlled for. They performed a classification analysis to study the practical implications of different procedures for handling missing data. For each data set, they created three different sub-samples of sizes 10%, 25% and 50% of the original data set. Each time 50% of the data was reserved for the subsequent out-of-sample classification analysis. In each sample, 5%, 10%, 20%, 35% and 50% of the elements were deleted, thus creating artificial missing data problems satisfying M C A R assumption. Mean substitution and stochastic complexity were used to create complete data matrices by imputing the missing values. Results showed that the stochastic complexity based approach performed well in recovering the original missing data. Studies with Non-randomly Miss ing Data Most missing-data treatment techniques require the assumption that data are missing at random. However, data sets in which values are randomly missing are rare in the fields of psychology and education. Research on the extent to which M D T s developed for randomly missing data may be useful for non-randomly missing data is therefore important. This area of research has received little attention. Azen et al. (1989) carried out a simulation study to compare the performance of three methods (listwise deletion, pairwise deletion, and expectation maximization) in estimating regression coefficients and missing values, for situations with varying proportions of missing data (5% and 25%) and varying magnitudes of correlations ( R 2 = .5 and .9). Two evaluation criteria were used. The first criterion, C i , was a measure of how well the M D T estimated regression coefficients: £(b , - p,-) C , = [2G2(2 + 2p - p 2]/[n(l + 2p)(l - p)] 31 The denominator of C\ is a normalizing factor equal to the expected value of the numerator conditioned on no missing data. The second criterion, C2 , was a measure of how well the M D T imputed the data in the sense that it measured how well each M D T estimated the mean vector and covariance matrix: C 2 = (3nf)"1ZZd2iymin d2,y where the summation is over all missing values, and d\} =nW, y + ( X * , y - X # , 7 ) 2 min d2ij = V(X y ) - CXHC'1WUCWX X ' , ; = E(X j ; IW,-, A) X # , y=E(X, yIW„ A # ) and A* is the estimated covariance matrix, min d2^ is that part of d2y which results if A was estimated perfectly, and is a constant over all MDTs. For estimating regression coefficients at 5% missing, analyses revealed that all three methods performed well for R = 0.5 condition. However, pairwise deletion was generally inferior for R 2 = 0.9 condition. E M and listwise deletion performed equally well regardless of the magnitude of correlations, when the percentage of missing data was only 5 per cent. When the percentage missing increased to 25, E M was generally the best. In addition, PW was competitive only for situations with weak correlations and/or little missing data. With a pattern of censored missing data, the PW method performed better than the E M method. For estimating regression coefficients at 25% missing, analyses of variance showed that the E M method was generally the preferred method. Pairwise deletion was competitive for weak correlations. In addition, an analysis of variance showed that E M method performed statistically better for the random and related patterns of missing data than it performed with censored data. A limitation of the study by Azen et al. (1989) is that they made use of only 50 replications. Efron (1980) suggested that although as few as 100 replications could provide a 32 stable estimate of various statistics, a tenfold increase in the number of replications would provide desirable accurate results. Also, Azen et al. (1989) did not consider two important imputation methods; mean substitution and regression imputation. There is need for studying the effectiveness of these methods in comparison to pairwise deletion and E M algorithm using a large number of replications (say, at least 1,000) in order to offer more reliable guidance to field researchers. Kromrey and Hines (1994) investigated the effects of non-randomly missing data in two predictor regression analyses. They examined differences in the effectiveness of mean substitution, simple regression, multiple regression, listwise deletion, and pairwise deletion on estimates of R 2 and regression coefficients. They used bootstrap samples drawn from three large sets of actual field data, representing Likert-type rating data, achievement test data, and psychological trait data. The study design was a 3 x 3 x 6 x 5 , with three between-subject factors (parent population, sample size, proportion of data missing) and one within-subjects factor (MDT). For comparison, they computed the effect sizes obtained from the M D T conditions relative to the complete sample condition. Kromrey and Hines found that the differences among the MDTs increased as the proportion of missing data increased. They also found that multiple regression imputation consistently yielded overestimates of R 2 . Conversely, the use of mean imputation consistently yielded underestimates of R 2 . The simple regression imputation procedure overestimated R 2 only in the psychological trait data, where the overestimation was consistent. Listwise deletion underestimated R 2 , except in the psychological trait data, where an inconsistency in the direction of the effect was realized. Although they found that no M D T yielded consistently best estimates of R 2 across the data sets, sample sizes, and proportion of missing data, pairwise deletion and the simple regression imputation appeared to perform better in most situations than the use of mean substitution and multiple regression imputation. In the estimation of regression coefficients, Kromrey and Hines (1994) found that listwise deletion provided the best overall performance. The imputation techniques (multiple 33 regression, simple regression, and mean substitution) did not perform wel l in estimating population R 2 as well as the regression coefficients when applied to non-randomly missing data. Whereas Kromrey and Hines provided support for the use of certain M D T s in situations where data are missing non-randomly, they stressed the importance of careful selection of missing data treatment because of the substantial differences in the effectiveness of the M D T s . The differences in the effectiveness of the treatments across different types of data highlighted the need for further research to identify the types of data matrices that may be more amenable to analysis by those methods. Whereas Kromrey and Hines (1994) offered important guidelines on the effectiveness of M D T s , there is need to extend their findings by including E M in the comparisons. A key limitation in the study by Kromrey and Hines (1994) is the extent to which the results may be generalized. The study was limited to missing data occurring on only one predictor in a 2-predictor model. Such a model is rare in psycho-educational research. There is need for further research to examine the use of M D T s when missing data occur systematically on more than one predictor. In a more recent study, Brockmeier, Kromrey, and Hines (1996) investigated within the context of a two-predictor multiple regression analysis with non-randomly missing data, the effectiveness of eight missing data treatments on multiple R 2 and each standardized regression coefficient. They recommended the use of stochastic regression in estimating population R 2 except when the proportion of missing data is high (60%). They also recommended the use of pairwise deletion and listwise deletion in estimating the sample estimate when percent missing is low (10%). When examining the standardized regression coefficients, none of the M D T s were found effective when missing data was 60%. They recommended the use of deletion and stochastic regression procedures to generate unbiased parameter estimates when percent missing is low. A limitation of the study is that they used a two-predictor model only. 34 Fraser and Halperin (1998) compared listwise deletion, pairwise deletion, and maximum likelihood method in the estimation of the mean vector and covariance matrix of a random sample from a multivariate normal distribution. A monotonic missing pattern was used, formed by three time periods in a longitudinal study. The parameters of interest were the mean, variance, and covariance. Specifically, they wanted to discover whether maximum likelihood would outperform the quick methods that were simpler and easier to obtain. There were four between-group factors in their design. These were number of variables in each time period (2, 3, 2) and (3, 2, 1), correlation among the variables (R 2 =.32 and .71), number of observations for variables in the first block (50 and 70), and attrition rate (20% and 40%). The within-factor was the missing data treatment (listwise deletion, pairwise deletion, and maximum likelihood). A split-plot factorial analysis was used for which only the interactions, which included the missing data methods, were discussed. Findings showed that attrition rate discriminated among the missing data methods for the mean and variance. For the mean vector, the interaction between estimation methods and the attrition rate was significant with an effect size in the medium to high range according to Cohen's (1988) guidelines. The other design factors did not differentiate among the methods. For the mean, pairwise deletion was recommended for 20% attrition, but maximum likelihood was recommended for 40% attrition. For the variance, pairwise deletion was recommended at both 20% and 40% attrition rates. The only significant interaction was that of estimation methods and attrition rate, with an effect size in the middle to higher range. For the covariance, none of the design factors were able to discriminate among the missing data methods. Studies in Structural Equations Modeling A number of researchers have conducted studies involving MDTs in the context of structural equation modeling. A structural model is specified in part by r| = Br| + T£ + £ 35 where r\ is a vector of endogenous variables, cj is a vector of exogenous variables, C, is a vector of unobserved, exogenous disturbances, and B and Y are matrices of coefficients. C, and ^ are assumed to be uncorrected. For convenience, all variables are assumed to have means of zero, thus eliminating the need for intercept terms in the model. To achieve identification, one must usually impose restrictions on B, T, and var(Q. Important special cases of the above equations include multivariate regression (B = 0), multiple regression (B = 0 and r\ is a scalar) and recursive systems (B is subdiagonal and var(Q is diagonal). Al l i son (1987) proposed a maximum likelihood method which he claimed to be both consistent and efficient in estimating parameters for the general linear structural relations model when data are missing. The method capitalizes on the ability of statistical software for structural equations modeling to estimate simultaneously the same model for two or more samples. For missing data problems, the sample is divided into subsamples, each having a different set of variables present. The model is then estimated simultaneously for all subsamples, constraining corresponding parameters to be equal across subsamples. To facilitate estimation of the above model with missing data, Al l i son (1987) modified the model to the equivalent form TJ* = B*r|* + C*, 'B r 0 0 allowing for the possibility that t| may not be directly measured. This is accomplished by adding equations that specify the effects of the latent variables on a set of observed indicators: y = A^r | +e. Here, y is a vector of observed variables, and £ is a vector representing random measurement error. A y is a matrix of coefficients that must usually be restricted in some way to achieve identification. where ri*=(r|, £ ) ' , £*=(£. £ ) ' , and B*= This class of models is greatly expanded by 36 To allow for missing variables, Allison (1987) suggested that any variable that is missing from any of the subsamples must be an indicator of a latent variable. Any missing covariances are set to zero, missing means to zero, and missing variance to one. The coefficients in the A y matrix are then used to "switch" the variable "on" or " o f f depending on whether it is present or absent in a particular subsample. Specifically, in subsamples with data missing for a particular variable y„ all elements in row i of A^ must be fixed at zero. One must also fix var(e,) = 1 and cov(e„ £,•) = 0 for i * j. These constraints ensure that the pseudo-values of zero and one in the sample covariance matrix for that subsample will be fitted exactly. In subsamples with data present for that variable y„ the treatment depends on whether or not the model allows for random error in the measurement of that variable. If the model does not allow for random error, one of the 'kij coefficients should be fixed at zero and var(e,) should be fixed at zero. If random error is allowed, no special constraints are needed; the X coefficient and the error variance should be left as free parameters to be estimated. Finally, all other parameters are constrained to be equal across subsamples. With the LISREL computer program, Allison (1987) demonstrated the superiority of maximum likelihood estimation when data are missing at random but not observed at random. Muthen, Kaplan, and Hollis (1987) gave a general latent variable model that includes the specification of a missing data mechanism. In their model, M C A R was not a prerequisite for unbiased estimation in large samples, as when using the traditional listwise or pairwise deletion approaches. Using artificial data, they found that likelihood estimation was superior to traditional estimation methods in situations involving data that were not missing completely at random. Arminger and Sobel (1990) constructed a nonlinear method, called pseudo-maximum likelihood estimation (PML), which can be used with missing data. They illustrated how maximum likelihood estimation of parameters using missing data (see Little and Rubin, 1987) can be extended directly to the P M L estimation of the same parameters. 37 Brown (1994) assessed the relative efficacy of five indirect methods for dealing with missing data in structural equation models using simulated data. The five methods studied were listwise deletion, pairwise deletion, mean substitution, hot-deck imputation and similar response pattern imputation. However, Brown's study was restricted to a cross-sectional design, and he did not consider other estimation methods, like the E M algorithm in his comparison although he recommended this for future research. Reinecke and Schimdt (1996) compared listwise deletion, pairwise deletion, and similar response pattern imputation with a non-iterative maximum likelihood procedure developed for nested missing values in panel data. The similar pattern method is used to impute the item non-responses in each wave while the non-iterative maximum likelihood procedure is used to consider unit non-response because of panel attrition. They found that listwise deletion method led to an unacceptable waste of information across and within panel waves. However, pairwise deletion method led to a reasonable amount of explained variances with the best model fit. The non-iterative maximum likelihood procedure produced reasonable estimates of the covariance matrix taking into account unit non-response, and imputing the item non-responses in each wave did not contribute to the explained variances and the overall model fit. Employing simulation methodology, Marsh (1998) addressed the use of sample covariance matrices constructed with pairwise deletion for data missing completely at random. In the study, 3 levels of sample size (N = 200, 500, 1,000) and 5 levels of percent missing (0, 1, 10, 25, and 50) were used. A population covariance matrix was generated from a simple population model in which three latent factors were defined by three measured variables such that each of the nine measured variables had a nonzero loading on one and only one latent factor. Population values were .7 for all factor loadings, .51 for all measured variable uniqueness, and .4 for all correlations among the three latent factors. Five sets of 100,000 cases were simulated from this population covariance matrix by G E N R A W , a program for data generation within LISREL 8 by Joreskog and Sorbom (1993). The cases differed only in respect to the five levels of missing 38 values to allow for comparisons. For each set, the probability that a particular data point was missing was specified to be 0%, 1%, 10%, 25%, or 50%. A random variable was paired with each data point using the random variable routine in SPSS, and this random variable was used to select values that were missing. The investigation emphasized the effects of missing values both on the %2 test statistic, subjective indices of fit, and parameter estimates and on how these effects interact with sample size. Three indices of fit, namely, relative noncentrality index (RNI), nonnormed fit index (NNFI) and root mean square error of approximation (RMSEA) were considered. Evaluation of parameter estimates was based on their means and standard deviations. To assess the relative size of the various effects and to provide a nominal test of statistical significance, two-way analyses of variance were conducted in which the effects of the three sample sizes and five levels of missing data were considered. One potentially serious limitation of the pairwise deletion approach is the occurrence of a non-positive definite covariance matrix. Of the 4,000 covariance matrices generated in the simulation study, only 27 non-positive definite covariance matrices were observed. This occurred only in the cell with N = 200 and percent missing of 50. Surprisingly, contrary to the findings of Brown (1994), mean estimates of factor loadings, of factor correlations, and of uniqueness were similar across all the cells of the design. In sum, estimates based on pairwise deletion were unbiased even when percent missing was large and the sample size was small. Marsh (1998) also found that consistent with expectations, the sizes of standard deviations varied inversely with sample size and directly with percent missing. Sample size, percent missing, and their interaction substantially influenced the magnitudes of observed standard deviations. With respect to the fit indices (RNI, NNFI, RMSEA) , the pattern of the effects was similar to that observed for the %2 test statistic. Results also showed that the %2 test statistics were largely successful in eliminating the effect of missing values, although there were 39 still small effects associated with sample size and the interaction of sample size and percent missing. Marsh (1998) recognized the potential limitation of generalizing the results from his study. Whereas a wide range of sample sizes and percent missing were considered, the results were based on only one population generating model, and the variance-covariance matrix constructed under pairwise deletion contained no misspecification error (i.e., only "true' models that should be able to explain the data were considered). Hence there is need to explore a wider range of conditions in further simulation research. One other limitation of the study was the use of M C A R , an assumption that is unreasonable in most applications. Summary Missing data can be of two types, namely unit non-response and item non-response. The distinction between unit non-response and item non-response was discussed. For example, unit non-response may occur when a respondent totally refuses to respond in a questionnaire. Item non-response may be a result of an instrument or question being broken off after being partly completed. Several methods of treating missing data were presented. These methods may generally be classified into two categories: deletion methods and imputation methods. Deletion methods include listwise deletion and pairwise deletion. Imputation methods include mean substitution, regression imputation and expectation maximization procedure. Factors that may influence the performance of MDTs were discussed. These included missing pattern, percent missing, non-normality, number of predictors and magnitude of correlation. In particular, it was noted that the effects of missing pattern and non-normality on parameter estimates after different treatments of missing data had not received enough attention by researchers. 40 MDTs were essentially developed for use under the assumption of M C A R . Therefore, previous researchers tended to evaluate MDTs using data sets under the M C A R condition, with little regard of the fact that the M C A R condition is difficult to obtain in real research settings. Two common systematic missing patterns were identified: monotonic and non-monotonic patterns. A number of limitations of previous studies were noted. Of importance were the small number of replications, and failure by researchers to use regression models with more than two predictors. It was also noted that an area that needed further research is the effectiveness of regression imputation and E M method when compared with MS and PW methods. In doing so, it is important to consider the role of other factors like pattern of missing data, size of R 2 , number of predictors, sample size and percent missing, the levels of which must be based on a sound rationale. Various criteria for the evaluation of MDTs were presented, indicating that different researchers used different criteria, the choice depending mainly on the parameter estimate of interest. Such criteria included effect size, Euclidean norm, RMSE, MSE, sampling means and standard deviations, as well as the average of the squared multiple correlation coefficients of congruence. Notable findings in previous research include that of Azen et al. (1989) who found that in the estimation of regression coefficients, with 5 percent missing, both E M and PW performed well when multiple R 2 = 0.5. However, PW was inferior to E M when multiple R 2 = 0.9. At 25 percent missing, E M performed better than PW, although PW was competitive when R 2 = 0.5. On the other hand, with a pattern of censored data, the E M method did not perform well in the estimation of parameters as it was biased. The bias was possibly due to the non-linearity of the regression relationship among variables. Kromrey and Hines (1994) found that RS and MS methods did not perform well in estimating population R 2 and regression coefficients when 41 applied to non-randomly missing data. Fraser and Halperin (1998), using a monotonic missing pattern, found that the P W method was better than maximum likelihood method for variance estimation at both 20% and 40% missing. 42 CHAPTER 3 METHOD To answer research questions presented earlier, MDTs were evaluated under various conditions using simulated data. In the present chapter, the advantages of using simulation methodology, issues in data generation, factors influencing the effectiveness of MDTs, the study design, the procedure, and analysis and evaluation criteria are discussed. Advantages of Simulation Methodology Simulation methodology may be used to study the properties of statistical techniques under diverse settings. The main goal of conducting a simulation study is to collect evidence for evaluating specific procedures. As noted by Efron (1980), computer intensive studies can both advance statistical theory, and provide improved methods of solution where no good theory exists. Examples of areas where such methodology can be applied include examination of robustness properties, assessment of small sample versus asymptotic agreement, or comparison of a statistical method with its competitors (Johnson, 1987). Rubin (1991), like Efron (1980), also noted that simulation methodology is an inherently statistical idea in that it capitalizes on the proposition that the fine detail available in an analytic solution is often not necessary. In practice, inferences about a parameter 6 can be as beneficially drawn from several hundred equally likely values of 6 , as from formulas giving the mean and variance of an infinite number of such draws. In fact, Rubin (1984) made an argument that several hundred random draws often may yield better practical inferences than traditional analytic solutions based on asymptotic approximations. First, asymptotic approximations can provide a less accurate picture of inferential uncertainty than simulated draws when the likelihood function is not nearly normal. Second, implemented simulation techniques allows the investigator to explore a variety of underlying models and thereby avoid fixation with possibly inappropriate models that have neat analytic or asymptotic solutions. 43 Issues in Data Generation A simulation study requires the consideration of how to generate the observations to be used in the study. In this study, multivariate random numbers from populations with specified skew, kurtosis, and intercorrelations were used. Although several random number generating methods are available, many of the generators currently in use are seriously flawed (L'Ecuyer, 1988, 1990, 1994). Therefore, it was important to select a random number generator with extreme caution. L'Ecuyer (1994) provided important requirements for a good random number generator. He argued that a good general-purpose generator must have specified statistical properties, namely, adequate period length, ease of implementation, efficiency, portability, and reproducibility. He recommended the use of higher-order congruential generators instead of multiplicative linear congruential generators. EQS (Bentler, 1989), a statistical program that uses a higher-order congruential generator, was used to generate data. Factors Influencing MDT Performance A number of factors may influence the performance of MDTs when used to treat missing data in research situations. In this study, the factors that were considered include Non-normality (NM), Sample Size (SZ), Proportion of Missing Data (PM), and Pattern of Missing Data (PT). However, there is a possibility that in a multiple regression context, the Number of Predictors (p), and the magnitude of multiple R 2 may also have an impact on the effectiveness of MDTs. For this reason, four studies were conducted. In the first study, there were four predictors under low R 2 condition; in the second study, there were four predictors under a high R 2 condition; in the third study, there were nine predictors under a low R 2 condition; and the fourth study involved the use of nine predictors under high R 2 condition. 44 To be closer to what researchers have encountered, the values used at different levels of the manipulated factors were largely based on the findings from a comprehensive review of papers in four journals, namely, American Educational Research Journal, Journal of Applied Psychology, Journal of Educational Psychology, and Journal of Educational Research. In the review, I found that a total of 177 studies in the 1990 to 1995 issues of these journals [excluding articles in the 1993 Journal of Applied Psychology, 78(4) to 78(6)] used multiple linear regression analysis. Of the 177 studies, 9.04% (16) were in the American Educational Research Journal, 46.33% (82) were in the Journal of Applied Psychology, 29.94% (53) were in the Journal of Educational Psychology, and 14.69% (26) were in the Journal of Educational Research. These studies were reviewed to help identify which values of the design variables are commonly found in psycho-educational research so that representative levels could be selected for use in this study. Table 1 is a breakdown of the number of studies in each journal that used multiple regression analysis. Table 1: Number of studies using multiple regression in four journals Y E A R 1990 1991 1992 1993 1994 1995 T O T A L AERJ 2 5 3 1 1 4 16 JAP 8 22 13 8 10 21 82 JEP 18 7 8 12 2 6 53 JER 6 5 4 6 3 2 26 T O T A L 34 39 28 27 16 33 177 Note: AERJ = American Educational Research Journal JAP = Journal of Applied Psychology JEP = Journal of Educational Psychology JER = Journal of Educational Research The following is a description of my findings based on the review of the four journals regarding proportion of missing data, sample size, non-normality, number of predictors, and magnitude of R ' that can be considered typical in psycho-educational research. 