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Analysis of Covariance with Heterogeneity of Regression and a Random Covariate Christopher J View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of New Mexico University of New Mexico UNM Digital Repository Psychology ETDs Electronic Theses and Dissertations Fall 11-15-2018 Analysis of Covariance with Heterogeneity of Regression and a Random Covariate Christopher J. McLouth University of New Mexico - Main Campus Follow this and additional works at: https://digitalrepository.unm.edu/psy_etds Part of the Psychology Commons Recommended Citation McLouth, Christopher J.. "Analysis of Covariance with Heterogeneity of Regression and a Random Covariate." (2018). https://digitalrepository.unm.edu/psy_etds/271 This Dissertation is brought to you for free and open access by the Electronic Theses and Dissertations at UNM Digital Repository. It has been accepted for inclusion in Psychology ETDs by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected]. i Christopher J. McLouth Candidate Psychology Department This dissertation is approved, and it is acceptable in quality and form for publication: Approved by the Dissertation Committee: Timothy E. Goldsmith, PhD, Chairperson Harold D. Delaney, PhD Davood Tofighi, PhD Li Li, PhD ii ANALYSIS OF COVARIANCE WITH HETEROGENEITY OF VARIANCE AND A RANDOM COVARIATE by CHRISTOPHER J. MCLOUTH B.A., Psychology, University of Michigan, 2008 M.S., Psychology, University of New Mexico, 2013 DISSERTATION Submitted in Partial Fulfillment of the Requirements for the Degree of Doctorate of Philosophy in Psychology The University of New Mexico Albuquerque, New Mexico December, 2018 iii ACKNOWLEDGEMENTS I would like to thank my advisor, Dr. Harold Delaney, for his wisdom, thoughtful guidance, and unwavering support and encouragement throughout this process. This project would not have been possible without it. I also want to thank my committee members, Drs. Goldsmith, Tofighi, and Li for their helpful contributions. I thank my wife, Laurie, whose love and support provided the encouragement to complete this dissertation. Finally, I thank my friends and family for being understanding and supporting my academic mission. iv ANALYSIS OF COVARIANCE WITH HETEROGENEITY OF REGRESSION AND A RANDOM COVARIATE by Christopher J. McLouth B.A., Psychology, University of Michigan, 2008 M.S., Psychology, University of New Mexico, 2013 Ph.D., Quantitative Psychology, University of New Mexico, 2018 ABSTRACT The analysis of covariance (ANCOVA) is a statistical technique originally developed by Fisher (1932) to increase the precision of the estimate of a treatment effect in experimental data. It is used when researchers have both qualitative and quantitative predictors of a continuous outcome. ANCOVA’s most basic assumptions are similar to those of analysis of variance (ANOVA) and other related linear models, but also include an additional assumption regarding the regression line relating the outcome and the covariate. This assumption, referred to as homogeneity of regression, requires that the within-group regression slopes be the same for all groups. Typically, one also assumes that the covariate is a fixed effect, but some methodologists question this practice. v Consequences of either employing a model allowing for heterogeneity of regression (an ANCOHET model) or presuming that the covariate is random can be dealt with fairly easily if both do not occur simultaneously. However, a problem arises when one simultaneously encounters both a random covariate and heterogeneity of regression: the interaction between the random covariate and the fixed factor of treatment will affect the apparent evidence for the main effect of treatment. The current study investigated the utility of different methods of testing for the main effect of treatment in the presence of heterogeneity of regression with a random covariate. Using a Monte Carlo simulation, a 2x3x5x2x3 design manipulated the number of groups, sample size per group, extent of heterogeneity of regression, presence of a group effect, and test location to investigate this issue. For each combination of these factors, different types of tests of the group effect were conducted, as explained in the Method section. Two error terms, the mean square for the interaction and an average of the mean square error for an ANCOHET model and the interaction mean square, performed poorly across all simulations for all metrics. On the other hand, the ANCOHET and ANCOVA error terms produced Type I error rates, power, confidence interval coverage rates and widths, as well as average standard errors under low and medium levels of heterogeneity of regression that were acceptable. When heterogeneity of regression was high or extreme, the ANCOHET approach underestimated the true standard error, whereas the ANCOVA error term did not. Using an approach to increase the ANCOHET standard error based on Chen (2006) did not result in a large enough increase to make up for this underestimation. In conclusion, with the low and moderate levels of heterogeneity of regression typically reported in the literature the ANCOHET test of the group effect can be recommended vi even with a random covariate. Under large or extreme levels of heterogeneity of regression, an error term from a standard ANCOVA should instead be used. vii TABLE OF CONTENTS LIST OF FIGURES ......................................................................................................... ix LIST OF TABLES ............................................................................................................ x INTRODUCTION............................................................................................................. 1 ANCOVA – An Overview .............................................................................................. 1 ANCOVA Assumptions .................................................................................................. 1 Regarding the Homogeneity of Regression Assumption ................................................ 2 Consequences of Violating the Homogeneity of Regression Assumption ..................... 2 Type I Error Rates in the Presence of Heterogeneity of Regression ............................... 6 Accuracy in Parameter Estimation .................................................................................. 7 To Fix or To Presume Random? ..................................................................................... 8 Consequences of Treating a Factor as Random .............................................................. 9 The Issue at Hand: Heterogeneity of Regression with a Random Covariate ................ 11 METHOD ........................................................................................................................ 12 How to Test for Treatment Main Effects ...................................................................... 12 Additional Error Terms ................................................................................................. 15 Simulation Design ......................................................................................................... 16 Data Generation............................................................................................................. 17 Non-Null Conditions to Determine Power .................................................................... 18 Confidence Interval Construction and Accuracy Estimation ........................................ 19 Average Standard Error compared to the True Standard Deviation ............................. 24 Homogeneity of Variance Assumption ......................................................................... 25 Justification for Levels of Simulation Factors .............................................................. 26 RESULTS ........................................................................................................................ 31 Rejection Rates for Two-Group, Null Conditions ........................................................ 31 Rejection Rates for Three-Group, Null Conditions ...................................................... 33 Power Results for Two-Group Conditions .................................................................... 34 Power Results for Three-Group Conditions .................................................................. 36 Confidence Interval Coverage ....................................................................................... 37 Confidence Interval Accuracy ....................................................................................... 38 Average Standard Error Compared to True Standard Deviation .................................. 40 viii DISCUSSION .................................................................................................................. 43 Study Findings .............................................................................................................. 47 Recommendations for Dealing with Heterogeneity of Regression ............................... 49 REFERENCES ................................................................................................................ 52 Appendix A ...................................................................................................................... 86 Appendix B .................................................................................................................... 104 Appendix C ...................................................................................................................
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