45 Proportion of Missing Data A review of the four journals mentioned above showed that most studies reported response rates at the unit level, but not at the item level. Of the 177 studies that used OLS regression analysis, 29.38% (52) reported response rates. I derived non-response rates based on these response rates. The mean non-response rate was 28.8 percent, with a standard deviation of 21.03 (N = 52). The distribution was multimodal, with the smallest mode being 22.86. The lower quarfile was 12.83, the median 22.86, and the upper quartile 41.38. Considering that these values included both unit non-response and item non-response, it is most likely that the second quartile is the upper bound and the first quartile is the lower bound for item non-response. Based on these findings, the levels for percent missing in the present study were 10%, 15%, and 20%. These values were similar to what other researchers had used. For example, Raymond and Roberts (1987) had three levels: 2%, 6%, and 10%, and Azen et al. had two levels: 5% and 25%. The number of data points deleted to create missing data patterns in the present study was computed as percent missing multiplied by the total number of data points in the predictors. Sample Size Of the 177 studies using multiple regression, 19 used data sets with more than 1,000 cases (e.g., High School and Beyond data with N = 14,825). Such data sets were found to severely skew the distribution of sample sizes in a positive direction. I eliminated sample sizes greater than 1,000 because most surveys would use values below this. However, this did not totally eliminate skew as the resulting distribution was still positively skewed, but not as severely skewed as when sample sizes greater than 1,000 were included. It was also difficult to ascertain the sample size used in a number of studies. In all, 148 sample sizes below 1,000 were reported. The mean sample size was 218 with a standard deviation of 192 (N = 148). The minimum sample size was 20 and the maximum, after eliminating those above 1,000, was 983, giving a 46 range of 963. The first quartile was 94, the median 153, and the third quartile 265. In order to have typical sample sizes in this study, samples of size 94, 153, and 265 were used. Non-normality Whereas the normality assumption is a prerequisite in multiple regression analysis, Micceri (1989) and Sawilosky and Blair (1992) observed that many data sets in real situations are skewed and/or kurtotic, thus violating the normality assumption. When investigating the effects of non-normality, it is extremely important to consider non-normality in the degree found in the literature. Before discussing the levels of non-normality used in this study, it is perhaps important to present an observation on how often researchers report violation of assumptions in their studies. Of the 177 articles that were reviewed, reports on regression diagnostics were contained in only five studies! This shows how researchers have often failed to report important issues before doing multiple regression analysis. Two of the five studies contained reports on violation of normality assumption, one had reports on multicollinearity diagnostics, one had reports on violation of homogeneity of variance assumption, and one had reports on outlier diagnostics. The study in which outlier diagnostics were reported also reported deletion of missing data. The review did not help in choosing appropriate levels of non-normality. So, I relied on previous reviews to choose levels of non-normality. After observing various empirical distributions, Pearson and Please (1975) found that typical non-normality had skew less than 0.8 and kurtosis between -0.6 and 0.6. However, Sawilosky and Blair (1992) who based their report on the findings of Micceri (1989) later provided eight real-world distributions different from what was reported by Pearson and Please (1975). Three types of psychometric distributions were identified as typical; discrete mass at zero with gap, extreme asymmetry, and extreme bimodality, with (skew, kurtosis) = (1.65, 0.98), (1.64, 1.52), and (-.08, -1.70), respectively; the values for kurtosis having been such that normal kurtosis is 0.0. Whereas these were considered typical distributions, Micceri (1989) noted that 47 typical kurtosis estimates ranged from -1.70 to 37.37. In their simulation study, Graham, Hofer, and MacKinnon (1996) used distributions with skew -.68 to 3.30 and kurtosis -.04 to 13.11. Based on previous findings, it is highly probable that typical distributions in psychometric scores would have skew hardly exceeding 3.0. However, typical distributions of psychometric scores seem to have kurtoses that vary widely, from -2.0 to approximately 40.0. Therefore these range of values were used as a guide in this study. The distributions used in this study were a normal distribution (skew = 0 and kurtosis = 0), a low level of non-normality (skew = 1 and kurtosis = 3), a medium level of non-normality (skew =1.8 and kurtosis = 6), and a high level of non-normality (skew = 3, kurtosis = 25). Number of Predictors In the reviews of four journals described earlier, of the 177 studies that used multiple regression analysis, 174 reported number of predictors. The mean number of predictors was 7.5 with a standard deviation of 5.94 (N = 174). The minimum value was 2 and the maximum was 40, giving a range of 38. The first quartile was 4, the median 5 and the third quartile 9. The distribution was unimodal, with a mode of 4. Based on these outcomes, this study included 4 and 9 predictor regression models. Size of Multiple R 2 Researchers usually want to explain maximum variance in the criterion variable. This is more particularly so when using multiple regression analysis. With this in mind, and using the 177 studies that employed multiple regression analysis, I recorded the maximum value of R 2 in each study. The value of R 2 was reported in 148 studies. The mean R 2 was .43 with a standard deviation of .22 (N = 148). The distribution of R 2 was bimodal. In other words, the studies that were reviewed could be categorized into two groups, those that had low R 2 from 0.01 to 0.30, and those that had moderate to high R 2 from 0.40 to 0.90. Seaman, Algina and Olejnik (1985) noted 48 that correlations encountered in behavioral science research often occur in the interval from .2 to .7, values that are consistent with those found in my review. Based on these findings, the two levels of R 2 used in this study were L O W (with R 2 approximately 0.2) and H I G H (with R 2 approximately 0.6). Pattern of Miss ing Data T w o patterns of missing data considered in this study were monotonic pattern and non-monotonic pattern. These patterns were selected because they represent two major categories of systematically missing data discussed in Little and Rubin (1987) and Little (1992). The monotonic pattern of missing data arises when the variables can be arranged so that for j = \,...,K - 1, Xjr is observed whenever Xy+i is observed. Little and Rubin (1987) noted that attrition from a panel survey leads to data of this form. To illustrate the mechanism of non-monotonic pattern of missing data, assume that Y is a measure of pupils' ability in Figure 2. If Xp represents father's income, then pupils of high ability may be too sensitive to respond, and pupils of low ability may not know their father's income. On the other hand, pupils with moderate ability may be in a position to respond to Xp. Figure 1 shows the monotonic pattern, and Figure 2 shows the non-monotonic pattern used in the present study. The criterion variable (Y) and p predictors, p = 4 or 9, were all continuous. In both figures, rectangles stand for available values. To create dependency of missing values in the set of predictors on the magnitude of Y-values, the data were sorted using Y before deletion, with the lowest score being at the top and the highest score being at the bottom. Deleting data using templates with configurations given in Table 2 and 3 respectively created the patterns in Figure 1 and 2. The same numbers of data points were deleted for both monotonic and non-monotonic patterns for corresponding X ' s . In other words, for each variable across pattern, the same number of data points was deleted. V A R I A B L E Y X i X 2 X p _ i Figure 1: Monotonic pattern of missing data V A R I A B L E Y X j X 2 X p _ ! Figure 2: Non-monotonic pattern of missing data 50 Table 2 Number of data points deleted to create missing patterns in the 4-predictor model Sample Size Percent Missing Number of Data Points Deleted Total x, X 2 X 3 X 4 94 10 0 8 15 15 38 15 0 12 22 22 56 20 0 15 30 30 75 153 10 0 13 24 24 61 15 0 18 37 37 92 20 0 24 49 49 122 265 10 0 22 42 42 106 15 0 31 64 64 159 20 0 42 85 85 212 Table 3 Number of data points deleted to create missing patterns in the 9-predictor model Sample Size Percent Missing Number of Data Points Deleted Total x, x2 X 3 X 4 x5 x6 X 7 x8 X 9 94 10 0 2 6 8 10 12 14 16 18 86 15 0 8 10 12 14 16 20 22 26 128 20 0 10 12 16 18 24 26 30 34 170 153 10 0 8 10 12 14 18 20 26 30 138 15 0 12 14 18 22 26 32 38 44 206 20 0 18 20 24 28 34 44 50 58 276 265 10 0 14 18 20 24 30 36 44 52 238 15 0 22 26 32 38 46 54 64 76 358 20 0 30 34 42 50 60 72 86 102 476 51 Study Design Four separate studies were conducted in this investigation because the design included 2 levels of population R 2 (approximately 0.2 and 0.6) and 2 levels of number of predictors (4 and 9). Study 1 was with four predictors under low R 2 condition, Study 2 was with four predictors under high R 2 condition, Study 3 was with nine predictors under low R 2 condition, and Study 4 was with nine predictors under high R 2 condition. A s shown in Table 4, there were four levels of non-normality, three levels of sample size, three levels of percent missing, and two levels of missing pattern. Each study was a 4 (non-normality) x 3 (percent missing) x 2 (missing pattern) fully crossed factorial design, across 3 levels of sample sizes, giving rise to 72 conditions. The within-factor was M D T including mean substitution, pairwise deletion, regression imputation, and E M method of treating missing data. Table 4 Design Factors and Levels for the Study Design Factors Distribution R Predictors Sample Size Pattern Percent Missing (Sk., Kurt.) (0, 0) Low (.2±.05) 4 94 Monotonic 10 0 .3) High ( .6± .05) 9 153 Non-monotonic 15 (1-8,6) 265 20 (3, 25) Procedure Population correlation matrices were used to generate multivariate normal random variables using data generation algorithms built in E Q S (Bentler, 1989), a widely used structural equations modeling program. Another reason for choosing E Q S is that it has a built-in procedure developed by Fleishman (1978), whose equations were extended to cater for multivariate data by Vale and Maurel l i (1983). The procedure makes it possible to generate data with specified skew and kurtosis, while keeping the correlations invariant. This is necessary i f comparisons are to be 52 made across different data sets. The correlations used in the study represent typical values found in educational and psychological research as identified in the review of studies. The two levels of number of predictors 4 and 9, gave rise to 5 and 10 variables, respectively, if Y is included. The use of 5 variables implies a 5 by 5 zero order correlation matrix and the use of 10 variables implies a 10 by 10 zero order correlation matrix. Considering that these correlation matrices are symmetric, one needs 10 bivariate correlations to construct a zero order correlation matrix for 5 variables, and 45 bivariate correlations to construct a zero order correlation matrix for 10 variables. In order to construct correlation matrices for each level of R 2 , a set of sixty correlations from .01 to .60 in increasing steps of .01 was used, from which 10 and 45 correlations, respectively, were randomly selected using a table of random digits. Only positive values were considered because the direction of relationship would not affect generalizability. The interval .01 to .60 was selected because in the review of 177 articles in four journals discussed earlier, the distribution of minimum correlations across all studies had a median of .01 and the distribution of maximum correlations had a median of .60. The randomly selected correlations were then rearranged from top to bottom of a lower triangular matrix, with the first column representing correlations between Y and X's and the remaining columns representing inter-correlations among the X's. Using matrix data input in SPSS computer software (SPSS Base System Reference Guide, Release 6.0, 1993), the variance in Y explained by the X's (i.e., multiple R 2) was obtained. If the resulting value of R 2 was not as desired (i.e., 0.20 ± .05 for L O W R 2 or 0.60 ± .05 for HIGH R 2), then another correlation was randomly chosen from the set of 60 to replace a single entry in the correlation matrix. The replaced element was dictated by whether the value of obtained R 2 was larger or smaller than desired. This procedure was repeated until the desired R 2 values were obtained within R 2 ± .05. Once the matrices were generated, they were examined for positive definiteness. The steps used in constructing correlation matrices are summarized as follows: 53 1. Start with a set of sixty correlations from .01 to .60 in increasing steps of .01. 2. Randomly select correlations in (1) to get the elements in the lower triangle of a 5 x 5 or a 10 x 10 correlation matrix. 3. For the correlation matrix generated in (2), determine the value of R 2 using matrix data input in SPSS. 4. Check whether the R 2 in (3) is as desired (within R 2 ± .05). If R 2 is as desired, then use the correlation matrix for data generation. If not, then randomly select another correlation from (1) to replace one value in the correlation matrix. The correlation to be replaced was selected on the basis of whether the value of R 2 in (4) was more or less than the desired value. 5. Repeat (4) until the desired R 2 is obtained. The population correlation matrix for the model, Y = .140X] + .206X 2 + .244X 3 + .137X 4 , having R 2 = 0.19264, was as follows: Y X i X 2 X 3 X 4 1 .26 1 .25 .14 1 .30 .26 .01 1 .23 .20 .16 .13 1 For the four predictor model, Y = .321X, + .253X 2 + .333X 3 + .220X 4 , having R 2 = 0.59437, the correlation matrix was as follows: Y x, x 2 x 3 X 1 52 1 51 .34 1 60 .32 .26 1 45 .03 .28 .45 1 54 The population correlation matrix for Y = .113Xj + .060X2 + .140X3 + .056X4 + .097X5 + .128X6 + .144X7 + .168X8 + .130X9, having R 2 = .21267 was as follows: Y X] X 2 X3 X4 X5 Xg X 7 X 8 X 9 r 1 1 .21 1 .15 .04 1 .21 .16 .01 1 .20 .19 .14 .13 1 .18 .10 .09 .02 .17 1 .19 .11 .13 .07 .08 .03 1 .22 .18 .15 .09 .16 .07 .06 1 .23 .05 .08 .01 .20 .13 .04 .05 .21 .03 .12 .14 .10 .14 .07 .02 For the nine predictor model, Y = .119Xi + ,304X2 + .053X3 + .264X4 + .053X5 + .336X6 -.284X7 + .260X8 + .024X9 having R 2 = 0.58590, the correlation matrix was as follows: Y x, x 2 x 3 x 4 x 5 Xg X 7 X 8 X9 " 1 .21 1 .45 .08 1 .35 .26 .32 1 .46 .21 .12 .31 1 .21 .18 .14 .16 .09 1 .55 .10 .39 .32 .37 .13 1 .13 .28 .41 .42 .23 .39 .30 1 .45 .11 .22 .34 .20 .48 .16 .18 1 .34 .20 .36 .07 .10 .12 .24 .19 .46 1 With the resulting correlation matrices, E Q S (Bentler, 1989) was used to generate 1000 random samples with (skew, kurtosis) = (0, 0), (1, 3), (1.8, 6), (3, 25) with three sample sizes of 94, 153, and 265. The seed for data generation was 524697. Simulation Program A simulation program was written that created the pattern of missing data, performed a simultaneous linear multiple regression analysis upon implementing a M D T , and appended the 55 parameter estimates and standard errors of regression coefficients in an output file. The routine for creating missing pattern was verified manually by examining hard copies of a set of data files. For MS, the mean of the variable was substituted for all the missing cases in that variable. For PW, the mean and variance for each variable were estimated using all available observations for that variable, and the covariances between pairs of variables were estimated using only cases that were complete in both variables. For RS, the missing values were predicted from a linear regression equation between X, and the predictor with no missing values, X ! . For E M , the initial means, variances, and covariances were based on the values under RS. Figure 3 shows a flowchart of the E M method for imputing the missing values in X , as implemented in the simulation program. The accuracy of MS and PW algorithms were checked using SPSS statistical program (SPSS Inc., 1993). The accuracy of RS was checked using B M D P A M (Dixon, 1988) with the command /ESTIMATE M E T H O D = REGR T Y P E = C O M P L E T E Similarly, the accuracy of the E M algorithm was checked using B M D P A M (Dixon, 1988) with the command /ESTIMATE M E T H O D = REGR T Y P E = M L Considering that the matrix X'X may not be positive definite under pairwise deletion method, the simulation program was tailored to flag such samples. The Gauss-Jordan decomposition approach, adapted from Numerical Recipes (Press, Teukolsky, Vetterling, & Flannery; 1992), was used for matrix inversion. As described in Marascuilo and Levin (1983), the matrix of predictors X was not augmented by a column of ones in the first column because the estimation of Po was not required. By partitioning the variance-covariance matrix of the original 56 data set into X x x and Z „ , the vector of regression coefficients was calculated using the equation A " ' A b = £ xx ~Lxy. The/? standard errors of b i , b 2,...,bp were calculated using lMSresX xx N-l where MSres denotes the estimate of the residual variance about the estimated regression equation for p predictor variables and N is the sample size. The parameter estimates from the simulation program were congruent to those from B M D P A M (Dixon, 1988) to six decimal places. Calculate ~Lmodel, the model variance-covariance matrix. Estimate and impute missing values in each X , from linear regression equation of X , on X j . Calculate the variance-covariance matrix, for Y , X j , . . predictors .Xp, for p No Regress the current X , on X i . Replace imputed values with new values using current regression equation. Yes Stop Figure 3: Flowchart for EM method used in the study 58 Evaluation Criteria According to Larsen and Marx (1981), four qualities of a good estimator are unbiasedness, efficiency, consistency, and sufficiency. Unbiasedness is one way to decide whether or not an estimator's distribution is suitably centered. In other words, i f 8 is a parameter to be estimated, then its estimator W should be somewhat "centered" with respect to 8. If it is not, the estimator w i l l tend either to overestimate or underestimate 8, a condition that is not desirable. A n estimator is said to be unbiased i f "on the average", it w i l l yield the true parameter value; that is, i f the underlying experiment is repeated with infinitely many samples of size T, the average value of the estimates from all those samples w i l l be equal to the true value. More formally, let Yu Y2, Yn be a random sample from/y(y; 6). A n estimator W = h(Yi, Y2, Yn) is said to be unbiased i f E(VV) = 8, for all 6. The term bias should be used to describe a situation in which there is a tendency for an estimator to consistently overestimate or underestimate parameters across a large number of samples. Unbiased estimators are not unique (Larsen & Marx, 1981). This implies that an unbiased estimator may not necessarily be the best. In order to select the best estimator, we need to compare their dispersions and choose the estimator with the smallest dispersion, so that the probability of its being close to the true 8 w i l l be large (Larsen & Marx, 1981). This property of a good estimator is called efficiency. Put differently, suppose Wi and W2 are two unbiased estimators of 8. Then, W\ is said to be more efficient than W2 i f the variability of W\ is less than that of W2. Although the concepts of unbiasedness and efficiency lead to the most basic characterizations of point estimates, consistency is another important property of a good estimator. Consistency is attractive because it says that as the sample size increases indefinitely, the distribution of the estimator becomes entirely concentrated at the parameter value (Goldberger, 1991). Roughly speaking, an estimator is consistent i f Wn, the n'h member of an 59 infinite sequence of estimators, Wj, W2,..., W„,..., lies arbitrarily close to the parameter being estimated, as n becomes large (Larsen & Marx, 1981). This has two immediate implications: (1) Wn is asymptotically unbiased and (2) the variance of Wn converges to 0. In mathematical terms, an estimator W„ = h(Y\, Y2, Y„) is said to be consistent (for ff) if it converges stochastically to 0, that is, if for all £ > 0 and 8 > 0 , there exists an n(e,8) such thatP(jW n -d\ < fi) > 1-8, for n > (e,<5). Sufficiency is the fourth property of a good estimator. It is related to the amount of "information" a given estimator contains. If Yu Y2, Y„ is a random sample from/Ky; ff), then the statistic W = h(Y\, I2. • • •> Y„) is said to be sufficient for 6 if, for all 6 and all possible sample points, the conditional probability density function of Y\, Y2, ..., Yn given w does not depend on 8, either in the function itself or in the function's domain. Whereas the present study was not concerned with the evaluation of estimators, there exists some correspondence between the properties of a good estimator and the evaluation criteria applied in the study. The two criteria used in the evaluation of MDTs in the present study were accuracy (precision) and bias of parameter estimates. As noted by Roth (1994), the accuracy of parameter estimates is usually the criterion for the evaluation of MDTs. The concept of accuracy of parameter estimates is not equivalent to that of efficiency of an estimator, although the two concepts are related because they both make use of dispersion procedures. For example, it is possible to investigate the relative accuracy of two estimators, the mean and the median. But when only one estimator is used as in the present study, it is the spread of the parameter estimates around a population value that is of interest, and the magnitude of the spread should depend on which M D T was used. Raymond and Roberts (1987) defined accuracy of parameter estimates as the amount of dispersion found around a population parameter. Judge, Hi l l , Griffiths, Liitkepohl, and Lee (1988) stressed the importance of considering the bias of estimators. Note that the bias of estimators is similar to the bias of parameter 60 estimates, but the two are not equivalent. The term bias in parameter estimates may be used to describe a situation in which there was a tendency for an M D T to consistently overestimate or underestimate parameters across a large number of samples following treatment of missing data by that M D T (Roth, 1994). In other words, analysis of bias in parameter estimates involves the examination of any consistent errors due to MDTs. Judge et al. (1988) noted that the estimator with the smallest dispersion (i.e., the most accurate), should be preferred even if it is biased. In a similar manner, the M D T under which the parameter estimate of interest has the smallest dispersion around the true value should be preferred even if it is biased. Although Finkbeiner (1979) used both accuracy and bias of parameter estimates for assessing the optimal performance of MDTs, the best M D T was that with the smallest dispersion regardless of bias. There is no consistency among researchers on which measure of dispersion is the most preferred in the evaluation of MDTs. Based on my literature review, previous researchers have used Euclidean distance, root mean square error (RMSE), mean square error (MSE), and residual sum of squares of deviations, among others. According to Judge et al. (1988) and Barford (1985), two important methods for establishing estimation accuracy are mean square error (MSE) and mean absolute error (MAE), the choice of which one to use being arbitrary. Either way we obtain a positive quantity. The smaller either quantity is, the greater the accuracy of the parameter estimate. Intuitively, both M S E and M A E represent the average loss or risk incurred in the estimation of the parameter. Lyon (1970) noted that M S E and M A E are linearly related, with M S E being about 1.25 times larger than M A E when sample size is greater than 4. The advantage of M S E over M A E is that M S E is more mathematically tractable, a characteristic not required in the present study. Shchigolev (1960/1965) noted that M A E offers an excellent option as a criterion for evaluating the accuracy of parameter estimates if the quality of the measurements is poor, a condition that is quite likely with missing data. For this reason, M A E was used as the criterion for M D T evaluation in the present study. The M A E was defined as 61 1 A — y A where B is the number of parameter estimates (1,000 in this study), 6b is the b& parameter estimate, and 6 is the population parameter. Bias was defined as follows: Bias= E(6)-9 A whereE(6) is the mean of the 1,000 parameter estimates per cell, and 6 is the true parameter value. In the third and last part of the analysis, the performance of MDTs was assessed with respect to actual coverage of confidence intervals when nominal alpha is .05. This was achieved by first constructing confidence intervals using each sample regression coefficient and its corresponding standard error. The formula used in the construction of the confidence intervals was [b,-- 1.96SE(b,)] < p\< [b, + 1.96SE(b,)], where b, is the sample estimate of p„ the population regression coefficient for predictor variable X„ and SE(b,) is the standard error of b,. A l l confidence intervals spanning the actual population regression coefficient used in data generation were counted and converted into percentages. Good coverage probabilities were those that were closest to 95%. The best M D T was selected on the basis of magnitude of mean absolute error of estimation. However, if the corresponding bias was the least, then the performance of that M D T was considered optimal. Analysis There were four analyses. First, as in Milligan (1980, 1981, 1989a) and Donoghue (1995), a fully crossed factorial analysis of variance was conducted in order to determine the effects of pattern, percent missing and non-normality on absolute error of estimation across different sample sizes. The independent variables were missing pattern, percent missing, and non-normality. Second, M A E for parameter estimates in each cell was calculated and graphed. 62 Third, bias in parameter estimates in each cell was calculated and graphed. Fourth, mean coverage probability was calculated and graphed. The graphs were used as an aid for evaluating the performance of MDTs. Given the very large amount of data (72,000 regression analyses for each MDT), many of the main effects and some two-way interactions were statistically significant (p < .001). The purpose of the A N O V A was to summarize the data and help to highlight important effects. Therefore, a measure of effect size was adopted in place of traditional significance testing. Usually, rf would be used in this context, where ^ 2 _ effect As noted by Donoghue (1995), this index has a disadvantage in large designs in that the denominator contains not only error variance and systematic variance of interest, but also irrelevant systematic variance of other factors in the design. The larger the design becomes, the more apparent this effect becomes. This defect was worsened by the fact that some factors had very large effects, thus obscuring the effects of other factors. For this reason, as recommended by Donoghue, the following alternate equation for effect size was used: 2 _ SSeffect 'lull ~ Considering that statistically significant effects may not necessarily be of practical significance, a practical criterion of 7J 2 ai t > .03 was selected. Although this criterion is somewhat arbitrary, it was chosen because computer intensive studies in other areas have used similar criteria (e.g., co2 > .03 was used by Anderson & Gerbing, 1984, and Gerbing & Anderson, 1985; n 2 a i t > .03 was used by Donoghue, 1995). 63 Summary In the present chapter, the advantages of simulation methodology were highlighted. The greatest advantage of simulation methodology is that it allows the investigator to manipulate design conditions that would otherwise be impossible to include in a study. Issues in data generation were also presented. It was noted that the choice of random number generator is crucial in any simulation study of this nature. Higher-order congruential generators are generally preferred to multiplicative linear congruential generators. Several factors were identified that may influence the performance of MDTs in a multiple linear regression context. These include sample size, missing pattern, percent missing, non-normality, size of multiple R 2 , and number of predictors. It was also noted that the selection of typical levels of design factors is necessary to enhance generalizability of findings. In the present study, the choice of levels of design factors was based on a review of four psycho-educational research journals. However, the review did not help in the choice of levels of non-normality. Previous research on typical levels of non-normality was used in this case. The design of the present study was discussed. Four separate studies were conducted in the investigation (2 levels of number of predictors by 2 levels of multiple R 2 ). In each study, the independent variables were missing pattern (2 levels), percent missing (3 levels), and non-normality (4 levels). The within-factor was M D T (mean substitution, pairwise deletion, regression imputation, and E M method). Upon generation of data using EQS (Bender, 1993) at 3 levels of sample size (N = 94, 153, 265), a customized simulation program was used to delete part of the data to create monotonic and non-monotonic patterns. The program was then used to impute the deleted data with different MDTs. The simulation program ran a simultaneous multiple linear regression analysis. The standard multiple regression output (multiple R2esumate, regression coefficients, and standard errors of the regression coefficients) was saved and imported into SPSS for analysis. 64 A 2 x 3 x 4 fully crossed factorial A N O V A was conducted using absolute error of estimation for R 2 e s u m ate a n d regression coefficients as the dependent variables. The design factors were missing pattern, percent missing, and non-normality. Analysis was at each level of sample size. The effect size (r| 2) was chosen as a criterion for practical significance. Accuracy, as measured by mean absolute error of estimation for R2estimate and mean absolute error of estimation for regression coefficients, was used to evaluate the M D T s . Bias in R2estimate and regression coefficients was used to determine which M D T s yielded estimates "close" to the true parameter. Lastly, coverage probability for regression coefficients was used to determine the Type I error rates under each M D T . Results are presented in the next chapter. 65 CHAPTER 4 RESULTS In this chapter, results from the four studies are presented in the following order: 1. Study 1: Four predictors under low R 2 condition 2. Study 2: Four predictors under high R 2 condition 3. Study 3: Nine predictors under low R 2 condition 4. Study 4: Nine predictors under high R 2 condition. Tables and graphs for mean absolute error are presented for monotonic and non-monotonic missing patterns, arranged in increasing order of sample size, percent missing, and non-normality. For ease of comparison, results in tables and graphs are consistent across studies, and graphical plots at each level of sample size have the same scale. Similarly, tables and graphs for bias in parameter estimates are provided in Appendix A and B , respectively. Study 1: Four Predictors under Low R 2 Condition Effects on Absolute Error for R 2 ^ ^ The effects of design variables on absolute error for R2estimate are listed in Table 5. Using r|2 ait > .03 as the criterion, the table shows that none of the effects was of practical significance under all M D T s with samples of size 94 and 153. Under sample size 265, the effect of percent missing on error of estimation was of practical significance under E M and R S , the largest effect being under E M . Relative Performance of M D T s in the Estimation of Population R 2 Table 6 contains means and standard deviations of absolute error for R2estimate- The influence of non-normality on the performance of M D T s at sample size 94 is revealed in Figure 4. The performance of all M D T s as measured by the mean absolute error of estimation deteriorated with increasing non-normality. Under monotonic missing pattern, P W treatment of 66 missing data was the worst and E M treatment the best. However, there was little differentiation in the performance of E M , RS and MS, especially when data were normal (skew = 0 and kurtosis = 0). This differentiation increased with increasing non-normality. Under non-monotonic pattern, it was again E M that outperformed other MDTs, followed by RS, MS and PW, in that order. This finding was similar to that for monotonic pattern, except that MDTs differed more in performance under non-monotonic pattern than under monotonic pattern. MDTs differed the least when data were normal, and differed most at the highest level of non-normality (skew = 3 and kurtosis = 25). E M and RS methods performed almost equally well. Table 5: Effects on absolute error for R 2 „ « i m a f P fulfilling the n 2„i, > .03 criterion Treatment Sample Size Effect df SS e f f e c t SS e r r o r 7? 2 a l t EM 94 153 265 Percent Missing 2 1.77 48.67 .035 MS 94 153 265 -PW 94 153 265 -RS 94 153 265 Percent Missing 2 1.77 50.24 .034 Note: EM = Expectation-maximization method, MS = Mean substitution, PW = Pairwise deletion, RS -Regression imputation, - = No effect of practical significance. The influence of non-normality on the performance of MDTs at sample size 153 is revealed in Figure 5. The findings were similar to those under sample size 94, except that mean absolute error of estimation under sample size of 153 was much smaller. Figure 6 shows the influence of non-normality on the performance of MDTs at sample size 265. In this case the mean absolute error of estimation was much smaller than under sample size 94 and 153. 67 Table A . l in Appendix A contains bias in R2estimate under low R 2 condition with four predictors. The bias in R2estimate across all sample sizes varied from - .04 to .06, a range of .10. Figure B . l in Appendix B is a set of typical graphical plots for bias in R2esumate ( N = 94) showing that all the M D T s had positive bias at 10% missing. A t this level, the smallest bias was under E M . A t 15% missing, the smallest bias was under E M for monotonic pattern. 68 Table 6 Means and standard deviations* of absolute error for R 2estimate Monotonic Missing Pattern Non-monotonic Missing Pattern N Percent Missing Normality (sk., kurt.) EM Imputation Mean Substitution Pairwise Deletion Regress. Imputation EM Imputation Mean Substitution Pairwise Deletion Regress. Imputation 94 10 (0,0) (1.3) (1.8,6) (3,25) 0589(045) 0628(050) 0701(057) 0885(080) 0598(046) 0639(051) 0714(058) 0912(082) 0635(050) 0684(053) 0761(060) 0965(087) 0594(046) 0634(050) 0708(058) 0898(081) 0532(041) 0550(041) 0607(045) 0706(059) 0585(046) 0622(047) 0691(053) 0852(072) 0616(047) 0657(049) 0726(056) 0894(077) 0548(042) 0568(042) 0624(047) 0719(060) 15 (0,0) (1,3) (1.8,6) (3,25) 0570(043) 0612(046) 0678(054) 0850(076) 0583(044) 0628(048) 0697(056) 0886(080) 0630(049) 0672(052) 0745(059) 0947(085) 0577(043) 0621(047) 0688(055) 0865(078) 0526(038) 0570(043) 0681(056) 0432(031) 0575(044) 0609(046) 0665(051) 0821(069) 0603(046) 0642(049) 0712(055) 0884(078) 0523(039) 0540(039) 0584(044) 0689(057) 20 (0,0) (1,3) (1.8,6) (3,25) 0557(041) 0602(044) 0666(052) 0824(072) 0576(042) 0623(046) 0691(055) 0869(077) 0620(048) 0664(050) 0735(057) 0931(085) 0565(041) 0612(045) 0678(053) 0842(074) 0496(036) 0515(037) 0554(042) 0667(054) 0561(042) 0595(044) 0647(050) 0792(067) 0599(045) 0636(048) 0709(054) 0910)108) 0506(037) 0524(038) 0563(042) 0672(055) 153 10 (0,0) (1,3) (1.8,6) (3,25) 0457(034) 0474(036) 0526(042) 0683(058) 0462(035) 0480(037) 0534(042) 0700(060) 0487(038) 0505(040) 0564(044) 0723(061) 0460(034) 0477(037) 0530(042) 0690(059) 0421(032) 0413(032) 0444(035) 0521(042) 0453(034) 0457(036) 0496(040) 0619(052) 0471(036) 0483(038) 0535(042) 0669(060) 0431(032) 0423(033) 0455(036) 0528(043) 15 (0,0) (1.3) (1.8,6) (3,25) 0448(033) 0468(034( 0514(039) 0662(056) 0453(033) 0475(035) 0525(040) 0685(058) 0483(037) 0500(039) 0555(044) 0710(060) 0451(033) 0472(035) 0520(040) 0671(057) 0432(031) 0418(032) 0442(034) 0528(040) 0460(034) 0455(035) 0487(039) 0602(050) 0461(036) 0472(037) 0523(041) 0661(060) 0439(032) 0423(033) 0447(035) 0531(041) 20 (0,0) (1,3) (1.8,6) (3,25) 0461(033) 0480(034) 0519(038) 0656(054) 0466(034) 0489(035) 0534(040) 0685(057) 0481(037) 0495(039) 0547(043) 0698(059) 0464(033) 0485(035) 0526(039) 0666(055) 0456(032) 0455(032) 0476(034) 0558(040) 0469(034) 0464(035) 0499(038) 0604(048) 0454(035) 0469(036) 0519(047) 0669(062) 0456(032) 0453(033) 0475(034) 0558(040) 265 10 (0,0) (1.3) (1.8,6) (3,25) 0346(027) 0351(027) 0383(030) 0483(038) 0348(027) 0353(027) 0386(030) 0491(039) 0359(027) 0372(029) 0413(032) 0521(043) 0347(027) 0352(027) 0385(030) 0487(039) 0341(025) 0338(025) 0359(027) 0415(031) 0349(026) 0352(027) 0377(028) 0453(036) 0346(027) 0357(027) 0397(030) 0488(039) 0344(026) 0341(026) 0363(027) 0417(032) 15 (0,0) (1,3) (1.8,6) (3,25) 0361(028) 0368(028) 0397(030) 0484(037) 0370(028) 0370(028) 0400(030) 0495(038) 0369(029) 0369(029) 0409(032) 0513(042) 0370(028) 0370(028) 0399(030) 0489(038) 0372(027) 0377(027) 0394(028) 0453(031) 0366(026) 0368(027) 0391(029) 0459(034) 0340(026) 0353(027) 0394(029) 0485(039) 0370(026) 0374(027) 0392(028) 0451(031) 20 (0,0) (1.3) (1.8,6) (3,25) 0393(029) 0400(029) 0424(030) 0496(036) 0390(029) 0398(029) 0425(031) 0508(037) 0355(027) 0367(028) 0405(031) 0504(041) 0392(029) 0400(029) 0425(031) 0502(037) 0422(028) 0427(028) 0443(029) 0508(032) 0388(027) 0389(027) 0411(028) 0482(034) 0337(026) 0353(027) 0394(029) 0490(039) 0415(028) 0420(028) 0437(029) 0506(032) * All values in the table are preceded by an omitted decimal point. Standard deviations are in parentheses. Standard errors of the means can be obtained by dividing each standard deviation by 1000 s 4. CC O CC O < < MONOTONIC PATTERN N = 9 4 , PM = 10 (1.3) (1.8,6) NON-NORMALITY (SKEW, KURTOSIS) N = 94, PM = 15 * (0,0) ( 1 , 3 ) ( 1 . 8 , 6 ) ( 3 , 2 5 ) NON-MONOTONIC PATTERN N = 94, PM = 10 (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 9 4 , PM = 15 (3, 25) (0,0) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) (3, 25) N = 94, PM = 20 (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) (3.25) N = 94, PM = 20 (0,0) (1.3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) LEGEND: MS; PW; RS: EM Figure 4: Mean absolute error for R2estimate across levels of non-normality ( N = 94) Note: N = Sample size, P M = Percent missing, R S Q = R2esumate 70 MONOTONIC PATTERN 0 N = 153, P M = 1 0 V) .10 , cc CC O .09 u_ DC O 0 8 CC CC UJ .07 , (0-0) (1.3) (1.8,6) (3,25) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) NON-MONOTONIC PATTERN N = 153, P M = 10 .10 o 5 03 J I (0.0) (1.3) (1.8,6) (3,25) NON-NORMALITY (SKEW, KURTOSIS) N = 153, P M = 1 5 N = 153, P M = 15 (0 ,0 ) (1 ,3 ) (1 .8 ,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) (3 ,25) CC CC o Li-CC o CC m < < UJ 5 (0 ,0 ) ( 1 , 3 ) (1 .8 ,6 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) (3, 25) Figure 5: Mean absolute error for R2estimate across levels of non-normality ( N = Note: N = Sample size, P M = Percent missing, R S Q = R2estimate 153) MONOTONIC PATTERN N = 265, P M = 10 (0, 0) (1, 3) (1.8, 6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) NONMONOTONIC PATTERN N = 265, P M = 10 (0.0) (1.3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) (1.8,6) N = 2 6 5 , PM = 15 N = 265, P M = 15 (0,0) (1,3) (1.S N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) (3, 25) (0,0) (1.3) (1.8,6) (3, 25) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 265, P M = 20 N = 265, P M = 20 (0, 0) (1, 3) (1.8, 6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) (0,0) (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) (3, 25) LEGEND: MS • P W ; RS; EM Figure 6: Mean absolute error for R2estjmate across levels of non-normality ( N = 265) Note: N = Sample size, P M = Percent missing, R S Q = R2estimate Effects on Absolute Error for Regression Coefficients The effects of design variables on absolute error for bi are listed in Table 7. For b 2 and b 4 , no effect of practical significance was observed. For b 3 , however, the effect of non-normality was of practical significance for all M D T s and at each level of sample size. Using r | 2 a i t > .03 as the criterion, the table shows that for b i , the effect of non-normality was of practical significance under E M and R S . The effect of pattern of missing data was of practical significance under M S and P W . Also , the effect of pattern of missing data was the strongest, with little differentiation between E M and RS where its effect was of practical significance. Table 7 Effects on absolute error for regression coefficients fulfilling the criterion of f i 2 ^ > .03* Sample size M D T b, b 2 b 3 b 4 94 E M _ N L (033) MS - - N L (034) -PW - - N L (034) -RS - - N L (034) -153 E M NL(031) - N L (036) MS PT (030) - N L (037) -PW PT (030) - N L (037) -RS NL(031) - N L (037) -265 E M N L (038) - NL(041) _ MS PT (080) - N L (043) -PW PT (079) - N L (043) -RS N L (041) - N L (042) -* Values of r | 2 a U are in parentheses preceded by an omitted decimal point. Note: E M = E M imputation, MS = Mean substitution, PW = Pairwise deletion, RS = Regression imputation, PT = Pattern, N L = Non-normality, - = No effect of practical significance. Relative Performance of M D T s in the Estimation of Regression Coefficients Table 8 contains mean absolute error and the corresponding standard deviation for b i . M D T s provided similar results for b 2 , b 3 , and b 4 , and for this reason, only results for bj are 73 presented. The influence of non-normality on the performance of MDTs at sample size 94 is revealed in Figure 7. The performance of all MDTs deteriorated with increasing non-normality as measured by mean absolute error for b\. This was regardless of missing pattern. At sample size 94, under monotonic pattern of missing data, MDTs overlapped in pairs; MS and PW formed one pair and E M and RS formed another pair. This outcome was observed at all levels of percent missing. Under monotonic pattern of missing data, E M and RS consistently outperformed MS and PW. At sample size 94 with non-monotonic pattern, again pairing of MDTs similar to those under monotonic pattern was observed, with E M and RS outperforming MS and PW. Whereas the mean absolute error of estimation under E M and RS was smaller with non-monotonic pattern than with monotonic pattern, the mean absolute error of estimation under MS and PW was smaller under monotonic pattern than under non-monotonic pattern. The influence of non-normality on the performance of MDTs at sample size 153 and 265 are revealed in Figures 8 and 9, respectively. The outcome was similar to that under sample size 94. In sum, the best performance was under E M and RS, regardless of missing pattern. However, at each level of percent missing, the mean absolute error of bi under E M and RS was smaller under non-monotonic pattern than under monotonic pattern. Also, at each level of percent missing, the mean absolute error of estimation under MS and PW was smaller with monotonic pattern than with non-monotonic pattern. Table A.2 in Appendix A shows bias in bi. Figure B.2 in Appendix B shows graphs for the bias in bi (N = 94). The bias across all conditions of the design variables had a range of .16. The smallest bias was under RS across all levels of the design variables. However, the bias under E M was very close to that under RS. Whereas E M and RS consistently underestimated Pi, MS and PW, which had equivalent estimators, consistently overestimated Pi. The bias under non-monotonic missing pattern was generally larger than that under monotonic missing pattern. The 74 maximum range in bias in estimating p2, P 3 , and p4 was .04, a relatively' smaller value compared to that of Pi. This indicates that the largest bias was for Pi Therefore, graphs for p2, P3, and p4 were omitted. 75 Table 8 Means and standard deviations* of absolute error of estimation for bi Monotonic Missing Pattern Non-monotonic Missing Pattern N Percent Missing Normality (sk., kurt.) EM Imputation Mean Substitution Pairwise Deletion Regress. Imputation EM Imputation Mean Substitution Pairwise Deletion Regress. Imputation 94 10 (0,0) (1,3) (1.8,6) (3,25) 0765(056) 0828(062) 0907(071) 1030(087) 0837(061) 0899(068) 0979(077) 1109(101) 0837(061) 0897(068) 0978(077) 1108(101) 0722(057) 0834(063) 0913(072) 1035(089) 0658(047) 0701(049) 0768(055) 0919(064) 0959(071) 1024(081) 1094(091) 1295(128) 0959(071) 1023(080) 1091(091) 1292(128) 0639(047) 0674(049) 0741(055) 0895(065) 15 (0,0) (1,3) (1.8,6) (3,25) 0749(055) 0816(061) 0896(070) 1021(088) 0860(063) 0920(070) 0997(079) 1128(106) 0860(063) 0921(069) 0998(078) 1128(105) 0755(056) 0820(062) 0897(070) 1019(089) 0629(046) 0680(048) 0742(053) 0913(067) 1045(076) 1103(086) 1168(095) 1380(134) 1044(076) 1101(086) 1164(095) 1372(133) 0607(046) 0656(048) 0720(053) 0912(070) 20 (0,0) (1,3) (1.8,6) (3,25) 0738(055) 0803(061) 0880(069) 1011(088) 0886(065) 0941(072) 1015(081) 1150(108) 0885(065) 0941(072) 1015(081) 1149(108) 0737(055) 0802(061) 0877070) 1001(089) 0623(045) 0673(049) 0747(055) 0906(071) 1109(079) 1160(089) 1220(099) 1430(137) 1108(079) 1158(089) 1214(099) 1421(137) 0598(045) 0648(049) 0726(054) 0920(075) 153 10 (0,0) (1,3) (1.8,6) (3,25) 0575(043) 0620(046) 0687(051) 0820(071) 0643(047) 0681(051) 0742(058) 0873(078) 0642(047) 0681(051) 0741(058) 0873(078) 0580(044) 0621(046) 0686(052) 0814(071) 0514(036) 0558(037) 0613(041) 0731(051) 0817(057) 0861(066) 0910(076) 1120(105) 0816(057) 0860(066) 0907(075) 1114(105) 0490(036) 0526(037) 0581(040) 0703(050) 15 (0,0) (1,3) (1.8,6) (3,25) 0564(042) 0604(045) 0666(051) 0800(070) 0668(048) 0707(053) 0762(061) 0895(082) 0667(048) 0705(053) 0761(061) 0893(083) 0568(042) 0606(045) 0664(051) 0790(070) 0502(034) 0548(036) 0593(040) 0709(051) 0919(061) 0962(070) 1005(080) 1215(109) 0918(061) 0961(070) 1001(079) 1207(109) 0471(034) 0513(035) 0565(040) 0695(052) 20 (0,0) (1,3) (1.8,6) (3,25) 0548(041) 0590(044) 0651(050) 0783(068) 0697(049) 0732(055) 0787(063) 0911(085) 0695(049) 0730(055) 0784(063) 0910(085) 0552(041) 0591(044) 0650(050) 0771(067) 0490(034) 0525(036) 0571(039) 0691(051) 0994(065) 1034(073) 1074(082) 1272(112) 0993(065) 1033(073) 1069(082) 1264(111) 0462(034) 0502(035) 0553(039) 0690(053) 265 10 (0,0) (1,3) (1.8,6) (3,25) 0439(033) 0466(036) 0517(039) 0640(053) 0491(036) 0513(039) 0560(044) 0674(059) 0491(036) 0513(039) 0560(044) 0674(059) 0441(033) 0465(036) 0515(040) 0633(053) 0390(028) 0434(030) 0471(032) 0593(040) 0710(046) 0763(052) 0796(058) 0993(081) 0710(046) 0761(052) 0793(058) 0988(081) 0359(027) 0391(028) 0431(031) 0561(039) 15 (0,0) (1,3) (1.8,6) (3,25) 0428(032) 0455(034) 0508(037) 0614(051) 0517(037) 0541(040) 0580(046) 0692(060) 0517(037) 0541(040) 0580(046) 0692(060) 0430(032) 0454(034) 0502(038) 0607(051) 0385(027) 0424(029) 0460(031) 0571(039) 0826(050) 0873(055) 0903(061) 1089(084) 0825(050) 0872(055) 0899(061) 1083(084) 0346(026) 0386(027) 0427(030) 0554(038) 20 (0,0) (1,3) (1.8,6) (3,25) 0415(031) 0441(033) 0488(036) 0602(048) 0545(039) 0567(042) 0604(047) 0708(063) 0545(039) 0566(042) 0604(047) 0708(063) 0412(031) 0436(033) 0480(036) 0585(049) 0385(027) 0419(028) 0448(031) 0558(039) 0915(052) 0955(057) 0982(063) 1150(086) 0915(052) 0954(057) 0977(063) 1143(086) 0350(026) 0389(027) 0423(030) 0551(039) * All values in the table are preceded by an omitted decimal point. Standard deviations are in parentheses. Standard errors of the means can be obtained by dividing each standard deviation by 100014. MONOTONIC PATTERN N = 9 4 , PM = 10 (0.0) (1.3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) (3, 25) NON-MONOTONIC PATTERN N = 94, PM = 10 (1.3) N O N - N O R M A L F T Y ( S K E W , K U R T O S I S ) (3,25) N = 94, PM = 15 (0.0) (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) (1.3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 94, PM = 20 (1 .8 ,6 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) ( 3 , 2 5 ) 94, PM = 20 (1 ,3 ) ( 1 . 8 , 6 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) ( 3 , 2 5 ) LEGEND: MS; PW; RS; EM Figure 7: Mean absolute error for bi across levels of non-normality (N = 94) Note: N = Sample size, P M = Percent missing. 77 MONOTONIC PATTERN N = 153, PM = 10 (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 153, PM= 15 (0,0) (1,3) (1.8,6) (3,25) NON-NORMALITY (SKEW, KURTOSIS) N = 153, PM = 20 (0,0) (1,3) (1.8,6) (3,25) NON-NORMALITY (SKEW, KURTOSIS) NON-MONOTONIC PATTERN N = 153, PM = 10 N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 153, PM = 15 (3. 25) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 153, PM = 20 (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) LEGEND: MS; PW; RS; EM Figure 8: Mean absolute error for b! across levels of non-normality (N = 153) Note: N = Sample size, P M = Percent missing 78 MONOTONIC PATTERN N =265, PM = 10 (0,0) (1,3) (18,6) NON-NORMALITY (SKEW, KURTOSIS) N = 265, PM = 15 (3, 25) (0.0) (1.3) (1.8,6) (3,25) NON-NORMALITY (SKEW, KURTOSIS) N =265, PM = 20 o o cc O m < z < m tr O Li-tr o tr tr O < UJ 5 NONMONOTONIC PATTERN N =265, PM = 10 (0,0) (1,3) (18,6) NON-NORMALITY (SKEW, KURTOSIS) N = 265, PM = 15 (0.0) (1,3) (1-8.6) (3,25) NON-NORMALITY (SKEW, KURTOSIS) N = 265, PM = 20 NON-NORMALITY (SKEW, KURTOSIS) (0,0) (1.3) (1.8,6) NON-NORMALITY (SKEW, KURTOSIS) LEGEND: MS P W ; RS; EM Figure 9: Mean absolute error for b] across levels of non-normality (N = 265) Note: N = Sample size, P M = Percent missing 79 Coverage Probability for p\ Coverage probabilities under monotonic pattern were found to differ from those under non-monotonic pattern only for (3]. M D T s were not differentiated in their coverage probabilities for p2, p3, and p4, whose graphs looked the same and the averages extremely close. For this reason, only the coverage probabilities for Pi are reported. Table 9 contains mean coverage probabilities for Pi, under various design conditions. These means are plotted for sample size 94, 153 and 265 in Figures 10, 11, and 12, respectively. The graphs show that coverage probability decreased with increasing non-normality at al l levels of sample size. Also , there was little differentiation in coverage probability between M S and P W , and between E M and R S , and this was regardless of missing pattern. However, the difference in coverage probability between the pairs M S / P W and E M / R S was larger under non-monotonic pattern than under monotonic pattern. A l l M D T s had coverage probabilities above 90% with normal data under monotonic missing pattern. M S and P W had poor coverage probabilities (below 90%); with the lowest coverage probability (below 65%) occurring at sample size 265 under non-monotonic pattern when percent missing was 20. 80 Table 9 Mean coverage probability (%) for fti Monotonic Missing Pattern Non-monotonic Missing Pattern N Percent Normality EM Mean Pairwise Regress. EM Mean Pairwise Regress. Missing (sk., kurt.) Imputation Substitution Deletion Imputation Imputation Substitution Deletion Imputation 94 10 (0,0) 96.3 93.6 93.5 95.9 98.8 89.3 88.9 98.8 (1.3) 95.5 93.2 93.2 94.9 98.7 87.7 87.6 98.6 (1.8,6) 93.2 90.9 90.7 92.8 97.9 84.5 84.4 97.6 (3,25) 87.8 84.3 83.8 87.4 94.8 78.0 78.2 95.1 15 (0,0) 97.4 93.3 93.2 97.0 99.1 85.3 85.2 99.0 (1.3) 96.0 92.0 91.8 95.0 99.1 83.7 83.5 99.1 (1.8,6) 93.7 90.0 89.8 93.6 98.6 81.8 81.7 98.4 (3,25) 89.1 83.8 83.9 88.9 95.3 75.0 74.4 95.2 20 (0,0) 98.0 93.4 92.6 97.6 99.5 83.3 82.8 99.4 (1.3) 96.7 91.9 91.0 96.2 99.0 80.6 80.5 99.0 (1.8,6) 94.7 89.8 89.4 94.5 98.3 79.1 79.1 98.2 (3,25) 89.4 84.4 83.4 89.3 95.0 73.2 73.0 94.6 153 10 , (0,0) 97.0 94.7 94.6 97.0 99.3 85.1 84.8 99.2 (1.3) 96.4 93.0 93.1 96.1 98.6 82.5 82.1 98.7 (1.8,6) 93.8 90.0 89.9 93.6 97.8 79.8 79.3 97.7 (3,25) 88.0 84.4 84.0 87.9 93.6 71.4 71.3 94.0 15 (0,0) 97.3 94.5 94.1 97.4 99.8 80.2 80.1 99.8 (1.3) 96.4 92.4 92.2 96.0 99.3 77.4 76.9 99.3 (1.8,6) 94.3 89.1 89.2 94.0 98.4 74.9 74.8 98.7 (3,25) 89.3 83.5 83.2 89.1 94.1 68.0 68.4 94.2 20 (0,0) 98.0 93.5 92.9 97.8 99.9 76.7 75.9 99.9 (1.3) 97.3 91.4 91.4 97.1 99.3 74.5 73.5 99.4 (1.8,6) 95.6 88.5 88.1 95.6 98.6 72.1 72.6 98.7 (3,25) 90.7 83.5 82.8 91.3 95.4 66.7 66.6 95.2 265 10 (0,0) 96.9 94.1 93.9 96.6 98.8 79.2 78.9 98.8 (1,3) 95.7 93.2 92.9 95.6 97.9 76.3 75.6 98.1 (1.8,6) 93.1 90.2 90.0 93.1 97.1 74.3 74.0 97.4 (3,25) 85.0 82.6 82.2 85.2 90.4 64.4 63.8 91.0 15 (0,0) 97.3 93.1 92.7 97.4 98.9 72.2 71.5 99.2 (1,3) 96.6 92.7 92.2 96.2 98.1 69.6 69.2 98.6 (1.8,6) 94.6 90.0 89.2 94.6 97.4 66.8 65.7 97.8 (3,25) 87.2 82.5 81.6 87.1 91.6 59.4 59.0 91.7 20 (0,0) 98.1 91.3 90.6 98.1 99.4 64.3 63.9 99.6 (1.3) 97.3 90.9 90.3 97.1 98.3 63.7 63.0 98.7 (1.8,6) 95.4 87.7 97.0 95.2 97.8 60.9 60.4 98.2 (3,25) 88.5 81.7 81.0 88.8 92.6 56.5 56.2 92.9 MONOTONIC PATTERN N = 94 , P M = 10 N = ( 1 . 3 ) ( 1 . 8 , 6 ) ( 3 , 2 5 ) NON-NORMALITY (SKEW, KURTOSIS) 9 4 , P M = 15 ( 0 , 0 ) ( 1 , 3 ) ( 1 -8 ,6 ) ( 3 , 2 5 ) NON-NORMALITY (SKEW, KURTOSIS) N = 9 4 , PM = 20 ( 0 , 0 ) ( 1 , 3 ) (1 .8 ,6 ) ( 3 , 2 5 ) NON-NORMALITY (SKEW, KURTOSIS) NON-MONOTONIC PATTERN N = 94 , P M = 10 ( 0 , 0 ) ( 1 , 3 ) ( 1 -8 . 6 ) ( 3 , 2 5 ) NON-NORMALITY (SKEW, KURTOSIS) N = 9 4 , PM = 15 ( 0 . 0 ) ( 1 , 3 ) (1 .8 ,6 ) ( 3 , 2 5 ) NON-NORMALITY (SKEW, KURTOSIS) N = 9 4 , P M = 20 ( 0 . 0 ) ( 1 , 3 ) ( 1 . 8 , 6 ) ( 3 , 2 5 ) NON-NORMALITY (SKEW, KURTOSIS) LEGEND: MS PW; RS; EM Figure 10: Mean coverage probability for fjj (N = 94) 82 MONOTONIC PATTERN N = 1 5 3 , P M = 1 0 100 ' 95 1 " All 90 1 BIL 85 1 < OB 80 1 CC CL 75 1 LU AG 70 1 cc LU 65 1 > O o 60 1 < 55 I LU 2 50 i (0. 0) 0 .3) (1.8,6) (3, N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 1 5 3 , P M = 1 5 (1,3) ( 1 8 , 6 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 1 5 3 , P M = 2 0 (1.3) (1.8.6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) NON-MONOTONIC PATTERN N = 153, PM = 10 (0. 0) (1,3) (1.8.6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) 1 5 3 , P M = 1 5 (0,0) (1,3) (1.8.6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 1 5 3 , P M = 2 0 (3,25) 0.3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) (3,25) L E G E N D : M S • PW; R S ; E M Figure 11: Mean coverage probability for [3] (N=153) 83 N = M O N O T O N I C P A T T E R N 2 6 5 , PM = 10 (1 .3 ) (1 .8 ,6 ) ( 3 , 2 5 ) NON-NORMALITY (SKEW, KURTOSIS) 2 6 5 , PM = 1 5 N = (1 .3 ) (1 .8 ,6 ) (3 ,25 ) NON-NORMALITY (SKEW, KURTOSIS) 265, PM = 20 (1 .3) (1 .8 ,6 ) (3 ,25 ) NON-NORMALITY (SKEW, KURTOSIS) O tr N O N - M O N O T O N I C P A T T E R N 265, PM = 10 ( C O ) (1.3) (1.8,6) (3,25) NON-NORMALITY (SKEW, KURTOSIS) N = 265, PM = 15 (1.3) (1-8,6) (3,25) NON-NORMALITY (SKEW, KURTOSIS) ;265, PM = 20 (1.3) (1.8,6) NON-NORMALITY (SKEW, KURTOSIS) LEGEND: -MS PW; RS; EM Figure 12: Mean coverage probability for PJ (N = 265) 84 Summary In the foregoing sections, effects on absolute error for R2estimate and Pi were reported, together with the relative performance of M D T s . In addition, M D T s were compared with respect to the coverage probabilities of population regression coefficients when nominal alpha is .05. There were no effects on absolute error for bi of practical significance at sample size 94 or 153. However, at sample size 265, the effect of percent missing was of practical significance under E M and RS only. Wi th respect to mean absolute error for R2estimate, the performance of all M D T s deteriorated with increasing non-normality. The best performance was under E M , the worst under P W . The smallest bias in R2estimate across different conditions of the design variables was not under a single M D T . There was more differentiation in the performance of M D T s with non-monotonic missing pattern than with monotonic missing pattern. In each case, M D T s differed more at higher levels of non-normality than at lower levels, except at the highest level of sample size with 20% missing. The effect of missing pattern on absolute error for bi was of practical significance under M S and P W , but only when sample size was moderate or large (153 or 265). Non-normality showed an effect of practical significance on absolute error for b i under E M and R S when sample size was moderate to large (153 to 265). The strongest effect on absolute error for b] was that of missing pattern under both M S and P W when sample size was 265. It appears that with respect to effects on absolute error for b i , M S and P W were alike in their performance. Also , E M and R S were alike in that the absolute error for bi under the two M D T s was affected by non-normality. A s for the relative performance of M D T s in estimating population regression coefficients, M D T s differed mainly on the coefficient of the predictor with no missing data (bi in this case). Therefore, only the findings regarding bi are reported. For b i , the performance of M D T s deteriorated with increasing non-normality. This was regardless of missing pattern. Overall , RS and E M outperformed M S and RS in estimating pi with respect to accuracy of parameter 85 estimates. The smallest bias in b | was also under RS . Whereas M S and P W consistently overestimated Pi, E M and RS consistently underestimated Pi. The performance of M S was undifferentiated from that of P W across the two patterns. However, E M and RS were more differentiated from M S and P W under non-monotonic than under monotonic missing pattern. Coverage probabilities decreased with increasing non-normality. Both E M and RS provided coverage probabilities that were closer to the nominal value of 95% than did M S and P W . In sum, the findings from this study suggest that with fewer predictors (four in this case) under low criterion-predictor relationships, E M and R S generally were superior to M S and P W with respect accuracy in parameter estimation. However, the smallest bias was under R S . How the outcomes w i l l change when R 2 is high was examined in the next study. 86 Study 2: Four Predictors under High R 2 Condition Effects on Absolute Error for R^sumate This study was identical to Study 1 in design and analysis, except that the data were generated using high multiple R 2 (0.59437). Table 10 contains the effects of design variables on absolute error for R2estimite that fulfilled the criterion of r | 2 a i t > .03, showing that at sample size 94, the strongest effect was that of missing pattern under E M . There was no effect of practical significance under MS at sample size 94. Percent missing affected the performance of E M and RS more than it affected MS and PW. On the other hand, non-normality affected the performance of MS and RS more than it affected E M and RS. At sample size 153, there was no effect of practical significance under M S . Under PW, the effect of non-normality was of practical significance. Under E M and RS, the effects of pattern and non-normality on absolute error were of practical significance. However, their interaction effect was also significant, and thus their main effects have to be qualified. These ordinal interaction effects are displayed in Figures 13 and 14, showing that absolute error were generally smaller under monotonic missing pattern than under non-monotonic missing pattern. However, the difference in absolute error with normal data was smaller than the difference in absolute error at the highest level of non-normality (skew=3 and kurtosis=25). At sample size 265, the effect of percent missing on absolute error for R2estimate under MS was of practical significance. Under PW, the effect of non-normality was of practical significance. Under E M and RS, the effects of pattern and non-normality on absolute error for R2estimate were of practical significance. However, their interaction effect was also significant, and thus their main effects have to be qualified as was the case at sample size 153. These ordinal interaction effects are displayed in Figures 15 and 16, showing that absolute errors were generally smaller under monotonic pattern than under non-monotonic pattern. However, the difference in absolute error across missing pattern was smaller for normal data than for non-normal data. 87 Relative Performance of MDTs in the Estimation of Population R 2 Table 11 contains means and standard deviations of absolute error in R2estimate under various design conditions. The means are plotted in Figures 17, 18, and 19 for sample sizes 94, 153 and 265, respectively. The figures show that the performance of all MDTs deteriorated with increasing non-normality as measured by the absolute bias in R2estimate estimate. Overall, PW had the best performance and E M the worst. MDTs were more differentiated at the highest level of percent missing (20%). Also, at each level of percent missing, MDTs were most differentiated for normal data, and least differentiated at the highest level of non-normality (skew = 3 and kurtosis = 25). Table A.3 in Appendix A contains bias in R2estimate under high R 2 condition with four predictors. Bias in R2 e s umate varied from -.25 to .04 across all sample size conditions, giving a range of .29. This suggests a substantial difference in bias among MDTs in the estimation of population R 2 . Figure B.3 in Appendix B is a set of graphical plots for the bias in R 2 e s u m a t e (N = 94), showing that all the MDTs, except PW, tended to underestimate R 2 . The smallest bias was under PW at all levels of the design variables. 88 Table 10 Effects on absolute error for R 2 ^ ; ^ fulfilling the rj 2^ > .03 criterion Sample size M D T Effect SSeffect SS r e sidu ai df 94 E M Percent missing 3.60 74.87 2 .046 Pattern 27.08 74.87 1 .266 MS - - - -PW Non-normality 2.78 57.82 3 .046 RS Pattern 21.43 74.51 1 .223 Percent missing 3.46 74.51 2 .044 Non-normality 3.87 74.51 3 .049 153 E M Pattern 35.68 51.73 1 .408 Percent missing 3.48 51.73 2 .063 Non-normality 2.25 51.73 3 .042 PatternxNon-normality 1.73 51.73 3 .032 M S - - - - _ PW Non-normality 1.62 25.21 3 .061 RS Pattern 30.48 52.03 1 .369 Percent missing 3.65 52.03 2 .066 Non-normality 4.07 52.03 3 .073 PatternxNon-normality 2.44 52.03 3 .045 265 E M Pattern 41.99 34.06 1 .552 Percent missing 3.94 34.06 2 .104 Non-normality 2.03 34.06 3 .056 PatternxNon-normality 2.94 34.06 3 .079 MS Percent missing 1.54 39.38 2 .038 PW Non-normality .97 15.13 3 .060 RS Pattern 37.32 34.68 1 .518 Percent missing 3.96 34.68 2 .102 Non-normality 3.66 34.68 3 .095 PatternxNon-normality 3.97 34.68 3 .103 Note: E M = E M imputation, MS = Mean substitution, PW = Pairwise deletion, RS = Regression imputation, - - No effect of practical significance. .24 .22 .20 .18 o rr 1 6 rr £ .14 CE g .12 rr UJ UJ .10 J O oo CD < .08 .06 (0, 0) (1,3) (1.8, 6) LEGEND MONOTONIC NON-MONOTONIC (3, 25) NON-NORMALITY (SKEW, KURTOSIS) Figure 13: Interaction effect of pattern and non-normality on absolute error for R 2 Note: RSQ = R 2 e s t i mate. , under E M (N= 153) O oo rr rr O U_ tr o rr rr UJ UJ O oo m < .08 .06 LEGEND MONOTONIC NON- MONOTONIC (0, 0) (1.3) (1.8, 6) (3, 25) NON-NORMALITY (SKEW, KURTOSIS) Figure 14: Interaction effect of pattern and non-normality on absolute error in R2estimate under RS (N = 153) Note: RSQ = R 2 e s t i m a t e . o co rx tr O LL rx O rx rx UJ LU O GO m < LEGEND MONOTONIC NON-MONOTONIC (0,0) (1,3) (1.8,6) NON-NORMALITY (SKEW, KURTOSIS) (3, 25) Figure 15 : Interaction effect of pattern and non-normality on absolute error for R 2 e s t i m a te under E M (N = 265) Note: RSQ = R2 e s,ima.e. .24 o co m < .22 .20 .18 .16 .14 .12 o co rx rx O LL rx O rx rx UJ UJ io J, Z) .08 .06 (0,0) (1,3) (1.8,6) NON-NORMALITY (SKEW, KURTOSIS) LEGEND MONOTONIC NON-MONOTONIC (3, 25) Figure 16: Interaction effect of pattern and non-normality on absolute error for R2estimate under RS (N = 265) Note: RSQ = R 2 e s l i m a te. 91 Table 11 Means and standard deviations' of absolute error for R ^ m ^ . Monotonic Missing Pattern Non-monotonic Missing Pattern N Percent Missing Normality (sk., kurt.) EM Imputation Mean Substitution Pairwise Deletion Regress. Imputation EM Imputation Mean Substitution Pairwise Deletion Regress. Imputation 94 10 (0,0) (1,3) (1.8,6) (3,25) 0630(049) 0666(051) 0721(056) 0884(067) 0608(047) 0644(050) 0707(054) 0902(067) 0538(040) 0585(044) 0674(051) 0913(083) 0615(047) 0654(047) 0714(055) 0898(068) 0967(055) 1091(059) 1108(064) 1511(087) 0620(048) 0648(051) 0691(055) 0878(069) 0515(039) 0548(043) 0624(049) 0796(076) 0788(054) 0900(059) 0941(063) 1400(087) 15 (0,0) (1,3) (1.8,6) (3,25) 0813(055) 0826(058) 0851(062) 0952(074) 0720(053) 0745(054) 0785(058) 0933(069) 0535(040) 0582(044) 0668(051) 0900(082) 0752(054) 0777(056) 0813(060) 0948(073) 1388(054) 1532(058) 1541(066) 1935(092) 0729(054) 0771(058) 0801(062) 1043(080) 0510(039) 0548(043) 0621(049) 0808(083) 1167(057) 1338(060) 1370(067) 1867(093) 20 (0,0) (1,3) (1.8,6) (3,25) 1063(061) 1060(066) 1069(071) 1113(085) 0880(059) 0897(061) 0922(066) 1020(077) 0533(040) 0582(044) 0671(053) 0927(115) 0957(060) 0969(064) 0990(069) 1075(082) 1797(052) 1924(058) 1930(067) 2281(095) 0870(060) 0923(065) 0961(069) 1234(088) 0517(040) 0559(045) 0629(052) 0831(080) 1597(055) 1765(059) 1788(068) 2245(097) 153 10 (0,0) (1,3) (1.8,6) (3,25) 0568(043) 0581(044) 0619(047) 0744(056) 0523(042( 0542(043) 0591(045) 0741(056) 0411(031) 0436(033) 0502(038) 0699(052) 0535(042) 0554(043) 0600(046) 0747(056) 1019(044) 1177(047) 1208(052) 1642(075) 0550(041) 0585(043) 0627(046) 0834(063) 0393(030) 0408(032) 0461(037) 0616(050) 0803(046) 0959(048) 1005(052) 1515(075) 15 (0,0) (1,3) (1.8,6) (3,25) 0799(049) 0794(052) 0810(054) 0873(064) 0671(048) 0680(049) 0712(051) 0817(061) 0409(031) 0433(033) 0498(038) 0691(051) 0713(049) 0720(050) 0748(053) 0843(062) 1467(042) 1625(046) 1656(052) 2091(077) 0696(047) 0745(050) 0790(053) 1059(073) 0393(031) 0410(033) 0462(037) 0618(052) 1230(045) 1417(048) 1465(054) 2012(077) 20 (0,0) (1,3) (1.8,6) (3,25) 1074(052) 1056(056) 1046(061) 1065(072) 0867(052) 0855(055) 0868(058) 0931(067) 0408(031) 0430(033) 0492(038) 0689(053) 0953(053) 0940(057) 0947(060) 1001(070) 1837(040) 1982(045) 2004(051) 2410(079) 0862(052) 0927(054) 0964(058) 1267(081) 0402(032) 0420(034) 0471(038) 0637(056) 1643(043) 1830(046) 1870(052) 2382(080) 265 10 (0,0) (1,3) (1.8,6) (3,25) 0541(036) 0536(037) 0546(040) 0612(048) 0479(035) 0477(036) 0496(039) 0594(046) 0307(024) 0328(026) 0378(029) 0533(040) 0495(035) 0493(037) 0509(040) 0602(047) 1067(033) 1243(036) 1297(041) 1783(063) 0516(034) 0560(038) 0599(042) 0848(060) 0299(024) 0314(026) 0354(029) 0484(039) 0832(036) 1012(038) 1078(043) 1648(063) 15 (0,0) (1,3) (1.8,6) (3,25) 0807(039) 0792(042) 0782(047) 0772(059) 0670(039) 0659(042) 0659(045) 0700(054) 0306(024) 0326(026) 0375(029) 0526(040) 0715(040) 0704(042) 0702(046) 0728(056) 1490(031) 1665(034) 1719(040) 2200(061) 0683(039) 0752(042) 0797(046) 1097(066) 0308(025) 0325(026) 0363(029) 0491(040) 1256(034) 1462(036) 1531(041) 2125(062) 20 (0,0) (1,3) (1.8,6) (3,25) 1096(040) 1076(045) 1056(051) 1003(066) 0880(042) 0859(046) 0847(051) 0839(061) 0305(024) 0324(026) 0372(029) 0519(040) 0969(041) 0951(046) 0936(051) 0913(064) 1879(030) 2037(033) 2082(039) 2509(060) 0868(042) 0954(044) 1001(049) 1336(071) 0325(026) 0343(028) 0376(030) 0505(042) 1678(032) 1879(034) 1941(040) 2474(061) * All values in the table are preceded by an omitted decimal point. Standard deviations are in parentheses. Standard errors of the means can be obtained by dividing each standard deviation by 100054. 92 o CO cc MONOTONIC PATTERN N = 94, PM = 10 ( 1 , 3 ) (1 .8 ,6 ) N O N - N O R M A L r T Y ( S K E W , K U R T O S I S ) N = 94, PM = 15 ( 3 . 2 5 ) ( 0 , 0 ) ( 1 , 3 ) (1 .8 ,6 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 94, PM = 20 (3, 25) ( 0 , 0 ) ' ( 1 , 3 ) ( 1 . 8 , 6 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) (3, 25) LEGEND: MS PW; NON-MONOTONIC PATTERN N = 94, PM = 10 (0, 0) ( 1 , 3 ) (1.8, 6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) 94, PM = 15 ( 1 , 3 ) (1 .8 ,6 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) = 94, PM = 20 ( 0 , 0 ) ( 1 , 3 ) ( 1 . 8 , 6 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) (3, 25) RS; EM Figure 17: Mean absolute error for R e s t i m a t e across levels of non-normality (N = 94) Note: N=Sample size, PM=Percent missing, RSQ = R 2 e s t i m a t e MONOTONIC PATTERN N = 153, PM = 10 ( 0 , 0 ) ( 1 , 3 ) (1 .8 ,6 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) 153, PM = 15 ( 0 . 0 ) ( 1 , 3 ) (1 .8 ,6 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 153, PM = 20 .06 .04 .02 ( 0 , 0 ) ( 1 , 3 ) (1 .8 ,6 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) tr O < z < O CO or or O O to < z < NON-MONOTONIC PATTERN N = 153, PM = 10 ( 0 , 0 ) ( 1 , 3 ) ( 1 . 8 , 6 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 153, PM = 15 (3, 25) (0, 0) ( 1 , 3) (1.8, 6 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 153, PM = 20 ( 0 , 0 ) ( 1 , 3 ) ( 1 . 8 , 6 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) (3 . 25 ) LEGEND: MS PW; RS; EM Figure 18: Mean absolute error for R 2 e sum ate across levels of non-normality (N = 153) Note: N=Sample size, PM=Percent missing, RSQ = R2estimate MONOTONIC PATTERN :265, P M = 10 (0,0) (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) (0.0) (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 265, P M = 20 (0,0) (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) (3,25) O CC o CO m < cc o NON-MONOTONIC PATTERN N = 265 , PM = 10 (0,0) (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 265, PM = 15 (0,0) (1,3) (18,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 265 , PM = 2 0 (0,0) (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) LEGEND: MS P W ; RS; EM Figure 19: Mean absolute error for R2estimate across levels of non-normality (N = 265) Note: N=Sample size, PM=Percent missing, RSQ = R 2 e s timate Effects on Absolute Error for Regression Coefficients The effects of design variables on absolute error for regression coefficients, which fulfilled the criterion T | 2 a i t > .03, are in Table 12. Percent missing had no effect of practical value. Miss ing pattern had an effect of practical significance only for bias in b i , the intensity of which increased with sample size. This was true for all M D T s . In addition, the effect of non-normality on absolute error for all regression coefficients was of practical significance under all M D T s , except E M . The strength of this effect also increased with sample size. Notice that for absolute error for bi under E M , it was only the effect of missing pattern that met the criterion r] 2 a i t > .03. Table 12 Effects on absolute error for regression coefficients fulfilling the criterion of r\2*u > .03* Sample size M D T b] bj bj b^ 94 E M PT (157) N L (050) N L (074) N L (033) MS PT (082) N L (055) NL(081) N L (036) N L (043) - - -PW PT (082) NL(056) N L (080) N L (037) N L (043) - - -RS PT (162) N L (052) N L (075) N L (039) N L (050) - - -153 E M PT (278) N L (056) N L (085) N L (040) MS PT(164) N L (057) N L (096) N L (049) N L (043) - - -PW PT (163) N L (057) N L (096) N L (049) N L (042) - - -RS PT (302) N L (060) N L (083) NL(051) N L (055) - - -265 E M PT (419) N L (069) N L (107) N L (037) M S PT (292) N L (069) NL(122) NL(.061) N L (045) PW PT(291) N L (068) NL(123) NL(061) N L (045) - - -RS PT (462) N L (073) NL(113) N L (059) N L (042) * Values of r) 2 a l t are in parentheses preceded by an omitted decimal point. Note: E M = E M imputation, MS = Mean substitution, PW = Pairwise deletion, RS = Regression imputation, PT = Pattern, N L = Non-normality, - = No effect of practical significance. 96 Relative Performance of M D T s in the Estimation of Regression Coefficients Table 13 contains cell means and standard deviations of absolute error for bi. The influence of non-normality on the performance of M D T s at sample size 94 is revealed in Figure 20. The figure shows that in general, the mean absolute error was smaller under monotonic pattern than under non-monotonic pattern, and that M D T s were more differentiated as percent missing increased. However, under both missing patterns, the M S and P W methods were undifferentiated. Overall , R S performed the best under both monotonic and non-monotonic missing patterns, and E M performed the worst. However, at 15% and 20% missing under non-monotonic pattern, E M performed better than M S and P W at the highest level of non-normality (skew = 3 and kurtosis = 25). Figure 21 shows mean absolute error for bi at sample size 153. Under monotonic missing pattern, R S performed the best and E M the worst. This was true with non-monotonic pattern as well , except that at 15% and 20% missing under non-monotonic pattern, E M performed better than both M S and P W at the highest level of non-normality (skew = 3 and kurtosis = 25). Also , at 15% and 20% missing under non-monotonic pattern, the performance of R S , M S and P W was undifferentiated except at the highest level of non-normality where R S outperformed the rest. Figure 22 is a set of graphs for mean absolute error for bi at sample size 265. The figure shows that results at this level of sample size were similar to those under sample size 153. Table A . 4 in Appendix A shows the bias in bi under high R 2 with four predictors ( N = 94). Figure B.4 in Appendix B shows typical graphs for the bias in bi. The bias in bj varied from -.177 to .195, giving a range of .37. The smallest bias was under RS across all levels of the design variables. However, the bias under E M was very close to that under R S . Whereas M S and P W consistently overestimated Pi, E M and R S consistently underestimated Pi. The range in bias for b2 to b 4 , with a maximum of .08, was far smaller than that for bj. Therefore, their graphical plots were omitted. 97 Table 13 Means and standard deviations' of absolute error for bi Monotonic Missing Pattern Non-monotonic Missing Pattern N Percent Normality EM Mean Pairwise Regress. EM Mean Pairwise Regress. Missing (sk., kurt.) Imputation Substitution Deletion Imputation Imputation Substitution Deletion Imputation 94 10 (0,0) 0721(053) 0670(051) 0668(051) 0604(046) 1346(067) 1098(072) 1097(072) 0944(054) (1,3) 0814(060) 0770(063) 0767(062) 0704(053) 1514(076) 1232(091) 1230(091) 1146(065) (1.8,6) 0924(068) 0885(075) 0883(074) 0819(061) 1564(085) 1329(107) 1325(106) 1226(071) (3,25) 1198(091) 1197(115) 1195(115) 1108(084) 1825(104) 1845(171) 1840(171) 1641(098) 15 (0,0) 0790(056) 0742(055) 0739(055) 0625(047) 1533(067) 1311(077) 1310(077) 1175(058) (1,3) 0873(063) 0844(067) 0842(067) 0722(054) 1643(076) 1429(096) 1427(096) 1346(069) (1-8,6) 0976(071) 0957(079) ' 0954(079) 0830(062) 1652(084) 1512(113) 1506(112) 1389(076) (3,25) 1242(092) 1277(123) 1274(123) 1117(085) 1800(102) 2008(177) 2000(176) 1705(102) 20 (0,0) 0890(062) 0820(061) 0815(061) 0669(050) 1696(069) 1468(082) 1466(082) 1410(062) (1,3) 0964(067) 0918(073) 0914(072) 0759(056) 1745(077) 1573(101) 1570(100) 1522(072) (1.8,6) 1059(074) 1026(084) 1023(084) 0860(063) 1741(085) 1652(117) 1642(116) 1533(079) (3,25) 1317(096) 1343(129) 1340(129) 1149(086) 1759(101) 2112(180) 2100(180) 1709(102) 153 10 (0,0) 0595(043) 0525(038) 0524(038) 0463(035) 1366(052) 1039(059) 1038(059) 0922(045) (1,3) 0664(047) 0587(046) 0586(046) 0543(040) 1523(060) 1158(075) 1156(075) 1123(053) (1.8,6) 0758(054) 0677(054) 0677(054) 0642(046) 1548(069) 1230(089) 1223(089) 1177(061) (3,25) 1021(073) 0954(084) 0954(083) 0914(067) 1776(089) 1673(142) 1667(141) 1581(083) 15 (0,0) 0703(048) 0597(043) 0595(043) 0493(037) 1582(053) 1269(064) 1267(064) 1184(047) (1,3) 0764(053) 0665(052) 0662(052) 0573(042) 1679(060) 1374(079) 1371(079) 1349(055) (1-8,6) 0947(059) 0747(061) 0745(061) 0669(048) 1673(069) 1436(094) 1427(094) 1368(063) (3,25) 1082(078) 1024(093) 1022(093) 0935(068) 1738(087) 1845(145) 1834(144) 1628(086) 20 (0,0) 0811(049) 0672(047) 0668(047) 0553(040) 1695(051) 1423(066) 1422(066) 1413(047) (1,3) 0858(055) 0728(056) 0724(056) 0621(045) 1743(058) 1514(081) 1511(081) 1529(056) (1.8,6) 0922(061) 0801(065) 0798(065) 0703(050) 1718(067) 1566(096) 1554(096) 1521(065) (3,25) 1134(080) 1062(097) 1061(097) 0956(068) 1699(085) 1940(147) 1926(147) 1665(087) 265 10 (0,0) 0537(036) 0424(032) 0424(032) 0383(029) 1378(040) 1023(047) 1023(047) 0903(036) (1,3) 0578(040) 0482(036) 0482(036) 0441(033) 1527(047) 1129(059) 1127(059) 1102(043) (1.8,6) 0639(045) 0548(042) 0548(042) 0511(038) 1541(056) 1179(070) 1174(070) 1141(051) (3,25) 0831(059) 0768(060) 0762(060) 0718(052) 1717(075) 1562(109) 1556(108) 1507(071) 15 (0,0) 0656(039) 0494(036) 0493(036) 0427(031) 1583(039) 1243(049) 1243(049) 1187(035) (1,3) 0682(043) 0548(041) 0548(040) 0473(035) 1672(046) 1333(061) 1331(061) 1346(043) (1.8,6) 0726(048) 0609(046) 0609(046) 0534(039) 1664(054) 1374(073) 1366(073) 1360(051) (3,25) 0883(062) 0813(066) 0813(066) 0724(053) 1686(075) 1725(111) 1715(111) 1577(073) 20 (0,0) 0783(042) 0578(039) 0576(039) 0502(033) 1729(038) 1411(050) 1411(050) 1429(035) (1,3) 0803(046) 0617(045) 0616(045) 0534(037) 1765(044) 1489(062) 1487(062) 1535(042) (1.8,6) 0837(051) 0672(050) 0672(050) 0587(041) 1734(053) 1522(074) 1513(073) 1520(050) (3,25) 0970(064) 0862(070) 0862(070) 0754(054) 1637(074) 1829(112) 1818(112) 1585(074) * All values in the table are preceded by an omitted decimal point. Standard deviations are in parentheses. Standard errors of the means can be obtained by dividing each standard deviation by 100VA. 98 MONOTONIC PATTERN N = 94, P M = 10 (0, 0) (1, 3) (1.8, 6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) (3,25) N = 94, P M = 15 (0.0) (1.3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 94 , P M = 20 (3. 25) (0, 0) (1, 3) (1.8, 6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) (3, 25) LEGEND: MS PW; NON-MONOTONIC PATTERN N = 94 , P M = 10 (1.8.6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) (3.25) N = 94 , P M = 15 (0,0) (1,3) (1.8,6) NON-NORMALITY (SKEW, KURTOSIS) N = 9 4 , P M = 2 0 (3, 25) LX O LX O LX LX U J U J H Z> _1 o CO CQ < z < LU (0,0) (1,3) (1.8,6) NON-NORMALITY (SKEW, KURTOSIS) (3,25) RS; EM Figure 2 0 : Mean absolute error for bi across levels of non-normality (N = 94) Note: N = Sample size, P M = Percent missing. MONOTONIC PATTERN N = 153, PM = 10 ( 1 , 3 ) ( 1 . 8 , 6 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 153, PM = 15 (3, 25) ( 1 , 3 ) ( 1 . 8 , 6 ) NON-NORMALITY (SKEW, KURTOSIS) N = 153, PM = 20 (3, 25) ( 0 , 0 ) ( 1 , 3 ) ( 1 . 8 , 6 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) (3, 25) NON-MONOTONIC PATTERN N = 153, PM = 10 tr o CC CC ( 0 , 0 ) ( 1 , 3 ) ( 1 . 8 , 6 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) I = 153, PM = 15 ( 1 , 3 ) ( 1 . 8 , 6 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) 153, PM = 20 (3, 25) ( 1 . 3 ) ( 1 . 8 . 6 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) (3, 25) LEGEND: MS PW; RS; EM Figure 21: Mean absolute error for bi across levels of non-normality (N = 153) Note: N = Sample size, P M = Percent missing. 100 MONOTONIC PATTERN N = 265, PM = 10 0) (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 265, PM = 15 (3, 25) (0,0) (1,3) (1.8,6) NON-NORMALITY (SKEW, KURTOSIS) N = 265, PM = 20 (0,0) (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) (3,25) NON-MONOTONIC PATTERN N = 265, PM = 10 (0,0) (1,3) (1.8.6) NON-NORMALITY (SKEW, KURTOSIS) = 265, PM = 15 (1.3) (1.8,6) NON-NORMALITY (SKEW, KURTOSIS) N = 265, PM = 20 CO cc o LL CC o cc cc UJ LU o CO CO < < UJ (1.3) (1.8,6) NON-NORMALITY (SKEW, KURTOSIS) (3, 25) LEGEND: MS PW; RS; EM Figure 22: Mean absolute error for bj across levels of non-normality (N = 265) Note: N = Sample size, P M = Percent missing. 101 Coverage Probability for MDTs mostly differed in coverage probabilities for Pi and not for p 2 , P3,...,Po. For this reason, only the results for Pi are reported. Table 14 contains mean coverage probabilities for Pi at different levels of sample size. The means at sample size 94 are displayed in Figure 23. The figure shows that coverage probability decreased with increasing non-normality for all MDTs under both monotonic and non-monotonic patterns of missing data. RS provided the best coverage probability under both monotonic and non-monotonic missing patterns, followed by E M , M S and PW, in that order. However, under monotonic missing pattern, MDTs differed most in coverage probability of p i at 20% missing, and least at 10% missing. A l l MDTs performed poorly under non-monotonic missing pattern. Under this pattern, coverage probabilities ranged from 41.8% at the highest level of percent missing (20%) and highest level of non-normality (skew = 3 and kurtosis=25) to 86.4% at the lowest level of percent missing (10%) and normal data. Thus, with non-monotonic missing pattern, the only result close to the nominal value was 86.4%, and this was under RS for normal data at 10% missing. Figure 24 shows coverage probabilities at sample size 153. RS provided the best coverage probability under both monotonic and non-monotonic patterns. However, all MDTs had poor coverage probabilities with non-monotonic missing pattern. At 10% missing, MDTs had the most differentiated coverage probabilities for normal data, and the least differentiated coverage probabilities at the highest level of non-normality (skew = 3 and kurtosis = 25). Figure 25 shows coverage probabilities at sample size 265. Under monotonic missing pattern, RS outperformed the other MDTs with respect to coverage probabilities, followed by MS, PW and E M , in that order. Again, all MDTs had poor coverage probabilities under non-monotonic pattern, ranging from 9.6% under E M to 58.1% under RS. 102 Table 14 Mean coverage probability (%) for fli Monotonic Missing Pattern Non-monotonic Missing Pattern N Percent Normality EM Mean Pairwise Regress. EM Mean Pairwise Regress. Missing (sk., kurt.) Imputation Substitution Deletion Imputation Imputation Substitution Deletion Imputation 94 10 (0,0) 92.2 92.3 91.1 94.9 71.8 69.2 66.5 86.4 (1.3) 88.3 88.6 86.5 91.0 65.0 64.2 62.2 76.8 (1.8,6) 83.9 83.0 82.0 87.3 61.7 61.5 59.2 73.6 (3,25) 69.8 69.4 68.4 72.5 55.8 50.3 48.6 60.0 15 (0,0) 91.6 90.2 88.4 95.8 69.1 58.2 55.2 82.2 (1.3) 86.2 86.5 83.7 91.7 65.4 55.4 52.4 74.6 (1.8,6) 82.4 80.7 78.2 87.2 63.1 56.1 52.6 71.3 (3,25) 69.3 67.6 65.6 74.6 59.6 46.7 44.7 62.0 20 (0,0) 90.3 87.8 83.7 95.6 65.2 51.6 46.9 75.4 (1,3) 85.3 82.5 80.4 90.7 64.2 50.0 46.3 70.0 (1.8,6) 81.3 78.7 75.4 86.5 63.0 49.4 45.6 68.9 (3,25) 69.0 67.0 64.4 74.2 63.3 44.3 41.8 65.1 153 10 (0,0) 91.4 93.5 92.5 96.4 52.1 58.0 54.5 78.2 (1.3) 86.7 89.6 88.0 92.6 44.7 53.8 50.9 63.6 (1.8,6) 80.4 83.3 81.4 86.9 44.7 52.0 49.9 59.5 (3,25) 65.6 69.1 67.0 70.6 43.9 43.5 41.1 48.4 15 (0,0) 88.5 90.1 87.1 95.7 42.8 43.4 38.5 67.5 (1,3) 81.4 85.0 82.8 91.5 41.0 41.5 38.0 56.1 (1.8,6) 77.2 80.2 76.2 85.5 44.2 42.4 40.2 55.1 (3,25) 65.0 66.5 64.6 71.3 49.5 36.9 34.1 52.0 20 (0,0) 86.1 87.3 82.9 94.8 41.3 33.6 29.7 57.6 (1.3) 79.0 82.7 78.6 90.5 43.4 33.5 30.2 51.0 (1.8,6) 74.9 77.5 74.0 84.2 44.8 36.7 32.7 51.0 (3,25) 63.6 66.3 64.5 72.2 53.4 34.6 30.7 54.1 265 10 (0,0) 87.0 90.0 88.7 94.5 21.9 36.4 33.7 58.1 (1.3) 82.3 86.5 84.4 89.4 20.3 33.8 31.2 43.7 (1.8,6) 76.4 81.4 78.0 84.2 23.7 34.0 31.9 42.3 (3,25) 61.4 66.5 65.0 68.8 28.2 26.7 25.3 33.8 15 (0,0) 80.5 86.8 84.2 93.0 13.2 20.9 18.4 34.2 (1,3) 75.9 84.0 79.5 88.7 15.6 21.0 18.6 31.3 (1.8,6) 72.9 77.5 73.9 83.4 20.8 23.8 21.6 32.4 (3,25) 61.2 65.5 61.1 69.8 32.6 22.1 19.7 35.0 20 (0,0) 74.5 79.8 76.2 91.1 9.6 13.7 11.4 22.1 (1-3) 70.2 78.0 72.7 86.0 14.9 15.1 13.1 23.0 (1.8,6) 66.6 74.0 68.3 81.3 20.5 17.9 15.5 28.5 (3,25) 57.9 62.7 58.1 69.7 36.7 19.3 17.4 38.7 103 MONOTONIC PATTERN N = 94 , P M = 10 (0,0) (1,3) (1.8,6) N O N - N O R M A L I T Y (SKEW, KURTOSIS) N = 94 , P M = 15 (0.0) (1,3) (1.8,6) N O N - N O R M A L I T Y (SKEW, KURTOSIS) N = 94 , P M = 2 0 N O N - N O R M A L I T Y ( S K E W , KURTOSIS) ca < m O cc Q_ LU a < CC LU > o o NON-MONOTONIC PATTERN N = 94 , P M = 10 (0,0) (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , KURTOSIS ) N = 9 4 , P M = 15 (0, 0) (1, 3) (1.8. 6) N O N - N O R M A L I T Y ( S K E W , KURTOSIS ) N = 94 , P M = 2 0 (3, 25) (0.0) (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , KURTOSIS) (3,25) LEGEND: MS PW; RS; EM Figure 23: Mean coverage probability for i3i (N = 94) Note: N = Sample size, P M = Percent missing. MONOTONIC PATTERN N = 153 , P M = 10 < CD O CC CD < cc UJ > O O z < (1,3) (1,8,6) N O N - N O R M A L I T Y ( S K E W , KURTOSIS ) 153 , P M = 15 < CQ O DC C L U J CD < tr U J > o o (0,0) (1,3) (1.8,6) N O N - N O R M A L I T Y (SKEW, KURTOSIS ) N = 1 5 3 , P M = 2 0 m < ca O cc Cc 1X1 CD < CC U J > 8 < L U (0,0) (1,3) ( 1 . 8 . 6 ) N O N - N O R M A L I T Y ( S K E W , KURTOSIS ) (3, 25) NON-MONOTONIC PATTERN N = 1 5 3 , P M = 10 100 >- 90 H _1 80 ca BA 70 O cc 60 C L U J 50 CD < 40 cc > 30 O O 20 z < 10 U J 5 0 (0 100 All 90 _j CO 80 < CO 70 O PR 60 U J CD 50 < CC 40 U J > O 30 o z 20 < U J 10 2 0 0) (1.3) (1.8.6) N O N - N O R M A L I T Y ( S K E W , KURTOSIS ) N = 1 5 3 , P M = 15 (0,0) (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , KURTOSIS ) N = 1 5 3 , P M = 2 0 (0,0) (1,3) (18,6) N O N - N O R M A L I T Y ( S K E W , KURTOSIS ) LEGEND: MS P W ; RS; EM Figure 24: Mean coverage probability for (3) (N = 153) Note: N = Sample size, P M = Percent Missing 105 MONOTONIC PATTERN N = 265, P M = 10 (0,0) (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 265, PM = 15 (3, 25) (0,0) (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 2 6 5 , PM = 20 (3,25) (0.0) (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) (3,25) NONMONOTONIC PATTERN N = 265, P M = 10 O O (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 2 6 5 , PM = 15 (1.8,6) (3, 25) N O N - N O R M A L I T Y ( S K E W , KURTOSIS ) N = 2 6 5 , PM = 20 (3,25) N O N - N O R M A L I T Y ( S K E W , KURTOSIS ) LEGEND: -MS •PW; RS; EM Figure 25: Mean coverage probability for Pi (N = 265) Note: N = Sample size, P M = Percent missing. 106 Summary In the previous sections, the effects of design variables on absolute error for parameter estimates were presented. Whereas pattern effects were of practical significance under E M and RS at sample sizes 153 and 265, respectively, the interaction effect of pattern and non-normality on absolute error for R2estimate was also of practical significance under E M and RS. However, this was not the case under MS and PW. Under MS, only the effect of percent missing was of practical significance at sample size 265. Under PW, it was only the effect of non-normality that was of practical significance at all levels of sample size. It appears that MS was the least sensitive to the effects of pattern and non-normality. PW was also insensitive to pattern effects, though it was affected by non-normality. With respect to absolute error for R2estimate, the performance of all MDTs deteriorated with increasing non-normality. Regardless of missing pattern, PW outperformed the other MDTs, and generally E M produced the largest mean absolute errors for R2esumate- However, the performance of MDTs was more differentiated under non-monotonic missing pattern than under monotonic pattern. MDTs were more differentiated at the highest level of percent missing (20%) than at the lowest level (10%). Also, under monotonic pattern, and at each level of percent missing, MDTs were most differentiated for normal data and least differentiated at the highest level of non-normality. With respect to R 2 e s u m a t e , m e smallest bias was under PW. This was at all levels of the design variables. Except for PW, the MDTs tended to underestimate population R 2 . Thus, in the estimation of R 2 , PW method was the most accurate and least biased. It was only for bi that missing pattern had an effect of practical significance. This was true for all MDTs. In addition, the effect of non-normality on absolute error for regression coefficients was of practical significance under all MDTs, except E M . Overall, RS was the most accurate under both monotonic and non-monotonic missing patterns, and E M the least accurate in estimating Pi. The mean absolute error for bi was smaller 107 under monotonic pattern than under non-monotonic pattern, and the M D T s became more differentiated as percent missing increased. In the estimation of p\, RS was the most accurate and least biased. RS generally provided the best coverage probability under both monotonic and non-monotonic missing patterns. However, all M D T s had poor coverage probabilities under non-monotonic missing pattern. Also , coverage probability generally decreased with increasing non-normality. Study 3: Nine Predictors under Low R2 Condition Effects on Absolute Error for R 2 ^ ™ ^ . The effects of design variables on absolute error for R2estimate are listed in Table 15. Using T\ 2 au ^ -03 as the criterion, the table shows that with a sample of 94, the effect of non-normality was of practical significance under each M D T , with the largest effect being that of non-normality under M S , and the smallest under P W . With sample size 153, the effect of non-normality on absolute error for R2estimate was of practical significance under all M D T s . The largest effect was that of non-normality under M S , and the smallest was under P W . With sample size 265, the effect of non-normality was of practical significance under all M D T s . The largest effect was that of non-normality under P W and the smallest under E M . Table 15 Effects on absolute error for R 2 , ^ ^ . fulfilling the criterion of T ) 2 ^ > .03 Sample size M D T Effect SS eff e c t residual df 94 E M Non-normality 3.57 72.01 3 .047 M S Non-normality 5.21 102.34 3 .048 PW Non-normality 4.10 105.30 3 .037 RS Non-normality 3.57 84.47 3 .041 153 E M Non-normality 1.55 33.82 3 .044 MS Non-normality 2.23 47.65 3 .045 PW Non-normality 1.95 54.62 3 .034 RS Non-normality 1.60 39.24 3 .039 265 E M Non-normality 0.62 19.90 3 .030 MS Non-normality 0.82 23.87 3 .033 PW Non-normality 1.05 27.91 3 .036 RS Non-normality 0.73 21.40 3 .033 Note: E M = E M imputation, MS = Mean substitution, PW = Pairwise deletion, RS = Regression imputation. Relative Performance of M D T s in the Estimation of Population R 2 Table 16 contains means and standard deviations of mean absolute error for R2 e s timate under various design conditions. The influence of non-normality on the performance of M D T s at sample size 94 is revealed in Figure 26. The performance of all M D T s deteriorated with 109 increasing non-normality. The smallest mean absolute bias was under E M for both monotonic and non-monotonic missing patterns, and at all levels of percent missing. A l so , E M and RS tended to give smaller absolute errors under non-monotonic pattern than under monotonic pattern. Under non-monotonic missing pattern, E M overlapped with RS and M S overlapped with P W . In sum, with a sample of 94, and regardless of missing pattern, E M performed the best. The influence of non-normality on the performance of M D T s at sample size 153 is revealed in Figure 27. The performance of all M D T s deteriorated with increasing non-normality as measured by the mean absolute error for R2estimate- For both monotonic and non-monotonic missing patterns, E M performed the best, followed by R S , M S and P W , in that order. Wi th sample size 153, E M performed the best regardless of missing pattern and percent missing. However, under non-monotonic missing pattern, the difference in performance between E M and R S became smaller with increasing percent missing. The influence of non-normality on the relative performance of M D T s at sample size 265 is revealed in Figure 28. The performance of all M D T s deteriorated with increasing non-normality, although mean absolute error was generally much smaller than that under sample size 94 and 153. With monotonic missing pattern, E M performed the best, followed by R S , M S and P W , in that order. M D T s differed most in performance at the highest level of non-normality (skew = 3 and kurtosis = 25). Also , the relative performance among E M , M S and R S became more and more undifferentiated with increasing percent missing. Wi th non-monotonic missing pattern at 10% missing, E M and R S performed the best. This was true at 15% missing, although M S performed equally well . A t 20% missing, M S and P W performed the best, and E M performed the worst. Table A . 5 in Appendix A contains bias in R2estimate under low R 2 condition with nine predictors. Bias in R 2 e s U mate varied from - .04 to .09 across all conditions of the design variables, giving a range of .13. These values were plotted in figure B.5 , a typical set of graphs for bias in R2estimate- A t sample size 94, the smallest bias was under R S followed by P W . Similar to the 110 results under low R 2 with four predictors, not a single MDT produced the smallest bias across all design conditions. I l l Table 16 Means and standard deviations* of absolute error for R ^ - i m ^ Monotonic Missing Pattern Non-monotonic Missing Pattern N Percent Missing Normality (sk., kurt.) EM Imputation Mean Substitution Pairwise Deletion Regress. Imputation EM Imputation Mean Substitution Pairwise Deletion Regress. Imputation 94 10 (0,0) (1.3) (1.8,6) (3,25) 0651(050) 0699(053) 0807(060) 1047(085) 0712(054) 0766(059) 0883(067) 1153(093) 0789(055) 0822(059) 0917(067) 1159(092) 0696(053) 0747(057) 0859(065) 1108(090) 0587(042) 0596(046) 0663(053) 0823(070) 0729(052) 0761(057) 0843(064) 1068(086) 0764(055) 0789(058) 0881(064) 1083(086) 0653(047) 0656(050) 0719(057) 0864(074) 15 (0,0) (1.3) (1.8,6) (3,25) 0588(043) 0632(048) 0732(056) 0972(081) 0665(050) 0716(055) 0828(064) 1103(091) 0754(054) 0783(057) 0872(064) 1093(089) 0643(048) 0688(053) 0792(062) 1041(087) 0521(039) 0538(041) 0604(048) 0764(065) 0691(051) 0719(055) 0803(061) 1015(082) 0719(052) 0745(055) 0833(061) 1033(084) 0581(043) 0584(045) 0648(052) 0806(070) 20 (0,0) (1,3) (1.8,6) (3,25) 0531(040) 0576(044) 0666(052) 0893(077) 0616(047) 0670(053) 0778(062) 1046(090) 0717(052) 0751(055) 0836(063) 1066(095) 0582(044) 0629(049) 0723(058) 0952(083) 0479(037) 0507(038) 0571(044) 0737(061) 0651(048) 0681(052) 0761(058) 0956(079) 0678(049) 0709(051) 0792(058) 0991(079) 0517(039) 0537(041) 0600(047) 0774(067) 153 10 (0,0) (1.3) (1.8,6) (3,25) 0467(035) 0504(036) 0563(041) 0738(058) 0506(039) 0547(040) 0611(045) 0806(065) 0569(042) 0596(043) 0653(048) 0827(065) 0499(039) 0539(040) 0601(044) 0786(063) 0396(030) 0410(031) 0457(034) 0559(046) 0492(037) 0517(039) 0574(044) 0722(059) 0552(041) 0577(042) 0630(046) 0775(061) 0445(033) 0453(034) 0498(038) 0582(049) 15 (0,0) (1,3) (1.8,6) (3,25) 0421(032) 0453(033) 0507(037) 0677(054) 0467(036) 0505(038) 0565(042) 0759(061) 0550(041) 0579(042) 0631(046) 0794(063) 0455(035) 0490(037) 0547(041) 0724(058) 0377(028) 0389(030) 0426(032) 0533(043) 0467(036) 0484(037) 0534(041) 0672(055) 0523(039) 0548(040) 0599(044) 0734(060) 0406(031) 0410(031) 0447(034) 0552(046) 20 (0,0) (1,3) (1.8,6) (3,25) 0410(030) 0440(032) 0489(036) 0653(050) 0449(035) 0491(036) 0549(041) 0738(059) 0530(039) 0559(040) 0609(044) 0770(060) 0438(033) 0472(034) 0524(038) 0694(054) 0389(029) 0405(030) 0433(032) 0544(042) 0447(034) 0464(035) 0504(039) 0629(053) 0500(038) 0526(038) 0575(042) 0720(061) 0393(030) 0406(031) 0439(033) 0559(043) 265 10 (0,0) (1,3) (1.8,6) (3,25) 0331(024) 0347(026) 0382(028) 0508(040) 0351(026) 0370(028) 0410(031) 0549(044) 0397(030) 0417(031) 0458(034) 0589(046) 0349(026) 0367(027) 0405(030) 0538(043) 0318(023) 0325(024) 0347(025) 0427(032) 0352(026) 0369(027) 0401(030) 0495(041) 0385(030) 0385(030) 0441(033) 0554(043) 0335(024) 0335(024) 0361(027) 0431(033) 15 (0,0) (1,3) (1.8,6) (3,25) 0335(025) 0346(026) 0376(028) 0489(038) 0350(026) 0366(028) 0401(030) 0529(043) 0388(029) 0405(031) 0444(033) 0576(045) 0348(026) 0362(027) 0395(029) 0513(041) 0352(025) 0368(026) 0378(026) 0461(033) 0352(026) 0360(027) 0382(029) 0470(038) 0373(029) 0390(029) 0426(032) 0530(041) 0346(025) 0355(026) 0369(027) 0460(033) 20 (0,0) (1,3) (1.8,6) (3,25) 0361(026) 0370(027) 0390(028) 0488(036) 0361(027) 0373(028) 0403(030) 0524(042) 0380(029) 0395(030) 0430(033) 0544(043) 0363(026) 0373(027) 0399(029) 0507(039) 0427(027) 0442(028) 0447(029) 0525(033) 0370(026) 0369(027) 0386(029) 0461(037) 0359(028) 0376(029) 0410(031) 0515(040) 0399(027) 0417(028) 0429(028) 0530(033) * All values in the table are preceded by an omitted decimal point. Standard deviations are in parentheses. Standard errors of the means can be obtained by dividing each standard deviation by lOW. 112 MONOTONIC PATTERN = 94, PM = 10 N = (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) 9 4 , P M = 15 (3. 26) (1,3) (18 ,6 ) (3,25) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) = 9 4 , P M = 2 0 (1.3) (1-8,6) (3,25) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) LEGEND: 'MS PW; NON-MONOTONIC PATTERN N = 9 4 , PM = 10 O co CC DC O LL CC o CC CC LU LU o co m < (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , KURTOSIS ) : 94 , P M = 15 O CO CC CC O LL CC o oc CC LU UJ I-3 < z < (0,0) (1,3) (1.8,6) (3,25) N O N - N O R M A L I T Y ( S K E W , KURTOSIS) O co CC CC o O CC D _l o CO CQ < z < (1,3) (1.8,6) (3,25) N O N - N O R M A L I T Y (SKEW, KURTOSIS ) RS; EM Figure 26: Mean absolute error for R 2 e s t i m a t e across levels of non-normality (N = 94) Note: N = Sample size, P M = Percent missing. 113 MONOTONIC PATTERN 153, PM = 10 (0 ,0 ) (1 ,3) (1 .8 ,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 153, PM = 15 (3 ,25) (0 ,0 ) (1 ,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 153, PM = 20 O O CO (0 .0 ) (1 ,3 ) (1 .8 ,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) o co cc CC o LL cc o CC cc O CO CO < z < LU 2 DC DC O I-o co LX LX o LL I X o I X tr UJ UJ O CO CO < z < til (3 ,25) NON-MONOTONIC PATTERN N = 153, PM = 10 (0,0) (1 ,3) (1.8,6) NON-NORMALITY (SKEW, KURTOSIS) N = 153, PM = 15 (3,25) (1 ,3) (1.8,6) NON-NORMALITY (SKEW, KURTOSIS) N = 153, PM = 20 (1.3) (1.8,6) NON-NORMALITY (SKEW, KURTOSIS) (3,25) LEGEND: MS PW; RS; EM Figure 27: Mean absolute error for R2estimate across levels of non-normality (N = 153) Note: N = Sample size, P M = Percent missing. 114 MONOTONIC PATTERN N = 2 6 5 , P M = 10 O co cc cc o u_ cc o cc DC o CO < LU 5 O CO CC o cc cc UJ D —I O CO CO < < (0, 0) ( 1 , 3) (1.8, 6) . N O N - N O R M A L I T Y (SKEW, KURTOSIS) N = 2 6 5 , P M = 15 (3, 25) ( 0 , 0 ) ( 1 . 3 ) ( 1 . 8 . 6 ) N O N - N O R M A L I T Y ( S K E W , KURTOSIS) N = 265, PM = 20 O co cc cc o u. cc o cc cc LU I-o CO m < z < LU (0,0) (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) (3, 25) DC tr O Ll-CC o LU < Z < LU 2 NON-MONOTONIC PATTERN 2 6 5 , P M = 10 (0, 0) N = ( 1 . 3 ) ( 1 - 8 . 6 ) N O N - N O R M A L I T Y ( S K E W , KURTOSIS) 2 6 5 , P M = 15 (3, 25) ( 0 , 0 ) ( 1 , 3 ) ( 1 . 8 , 6 ) N O N - N O R M A L I T Y ( S K E W , KURTOSIS) 265, PM = 20 ( 3 , 2 5 ) (1,3) (1.8,6) NON-NORMALITY (SKEW, KURTOSIS) (3,25) LEGEND: .MS PW; RS; EM Figure 28: Mean absolute error for R 2 e sumate across levels of non-normality (N = 265) Note: N = Sample size, P M = Percent missing. 115 Effects on Absolute Error for Regression Coefficients The effects of design variables on absolute error for regression coefficients are listed in Table 17. N o effects fulfilled r i 2 a i t > .03 as the criterion for 02, . . . , b 9 . Using the same criterion, the table shows that the effect of non-normality was of practical significance under R S only at sample size 94 and 153. The strength of the effect of non-normality increased with increasing sample size. With sample size 265, the effect of pattern (PT) on absolute error for bi was of practical significance under all M D T s . Pattern effects were strongest under E M algorithm. In addition to pattern effects, the effect of non-normality was of practical significance under R S only at all levels of sample size. Table 17 Effects on absolute error for regression coefficients fulfilling the criterion of r | 2 a i i > .03* Sample size M D T b] bj bj b_4 bj bj bj bj b_9 94 E M MS PW RS NL(033) -153 E M MS PW RS NL(046) --265 E M MS PW RS PT(101) -PT(039) -PT(038) -NL(071) -PT(045) --* Values of T ) 2 ^ are in parentheses preceded by an omitted decimal point. Note: E M = E M imputation, MS = Mean substitution, PW = Pairwise deletion, RS = Regression imputation, PT = Pattern, NL= Non-normality, - = No effect of practical significance. Relative Performance of M D T s in the Estimation of Regression Coefficients Table 18 contains means and standard deviations of absolute error for b] under various design conditions. The influence of non-normality on the performance of M D T s at sample size 116 94 is revealed in Figure 29. The performance of all M D T s deteriorated with increasing non-normality as measured by the mean absolute error for b i . This was regardless of missing pattern. Wi th sample size 94 under monotonic missing pattern, E M and RS both outperformed M S and P W , with E M and R S performing at the same level when percent missing was the lowest (10%). However, RS performed better at moderate to high levels of percent missing (15% to 20%). A s for M S and P W , there was little differentiation in performance across all levels of percent missing. Two pairings were apparent, that of M S and P W and that of E M and R S . The gap between the two pairings increased with percent missing. Wi th non-monotonic missing pattern, RS and E M again outperformed M S and P W . However, RS performed the best under conditions of low to moderate levels of non-normality. A t the extreme level of non-normality (skew = 3 and kurtosis = 25), E M performed slightly better than R S . The mean absolute error for b) under M S and P W was larger under non-monotonic than under monotonic missing pattern. The M D T s differed most in their performance for normal data. Figure 30 is a set of graphical plots of the mean absolute error at sample size 153. Under monotonic missing pattern, the results were similar to those under sample size 94, where E M and RS both outperformed M S and P W . There was little differentiation in the performance of M S and P W at all levels of percent missing. A s for E M and R S , difference in performance increased with percent missing, with R S doing better than E M as percent missing increased. Wi th sample size 153 under non-monotonic pattern, RS performed the best, followed by E M , with M S and P W tying in third place. However, E M outperformed RS under conditions of extreme non-normality (skew=3 and kurtosis=25). M D T s differed most for normal data, and least at the highest level of non-normality. Figure 31 is a graphical plot of mean absolute error at sample size 265. Wi th monotonic missing pattern, R S performed the best, although differentiation between E M and RS was smallest at the lowest level of percent missing (10%) and largest at the highest level of percent missing (20%). M S and P W were at par in third place. 117 Wi th non-monotonic missing pattern, R S performed the best. The M D T s differed least at the extreme level of non-normality (skew = 3 and kurtosis = 25), and most with normal data. Whereas M S and P W performed equally at all levels of percent missing, their performance relative to E M was differentiated most at the highest level of percent missing (20%), and differentiated least at the lowest level of percent missing (10%). In sum, the effects of missing pattern and non-normality on absolute error for regression coefficients were of practical significance only for b i . Mean absolute error increased with increasing non-normality. The M D T s were most differentiated with respect to mean absolute error for b- only. The best overall performance in the estimation of [}• was under R S , followed by E M . The performance of M S and P W was undifferentiated under both monotonic and non-monotonic missing pattern. Also , differentiation among M D T s was less prominent under monotonic pattern than under non-monotonic pattern. Under non-monotonic pattern, the M D T s differed most with normal data. Table A . 6 in Appendix A shows the bias in b\. Figure B.6 is a typical set of graphs of bias in bi, showing that the smallest bias across all levels of the design variables was under P W . The bias in bi varied from -.08 to .09 across al l design conditions, giving a range of .17. Whereas E M and M S consistently overestimated Pi, P W and RS consistently underestimated Pi. The bias for b2 to b 9 was much smaller compared to that for bi, with a maximum range of .05. Therefore, the graphs for bias in b2 to bo were omitted. 118 Table 18 Means and standard deviations* of absolute error for b Monotonic Missing Pattern Non-monotonic Missing Pattern N Percent Missing Normality (sk., kurt.) EM Imputation Mean Substitution Pairwise Deletion Regress. Imputation EM Imputation Mean Substitution Pairwise Deletion Regress. Imputation 94 10 (0,0) (1,3) (1.8,6) (3,25) 0682(050) 0739(055) 0815(063) 0929(084) 0784(059) 0833(065) 0913(073) 1035(098) 0785(059) 0835(065) 0915(073) 1037(098) 0696(052) 0746(057) 0822(064) 0938(084) 0686(041) 0729(041) 0759(042) 0860(043) 0880(067) 0927(077) 0989(087) 1142(118) 0883(067) 0928(077) 0989(087) 1142(118) 0607(041) 0659(042) 0711(043) 0903(054) 15 (0,0) (1,3) (1.8,6) (3,25) 0683(048) 0736(053) 0802(061) 0916(082) 0808(060) 0852(067) 0923(075) 1045(099) 0810(060) 0856(067) 0928(075) 1050(100) 0679(051) 0727(055) 0796(062) 0908(081) 0709(039) 0751(039) 0775(041) 0866(042) 0923(069) 0966(079) 1027(090) 1193(123) 0928(070) 0968(080) 1026(090) 1192(123) 0614(039) 0682(040) 0728(044) 0955(061) 20 (0,0) (1,3) (1.8,6) (3,25) 0664(047) 0714(051) 0780(059) 0891(079) 0822(061) 0859(061) 0935(076) 1062(102) 0827(062) 0869(069) 0944(077) 1073(103) 0653(049) 0698(053) 0765(061) 0880(080) 0711(037) 0743(037) 0759(038) 0845(044) 0960(072) 1000(082) 1052(092) 1224(125) 0969(073) 1004(083) 1053(093) 1226(126) 0655(039) 0723(041) 0758(044) 0969(062) 153 10 (0,0) (1,3) (1.8,6) (3,25) 0552(037) 0568(041) 0622(045) 0727(061) 0607(045) 0628(049) 0688(053) 0798(070) 0608(045) 0630(049) 0690(053) 0801(071) 0547(039) 0547(039) 0624(046) 0726(063) 0630(036) 0680(036) 0699(038) 0829(038) 0708(053) 0755(059) 0807(065) 0970(092) 0709(053) 0755(059) 0807(065) 0968(092) 0483(033) 0537(034) 0577(037) 0802(044) 15 (0,0) (1,3) (1.8,6) (3,25) 0544(037) 0564(039) 0613(042) 0712(058) 0620(046) 0645(049) 0700(054) 0803(072) 0622(047) 0646(050) 0702(055) 0806(072) 0529(038) 0549(040) 0600(074) 0696(059) 0651(034) 0695(035) 0711(036) 0821(037) 0766(056) 0811(061) 0857(068) 1018(096) 0769(056) 0812(061) 0858(068) 1015(096) 0523(033) 0592(034) 0627(036) 0868(049) 20 (0,0) (1,3) (1.8,6) (3,25) 0544(036) 0565(038) 0608(042) 0710(057) 0627(047) 0653(051) 0709(056) 0820(073) 0630(048) 0657(051) 0713(056) 0823(074) 0507(037) 0529(039) 0578(043) 0684(058) 0679(034) 0712(034) 0724(035) 0811(037) 0819(059) 0862(064) 0906(071) 1067(099) 0822(059) 0863(064) 0906(071) 1065(099) 0576(033) 0647(035) 0676(037) 0880(050) 265 10 (0,0) (1,3) (1.8,6) (3,25) 0408(030) 0435(031) 0481(034) 0580(043) 0445(034) 0480(037) 0527(041) 0643(053) 0445(034) 0481(037) 0528(041) 0644(053) 0390(030) 0425(031) 0470(035) 0574(045) 0598(031) 0664(032) 0682(033) 0803(035) 0591(042) 0637(048) 0672(054) 0828(073) 0591(042) 0636(048) 0671(054) 0826(073) 0398(026) 0475(028) 0516(030) 0742(038) 15 (0,0) (1,3) (1.8,6) (3,25) 0419(029) 0440(031) 0481(034) 0574(042) 0458(035) 0498(038) 0544(043) 0659(055) 0459(035) 0499(038) 0545(043) 0660(055) 0377(029) 0409(031) 0453(034) 0553(044) 0638(029) 0690(030) 0702(032) 0798(034) 0663(045) 0709(051) 0737(057) 0891(077) 0664(045) 0709(051) 0735(057) 0888(077) 0466(027) 0551(029) 0582(031) 0798(040) 20 (0,0) (1,3) (1.8,6) (3,25) 0431(029) 0452(030) 0486(033) 0574(040) 0473(036) 0510(040) 0553(044) 0667(056) 0475(036) 0512(040) 0555(045) 0668(057) 0369(028) 0398(030) 0441(033) 0538(041) 0659(028) 0697(029) 0700(031) 0777(033) 0721(047) 0764(053) 0790(059) 0939(078) 0722(047) 0764(053) 0788(059) 0935(078) 0534(027) 0611(029) 0630(032) 0821(042) * All values in the table are preceded by an omitted decimal point. Standard deviations are in parentheses. Standard errors of the means can be obtained by dividing each standard deviation by 1000^. 119 MONOTONIC PATTERN N = 9 4 , P M = 1 0 ( 1 , 3 ) ( 1 8 , 6 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 9 4 , P M = 1 5 ( 1 . 3 ) ( 1 . 8 , 6 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 9 4 , P M = 2 0 (3, 25) ( 1 . 3 ) ( 1 8 , 6 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) CC O l i -CC o cc cc o CO CQ < < NONMONOTONIC PATTERN 9 4 , P M = 1 0 N = (1 .3 ) ( 1 8 , 6 ) (3 ,25 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) 9 4 , P M = 1 5 (1 ,3) ( 1 8 , 6 ) (3 ,25) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 9 4 , P M (0 .0) (1 .3) ( 1 8 , 6 ) (3 ,25) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) LEGEND: MS PW; RS; EM Figure 29: Mean absolute error for bi across levels of non-normality (N = 94) Note: N = Sample size, P M = Percent missing. 120 MONOTONIC PATTERN 153, PM = 10 ( 1 , 3 ) ( 1 . 8 , 6 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) 1 5 3 , PM = 15 (3. 25) ( 1 , 3 ) ( 1 . 8 , 6 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) = 1 5 3 , PM=20 (3, 25) (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) (3, 25) m t r O < LU 2 CC O Ll-CC O DC O CO m < NON-MONOTONIC PATTERN N = 1 5 3 , PM = 10 N = ( 1 . 3 ) ( 1 . 8 , 6 ) ( 3 , 2 5 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) 1 5 3 , PM = 1 5 N = ( 1 . 3 ) ( 1 - 8 , 6 ) ( 3 , 2 5 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) 1 5 3 , PM = 2 0 cc O LL CC O cc cc o CO CO < ( 1 , 3 ) ( 1 - 8 , 6 ) ( 3 , 2 5 ) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) LEGEND: MS PW; RS; EM Figure 30: Mean absolute error for bi across levels of non-normality (N = 153) Note: N = Sample size, P M = Percent missing MONOTONIC PATTERN N = 2 6 5 , P M = 10 (0,0) (1,3) (1.8,6) (3,25) N O N - N O R M A L I T Y (SKEW, KURTOSIS ) N = 2 6 5 , P M = 15 N = (1.3) (1.8,6) (3.25) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) 2 6 5 , P M = 2 0 (1.3) (1.8,6) (3,25) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) co CC O LL cc o CC CC LU LU o CO CQ < NON-MONOTONIC PATTERN 2 6 5 , P M = 10 (1,3) (1.8,6) (3,25) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) 2 6 5 , P M = 1 5 (0.0) N = (1.3) (1.8,6) (3,25) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) 2 6 5 , P M = 2 0 (1.3) (1.8,6) (3,25) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) LEGEND? MS P W ; RS; EM Figure 31: Mean bias in b] across levels of non-normality (N = 265) Note: N = Sample size, P M = Percent missing 122 Coverage Probability for $± Coverage probabilities under monotonic pattern differed from those under non-monotonic pattern only for Pi. In addition, M D T s were not differentiated in their coverage probabilities for p 2 , p 3 , . . . , ^ 9 . For this reason, only the findings for Pi are reported. Table 19 contains mean coverage probabilities for Pi. These coverage probabilities are displayed in Figures 32, 33 and 34 for sample sizes 94, 153 and 265, respectively. The graphs show that coverage probability decreased with increasing non-normality for all M D T s . A t sample size 94, all M D T s had coverage probability above the criterion level of 95% for normal data when missing pattern was monotonic. This suggests that M D T s worked well with monotonic pattern. With non-monotonic pattern, E M and RS were least differentiated with respect to coverage probability which, with the exception of coverage probability at extreme non-normality (skew = 3, kurtosis = 25), were above 95%. This was maintained at all levels of percent missing. For normal data, P W produced coverage probability above 95%, and this was maintained across all levels of percent missing. The coverage probability for M S was the worst, starting below 92% for normal data, and deteriorating with increasing non-normality to 81%. Coverage probabilities for p, at sample size 153 are plotted in Figure 33. The performance of E M , R S , and P W was undifferentiated at al l levels of percent missing under monotonic pattern. Wi th non-monotonic pattern, the difference in coverage probabilities for E M and R S was undifferentiated, except at the highest level of non-normality when R S produced a smaller coverage probability than E M . The difference in coverage probabilities among M D T s was largest at extreme non-normality. The graphs also show that the M D T s were most differentiated with respect to coverage probability under non-monotonic pattern than under monotonic pattern. These findings are similar to those under sample size 94. The performance of M D T s at sample size 265 was similar to that under sample sizes 94 and 153. 123 In sum, al l M D T s produced coverage probabilities above 95% for normal data under monotonic missing pattern, with M S producing the smallest coverage probability. However, coverage probabilities decreased with increasing non-normality. The M D T s were most differentiated under non-monotonic missing pattern than under monotonic missing pattern. 124 Table 19 Mean coverage probability (%) for bi Monotonic Missing Pattern Non-monotonic Missing Pattern N Percent Normality EM Mean Pairwise Regress. EM Mean Pairwise Regress. Missing (sk., kurt.) Imputation Substitution Deletion Imputation Imputation Substitution Deletion Imputation 94 10 (0,0) 98.3 95.9 97.7 97.9 99.9 91.7 95.0 99.8 (1.3) 98.0 94.3 96.6 97.0 99.9 88.9 92.7 99.7 (1.8,6) 96.5 92.1 94.6 95.6 99.8 86.8 90.2 99.4 (3,25) 92.4 89.2 92.2 91.8 97.3 82.8 86.8 96.3 15 (0,0) 98.9 96.4 98.4 98.6 100.0 89.4 95.6 99.9 (1,3) 98.7 94.7 97.6 98.0 99.9 88.1 93.4 99.7 (1.8,6) 97.9 92.4 96.7 96.6 100.0 85.9 90.5 99.9 (3,25) 94.1 89.4 93.4 93.5 97.6 81.3 87.8 96.6 20 (0,0) 99.3 95.9 98.9 98.8 100.0 88.6 96.5 100.0 (1,3) 99.0 94.3 98.3 98.0 100.0 86.3 94.7 100.0 (1.8,6) 98.0 91.6 96.7 97.5 100.0 84.4 91.6 99.6 (3,25) 95.2 89.4 94.5 94.4 99.5 80.7 88.8 96.8 153 10 (0,0) 98.4 96.2 97.9 98.0 99.8 89.5 93.5 99.4 (1,3) 98.2 95.6 97.3 97.4 99.4 87.1 92.1 99.3 (1.8,6) 96.4 92.4 95.4 95.6 99.0 84.6 89.4 98.9 (3,25) 91.8 88.2 91.3 91.3 95.0 78.9 82.9 94.2 15 (0,0) 99.0 96.0 98.3 98.5 99.8 87.8 93.9 99.5 (1,3) 98.7 94.4 97.7 98.2 99.6 84.6 92.1 99.6 (1.8,6) 97.3 92.4 96.1 96.5 99.3 82.0 90.2 99.2 (3,25) 93.2 87.9 92.5 92.4 96.3 77.1 83.6 93.9 20 (0,0) 99.3 95.6 98.6 98.9 96.2 86.4 94.2 99.8 (1,3) 98.6 94.1 98.0 98.5 95.6 82.8 93.5 99.5 (1.8,6) 98.3 92.0 97.1 97.5 92.4 79.6 91.1 99.4 (3,25) 95.0 87.5 93.8 93.9 88.2 75.5 84.6 94.1 265 10 (0,0) 98.8 96.1 97.7 98.2 97.7 88.3 92.7 99.4 (1,3) 96.8 94.1 96.2 96.7 95.9 84.1 88.2 98.9 (1.8,6) 95.8 91.5 94.3 94.8 94.3 80.8 84.9 97.6 (3,25) 90.7 84.7 88.5 90.1 84.7 72.3 75.9 86.3 15 (0,0) 99.5 95.6 98.2 99.2 98.1 83.4 91.3 99.3 (1,3) 97.6 93.3 96.4 97.3 96.9 79.5 87.0 98.4 (1.8,6) 96.1 91.0 94.8 96.0 95.0 76.4 84.3 97.4 (3,25) 91.4 83.8 90.7 91.4 86.2 68.5 75.9 84.1 20 (0,0) 99.4 95.4 98.4 99.4 98.8 79.2 91.5 99.4 (1,3) 98.5 92.7 97.0 98.0 97.9 75.7 88.1 98.5 (1.8,6) 97.4 90.4 95.6 96.7 96.1 73.6 86.0 97.2 (3,25) 92.2 83.6 92.3 92.8 89.0 65.6 77.7 83.8 125 MONOTONIC PATTERN N = 94, PM = 10 (0, 0) (1, 3) (1.8, 6) N O N - N O R M A L I T Y ( S K E W , KURTOSIS) N = 94, PM = 15 (3, 25) (0,0) (1,3) (18,6) N O N - N O R M A L I T Y (SKEW, KURTOSIS ) N = 94, PM = 20 (3,25) (0,0) (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , KURTOSIS) NON-MONOTONIC PATTERN 94, PM (1,3) (1.8,6) N O N - N O R M A L r T Y ( S K E W , K U R T O S I S ) = 94, PM = 15 (1.3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , KURTOSIS ) : 94, PM = 20 (3, 25) (1.3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) LEGEND: MS •PW; RS; EM Figure 32: Mean coverage probability for P i (N = 94) Note: N = Sample size, P M = Percent missing 126 MONOTONIC PATTERN NON-MONOTONIC PATTERN N = 153, PM = 10 N = 153, PM = 10 ( 1 , 3 ) ( 1 . 8 , 6 ) NON-NORMALITY (SKEW, KURTOSIS) (3, 25) o > o z < LL1 2 ( 0 , 0 ) ( 1 , 3 ) ( 1 . 8 , 6 ) NON-NORMALITY (SKEW, KURTOSIS) N = 153, PM = 15 N = 153, PM = 15 (0. 0) ( 1 , 3) (1.8, 6) NON-NORMALITY (SKEW, KURTOSIS) N = 153, PM = 20 (3, 25) ( 0 , 0 ) ( 1 , 3 ) ( 1 8 , 6 ) NON-NORMALITY (SKEW, KURTOSIS) O cc s cc o o ( 0 , 0 ) ( 1 , 3 ) ( 1 8 , 6 ) NON-NORMALITY (SKEW, KURTOSIS) N = 153, PM = 20 100 BIL 90 < m O 85 cc 0 _ LU 80 < cc 75 LU > o 70 o AN 65 LU 2 60 ( 1 . 3 ) ( 1 . 8 , 6 ) NON-NORMALITY (SKEW, KURTOSIS) LEGEND: MS PW; RS; EM Figure 33: Mean coverage probability for (j, (N = 153) Note: N = Sample size, PM = Percent missing. 127 MONOTONIC PATTERN ( 0 , 0 ) ( 1 , 3 ) (1.C NON-NORMALITY (SKEW, KURTOSIS) (3, 25) N = 2 6 5 , P M = 15 ( 0 , 0 ) ( 1 , 3 ) (1 .8 ,6 ) NON-NORMALITY (SKEW, KURTOSIS) N = 2 6 5 , P M = 2 0 (0. 0) ( 1 , 3) (1.8, 6) NON-NORMALITY (SKEW, KURTOSIS) (3, 25) NON-MONOTONIC PATTERN N = 2 6 5 , P M = 10 ( 0 , 0 ) ( 1 , 3 ) ( 1 . 8 , 6 ) ( 3 , 2 5 ) N O N - N O R M A L I T Y ( S K E W , KURTOSIS ) N = 2 6 5 , P M = 15 ( 0 , 0 ) ( 1 , 3 ) ( 1 . 8 , 6 ) NON-NORMALITY (SKEW, KURTOSIS) (3, 25) N = 2 6 5 , P M = 2 0 ( 0 , 0 ) ( 1 , 3 ) ( 1 . 8 , 6 ) NON-NORMALITY (SKEW, KURTOSIS) ( 3 , 2 5 ) LEGEND: MS; PW; RS; EM Figure 34: Mean coverage probability for (N = 265) Note: N = Sample size, P M = Percent missing. 128 Summary The effect of non-normality on absolute error for R2esnmate was of practical significance under each M D T at all levels of sample size. The performance of all M D T s with respect to error for R2estimate decreased with increasing non-normality. E M performed the best under sample size 94 and 153, followed by RS , M S and P W , in that order. A t sample size 265, the absolute error for R2estimate was much smaller than at either sample size 94 or 153. The superiority of E M was for both monotonic and non-monotonic missing patterns. Also , at sample size 265, the relative performance among E M , M S and RS became more and more undifferentiated with increasing percent missing. The effects of design variables on absolute error for b] were also presented in the previous sections. Using r | 2 ai t > .03 as the criterion, the effect of non-normality was of practical significance under RS only. With sample size 265, the effect of pattern was of practical significance under all M D T s . Pattern effects were strongest under E M algorithm, and weakest under P W . In addition to pattern effects, the effect of non-normality was of practical significance under R S . N o effects fulfilled the r | 2 a I t > .03 criterion for b2,-. -,bg. The accuracy in parameter estimation under all M D T s deteriorated with increasing non-normality. This was regardless of pattern of missing data. Generally, E M and RS were more accurate than M S and P W . However, RS tended to perform better at moderate to high levels of percent missing (15% to 20%). The smallest bias with respect to R 2 was not under a single M D T across levels of the design variables. A t sample size 94, the smallest bias was under RS followed by P W . However, at sample size 153 and 265, the smallest bias was under E M . In the estimation of p i , RS was the most accurate under al l design conditions. However, the smallest bias in the estimation of Pi was under P W . This occurred at all levels of the design variables. E M and M S consistently overestimated Pi, and P W and R S consistently overestimated p, . Coverage probabilities under all M D T s were above the criterion level of 95% with monotonic missing pattern for normal data, and decreased with increasing non-normality. However, M D T s differed 129 in coverage probability only for P,. In addition, the graphs for p 2, p 3,...,p 9 were similar to those for Pi. For this reason, only the results for P- were reported. E M generally had the best coverage probability and M S the worst. Study 4: Nine Predictors and High R 2 Condition Effects on Absolute Error for R 2„, i m,,„. The effects of design variables on absolute error for R estimate are listed in Table 20. Using T| a i t > .03 as the criterion, the table shows that with a sample of 94, only the effect of non-normality was of practical significance under each M D T , with the largest effect being that of non-normality under pairwise deletion. For sample size 153, the effect of percent missing on absolute error was of practical significance under E M , M S , and R S . Wi th pairwise deletion, it was the effect of non-normality that was of practical significance. The largest effect was that of percent missing under mean substitution. For sample size 265, the effect of pattern was of practical significance under E M algorithm only. The effect of percent missing was of practical significance under E M , M S , and R S . A s with sample of 153, the effect of non-normality was of practical significance only under P W . The largest effect was that of percent missing under E M . Table 20 Effects on absolute error for R2^,-,^ fulfilling the T | 2 ^ T > .03 criterion Sample size M D T Effect SS eff e c t SSresidual df Tfalt 94 E M Non-normality 3.12 92.12 3 .033 MS Non-normality 3.65 76.41 3 .046 PW Non-normality 6.77 63.89 3 .096 RS Non-normality 4.26 90.91 3 .045 153 E M % Missing 7.23 76.64 2 .086 MS % Missing 3.36 63.42 2 .420 PW Non-normality 3.86 38.68 3 .091 RS % Missing 5.99 73.29 2 .076 265 E M % Missing 10.34 59.40 2 .148 Pattern 2.18 59.40 1 .035 MS % Missing 6.07 51.73 2 .105 PW Non-normality 2.24 22.22 3 .091 RS % Missing 9.28 57.49 2 .139 Note: E M = E M imputation, MS = Mean substitution, PW = Pairwise deletion, RS = Regression imputation. 131 Relative Performance of MDTs in the Estimation of Population R 2 Table 21 contains cell means and standard deviations of mean absolute errors for R2estimate under various design conditions. The influence of non-normality on the performance of MDTs at sample size 94 is revealed in Figure 35. Although the performance of all MDTs deteriorated with increasing non-normality, this deterioration was most pronounced after skew =1.8 and kurtosis = 6. This was observed for both monotonic and non-monotonic missing pattern and for all levels of percent missing. The relative performance of MDTs was undifferentiated at the lowest level of percent missing (10%), under both monotonic and non-monotonic missing patterns. However, at higher levels of percent missing (15% and 20%), PW yielded the best estimates and E M the worst for normal data, a trend that continued up to skew = 1.8 and kurtosis = 6. At the extreme level of non-normality (skew = 3 and kurtosis = 25), differentiation of performance of MDTs was least. Whereas mean absolute error for R2estimate increased with percent missing for E M , MS, and RS at each level of non-normality, the magnitude of the error under PW at each level of non-normality changed marginally for both missing patterns as well as under all conditions of percent missing. Figures 36 and 37 show the influence of non-normality on the performance of MDTs at sample sizes 153 and 265, respectively. The results were similar to those under sample size 94. In sum, the four MDTs differed most in their performance as percent missing increased, though their performance was most differentiated for normal data. MDTs were also differentiated more under non-monotonic missing pattern. PW had the best performance and E M the worst. Table A.7 in Appendix A contains bias in R2estimate under high R 2 condition with nine predictors. Bias in R2estimate varied from -.13 to .07 across all sample size conditions, giving a range of .20. The values in Table A.7 were plotted in Figure B.7, a typical set of graphs for bias in R2estimate- Although there was no single M D T under which the bias was smallest at all levels of the design variables, PW generally had the smallest bias. Whereas PW consistently overestimated R 2 , E M , MS, and RS tended to underestimateR . 132 Table 21 Means and standard deviations' of absolute error for R 2 e s t i m a t B Monotonic Missing Pattern Non-monotonic Missing Pattern N Percent Missing Normality (sk., kurt.) EM Imputation Mean Substitution Pairwise Deletion Regress. Imputation EM Imputation Mean Substitution Pairwise Deletion Regress. Imputation 94 10 (0,0) (1,3) (1.8,6) (3,25) 0520(042) 0583(045) 0665(049) 0963(070) 0524(041) 0593(044) 0683(050) 0975(069) 0545(039) 0603(042) 0698(049) 0998(070) 0525(041) 0591(044) 0678(049) 0958(069) 0547(043) 0583(046) 0634(049) 0887(070) 0535(040) 0566(043) 0625(047) 0867(062) 0535(039) 0581(041) 0665(047) 0951(069) 0538(041) 0573(045) 0631(048) 0897(073) 15 (0,0) (1,3) (1.8,6) (3,25) 0668(051) 0688(055) 0731(058) 0966(075) 0627(049) 0666(053) 0725(056) 0992(073) 0529(038) 0579(042) 0674(048) 0963(069) 0641(050) 0678(054) 0732(057) 0985(074) 0751(051) 0779(057) 0779(059) 1022(081) 0618(047) 0632(049) 0667(051) 0894(066) 0522(038) 0568(041) 0652(046) 0944(074) 0679(049) 0713(054) 0748(057) 1020(082) 20 (0,0) (1,3) (1.8,6) (3,25) 0878(043) 0871(063) 0876(066) 1053(081) 0775(056) 0781(060) 0807(062) 1026(076) 0520(039) 0563(042) 0653(048) 0938(068) 0825(058) 0829(062) 0847(064) 1052(079) 1005(059) 1014(066) 1011(069) 1223(093) 0763(054) 0750(056) 0756(057) 0949(071) 0514(038) 0554(042) 0648(048) 0964(081) 0913(058) 0936(064) 0952(067) 1236(099) 153 10 (0,0) (1,3) (1.8,6) (3,25) 0502(039) 0527(041) 0567(044) 0753(059) 0483(038) 0515(040) 0565(043) 0762(058) 0409(031) 0451(033) 0523(038) 0754(055) 0490(039) 0521(041) 0569(043) 0764(058) 0567(040) 0607(042) 0637(045) 0813(064) 0478(036) 0497(037) 0534(039) 0686(053) 0401(031) 0441(033) 0510(037) 0725(056) 0508(038) 0544(040) 0584(042) 0781(062) 15 (0,0) (1,3) (1.8,6) (3,25) 0684(046) 0692(050) 0705(053) 0828(065) 0621(045) 0633(048) 0655(051) 0806(062) 0401(031) 0441(033) 0513(037) 0742(055) 0644(046) 0659(049) 0680(052) 0822(064) 0801(046) 0827(051) 0836(055) 1003(075) 0620(043) 0628(044) 0641(046) 0779(061) 0392(030) 0432(033) 0499(037) 0707(055) 0705(045) 0741(049) 0760(052) 0971(074) 20 (0,0) (1,3) (1.8,6) (3,25) 0951(052) 0937(058) 0928(063) 0979(077) 0843(052) 0833(057) 0832(060) 0913(070) 0398(030) 0436(033) 0505(037) 0727(054) 0886(052) 0878(058) 0879(061) 0956(074) 1094(050) 1113(057) 1101(064) 1254(088) 0829(050) 0821(053) 0814(056) 0930(070) 0387(031) 0428(033) 0499(037) 0704(058) 0980(050) 1014(056) 1017(062) 1225(087) 265 10 (0,0) (1,3) (1.8,6) (3,25) 0481(033) 0489(035) 0507(039) 0601(049) 0448(032) 0456(035) 0480(038) 0594(047) 0311(023) 0335(025) 0385(029) 0572(043) 0457(032) 0466(035) 0490(038) 0602(048) 0588(035) 0617(039) 0628(044) 0790(059) 0459(033) 0470(036) 0494(038) 0621(047) 0306(023) 0330(025) 0381(029) 0557(041) 0503(034) 0535(037) 0557(041) 0743(056) 15 (0,0) (1,3) (1.8,6) (3,25) 0733(039) 0730(043) 0724(048) 0743(059) 0657(039) 0654(043) 0655(047) 0703(055) 0307(023) 0331(025) 0380(029) 0558(042) 0680(039) 0679(043) 0680(047) 0724(057) 0872(040) 0904(045) 0901(051) 1049(072) 0666(040) 0664(043) 0665(046) 0769(059) 0301(023) 0326(025) 0375(028) 0542(040) 0767(040) 0802(044) 0809(049) 1003(070) 20 (0,0) (1,3) (1.8,6) (3,25) 1008(041) 0995(047) 0978(052) 0930(069) 0899(042) 0884(047) 0868(052) 0839(065) 0304(023) 0327(025) 0375(028) 0548(041) 0940(042) 0928(047) 0915(053) 0887(067) 1156(041) 1189(048) 1187(055) 1324(080) 0895(043) 0896(048) 0887(052) 0965(068) 0302(023) 0324(025) 0370(028) 0533(042) 1047(042) 1095(048) 1103(054) 1288(079) * All values in the table are preceded by an omitted decimal point. Standard deviations are in parentheses. Standard errors of the means can be obtained by dividing each standard deviation by 1000^ . 133 cc o o cc cc m < MONOTONIC PATTERN 94, PM = 10 o CO CC CC O C .3) (1 .8 ,6) (3 ,25) NON-NORMALITY (SKEW, KURTOSIS) N = 94, PM = 15 (0,0) (1.3) (1.8,6) (3,25) NON-NORMALITY (SKEW, KURTOSIS) N = 94, PM = 20 (0 ,0 ) (1 ,3 ) (1 .8 ,6) (3 ,25 ) N O N - N O R M A L I T Y ( S K E W , KURTOSIS ) O co cc cc O u. cc O NON-MONOTONIC PATTERN N = 94, PM = 10 o CO cc cc o Ll-CC o cc 3 _l o CO CO < z < C .3) (1 .8 ,6 ) (3, 25) NON-NORMALITY (SKEW, KURTOSIS) = 94, PM = 15 (1 .3 ) (1 .8 .6) (3 ,25) NON-NORMALITY (SKEW, KURTOSIS) N = 94, PM = 20 ( 0 . 0 ) ( 1 , 3 ) (1 .8 .6) (3 ,25) N O N - N O R M A L I T Y ( S K E W , KURTOSIS ) LEGEND: MS PW; RS; EM Figure 35: Mean absolute error for R 2 e stim a te across levels of non-normality (N = 94) Note: N = Sample size, P M = Percent missing, RSQ = R2 e s timate MONOTONIC PATTERN N = 153, PM = 10 (1.3) (1 .8,6) (3, 25) NON-NORMALITY (SKEW, KURTOSIS) N = 153, PM (0,0) (1,3) (1 .8,6) NON-NORMALITY (SKEW, KURTOSIS) N = 153, PM = 20 (3,25) (1.8,6) NON-NORMALITY (SKEW, KURTOSIS) (3,25) LEGEND: MS DC DC o a. DC O CC cc m < O co DC DC o Li-CC O CC DC LU o CO m < z < LU NON-MONOTONIC PATTERN 153 , P M = 10 (1.8,6) NON-NORMALITY (SKEW, KURTOSIS) N = 153 , P M = 15 (0, 0) (1, 3) (1.8, 6) NON-NORMALITY (SKEW, KURTOSIS) N = 153 , P M = 2 0 (0,0) (1,3) (1.8,6) NON-NORMALITY (SKEW, KURTOSIS) PW; RS; Figure 36: Mean absolute error for R estimate across levels of non-normality (N = 153) Note: N = Sample size, P M = Percent missing, RSQ = R 2 e s t imate 135 MONOTONIC PATTERN N = 265, PM = 10 cc O l l -OC o CC DC O CO cc cc o o DC CC o CO CO < ( 1 . 8 , 6 ) NON-NORMALITY (SKEW, KURTOSIS) N = 265, PM = 15 ( 3 , 2 5 ) O CO DC DC O l i -CC O CC .13 .12 .1 1 .10 .09 .08 .07 .06 .05 .04 .03 .02 ( 0 , 0 ) ( 1 , 3 ) ( 1 . 8 , 6 ) ( 3 , 2 5 ) NON-NORMALITY (SKEW, KURTOSIS) 265, PM = 20 ( 3 , 2 5 ) NON-NORMALITY (SKEW, KURTOSIS) NON-MONOTONIC PATTERN N = 265, PM = 10 (0 ,0 ) ( 1 , 3 ) (1 .8 ,6 ) NON-NORMALITY (SKEW, KURTOSIS) N = 265, PM = 15 (3 ,25 ) (3 ,25) NON-NORMALITY (SKEW, KURTOSIS) N = 265, PM = 20 o co cc cc o LL DC O DC ce (0 ,0 ) ( 1 , 3 ) (1 .8 ,6 ) NON-NORMALITY (SKEW, KURTOSIS) (3, 25) LEGEND: MS PW; RS; EM Figure 37: Mean absolute error for R e s t i m a t e across levels of non-normality (N = 265) Note: N = Sample size, P M = Percent missing, RSQ = R 2 e 5 U mate 136 Effects on Absolute Error for Regression Coefficients Table 22 shows the effects of design variables on absolute error in b i , b 2 , . . . , b c , . The effect of missing pattern was strongest for b i , the regression coefficient of the predictor with no missing data. More specifically, under E M , the effect of missing pattern on absolute error for b] was of practical significance at all levels of sample size. Under M S and P W , pattern effects were significant only for sample size 265. There were no effects of practical significance for bs. Under M S and P W , the effect of non-normality on absolute error of b i was of practical significance under sample size 265 only. However, under R S , the effect of non-normality was of practical significance at all levels of sample size. The effect of non-normality on absolute error for b] under E M was not of practical significance. Only non-normality had an effect of practical significance for b 2 to be,. Table 22 Effects on absolute error for regression coefficients fulfilling the criterion of r\2^_ > .03* Sample size M D T b, b 2 b 3 b 4 b 5 bfi b 7 b 8 b 9 94 E M PT(042) NL(063) NL(038) NL(050) NL(068) NL(042) NL(045) MS - NL(061) NL(032) NL(048) NL(073) NL(039) NL(045) -PW - NL(060) NL(032) NL(047) NL(074) NL(039) NL(043) -RS NL(038) NL(066) NL(034) NL(051) NL(071) NL(038) NL(047) -153 E M PT(182) NL(057) NL(036) NL(054) NL(087) NL(056) NL(056) MS - NL(056) NL(031) NL(056) NL(089) NL(050) NL(061) -PW - NL(056) NL(030) NL(054) NL(089) NL(050) NL(060) -RS PT(058) - - - - - -NL(054) NL(061) NL(033) NL(058) NL(089) NL(048) NL(057) -265 E M PT(382) - - - _ _ - NL(082) NL(047) NL(071) NL(098) NL(069) NL(073) NL(039) MS PT(050) - - - - - -NL(031) NL(083) NL(041) NL(063) NL(105) NL(059) NL(077) NL(035) PW PT(050) - - - - _ _ _ NL(031) NL(084) NL(041) NL(063) NL(105) NL(060) NL(077) NL(034) RS PT(182) - - - - - - _ NL(088) NL(089) NL(045) NL(071) NL(100) NL(058) NL(075) NL(038) * Values of r i 2 a l t are in parentheses preceded by an omitted decimal point. Note: E M = E M imputation, MS = Mean substitution, PW = Pairwise deletion, RS = Regression imputation, PT = Pattern, N L = Non-normality, - = No effect of practical significance. 137 Relative Performance of MDTs in Estimation of Table 23 contains means and standard deviations of absolute error for in bi at various design conditions. The means at sample size 94 are plotted in Figure 38. The figure shows that mean absolute error for bi increased with non-normality. RS performed the best, regardless of missing pattern. However, under monotonic pattern of missing data, MDTs were least differentiated in performance at 10% missing, the degree of differentiation increasing with percent missing. RS consistently gave the best performance, followed by E M . With non-monotonic pattern, again RS outperformed the other MDTs, although its performance relative to E M was most prominent with normal data. At the extreme level of non-normality, little difference in performance was observed between RS and E M . Whereas E M consistently outperformed both MS and PW at 10% missing under non-monotonic missing pattern, MS and PW had a slight advantage over E M at 15% and 20% missing at low to moderate levels of non-normality. MDTs were more differentiated under non-monotonic pattern than under monotonic missing pattern. Figure 39 shows mean absolute error for bi at sample size 153 across levels of non-normality. Under monotonic pattern, the MDTs differed more in performance as percent missing increased. Unlike the finding at sample size 94, where MS and PW performed poorer than E M with monotonic pattern, little differentiation in performance was observed between M S , PW and E M at sample size 153 under low to moderate percent missing. At 20% missing, MS and PW did better than E M at all levels of non-normality. Under non-monotonic pattern, RS performed the best, followed by MS and PW. E M performed the worst. The MDTs differed most in performance with normal data, the differentiation becoming more blurred with increasing non-normality. At the most extreme non-normality and 20% missing, the performance of all MDTs was least differentiated. Graphical plots of mean absolute error in bi at sample size 265 are displayed in Figure 40, showing that under monotonic pattern, RS was best, followed by MS and PW. E M performed 138 the worst. Under non-monotonic pattern, however, RS only excelled with normal data at ten percent missing. The performance of M S and P W was best. However, the relative performance of M S and RS as well as P W and RS was least differentiated with normal data, and most differentiated at the highest level of non-normality (skew = 3, kurtosis = 25). In sum, RS performed the best under monotonic missing pattern at sample size of 265. Under non-monotonic pattern, M S and P W outperform RS at moderate to high levels of percent missing (15% to 20%). Table A . 8 shows the bias in bi under nine predictors and low R 2 condition. Figure B.8 is the corresponding set of graphs for bias in bi for N = 94, showing that for non-monotonic pattern, the smallest bias across all levels of the design variables was under M S and P W . The bias in b, varied from - .10 to .07 across all design conditions, giving a range of .17. Whereas M S and P W consistently overestimated p i , E M and RS consistently underestimated Pi. The bias for b 2 to bo were much smaller compared to that for b i , with a maximum range of .10 under b 7 . Therefore, the graphs for bias in b 2 to bg were omitted. 139 Table 23 Means and standard deviations* of absolute error for b Monotonic Missing Pattern Non-monotonic Missing Pattern N Percent Missing Normality (sk, kurt.) EM Imputation Mean Substitution Pairwise Deletion Regress. Imputation EM Imputation Mean Substitution Pairwise Deletion Regress. Imputation 94 10 (0,0) (1,3) (1.8,6) (3,25) 0573(041) 0639(046) 0696(052) 0784(071) 0625(046) 0669(052) 0728(058) 0816(077) 0625(046) 0670(052) 0730(058) 0817(078) 0579(043) 0631(047) 0687(052) 0770(068) 0682(040) 0761(039) 0793(040) 0932(043) 0726(053) 0796(063) 0863(074) 1004(104) 0727(053) 0797(063) 0865(074) 1006(104) 0587(041) 0663(041) 0717(044) 0929(057) 15 (0,0) (1,3) (1.8,6) (3,25) 0591(040) 0647(044) 0699(051) 0797(068) 0654(049) 0693(054) 0747(060) 0833(080) 0654(049) 0696(054) 0752(060) 0840(080) 0575(043) 0618(046) 0664(051) 0747(064) 0835(038) 0891(038) 0903(038) 0992(038) 0791(059) 0863(070) 0928(082) 1101(115) 0794(059) 0867(070) 0931(082) 1106(115) 0605(038) 0699(037) 0740(040) 0970(050) 20 (0,0) (1,3) (1.8,6) (3,25) 0609(040) 0653(043) 0701(048) 0783(063) 0677(051) 0709(055) 0760(061) 0844(083) 0679(052) 0716(056) 0767(062) 0854(084) 0567(043) 0609(045) 0660(049) 0741(063) 0865(035) 0915(036) 0912(037) 0981(039) 0834(062) 0893(073) 0953(085) 1136(118) 0841(062) 0901(073) 0961(085) 1146(118) 0686(038) 0781(039) 0806(041) 1009(055) 153 10 (0,0) (1,3) (1.8,6) (3,25) 0477(033) 0503(035) 0550(039) 0639(053) 0496(036) 0519(038) 0564(043) 0634(056) 0496(036) 0519(038) 0565(043) 0634(056) 0452(033) 0476(035) 0521(038) 0593(051) 0794(035) 0873(034) 0882(036) 0989(036) 0580(044) 0644(051) 0709(058) 0871(086) 0581(044) 0645(051) 0711(058) 0872(087) 0498(032) 0582(034) 0624(035) 0864(042) 15 (0,0) (1,3) (1.8,6) (3,25) 0505(034) 0531(036) 0574(039) 0661(052) 0509(037) 0534(039) 0578(044) 0657(058) 0510(037) 0535(040) 0579(044) 0657(058) 0452(033) 0478(035) 0524(039) 0608(050) 0855(031) 0917(031) 0915(033) 0991(033) 0621(047) 0687(053) 0741(061) 0921(090) 0622(047) 0687(053) 0743(061) 0922(090) 0564(033) 0656(034) 0684(035) 0909(043) 20 (0,0) (1,3) (1.8,6) (3,25) 0553(035) 0576(036) 0616(038) 0689(051) 0517(039) 0540(041) 0582(046) 0661(060) 0520(039) 0542(042) 0588(046) 0655(060) 0450(034) 0476(034) 0519(038) 0600(049) 0922(029) 0965(028) 0956(031) 0992(034) 0672(050) 0735(056) 0790(064) 0974(093) 0675(050) 0737(056) 0792(064) 0976(094) 0625(031) 0716(033) 0737(036) 0939(043) 265 10 (0,0) (1,3) (1.8,6) (3,25) 0370(027) 0391(028) 0425(030) 0508(038) 0350(027) 0379(029) 0415(033) 0508(043) 0350(027) 0379(029) 0415(033) 0510(043) 0323(025) 0346(026) 0383(029) 0470(037) 0792(028) 0881(028) 0890(030) 1002(030) 0472(034) 0532(041) 0572(047) 0729(047) 0473(034) 0473(034) 0574(047) 0730(066) 0424(027) 0527(029) 0562(030) 0815(035) 15 (0,0) (1,3) (1.8,6) (3,25) 0428(028) 0437(029) 0460(031) 0534(037) 0367(028) 0395(031) 0435(034) 0527(044) 0368(028) 0395(031) 0435(034) 0528(044) 0333(025) 0355(026) 0387(029) 0470(037) 0868(025) 0929(025) 0930(027) 1001(029) 0533(037) 0587(045) 0625(051) 0793(071) 0534(037) 0587(045) 0627(051) 0793(071) 0522(027) 0627(028) 0653(030) 0886(035) 20 (0,0) (1,3) (18,6) (3,25) 0481(029) 0488(030) 0506(032) 0574(036) 0372(029) 0400(031) 0443(035) 0535(045) 0373(029) 0401(031) 0443(035) 0537(045) 0334(025) 0357(026) 0390(028) 0475(035) 0913(022) 0958(023) 0953(025) 0997(028) 0573(040) 0628(047) 0664(053) 0834(073) 0574(040) 0630(047) 0666(053) 0835(073) 0604(025) 0704(026) 0722(029) 0922(035) * All values in the table are preceded by an omitted decimal point. Standard deviations are in parentheses. Standard errors of the means can be obtained by dividing each standard deviation by 1000 v \ 140 MONOTONIC PATTERN N = 94, PM = 10 (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , KURTOSIS) = 94, PM = 15 0) (1,3) (1.8,6) N O N - N O R M A L I T Y (SKEW, KURTOSIS) N = 94, PM = 20 (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , KURTOSIS) LEGEND: MS PW; NON-MONOTONIC PATTERN N = 94, PM = 10 >. 0) (1,3) (1.8,6) NON-NORMALITY (SKEW, KURTOSIS) N = 94, PM = 15 (3, 25) C. 0) (1, 3) (1.8, 6) NON-NORMALITY (SKEW, KURTOSIS) N = 94, PM = 20 CD CC O LL CC O CC CC LU LU I -D _ l o co m < (0,0) (1,3) (1.8,6) NON-NORMALITY (SKEW, KURTOSIS) RS; EM Figure 38: Mean absolute error for b) across levels of non-normality (N = 94) Note: N = Sample size, P M = Percent missing. 141 N = MONOTONIC PATTERN 153, PM = 10 CC o LL tr O CC CC D o C O CO < z < CC o Li-CC o CC o CO m < < LU 5 N = 0 .3 ) ( 1 . 8 . 6 ) NON-NORMALITY (SKEW, KURTOSIS) 153, PM = 15 I. 0) ( 1 . 3 ) ( 1 . 8 , 6 ) NON-NORMALITY (SKEW, KURTOSIS) N = 153, PM =2 0 (1 .3) (1 .8,6) NON-NORMALITY (SKEW, KURTOSIS) NON-MONOTONIC PATTERN N = 153, PM = 10 ( 0 . 0 ) ( 1 . 3 ) ( 1 . 8 . 6 ) NON-NORMALITY (SKEW, KURTOSIS) N = 153, PM = 15 (0 .0 ) (1 , 3 ) (1 . 8 ,6 ) NON-NORMALITY (SKEW, KURTOSIS) N = 153, PM = 20 (3, 25) m cc O O CC O co < (CO) (1,3) (1.8,6) NON-NORMALITY (SKEW, KURTOSIS) (3,25) LEGEND : MS PW; RS; •EM Figure 39: Mean absolute error for bi across levels of non-normality (N = 153) Note: N = Sample size, P M = Percent missing. MONOTONIC PATTERN 2 6 5 , PM = 10 CO cc o l l -CC o cc o (1 .3 ) (1 .8 ,6) NON-NORMALITY (SKEW, KURTOSIS) = 2 6 5 , PM = 15 (3, 26) —I o CO CO < z < ( 1 . 3 ) (1 .8 ,6) NON-NORMALITY (SKEW, KURTOSIS) 2 6 5 , PM = 20 (3 ,25) (0,0) ( 1 . 3 ) (1 .8 ,6 ) NON-NORMALITY (SKEW, KURTOSIS) ( 3 , 2 5 ) NON-MONOTONIC PATTERN 265, PM = 10 CO cc o l i -CC o cc 3 - J o CO CO < z < 0 . 3 ) (1 .8 ,6) NON-NORMALITY (SKEW, KURTOSIS) 2 6 5 , PM = 15 (3 ,25) N = ( 1 . 3 ) (1 .8 ,6 ) NON-NORMALITY (SKEW, KURTOSIS) 2 6 5 , PM = 20 (3 ,25) CO CC o l i -ce O i-D _ l o CO CO < z < ( 1 , 3 ) (1 .8 ,6 ) NON-NORMALITY (SKEW, KURTOSIS) (3 ,25) L E G E N D : MS PW; RS; E M Figure 40: Mean absolute error for b! across levels of non-normality (N = 265) Note: N = Sample size, P M = Percent missing. 143 Coverage Probability for p\ The M D T s differed in their coverage probabilities for p i only. The graphs for coverage probability of Pi, p 2 , P 3 , . . . , P Q were similar to those of Pi. For this reason, only the findings for pi are reported. Table 24 contains coverage probabilities at various levels of the design variables. Figure 41 is a set of line graphs at sample size 94. The graphs show that coverage probability decreased with increasing non-normality. Coverage probabilities under all M D T s were above the criterion level of 95% for normal data. This suggests that all M D T s worked well for normal data irrespective of missing pattern or percent missing. M D T s were most differentiated under non-monotonic pattern. E M and R S were the best and M S the worst. Figure 42 shows coverage probabilities for Pi at sample size 153, indicating that coverage probabilities decreased with increasing non-normality. The relative performance of M D T s was most differentiated under non-monotonic than under monotonic missing pattern. RS had the best coverage probabilities and M S the worst. Figure 43 is a set of graphical plots of mean coverage probabilities for Pi at sample size 265, showing that coverage probability decreased with increasing non-normality. The graphs also show that the M D T s were most differentiated under non-monotonic than under monotonic missing pattern. A l l M D T s performed close to the criterion level of 95% for normal data under monotonic pattern. Under non-monotonic pattern, all M D T s had coverage probabilities below the expected value. A t ten percent missing, RS had coverage probability closest to the expected value, except at the highest level of non-normality (skew = 3 and kurtosis = 25). A t this point, P W had the best coverage probability. This outcome was also observed at fifteen percent missing. A t 20% missing, P W had the best coverage probability and E M the worst (below 65%). 144 Table 24 Mean coverage probability (%) for fti Monotonic Missing Pattern Non-monotonic Missing Pattern N Percent Missing Normality (sk, kurt.) EM Imputation Mean Substitution Pairwise Deletion Regress. Imputation EM Imputation Mean Substitution Pairwise Deletion Regress. Imputation 94 10 (0,0) 97.9 95.9 97.1 97.4 98.6 89.5 92.2 97.2 (1,3) 96.3 93.4 95.3 95.6 96.5 86.6 89.3 96.2 (1.8,6) 93.8 90.7 92.6 93.6 94.3 82.9 86.1 93.5 (3,25) 89.2 87.2 89.4 89.9 85.3 76.6 79.3 84.4 15 (0,0) 98.7 95.7 97.2 98.4 98.8 86.1 91.1 98.7 (1,3) 97.2 93.5 96.2 96.7 96.3 83.9 88.3 97.4 (1.8,6) 95.8 91.0 94.0 94.9 94.8 81.6 85.0 95.9 (3,25) 90.6 86.8 90.0 91.2 85.9 75.1 79.4 84.5 20 (0,0) 99.3 95.6 97.6 98.4 99.3 85.7 91.8 98.6 (1,3) 97.9 93.0 96.2 97.3 98.2 83.0 89.2 97.8 (1.8,6) 97.3 90.9 94.4 96.0 96.1 81.2 87.2 96.1 (3,25) 92.5 86.5 91.2 92.5 88.5 74.9 81.7 85.7 153 10 (0,0) 97.9 96.4 97.2 97.3 89.0 88.5 91.0 97.5 (1,3) 96.5 94.5 95.4 96.3 82.2 84.4 86.9 94.4 (1.8,6) 93.8 90.6 93.2 93.9 79.9 81.5 83.5 91.7 (3,25) 87.5 85.2 87.9 88.4 64.5 73.3 76.1 73.6 15 (0,0) 98.2 95.6 97.9 98.0 90.1 85.9 90.3 96.8 (1,3) 96.4 94.4 96.7 97.0 85.1 81.7 86.7 93.7 (1.8,6) 93.9 91.0 93.8 95.3 81.7 79.0 83.8 90.6 (3,25) 87.4 85.0 89.8 • 89.3 70.1 72.4 76.4 71.9 20 (0,0) 98.2 95.2 98.1 98.4 90.8 84.8 90.5 97.2 (1,3) 96.3 93.2 97.1 97.2 85.2 80.6 87.3 93.5 (1.8,6) 94.6 90.5 95.2 95.2 81.5 77.4 83.8 91.0 (3,25) 88.6 85.7 91.9 90.6 72.8 70.5 77.3 74.4 265 10 (0,0) 96.4 95.7 97.0 97.5 67.5 87.0 89.1 93.3 (1,3) 95.2 93.7 95.4 96.1 53.9 81.4 84.2 87.3 (1.8,6) 92.8 91.3 92.9 93.8 51.6 77.3 79.7 81.8 (3,25) 84.6 84.1 87.1 87.0 38.6 68.1 70.3 56.2 15 (0,0) 96.1 95.6 97.6 97.6 61.8 81.4 87.1 91.3 (1,3) 94.3 94.0 96.3 97.2 51.6 77.1 81.4 82.2 (1.8,6) 91.1 90.9 94.7 94.4 50.2 73.8 78.1 77.2 (3,25) 84.0 84.0 88.7 87.0 41.8 65.4 67.9 53.8 20 (0,0) 95.3 95.7 98.3 98.1 59.4 80.1 87.3 89.1 (1,3) 92.6 93.4 97.1 97.6 50.3 74.7 82.0 79.1 (1.8,6) 90.8 91.0 95.2 95.4 51.3 71.2 78.7 74.5 (3,25) 83.8 95.4 90.2 89.4 47.1 64.1 69.5 53.4 145 MONOTONIC PATTERN N = 9 4 , P M = 1 0 (0.0) (1.3) (1.8,6) NON-NORMAUTY (SKEW, KURTOSIS (3,25) N = 9 4 , P M = 1 5 (0 ,0) (1 ,3) (1.8.6 NON-NORMALITY (SKEW, KURTOSIS) N = 9 4 , P M = 2 0 (0 ,0) (1,3) (1.8,6) NON-NORMAUTY (SKEW, KURTOSIS) (3 ,25) NON-MONOTONIC PATTERN N = 9 4 , P M = 1 0 (1 .3 ) (1 .8 ,6) NON-NORMALITY (SKEW, KURTOSIS) (3, 25) N = 9 4 , P M = 1 5 (0 .0 ) (1 ,3 ) (1 .8 ,6) NON-NORMAUTY (SKEW, KURTOSIS) N = 9 4 , P M = 2 0 (1 .3 ) (1 .8 ,6) NON-NORMALITY (SKEW, KURTOSIS) LEGEND: MS P W ; RS; EM Figure 41: Mean coverage probability for p\ (N = 94) Note: N = Sample size and P M = Percent Missing. 146 MONOTONIC PATTERN N = 153 , P M = 10 >•") (1,3) (1.8.6) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 153 , P M = 15 (0. 0) (1.3) (1.8,6) NON=NORMALITY (SKEW, KURTOSIS) N = 153, P M = 20 (1 ,3) (1.8,6) (3 ,25) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) o DC 13 < O o z < CO < CO o cc CD < DC LU > o o z < LU NON-MONOTONIC PATTERN N = 153, P M = 10 (1 .3 ) (1.1 N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) N = 153, P M = 15 (3 ,25) (1 .3) (1.8; 6) NON-NORMALITY (SKEW, KURTOSIS) N = 153, P M = 2 0 (3, 25) (1 .3) (1.8,6) (3 ,25) N O N - N O R M A L I T Y ( S K E W , K U R T O S I S ) LEGEND: MS PW; RS; EM Figure 42: Mean coverage probability (N = 153) Note: N = Sample size and P M = Percent Missing 147 CO < CQ o cc CL LU CD < cc LU > O o CQ < CQ o DC CL LU CD < CC LU > o o z < LU 5 CO < CO o cc CD < cc LU > o o z < U J MONOTONIC PATTERN N = 265, P M = 10 (0,0) 0,3) (1.8,6) N O N - N O R M A L I T Y (SKEW, KURTOSIS ) N = 265, P M = 15 (3, 25) (0,0) (1.3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , KURTOSIS) N = 265, P M = 20 (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , KURTOSIS) (3. 25) CQ < CQ O CC CL LU CD < cc LU > O o NONMONOTONIC PATTERN N = 265, P M = 10 (1,3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , KURTOSIS ) N = 265, P M = 15 (3725) (1.3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , KURTOSIS ) N = 265, P M = 20 (3. 25) (1.3) (1.8,6) N O N - N O R M A L I T Y ( S K E W , KURTOSIS ) (3, 25) LEGEND: MS PW; RS; EM Figure 43: Mean coverage probability (N = 265) Note: N = Sample size and P M = Percent Missing 148 Summary In the previous sections, the effects of pattern of missing data, percent missing and non-normality on absolute error for parameter estimates were reported. The effects of non-normality on absolute error for R2es-imate were the most prominent across M D T s at sample size 94. However, at sample size 153, the effect of percent missing was the most dominant across M D T s . The effect of missing pattern only showed at sample size 265 under E M . In the estimation of population R 2 , M D T s differed marginally when 10% of the data was missing, regardless of missing pattern. M D T s were differentiated more under non-monotonic than under monotonic missing pattern. In addition, M D T s were most differentiated for normal data, and also when percent missing was highest. Overall , the performance of P W was the best and E M the worst. The effects of design factors on regression coefficients were also reported in the previous sections. O f all factors, the effect of missing pattern on absolute error for bi was the strongest. For the remaining regression coefficients, except for b 5 , it was the effect of non-normality that was of practical significance, with the strongest effect being at sample size 265. Wi th respect to mean absolute error for regression coefficients, the M D T s differed most in the estimation of p - , and least in the estimation of p 2 , p3 , . . .po . In general, RS performed the best with respect to accuracy. However, with non-monotonic missing pattern at sample size 265, and 15% to 20% missing, M S and P W tended to outperform the other M D T s . Also , M D T s were more differentiated under non-monotonic pattern than under monotonic pattern of missing data. The smallest bias in R 2 at different levels of the design variables was not under a single M D T . However, the bias under P W was in most cases the smallest. Whereas P W consistently overestimated R 2 , E M , M S , and RS tended to underestimate R 2 . For Pi, the smallest bias was under M S and P W with non-monotonic pattern. M S and P W consistently overestimated Pi,and E M and R S consistently underestimated Pi. 149 With respect to coverage probability of p i , all M D T s performed well under monotonic missing pattern for normal data. Coverage probabilities decreased with increasing non-normality. Both E M and RS had similar coverage probabilities at sample size 94. A t sample size 153, the performance of R S was the best and M S the worst. For non-normal data under non-monotonic missing pattern at sample size 265, all M D T s generally had poor coverage probabilities, the worst being under E M at extreme non-normality. 150 CHAPTER 5 DISCUSSION In the present study, the relative performance of MS (mean substitution), PW (pairwise deletion), RS (regression imputation) and E M (expectation-maximization) methods was investigated in four studies under different conditions of sample size, missing pattern, percent missing and non-normality. Four similar studies were conducted using multiple regression models. Study 1 had four predictors with multiple R2 of .19, Study 2 four predictors with multiple R 2 of .59, Study 3 nine predictors with multiple R 2 of .21, and Study 4 nine predictors with R 2 of .58. What follows in this final chapter is a comparative discussion of findings from the four studies in relation to existing literature. Effects on Absolute Error of Parameter Estimates The effects of missing pattern, percent missing and non-normality on absolute error in R 2 es t ima te were of practical significance upon treatment of missing values with MS, PW, RS and E M . As expected, there were sizeable effects for missing pattern, percent missing and non-normality. Mean absolute error was generally smaller under monotonic pattern, and smallest at 10% missing. In addition, mean absolute error was smallest for normal data. When the number of predictors was four and multiple R 2 was .19, percent missing seemed to be the most important factor related to absolute error for R 2 e s t i i r a t e - This was especially so under E M and RS. However, this effect was only of practical significance with large samples (N = 265). When the number of predictors was four and multiple R 2 was .59, the effect of percent missing was of practical significance under MS at sample size 265, and under RS and E M at all levels of sample size. Whereas non-normality seemed to have an influence on absolute error mainly under PW, it was the ordinal interaction effect of missing pattern and non-normality that had effects of practical significance under RS and E M . Under RS and E M , mean absolute error 151 2 for R estimate was smaller for monotonic pattern than for non-monotonic pattern. However, the difference in mean absolute error across patterns was much greater at higher levels of non-normality. When the number of predictors was nine and multiple R 2 was .21, the effect of non-normality on absolute error for R2estimate seemed to be the most important, and this was under all M D T s . A t sample size 94 and 153, the effect of non-normality was strongest under M S . However, at sample size 265, the strongest effect was under P W . Under multiple R 2 of about .58 with nine predictors, non-normality seemed to be an important factor only at sample size 94. This was with the exception of P W that was affected by non-normality at all levels of sample size. A t sample sizes 153 and 265, percent missing appeared to be the most important factor. However, with E M , pattern effects dominated at sample size 265. The above results show that percent missing was the most important factor when number of predictors was four. However, with nine predictors, non-normality became the most important factor. This was regardless of the strength of the criterion-predictor relationship. Miss ing pattern effect was the most important under E M . The effect of non-normality on absolute error for regression coefficients appeared to dominate under all M D T s . Miss ing pattern only affected b i , the coefficient of the variable with no missing data. The strength of these effects tended to get larger with increasing sample size. Relative Performance of M D T s Table 25 shows the best M D T s in the estimation of R 2 and P i with respect to maximizing accuracy as measured by the mean absolute error of estimation, as well as M D T s that provided coverage probability closest to the expected value of 95% for each study. Table 26 shows the least biased M D T s in the estimation of R 2 and Pi. The tables indicate that under four predictors 152 and low R 2 condition, E M was the best with respect to accuracy in the estimation of population R 2 . This finding also held true under nine predictors and low R 2 condition. However, there was no M D T that was uniformly least biased under the low R 2 condition. Under high population R 2 (about .60), P W had the most accurate and least biased estimates, with a tendency to overestimate R 2 . This result was consistent with that of Azen et al. (1989) who found that although E M was generally the preferred method, P W performed better than E M for systematically missing data when population R 2 was 0.5. The limitation with P W is the fact that it can yield correlations outside the acceptable range. In a similar manner, covariance matrices based on P W may not be positive definite. Since many analyses based on the covariance matrix, including multiple regression, require a positive-definite matrix, modifications may be necessary when this condition is not satisfied. Table 25: Best M D T s in the estimation of parameters and coverage probability 4 Predictors 9 Predictors R 2 l x estimate b, Coverage Probability R 2 1 V estimate bi Coverage Probability L o w R 2 E M RS E M / R S E M RS E M High R 2 P W RS RS P W RS R S Table 26: Least biased M D T s in the estimation of parameters 4 Predictors 9 Predictors R 2 l v estimate b, R 2 I x estimate bi L o w R 2 None R S None P W High R 2 P W RS P W M S / P W 153 Kromrey and Hines (1994) reported that P W and RS performed better in most situations than M S in the estimation of R 2 . The findings in the present study seem to be in agreement with that of Kromrey and Hines, but only as far as the relative accuracy of P W and M S is concerned. A s for the relative accuracy of RS and M S , the present study showed that R S outperformed M S under the condition of low R 2 . However, under the condition of high R 2 , M S generally outperformed R S . Generally, the mean absolute error for R2estimate increased with increasing non-normality, a finding consistent with that of Graham, Hofer and MacKinnon (1996) who showed that estimates obtained for skewed data were somewhat less accurate than for normal data. M D T s also appeared to be more differentiated under non-monotonic than under monotonic pattern, suggesting that the choice of M D T may be more crucial when the missing pattern is non-monotonic. A l so , M D T s were most differentiated at the highest level of percent missing (20%), suggesting that the choice of M D T was most important at this level. This finding is consistent with that of Kromrey and Hines (1994) and Raymond and Roberts (1987) who found that the differences among M D T s increased as the proportion of missing data increased, although Raymond and Roberts (1987) used randomly deleted data. Therefore, the choice of M D T s is less critical i f the amount of missing data is small (about 10%). In the present study, E M had the best overall performance under low multiple R 2 condition (R 2 about .2), and P W had the best performance under high R 2 condition (R 2 about .6). However, under low R 2 condition, at sample size 265, and 15% to 20% missing data, P W and M S also performed well . N o M D T consistently provided the smallest bias under the low R 2 condition under four predictors and nine predictors. Under high R 2 condition, P W consistently provided the smallest bias regardless of number of predictors. Whereas P W had the tendency to overestimate R 2 , the other M D T s had the tendency to underestimate R 2 . In the estimation of regression coefficients, the greatest accuracy was under RS for each study. However, R S had the least biased estimates under four predictors only. Wi th nine 154 predictors, P W had the least biased estimates. The bias under M S with nine predictors and high R 2 condition was least differentiated from that of P W . In the estimation of regression coefficients, the performance of all M D T s deteriorated with increasing non-normality, and this was regardless of missing pattern. This result may be due to the fact that M D T s are based on normal theory, and as such, their performance is expected to deteriorate with increasing non-normality. Also , the mean absolute error for bi tended to be smaller under monotonic pattern than under non-monotonic pattern. There was a tendency for M D T s to differ more at higher levels of percent missing than at lower levels. O f particular interest was the fact that the M D T s were most differentiated with respect to accuracy in the estimation of (3i, the regression coefficient of the predictor with no missing values. Some researchers (e.g., Little, 1988; Muthen, Kaplan, & Hol l i s , 1987) have suggested the superiority of the E M method in the treatment of missing data. Al l i son (1987) reported that even with non-normal data, maximum-likelihood estimates should have reasonably good properties relative to competing estimators. Graham et al. (1996) found that estimates obtained with non-normal data using E M treatment were satisfactory. However, they were cautious to make no claim that the parameter estimates based on the E M method would be adequate when data were more non-normal than in their study. In their study, five variables were used, with univariate skew ranging from -.68 to 3.30 and univariate kurtosis ranging from -.25 to 13.11. In the present study, E M outperformed RS in the estimation of population R 2 under low R 2 (about .2). However, in the estimation of Pi, RS had more accurate estimates than E M regardless of size of correlation. R S also had the least bias with four predictors. Both RS and E M underestimated Pi, and this can be explained by the fact that the marginal distribution of the predictor is distorted because the imputations do not reflect variation in the distribution of the predictor given the criterion. Wi th nine predictors under high R 2 condition, P W and M S had the least bias, both M D T s overestimating Pi. 155 One possible explanation for the overwhelmingly favoured E M method (see Beale & Little, 1975; Gleason & Staelin, 1975; Little, 1979), is that the proportions of missing data investigated in previous studies were often disturbingly high, ranging from about 10 percent to 40 percent. Wi th 40 percent of the data missing, one would have to question seriously the appropriateness of conducting any analysis. The finding in the present study that the RS method performed about as well as the E M method is important in the sense that RS is less complex than E M method. With regard to comparisons of the accuracy of parameter estimates after different treatments of missing data, there is need to exercise some caution because different criteria were used in different studies. M D T s and Coverage Probabilities The relative effectiveness of M D T s in producing coverage probabilities that captured (31 at alpha = .05 under various design conditions was investigated. Overall , coverage probabilities under RS were closest to the nominal value. However, E M too had coverage probabilities close to the nominal value under low R 2 condition. It was also found that M D T s produced the best coverage probabilities with normal data, and the coverage probabilities decreased with increasing non-normality. This expected result might be attributed to the fact that the confidence intervals constructed were based on the assumption of normality. If this assumption was violated, then Type I error rates were likely to increase. Coverage probabilities under monotonic missing pattern were generally closer to the nominal value of 95% than under non-monotonic missing pattern, with M D T s being less differentiated under monotonic than non-monotonic missing pattern. Generally, for non-normal data, coverage probabilities fell below the nominal value with non-monotonic missing pattern. As reflected by the estimation and coverage probability for (3i, the E M method broke down under non-monotonic pattern at sample size 265. 156 Implications of the Study Considering that the choice of M D T in a multiple regression context not only depends on the magnitude of R 2 and number of predictors, but also on the parameter estimate of interest, the following implications for researchers seem pertinent: (1) The choice of M D T should be of little concern to researchers i f the proportion of missing data is 10 percent or less and the missing pattern is monotonic. In particular, in the estimation of R 2 and regression coefficients, as well as coverage probability, the results showed that M D T s were more differentiated with increasing percent missing. However, the differentiation was much larger under non-monotonic missing pattern than under monotonic missing pattern, implying that the choice of M D T is more crucial when the missing pattern is non-monotonic. (2) The accuracy of parameter estimates after treating missing data was found to deteriorate with increasing non-normality. The results showed that the mean absolute error of parameter estimates was largest at the highest level of non-normality, and smallest for normal data. Coverage probability was found to decrease with increasing non-normality. In general, coverage probability was closest to nominal value for normal data. (3) Wi th the estimation of R 2 as the goal of analysis, use of E M is recommended i f the anticipated R 2 is low (about .2), regardless of number of predictors. However, i f the anticipated R 2 is high (about .6), use of P W is recommended, regardless of number of predictors. Although P W performed the worst and E M the best under four predictors and low R 2 condition, E M was the best and P W the worst under four predictors and high R 2 condition. Under nine predictors and low R 2 condition, E M was the best, except at sample size 265 and 20 percent missing under non-monotonic pattern where M S performed the best. Under nine predictors and high R 2 condition, results showed 157 that P W had the best performance in terms of both accuracy and bias, and E M had the worst performance at all levels of the design variables. (4) Wi th the estimation of regression coefficients as the goal of analysis, the choice of M D T is most crucial for the variable with the least amount of missing data. Based on the accuracy and coverage probability of regression coefficients, the RS method is recommended regardless of the number of predictors or the magnitude of the anticipated R 2 . However, P W and M S methods tended to give less biased estimates than R S when the number of predictors is large (i.e., nine predictors). Researchers should avoid the use of E M when the missing pattern is non-monotonic and sample size is moderate to large (153 and 265) under nine predictors and high R 2 condition because the method breaks down. A contribution of the present study is that it addressed important and meaningful research questions on the relative effectiveness of selected M D T s . This should advance our understanding of the performance of the M D T s under various design conditions, enabling researchers to select M D T s more appropriately. In the present study, use was made of simulated data. It is important that real data sets are used in future research to validate these results. A l so , four and nine predictors were used in the study. A future investigator could modify the number of predictors. Finally, whereas the 95% confidence was used in the calculation of coverage probabilities in the present study, a future investigator could use the 99% confidence interval to extend generalization of results. Strengths and Limitations of the Present Study A s noted by Kromrey and Hines (1994), a critical issue in the generalizability of results involving M D T s is the consideration of regression models with more predictor variables in which missing data occur on more than one predictor. However, previous researchers tended to use 158 two-predictor regression models in which missing data occurred on only one predictor. Such situations are rare in psycho-educational research. In the present study, four and nine predictor models were used, with data missing on all but one predictor, thus allowing for more realistic generalizations. Also , unlike some of the previous studies with few replications, 1000 replications was utilized in the present study. This means that the results of the study more accurately represent the relative performance of M D T s that were examined. Knowledge of coverage probabilities is an important component in the evaluation of any statistical procedure. Previous studies on M D T s failed to investigate this component. In the present study, this vital component was included. Another important aspect of the present study was that levels of independent variables were determined by consulting the literature. This made the results of the present study more generalizable and more consistent with reality. However, a limitation of the present study was that the levels of non-normality were restricted by the parabolic parameter space as presented in Fleishman (1978). This implies that it was not possible to combine a level of skew with any other level of kurtosis. Such a restriction may sacrifice generalizability, but only as far as levels of non-normality are concerned. In his review of missing data research, Roth (1994) stressed the conspicuous deficiency of studies contrasting simple M D T s like mean substitution and pairwise deletion to the more complex ones like expectation-maximization ( E M ) or other maximum likelihood methods. Also , data were randomly deleted in most previous research, and there has been little work with different patterns of missing data. A n attempt was made in the present study to address these issues by including the E M method, and comparing M D T s within and across different patterns of missing data. In conclusion, it was found that percent missing, missing pattern, and multivariate non-normality had an effect on absolute error for standard multiple linear regression estimates. Under 159 low R 2 condition, E M was the best in estimating population R 2 . 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APPENDIX A BIAS IN PARAMETER ESTIMATES 170 Table A.1: Bias in Ruinate: 4 predictors and low R 2 condition Monotonic Missing Pattern Non-monotonic Missing Pattern N Percent Normality EM Mean Pairwise Regress. EM Mean Pairwise Regress. Missing (sk., kurt.) Imputation Substitution Deletion Imputation Imputation Substitution Deletion Imputation 94 10 (0,0) .014266 .015673 .026846 .014973 .005228 .013706 .022897 .007649 (1,3) .018106 .019812 .031224 .018943 .007193 .018463 .025382 .009785 (1.8,6) .024917 .026993 .037877 .025909 .011700 .024425 .030684 .014437 (3,25) .044387 .048456 .055844 .046140 .018653 .041704 .039049 .021012 15 (0,0) .004107 .006573 .023575 .005265 -.006121 .006203 .017708 -.003531 0,3) .007670 .010544 .027176 .008997 -.004370 .010777 .019654 -.001796 (1.8,6) .014246 .017606 .032999 .015753 -.000070 .016326 .024421 .002472 (3,25) .032844 .038732 .048037 .035135 .004499 .030531 .027681 .006057 20 (0,0) -.006420 -.002734 .019325 -.004837 -.017661 -.001691 .012267 -.015170 (1,3) -.003002 .001253 .022340 -.001200 -.016012 .002508 .013185 -.013770 (1.8,6) .003556 .008459 .027135 .005566 -.011744 .007848 .016764 -.009607 (3,25) .020912 .029022 .039035 .023670 -.008049 .020940 .014695 -.007253 153 10 (0,0) .000732 .001733 .015641 .001221 -.007253 -.007671 -.000044 .011790 (1,3) .003823 .005051 .018485 .004405 -.005508 -.006549 .003639 .012977 (1.8,6) .008844 .010328 .022917 .009531 -.004189 -.003493 .007838 .016326 (3,25) .024834 .027645 .037445 .026056 -.001080 -.000199 .020305 .022359 15 (0,0) -.009078 -.007223 .013557 -.008211 .001855 -.019266 -.007887 .007804 (1,3) -.005969 -.003768 .016143 -.004966 -.016849 -.018915 -.005019 .008681 (1.8,6) -.001081 .001506 .020180 .000068 -.016576 -.016417 -.001591 .011663 (3,25) .014051 .018620 .033150 .015916 -.014807 .008383 .015062 -.013461 20 (0,0) -.018985 -.016116 .010855 -.017809 -.029701 -.015390 .003777 -.027609 (1,3) -.015807 -.012512 .013049 -.014487 -.029573 -.013031 .004287 -.027720 (1.8,6) -.010946 -.007183 .016538 -.009472 -.027707 -.010339 .006982 -.025957 (3,25) .002832 .009186 .027708 .005117 -.026062 -.000917 .009241 -.025397 265 10 (0,0) -.007386 -.006653 .008374 -.007020 -.015480 -.008296 .004675 -.013391 (1,3) -.005643 -.004833 .010253 -.005246 -.015897 .006384 .005419 -.013636 (1.8,6) -.002768 -.001867 .013036 -.002336 -.015145 -.004821 .007477 -.012881 (3,25) .008313 .009743 .023768 .008959 -.014766 .002636 .011517 -.012973 15 (0,0) -.017002 -.015550 .007074 -.016337 -.026665 -.016087 .001471 -.024455 (1,3) -.015346 -.013766 .008743 -.014634 -.027575 -.014810 .001919 -.025484 (1.8,6) -.012460 -.010733 .011215 -.011693 -.027056 -.013548 .003764 -.025039 (3,25) -.001774 .000732 .020888 -.000691 -.028262 -.008132 .006587 -.027139 20 (0,0) -.026380 -.024027 .005421 -.025409 -.037143 -.023661 -.001555 -.035151 (1,3) -.024784 -.022240 .006816 -.023751 -.038160 -.022753 -.001321 -.036445 (1.8,6) -.021948 -.019174 .008922 -.020844 -.037845 -.021912 .000417 -.036238 (3,25) -.011708 -.007810 .017423 -.010320 -.039238 -.017029 .001553 -.038617 171 Table A.2: Bias in bi: 4 predictors and low R2 condition Monotonic Missing Pattern Non-monotonic Missing Pattern N Percent Missing Normality (sk., kurt.) EM Imputation Mean Substitution Pairwise Deletion Regress. Imputation EM Imputation Mean Substitution Pairwise Deletion Regress. Imputation 94 10 (0,0) -.003553 .017797 .017631 .001535 -.015127 .063690 .063519 -.004284 (1,3) -.004239 .017267 .017107 .000842 -.023164 .067521 .067273 -.013102 (1.8,6) -.004028 .017329 .017177 .000737 -.025374 .069068 .068474 -.015662 (3,25) -.004694 .020150 .019912 -.000223 -.040202 .092975 .092001 -.034090 15 (0,0) -.002918 .028086 .027854 .003538 -.016335 .079435 .079254 -.007360 (1,3) -.003073 .028157 .027920 .003369 -.023698 .082609 .082403 -.015818 (1.8,6) -.002566 .028646 .028395 .003654 -.024992 .084241 .083585 -.017569 (3,25) -.003445 .033081 .032771 .002614 -.036547 .107390 .106197 -.034251 20 (0,0) -.004045 .037464 .037158 .003816 -.019630 .089585 .089379 -.012553 (1,3) -.004454 .037401 .037099 .003539 -.025813 .092595 .092344 -.020017 (1.8,6) -.003874 .037881 .037598 .003947 -.026716 .094172 .093448 -.021432 (3,25) -.005430 .042480 .042179 .002244 -.036299 .116431 .115084 -.036463 153 10 (0,0) -.006552 .015227 .015104 -.001045 -.017654 .061592 .061544 -.006370 (1,3) -.007023 .015361 .015234 -.001416 -.024650 .066959 .066828 -.014007 (1.8,6) -.006630 .016109 .015986 -.001080 -.026071 .069333 .068876 -.015805 (3,25) -.005940 .018269 .018112 -.000828 -.039287 .091827 .091088 -.032804 15 (0,0) -.007904 .024452 .024255 -.000362 -.020207 .076926 .076864 -.010307 (1,3) -.008693 .024608 .024398 -.001026 -.025955 .081724 .081608 .026066 (1.8,6) -.008465 .025439 .025229 -.000875 .084090 .083573 -.017916 -.035630 (3,25) -.008599 .028423 .028191 -.001100 .105499 .104583 -.032104 20 (0,0) -.007575 .033347 .033133 .000565 -.021086 .087511 .087449 -.013943 (1,3) -.008272 .033511 .033312 -.000050 -.025291 -.091779 .091682 .019459 (1.8,6) -.008249 .034237 .034051 -.000098 -.024773 .094038 .093512 -.019415 (3,25) -.008856 .038024 .037844 -.000414 -.032505 .113444 .112478 -.031413 265 10 (0,0) -.006541 .015030 .014940 -.000822 -.006541 .015030 .014940 -.000822 (1,3) -.006653 .015191 .015100 -.000898 -.006653 .015191 .015100 -.000898 (18,6) -.006538 .015429 .015344 -.000805 -.006538 .015429 .015344 -.000805 (3,25) -.005659 .017387 .017280 .000014 -.005659 .017387 .017280 .000014 15 (0,0) -.007400 .023945 .023794 -.000024 -.007400 .023945 .023794 .000024 (1,3) -.007536 .024159 .024016 -.000128 -.007536 -.024159 .024016 -.000128 (1.8,6) -.007468 .024466 .024326 -.000097 -.007468 -.024466 .024326 .000097 (3,25) -.006843 .026869 .026731 .000674 -.006843 -.026869 .026731 .000674 20 (0,0) -.008453 .032562 .032388 .000090 -.008453 .032562 .032388 .000090 (1,3) -.008719 .032793 .032638 -.000087 -.008719 .032793 .032638 -.000087 (1.8,6) -.008777 .033140 .032993 -.000147 -.008777 .033140 .032993 -.000147 (3,25) -.008820 .035666 .035538 .000063 -.008820 .035666 .035538 .000063 172 Table A.3: Bias in R2estimate: 4 predictors and hiah R2 condition Monotonic Missing Pattern Non-monotonic Missing Pattern N Percent Normality EM Mean Pairwise Regress. EM Mean Pairwise Regress. Missing (sk., kurt.) Imputation Substitution Deletion Imputation Imputation Substitution Deletion Imputation 94 10 (0,0) -.043071 -.032549 .006910 -.035688 -.095194 -.037159 -.001864 -.073098 (1,3) -.038364 -.026508 .012208 -.030401 -.107712 -.036615 .000183 -.085818 (1.8,6) -.032025 -.019128 .018052 -.023750 -.107743 -.032931 .004372 -.086825 (3,25) -.006893 .011155 .039464 .002736 -.145121 -.036613 .010853 -.131758 15 (0,0) -.072550 -.055330 .003996 -.061662 -.138762 -.057195 -.009321 -.116293 (1,3) -.068037 -.049376 .008624 -.056655 -.153030 -.060504 -.008062 -.133318 (1.8,6) -.061726 -.041904 .013712 -.050136 -.153357 -.058508 -.004159 -.135452 (3,25) -.037447 -.011666 .032307 -.024891 -.190371 -.069734 -.001821 -.182862 20 (0,0) -.102963 -.078183 .000198 -.089139 -.179676 -.077683 -.016991 -.159654 (1,3) -.098870 -.072418 .003744 -.084586 -.192358 -.082682 -.015700 -.176377 (1.8,6) -.093023 -.065212 .007473 -.078548 -.192811 -.081983 -.012635 -.178327 (3,25) -.070800 -.035847 .020160 -.055429 -.226364 -.098519 -.012941 -.222622 153 10 (0,0) -.048001 -.038459 .003937 -.041188 -.101753 -.042653 -.004898 -.079005 (1,3) -.044315 -.033656 .007585 -.036928 -.117484 -.045083 -.004322 -.095044 (1.8,6) -.039674 -.028173 .011506 -.031941 -.120161 -.043950 -.001821 -.098610 (3,25) -.021813 -.006575 .029001 -.012702 -.162644 -.055057 .002803 -.149232 15 (0,0) -.077549 -.061168 .001876 -.066905 -.146712 -.063369 -.011451 -.122958 (1,3) -.074276 -.056434 .005151 -.063061 -.162431 -.067912 -.011314 -.141598 (1.8,6) -.070005 -.051070 .008595 -.058485 -.165459 -.068711 -.009097 -.146305 (3,25) -.052585 -.028829 .023926 -.039816 -.208631 -.089476 -.007394 -.200648 20 (0,0) -.106965 -.083823 -.000534 -.093800 -.183666 -.082938 -.017718 -.164268 (1,3) -.103710 -.078908 .002210 -.090011 -.198169 -.089170 -.017852 -.182995 (1.8,6) -.099457 -.073376 .005050 -.085493 -.200403 -.090644 -.015960 -.186945 (3,25) -.083699 -.051473 .017084 -.068645 -.240913 -.116290 -.016857 -.238137 265 10 (0,0) -.051019 -.042208 .001108 -.044514 -.106710 -.047061 -.007823 -.083144 (1,3) -.048718 -.039454 .003460 -.042001 -.124264 -.051691 -.008146 -.101166 (1.8,6) -.046054 -.036391 .006005 -.039181 -.129720 -.053893 -.006644 -.107582 (3,25) -.032769 -.020787 .018571 -.024861 -.178230 -.074161 -.005797 -.164564 15 (0,0) -.080257 -.065187 -.000349 -.070342 -.149003 -.066694 -.013685 -.125639 (1,3) -.078058 -.062408 .001715 -.067940 -.166520 -.073567 -.014329 -.146217 (1.8,6) -.075346 -.059222 .003932 -.065106 -.171866 -.076811 -.013067 -.153113 (3,25) -.062479 -.043555 .015243 -.051156 -.219971 -.105540 -.014288 -.212453 20 (0,0) -.109551 -.087571 -.002185 -.096749 -.187856 -.086303 -.019511 -.167753 (1,3) -.107484 -.084790 -.000373 -.094517 -.203705 -.094695 -.020210 -.187895 (1.8,6) -.104830 -.081495 .001530 -.091743 -.208198 -.098943 -.018821 -.194088 (3,25) -.092830 -.065960 .011134 -.078886 -.250940 -.131747 -.021038 -.247438 173 Table A.4: Bias in bi: 4 predictors and high R2 condition Monotonic Missing Pattern Non-monotonic Missing Pattern Percent Normality EM Mean Pairwise Regress. EM Mean Pairwise Regress. Missing (sk., kurt.) Imputation Substitution Deletion Imputation Imputation Substitution Deletion Imputation 10 (0,0) -.044364 .026069 .025725 -.021430 -.132160 .100736 .100575 -.088402 (1.3) -.044122 .026840 .026485 -.021743 -.148362 .110025 .109789 -.108142 (1.8,6) -.043598 .027679 .027321 -.021862 -.150516 .114023 .113351 -.111796 (3,25) -.035885 .035400 .035054 -.018999 -.168047 .158559 .157811 -.145154 15 (0,0) -.058228 .042136 .041695 -.029849 -.152025 .125779 .125605 -.114983 (1.3) -.057623 .043638 .043176 -.029758 -.162449 .134388 .134122 -.131328 (1.8,6) -.056943 .044909 .044435 -.029670 -.161765 .138342 .137455 -.132865 (3,25) -.050522 .054138 .053716 -.027550 -.164706 .181136 .180042 -.153287 20 (0,0) -.073890 .056536 .055978 -.041796 -.168786 .142991 .142765 -.139666 (1,3) -.073106 .058172 .057663 -.041416 -.173436 .151206 .150919 -.150438 (1.8,6) -.071935 .059611 .059151 -040749 -.171206 .155421 .154344 -.149302 (3,25) -.066650 .069425 .069051 -.039302 -.161881 .195152 .193755 -.155670 10 (0,0) -.046467 .024243 .024096 -.022734 -.136399 .100432 .100355 -.091043 (1,3) -.046337 .024550 .024394 -.023342 -.151974 .110715 .110517 -.110951 (1.8,6) -.045778 .025053 .024904 -.023499 -.153833 .114722 .114057 -.114661 (3,25) -.039826 .029514 .029379 -.021443 -.171656 .153399 .152608 -.149686 15 (0,0) -.062653 .039902 .039632 -.031998 -.158198 .125311 .125201 -.118249 (1,3) -.063226 .040630 .040343 -.033101 -.167713 .134493 .134260 -.134406 (1.8,6) -.063030 .041503 .041223 -.033529 -.166458 .138504 .137603 -.135336 (3,25) -.058190 .047533 .047246 -.032212 -.167809 .174731 .173572 -.155699 20 (0,0) -.075770 .051262 .050928 -.043572 -.169523 .141382 .141242 -.141259 (1,3) -.076477 .051749 .051429 -.044743 -.174145 .149399 .149108 -.152651 (1.8,6) -.076460 .052451 .052159 -.045344 -.171165 .153037 .151970 -.151277 (3,25) -.073103 .058327 .058094 -.044910 -.162976 .185987 .184636 -.159007 10 (0,0) -.046282 .024345 .024173 -.021832 -.137794 .101371 .101335 -.090241 (1,3) -.045909 .024531 .024356 -.021873 -.152678 .111219 .111081 -.110171 (1.8,6) -.045558 .024702 .024540 -.021889 -.154090 .115099 .114524 -.113651 (3,25) -.041959 .026978 .026785 -.020685 -.169893 .149754 .149025 -.147836 15 (0,0) -.061748 .037609 .037369 -.031900 -.158278 .123975 .123940 -.118681 (1,3) -.061452 .038125 .037896 -.031964 -.167210 .132569 .132390 -.134581 (1.8,6) -.061205 .038441 .038228 -.032089 -.166410 .136006 .135228 -.135922 (3,25) -.058529 .041565 .041341 -.031495 -.166441 .167875 .166842 -.155049 20 (0,0) -.076647 .049888 .049602 -.043747 -.172854 .140906 .140858 -.142859 (1,3) -.076710 .050160 .049903 -.044036 -.176512 .148401 .148209 -.153502 (1.8,6) -.076655 .050354 -.050126 -.044308 -.173384 .151357 .150425 -.152026 (3,25) -.074796 .053468 .053238 -.044188 -.161446 .179177 .177942 -.156020 174 Table A.5: Bias in Ro ta te : 9 predictors and low R2 condition Monotonic Missing Pattern Non-monotonic Missing Pattern N Percent Normality EM Mean Pairwise Regress. EM Mean Pairwise Regress. Missing (sk., kurt.) Imputation Substitution Deletion Imputation Imputation Substitution Deletion Imputation 94 10 (0,0) .052831 .059312 .050712 .045430 .053235 .056189 .043350 .034993 0.3) .057554 .061613 .055008 .049185 .057184 .057450 .043191 .034861 (1.8,6) .068053 .069384 .065000 .058658 .065366 .064697 .048981 .040809 (3,25) .093996 .088722 .088315 .080302 .088336 .078952 .054953 .049918 15 (0,0) .042348 .051091 .038944 .032150 .045819 .046922 .030176 .021457 (1,3) .048042 .052648 .043937 .036539 .049473 .047485 .029153 .021458 (1.8,6) .059131 .059179 .054209 .046359 .058149 .052881 .034892 .027905 (3,25) .084774 .075178 .076173 .066702 .078671 .061385 .036271 .034405 20 (0,0) .033224 .039240 .027276 .020160 .036377 .034118 .014620 .007609 (1,3) .039237 .039235 .032185 .024548 .040572 .033331 .013703 .008001 (1.8,6) .050370 .043741 .042017 .034070 .048783 .037227 .018858 .013979 (3,25) .075960 .054680 .062433 .053163 .066895 .040351 .018218 .018735 153 10 (0,0) .024081 .037727 .022876 .018704 .023693 .035168 .015583 .007499 (1,3) .027003 .039441 .025580 .021066 .026480 .036124 .014916 .006825 (1.8,6) .033505 .044340 .031796 .027030 .032271 .040693 .019070 .011250 (3,25) .055026 .062047 .052002 .045564 .049620 .053392 .022516 .017264 15 (0,0) .013263 .032734 .010793 .005506 .014975 .028709 .001444 -.006379 (1,3) .016091 .033953 .013235 .007658 .017794 .028972 .000218 -.006835 (1.8,6) .022644 .038081 .019311 .013459 .023191 .032933 .003764 -.002739 (3,25) .043891 .053315 .038164 .030784 .038522 .042412 .003946 .001574 20 (0,0) .002158 .026110 -.002039 -.007942 .005806 .020213 -.013383 -.020132 (1,3) .005529 .026781 .000791 -.005350 .008749 .019928 -.014600 -.020100 (1.8,6) .012498 .030075 .007104 .000792 .014139 .022724 -.011206 -.015933 (3,25) .033418 .041906 .024703 .017326 .026310 .029411 -.013422 -.013770 265 10 (0,0) .004579 .020329 .003728 .000501 .003144 .018141 -.003967 -.011415 (1,3) .006521 .021854 .005526 .002107 .005299 .018829 -.004977 -.012436 (1.8,6) .010616 .025318 .009463 .005910 .008633 .021799 -.002985 -.010177 (3,25) .025436 .038053 .023430 .018729 .019797 .030759 -.003701 -.008634 15 (0,0) -.005299 .017434 -.007048 -.011483 -.005627 .014069 -.017567 -.024917 (1,3) -.003380 .018667 -.005359 -.009935 -.003845 .014442 -.019538 -.026099 (1.8,6) .000668 .021685 -.001566 -.006268 -.000708 .017014 -.017849 -.023887 (3,25) .015257 .032797 .011519 .005518 .008221 .023300 -.020797 -.023545 20 (0,0) -.015384 .013401 -.018555 -.023676 -.014619 .008777 -.031385 -.037650 (1,3) -.013325 .014295 -.016832 -.022071 -.013399 .008754 -.033816 -.038862 (1.8,6) -.009165 .016764 -.013045 -.018375 -.010731 .010739 -.032510 -.036887 (3,25) .005661 .026275 -.000244 -.006610 -.034483 .015989 -.036778 -.037600 175 Table A.6: Bias in bi: 9 predictors and low R2 condition Monotonic Missing Pattern Non-monotonic Missing Pattern Percent Normality EM Mean Pairwise Regress. EM Mean Pairwise Regress. Missing (sk., kurt.) Imputation Substitution Deletion Imputation Imputation Substitution Deletion Imputation 10 • (0,0) .009833 .009915 -.006128 -.012206 .046551 .046828 -.030872 -.046161 (1,3) .012007 .012088 -.004135 -.009414 .051793 .051944 -.040177 -.052405 (1.8,6) .013377 .013477 -.002674 -.007032 .054007 .053769 -.042492 -.053081 (3,25) .017202 .017423 -.000692 -.002672 .071107 .070396 -.069777 -.067768 15 (0,0) .015750 .015961 -.008358 -.019312 .058620 .059061 -.041426 -.055151 (1,3) .017836 .018095 -.006379 -.016617 .063432 .063626 -.051031 -.060677 (1.8,6) .019042 .019393 -.004937 -.013997 .065537 .065216 -.053646 -.061220 (3,25) .023199 .023591 -.002933 -.009596 .083412 .082471 -.081168 -.072703 20 (0,0) .020966 .021316 -.010513 -.022902 .066610 .067359 -.049185 -.056977 (1,3) .022899 .023283 -.008691 -.020235 .071151 .071629 -.058033 .061282 (1.8,6) .024120 .024609 -.007232 -.017759 .073683 .073595 -.059195 -.060863 (3,25) .028878 .029354 -.005511 -.014754 .091164 .090313 -.082623 -.070609 10 (0,0) .009867 .009773 -.005260 -.015038 .045468 .045503 -.031524 -.054054 (1,3) .011752 .011684 -.003581 -.012758 .050471 .050409 -.041568 -.061210 (1-8,6) .013083 .013041 -.002348 -.010711 .051916 .051574 -.044378 -.061964 (3,25) .016113 .016123 .000048 -.006159 .069493 .068918 -.071630 -.076080 15 (0,0) .015659 .015460 -.007083 -.019716 .056634 .056676 -.042634 -.059257 (1,3) .017369 .017194 -.005917 -.018070 .061270 .061194 -.052341 -.065164 (18,6) .018684 .018536 -.004998 -.016479 .062941 .062529 -.054350 -.065314 (3,25) .021626 .021543 -.003385 -.012894 .080118 .079382 -.079808 -.075771 20 (0,0) .020790 .020510 -.009701 -.025059 .065799 .065861 -.051334 -.063507 (1,3) .022727 .022481 -.008400 -.023304 .070028 .069976 -.059848 -.067817 (1.8,6) .024085 .023885 -.007472 -.021714 .071950 .071494 -.061095 -.067414 (3,25) .027615 .027512 -.005445 -.018000 .088507 .087753 -.081332 -.074605 10 (0,0) .010831 .010839 -.004703 -.016976 .045312 .045329 -.031007 -.056489 (1,3) .011962 .011955 -.003757 -.015561 .050204 .050133 -.041230 -.063826 (1.8,6) .012407 .012397 -.003389 -.014727 .051504 .051205 -.043945 -.064762 (3,25) .015673 .015649 -.000770 -.010625 .068176 .067690 -.068839 -.076688 15 (0,0) .016277 .016234 -.007415 -.023142 .056651 .056693 -.042119 -.061757 (1,3) .017399 .017356 -.006549 -.021787 .061140 .061079 -.051694 -.067169 (18,6) .017859 .017824 -.006119 -.020807 .062256 .061927 -.053800 -.067625 (3,25) .021071 .021028 -.003894 -.017447 .078447 .077838 -.075765 -.076355 20 (0,0) .020799 .020700 -.010693 -.027943 .064524 .064590 -.050922 -.064597 (1,3) .021857 .021772 -.009978 -.026779 .068494 .068452 -.058832 -.068192 (18,6) .022322 .022261 -.009663 -.025880 .069789 .069426 -.059756 -.067639 (3,25) .025640 .025576 -.007823 -.023140 .085074 .084377 -.077983 -.073861 176 Table A.7: Bias in R^imata: 9 predictors and high R2 condition Monotonic Missing Pattern Non-monotonic Missing Pattern N Percent Normality EM Mean Pairwise Regress. EM Mean Pairwise Regress. Missing (sk., kurt.) Imputation Substitution Deletion Imputation Imputation Substitution Deletion Imputation 10 (0,0) -.017995 -.009956 .025089 -.013499 -.028137 -.010155 .022230 -.019576 (1,3) -.011074 -.001459 .030886 -.005921 -.026974 -.003873 .026239 -.017799 (1.8,6) -.001342 .009484 .039016 .004122 -.021177 .004594 .032855 -.012141 (3,25) .026631 .043122 .065413 .032964 -.005060 .037215 .048783 .000882 15 (0,0) -.051404 -.039536 .019039 -.043741 -.065173 -.038214 .015042 -.052603 (1,3) -.044442 -.030618 .024243 -.036097 -.064187 -.031898 .018343 -.052256 (1.8,6) -.034444 -.019071 .031202 -.025821 -.057584 -.022675 .023618 -.046445 (3,25) -.009133 .012538 .053658 -.000012 -.046609 .004988 .033044 -.041798 20 (0,0) -.081299 -.064520 .009862 -.072634 -.096991 -.062968 .005055 -.085631 (1,3) -.074832 -.055274 .014020 -.065539 -.095711 -.056517 .007326 -.085894 (1.8,6) -.065674 -.043592 .019326 -.056088 -.089486 -.047601 .010928 -.080946 (3,25) -.041498 -.010947 .037520 -.032317 -.084568 -.025069 .011960 -.083590 10 (0,0) -.036562 -.029355 .015445 -.031404 -.048530 -.029235 .013361 -.037181 (1,3) -.032284 -.023989 .019739 -.026709 -.048734 -.025062 .016540 -.037021 (1.8,6) -.026089 -.016858 .025384 -.020246 -.044695 -.019270 .021409 -.033354 (3,25) -.003966 .009549 .046896 .003173 -.038882 .000362 .035683 -.030228 15 (0,0) -.063702 -.053764 .011637 -.057242 -.077228 -.053296 .008996 -.065527 (1,3) -.059797 -.048354 .015178 -.052861 -.077823 -.049514 .011582 -.066571 (1.8,6) -.053840 -.041053 .019909 -.046613 -.073998 -.043663 .015786 -.063397 (3,25) -.032704 -.014338 .037531 -.024599 -.071683 -.028357 .025573 -.065662 20 (0,0) -.093607 -.081002 .006842 -.085963 -.108987 -.079821 .003031 -.096971 (1,3) -.090231 -.075987 .009479 -.082191 -.109711 -.076410 .004875 -.099037 (1.8,6) -.084793 -.069016 .013109 -.076503 -.106715 -.071468 .007742 -.097020 (3,25) -.065806 -.043205 .026572 -.056964 -.108682 -.060447 .014062 -.104516 10 (0,0) -.043399 -.037496 .009098 -.038952 -.056210 -.039073 .007455 -.045490 (1,3) -.041188 -.034554 .011784 -.036420 -.058064 -.036977 .009473 -.046868 (1.8,6) -.037866 -.030653 .015164 -.032911 -.056347 -.033968 .012447 -.045487 (3,25) -.020277 -.010110 .031287 -.014378 -.057637 -.024000 .024606 -.049238 15 (0,0) -.072425 -.063715 .006886 -.066362 -.086826 -.064416 .004503 -.075533 (1,3) -.070719 -.061155 .009166 -.064405 -.089589 -.063159 .005985 -.078775 (18,6) -.067832 -.057591 .012047 -.061379 -.088402 -.060668 .008435 -.078224 (3,25) -.050912 -.036962 .026188 -.043600 -.094420 -.055867 .017096 -.088534 20 (0,0) -.100549 -.089379 .004104 -.093538 -.115445 -.088847 .000707 -.104511 (1,3) -.099016 -.086914 .005969 -.091833 -.118679 -.088481 .001677 -.109063 (1.8,6) -.096217 -.083300 .008242 -.088963 -.118034 -.086515 .003559 -.109263 (3,25) -.080272 -.062953 .020309 -.072286 -.127192 -.084163 .009566 -.122996 177 Table A.8: Bias in bi: 9 predictors and high R2 condition Monotonic Missing Pattern Non-monotonic Missing Pattern Percent Normality EM Mean Pairwise Regress. EM Mean Pairwise Regress. Missing (sk., kurt.) Imputation Substitution Deletion Imputation Imputation Substitution Deletion Imputation 10 (0,0) -.011287 .001953 .001858 -.012416 -.057029 .026934 .026994 -.040914 (1,3) -.007595 .005145 .004984 -.009720 -.065066 .031838 .031892 -.049455 (1.8,6) -.004372 .007152 .006943 -.007985 -.065198 .034185 .034207 -.050841 (3,25) .004262 .014062 .014011 -.003194 -.077835 .047597 .047385 -.074398 15 (0,0) -.026645 .007626 .007540 -.014560 -.080429 .041427 .041620 -.051136 (1,3) -.023158 .010240 .010120 -.012398 -.085836 .047062 .047222 -.061345 (18,6) -.020280 .011878 .011760 -.011009 -.084729 .049653 .049714 -.062781 (3,25) -.010855 .018699 .018673 -.006710 -.093152 .066573 .066442 -.088411 20 (0,0) -.034395 .009464 .009587 -.020381 -.084472 .048176 .048583 -.063265 (1,3) -.031407 .011364 .011453 -.018546 -.089198 .053214 .053615 -.072875 (1.8,6) -.029057 .012553 .012714 -.017468 -.087593 .056135 .056588 -.073021 (3,25) -.020205 .019898 .020006 -.013120 -.093497 .074165 .074469 -.094165 10 (0,0) -.019732 .003859 .003686 -.009888 -.077828 .031150 .031165 -.041626 (1,3) -.017092 .006027 .005900 -.008104 -.086020 .035997 .036049 -.052341 (1.8,6) -.015059 .007503 .007415 -.006893 -.086306 .037559 .037587 -.054869 (3,25) -.006938 .012914 .012878 -.002294 -.096188 .053765 .053806 -.080782 15 (0,0) -.028588 .006187 .005901 -.013086 -.084891 .037233 .037215 -.052358 (1,3) -.026289 .008288 .008040 -.011926 -.091186 .042616 .042658 -.062658 (1-8,6) -.024436 .009665 .009457 -.011232 -.090582 .044540 .044591 -.064355 (3,25) -.016528 .015051 .014888 -.007673 -.097244 .062523 .062738 -.087630 20 (0,0) -.039642 .008587 .008156 -.016926 -.091750 .045712 .045734 -.060522 (1,3) -.037745 .010272 .009892 -.016041 -.096106 .050882 .050965 -.070347 (1.8,6) -.035971 .011498 .011195 -.015570 -.095059 .053045 .053113 -.071748 (3,25) -.029493 .016256 .016091 -.013042 -.097851 .072051 .072274 -.091367 10 (0,0) -.021870 .004424 .004406 -.008603 -.078951 .030216 .030201 -.039014 (1,3) -.019896 .005821 .005806 -.007473 -.088048 .034897 .034864 -.050516 (1.8,6) -.018535 .006777 .006772 -.006806 -.088706 .036440 .036395 -.053299 (3,25) -.012584 .010609 .010603 -.003608 -.099820 .052223 .052186 -.079756 15 (0,0) -.032584 .007716 .007656 -.012830 -.086684 .039610 .039605 -.051024 (1,3) -.030834 .009050 .008999 -.011945 -.092939 .044699 .044642 -.062034 (1.8,6) -.029581 .009929 .009902 -.011423 -.092888 .046124 .046108 -.064270 (3,25) -.024657 .013471 .013453 -.009254 -.099939 .062823 .062794 -.088026 20 (0,0) -.041984 .009583 .009470 -.017201 -.091298 .045849 .045860 -.059978 (1,3) -.040564 .010794 .010703 -.016522 -.095787 .050734 .050730 -.070199 (18,6) -.039316 .011600 .011551 -.016131 -.095224 .052342 .052321 -.071788 (3, 25) -.035205 .015281 .015263 -.014199 -.020604 .023871 .023883 -.021451 178 APPENDIX B GRAPHS FOR BIAS IN PARAMETER ESTIMATES 179 M O N O T O N I C P A T T E R N SZ: 94 PM 10 FT: A I (0.0) (1,3) (1.8,6) Normality Level SZ: 94 PM: 15 PT : A (0,0) (1,3) (1.8,6) (3,25) Normality Level SZ: 94 PM: 20 PT: A O co rx (0,0) L E G E N D : (1.3) (1.8,6) Normality Level (3,25) E M MS N O N - M O N O T O N I C P A T T E R N SZ: 94 PM: 10 FT: B (0,0) (1,3) (1.8,6) Normality Level (3,25) SZ: 94 F M 15 FT: B (0,0) (1,3) (1.8,6) Normality Level SZ: 94 FM 20 FT: B (3 ,25) O CO DC S - .02 PW RS Figure B. 1: Bias in R 2 e s t imate under four predictors and low R 2 condition Note: SZ = Sample size, P M = Percent missing. Values in parentheses under Normality Level represent skew and kurtosis, respectively. 180 M O N O T O N I C P A T T E R N S Z : 94 P M : 10 P T : A 3 -.01 (0.0) (1,3) (1,8,6) (3,25) Norma] ity Level N O N - M O N O T O N I C P A T T E R N S Z : 94 P M : 10 P T : B (0,0) (1,3) (1.8,6) (3,25) Normality Level S Z : 94 P M : 15 P T : A S Z : 94 P M : 15 P T : B (0,0) (1,3) (1.8,6) Normal ity Level (3,25) (0,0) (1,3) (1.8,6) (3,25) Normal ity Level S Z : 94 P M : 2 0 P T : A (0.0) L E G E N D : (1.3) (1.8,6) Normality Level (3,25) E M MS S Z : 94 P M : 2 0 P T : B PW (0.0) 0,3) (1.8,6) Normality Level RS (3,25) Figure B.2: Bias in bi under four predictors and low R 2 condition Note: SZ = Sample size, P M = Percent missing, A = monotonic pattern, B = Non-monotonic pattern. Values in parentheses under Normality Level represent skew and kurtosis, respectively. 181 MONOTONIC PATTERN SZ: 94 PM: 10 PT: A (0,0) (1,3) (1.8,6) Normality Leve l SZ: 94 PM: 15 PT: A (3,25) (0,0) (1,3) (1 .8 ,6) Normali ty Level SZ: 94 PM: 20 PT: A (3,25) (1.8,6) Normal ity Level (3,25) NON-MONOTONIC PATTERN SZ: 94 PM: 10 PT: B a -2o (0,0) (1,3) (1.8,6) Normali ty Level (3,25) SZ: 94 PM: 15 PT: B (0.0) (1,3) (1.8,6) Norma) ity Level (3 ,25) SZ: 94 PM: 20 PT: B Normali ty Level L E G E N D : E M MS PW RS Figure B.3: Bias in Romaic under four predictors and low R 2 condition Note: SZ = Sample size, P M = Percent missing. Values in parentheses under Normality Level represent skew and kurtosis, respectively. 182 M O N O T O N I C P A T T E R N S Z : 94 P M : 10 P T : A (0.0) (1,3) (1.8,6) Normal ity Level (3,25) N O N - M O N O T O N I C P A T T E R N S Z : 94 P M : 10 P T : B (0,0) (1,3) (1.8,6) Normali ty Level S Z : 94 P M : 15 P T : A S Z : 94 P M : 15 P T : B («,") (1.3) (1.8,6) (3,25) Normali ty Level S Z : 94 P M : 2 0 P T : A (0,0) (1,3) (1.8,6) (3,25) Normality Level L E G E N D : E M 1^ MS (0,0) (1,3) (1.3,6) (3,25) Normali ty Level S Z : 94 P M : 2 0 P T : B (0.0) (1,3) (1.8,6) Normali ty Level PW RS Figure B.4: Bias in bi under four predictors and high R 2 condition Note: SZ = Sample size, P M = Percent missing. Values in parentheses under Normality Level represent skew and kurtosis, respectively. 183 M O N O T O N I C P A T T E R N SZ: 94 PM 10 PT: A (0.0) (1,3) (1.8,6) Normality Level SZ: 94 Plvt 15 PT: A (3,25) m 02 (0,0) (1,3) (1.8,6) Normality Level SZ: 94 PM 20 PT: A (3,25) (0,0) L E G E N D : (1.3) (1.8,6) Normality Level (3,25) E M MS N O N - M O N O T O N I C P A T T E R N SZ: 94 PM 10 PT: B (0,0) (1,3) (1.8,6) N n r m a l i t v I Avpl SZ: 94 PM 15 PT: B (0,0) (1,3) (1.8,6) Normality Level 94 PM 20 PT: B (3,25) (0,0) PW (1,3) (1.8,6) Normality Level RS (3.25) Figure B.5: Bias in R2estimate under nine predictors and low R 2 condition Note: SZ = Sample size, P M = Percent missing. Values in parentheses under Normality Level represent skew and kurtosis, respectively. 184 MONOTONIC PATTERN S Z : 94 P M : 10 P T : A (0,0) (1,3) (1 .8 ,6) (3 ,25) Normality Level NON-MONOTONIC PATTERN S Z : 94 P M : 10 P T : B I (0,0) (1,3) (1.8,6) Normali ty Level S Z : 94 P M : 15 P T : A S Z : 94 P M : 15 P T : B (1,3) (1.8,6) (3,25) Normal ity Level 1 (0.0) (1.3) (1.8,6) Normali ty Level S Z : 94 P M : 2 0 P T : A (0,0) LEGEND: (1,3) (1.8,6) Normal ity Level EM (3 ,25) MS S Z : 94 P M : 2 0 P T : B (0,0) PW (1,3) (1.8,6) Normal ity Level RS (3 ,25) Figure B.6: Bias in b) under nine predictors and low R 2 condition Note: SZ = Sample size, PM = Percent missing. Values in parentheses under Normality Level represent skew and kurtosis, respectively. 185 M O N O T O N I C P A T T E R N S Z : 94 P M : 10 PT: A (0,0) (1.3) (1.8,6) (3.25) Normality Level S Z : 94 P M : 15 PT: A (0,0) (1,3) (1-8.6) Normality Level S Z : 94 P M : 20 PT: A (1,3) (1.8.6) Normality Level L E G E N D : E M MS N O N M O N O T O N I C P A T T E R N S Z : 94 P M : 10 PT: B (0,0) (1.3) (1.8,6) Normality Level S Z : 94 P M : 15 PT: B (0,0) (1,3) (1.8,6) Normality Level S Z : 94 P M : 20 PT: B (3,25) (1,3) (1.8,6) Normality Level PW RS Figure B.7: Bias in R 2 e s t imate under nine predictors and high R 2 condition Note: SZ = Sample size, P M = Percent missing. Values in parentheses under Normality Level represent skew and kurtosis, respectively. 186 M O N O T O N I C P A T T E R N SZ: 94 PM 10 PT: A (0.0) (1,3) (1.8,6) Normality Level (3,25) SZ: 94 PM 15 PT: A (0,0) (1,3) (1.8,8) Normality Level SZ: 94 PM 20 PT: A (3,25) (0,0) (1.3) (1.8,6) Normality Level (3,25) L E G E N D : E M 31 MS N O N - M O N O T O N I C P A T T E R N SZ: 94 PM 10 PT: B .02 0.00 -.02 -.04 CO < CO -.10 (0,0) (1,3) (1.8,6) Normality Level SZ: 94 PM 15 PT: B (0,0) (1,3) (1.8,6) (3 Normality Level SZ: 94 FM 20 PT: B (0,0) (1,3) (1.8,6) Normality Level PW RS Figure B.8: Bias in b, under nine predictors and high R 2 condition Note: SZ = Sample size, P M = Percent missing. Values in parentheses under Normality Level represent skew and kurtosis, respectively. APPENDIX C ARTICLES USED IN THE REVIEW Aie l lo , J. R., & Kolb , K . J. (1995). Electronic performance monitoring and social context: Impact on productivity and stress. Journal of Applied Psychology. 80(3). 339-353. 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The accuracy of parameter estimates and coverage probability of population values in regression models… Othuon, Lucas Onyango A. 1998
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Title | The accuracy of parameter estimates and coverage probability of population values in regression models upon different treatments of systematically missing data |
Creator |
Othuon, Lucas Onyango A. |
Date Issued | 1998 |
Description | Several methods are available for the treatment of missing data. Most of the methods are based on the assumption that data are missing completely at random (MCAR). However, data sets that are MCAR are rare in psycho-educational research. This gives rise to the need for investigating the performance of missing data treatments (MDTs) with non-randomly or systematically missing data, an area that has not received much attention by researchers in the past. In the current simulation study, the performance of four MDTs, namely, mean substitution (MS), pairwise deletion (PW), expectation-maximization method (EM), and regression imputation (RS), was investigated in a linear multiple regression context. Four investigations were conducted involving four predictors under low and high multiple R² , and nine predictors under low and high multiple R² . In addition, each investigation was conducted under three different sample size conditions (94, 153, and 265). The design factors were missing pattern (2 levels), percent missing (3 levels) and non-normality (4 levels). This design gave rise to 72 treatment conditions. The sampling was replicated one thousand times in each condition. MDTs were evaluated based on accuracy of parameter estimates. In addition, the bias in parameter estimates, and coverage probability of regression coefficients, were computed. The effect of missing pattern, percent missing, and non-normality on absolute error for R² estimate was of practical significance. In the estimation of R², EM was the most accurate under the low R² condition, and PW was the most accurate under the high R² condition. No MDT was consistently least biased under low R² condition. However, with nine predictors under the high R² condition, PW was generally the least biased, with a tendency to overestimate population R². The mean absolute error (MAE) tended to increase with increasing non-normality and increasing percent missing. Also, the MAE in R² estimate tended to be smaller under monotonic pattern than under non-monotonic pattern. MDTs were most differentiated at the highest level of percent missing (20%), and under non-monotonic missing pattern. In the estimation of regression coefficients, RS generally outperformed the other MDTs with respect to accuracy of regression coefficients as measured by MAE . However, EM was competitive under the four predictors, low R² condition. MDTs were most differentiated only in the estimation of β₁, the coefficient of the variable with no missing values. MDTs were undifferentiated in their performance in the estimation for b₂,...,bp, p = 4 or 9, although the MAE remained fairly the same across all the regression coefficients. The MAE increased with increasing non-normality and percent missing, but decreased with increasing sample size. The MAE was generally greater under non-monotonic pattern than under monotonic pattern. With four predictors, the least bias was under RS regardless of the magnitude of population R². Under nine predictors, the least bias was under PW regardless of population R². The results for coverage probabilities were generally similar to those under estimation of regression coefficients, with coverage probabilities closest to nominal alpha under RS. As expected, coverage probabilities decreased with increasing non-normality for each MDT, with values being closest to nominal value for normal data. MDTs were most differentiated with respect to coverage probabilities under non-monotonic pattern than under monotonic pattern. Important implications of the results to researchers are numerous. First, the choice of MDT was found to depend on the magnitude of population R², number of predictors, as well as on the parameter estimate of interest. With the estimation of R² as the goal of analysis, use of EM is recommended if the anticipated R² is low (about .2). However, if the anticipated R² is high (about .6), use of PW is recommended. With the estimation of regression coefficients as the goal of analysis, the choice of MDT was found to be most crucial for the variable with no missing data. The RS method is most recommended with respect to estimation accuracy of regression coefficients, although greater bias was recorded under RS than under PW or MS when the number of predictors was large (i.e., nine predictors). Second, the choice of MDT seems to be of little concern if the proportion of missing data is 10 percent, and also if the missing pattern is monotonic rather than non-monotonic. Third, the proportion of missing data seems to have less impact on the accuracy of parameter estimates under monotonic missing pattern than under non-monotonic missing pattern. Fourth, it is recommended for researchers that in the control of Type I error rates under low R² condition, the EM method should be used as it produced coverage probability of regression coefficients closest to nominal value at .05 level. However, in the control of Type I error rates under high R² condition, the RS method is recommended. Considering that simulated data were used in the present study, it is suggested that future research should attempt to validate the findings of the present study using real field data. Also, a future investigator could modify the number of predictors as well as the confidence interval in the calculation of coverage probabilities to extend generalization of results. |
Extent | 8836458 bytes |
Subject |
Regression analysis Statistics Error analysis (Mathematics) |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2009-06-24 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0054612 |
URI | http://hdl.handle.net/2429/9556 |
Degree |
Doctor of Philosophy - PhD |
Program |
Special Education |
Affiliation |
Education, Faculty of Educational and Counselling Psychology, and Special Education (ECPS), Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 1998-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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