Analysis of Data from a Controlled Repeated Measurements Design with Baseline-Dependent Dropouts
Abstract
Abstract. Differences in mean rates of change are of primary interest in many controlled treatment evaluation studies. Generalized linear mixed model (GLMM) procedures are widely conceived to be the preferred method of analysis for repeated measurement designs when there are missing data due to dropouts, but systematic dependence of the dropout probabilities on antecedent or concurrent factors poses a problem for testing the significance of differences in mean rates of change across time in such designs. Controlling for the dependence of dropout probabilities on baseline values poses a special problem because a theoretically correct GLMM random-effects model does not permit including the same baseline score as both covariate and dependent variable. Monte Carlo methods are used herein to evaluate the actual Type 1 error rates and power resulting from two commonly-illustrated GLMM random-effects model formulations for testing the GROUPS × TIMES linear interaction effect in group-randomized repeated measurements designs. The two GLMM model formulations differ by either including or not including baseline scores as a covariate in the attempt to control for imbalance caused by the baseline-dependent dropouts. Results from those analyses are compared with results from a simpler two-stage analysis in which dropout-weighted slope coefficients fitted separately to the available repeated measurements for each subject serve as the dependent variable for an ordinary ANCOVA test for difference in mean rates of change. The Monte Carlo results confirm modestly superior Type 1 error protection but quite superior power for the simpler two-stage analysis of dropout-weighted slope coefficients as compared with those for either of the more mathematically complex GLMM analyses.
References
Ahn, C. , Tonidandel, S. , Overall, J.E. (2000). Issues in the use of SAS.PROC.MIXED to test the significance of treatment effects in controlled clinical trials. Journals of Biopharmaceutical Statistics, 10, 265– 289Davidian, M. , Giltinan, D.M. (1995). Nonlinear models for repeated measurements data . London: Chapman and HallDiggle, P. , Kenward, M.G. (1994). Informative dropouts in longitudinal data analysis. Applied Statistics, 43, 49– 94Elkins, I. , Gibbons, R.D. , Shea, M.T. , Sotsky, S.M. , Watkins, J.T. , Pilkonis, P.A. , Hedeker, D. (1995). Initial severity and differential treatment outcome in the NIMH treatment of depression collaborative research program. Journal of Consulting and Clinical Psychology, 63, 841– 847Elkins, I. , Parloff, M.B. , Hadley, S.W. , Autry, J.H. (1985). NIMH treatment of depression collaborative research program: Background and research plan. Archive of General Psychiatry, 42, 305– 316Elkins, I. , Shea, M.T. , Watkins, J.T. , Imber, S.D. , Sotsky, S.M. , Collins, J.F. , ( 1989). NIMH treatment of depression collaborative research program: I. General effectiveness of treatment. Archives of General Psychiatry, 46, 971– 982Frison, L.J. , Pocock, S.J. (1997). Linearly divergent treatment effects in clinical trials with repeated measures: Efficient analysis using summary statistics. Statistics in Medicine, 16, 2855– 2872Geisser, S. , Greenhouse, S.W. (1958). An extension of Box's results on the use of the F distribution in multivariate analysis. Annals of Mathematical Statistics, 29, 885– 891Hedeker, D. , Gibbons, R.D. (1997). Application of random-effects pattern-mixture models for missing data in longitudinal studies. Psychological Methods, 2, 64– 78Hollon, S.D. , Shelton, R.C. , Loosen, P.T. (1991). Cognitive therapy and pharmacotherapy for depression. Journal of Consulting and Clinical Psychology, 59, 88– 99Huynh, H. , Feldt, L.S. (1970). Conditions under which mean square ratios in repeated measurements designs have exact F distributions. Journal of the American Statistical Association, 65, 1583– 1589Jacobson, N.S. , Hollon, S.D. (1996). Cognitive-behavior therapy versus pharmacotherapy: Now that the jury's returned its verdict, it's time to present the rest of the evidence. Journal of Consulting and Clinical Psychology, 64, 74– 80Klein, D.F. (1990). NIMH research on treatment of depression. Archives of General Psychiatry, 47, 682– 684Klein, D.F. , Ross, D.C. (1993). Reanalysis of the national institute of mental health treatment of depression collaborative research program general effectiveness report. Neuropsychopharmacology, 8, 241– 251Kraemer, H.C. , Thiemann, S. (1989). A strategy to use soft data effectively in randomized controlled clinical trials. Journal of Consulting and Clinical Psychology, 57, 148– 154Lecoutre, E. (1991). A correction for the e approximation in repeated measures designs with two or more independent groups. Journal of Educational Statistics, 16, 371– 372Lindquist, E.F. (1953). Design and analysis of experiments in psychology and education . Boston: Houghton Mifflin CompanyLittell, R.C. , Milliken, G.A. , Stroup, W.W. , Wolfinger, R.D. (1996). SAS systems for mixed models . Cary, NC: SAS InstituteLittle, R.J.A. (1993). Pattern-mixture models for multivariate incomplete data. Journal of the American Statistical Association, 88, 125– 134Little, R.J.A. (1994). A class of pattern mixture models for normal incomplete data. Biometrics, 81, 471– 482Little, R.J.A. (1995). Modeling the dropout mechanism in repeated measures studies. Journal of the American Statistical Association, 90, 1112– 1121Little, R.J.A. , Raghunathan, T. (1999). On summary measures analyses of the linear mixed effects model for repeated measures when data are not missing completely at random. Statistics in Medicine, 18, 2465– 2478Little, R.J.A. , Rubin, D.B. (1987). Statistical analysis with missing data . New York: WileyMurray, D.M. , Hannan, P.J. , Wolfinger, R.D. , Baker, W.L. , Dwyer, J.H. (1998). Analysis of data from group-randomized trials with repeated observations on the same groups. Statistics in Medicine, 17, 1581– 1600Overall, J.E. (1997). Dropouts and a random regression model. Journal of Biopharmaceutical Statistics, 7, 383– 402Overall, J.E. , Ahn, C. , Shivakumar, C. , Kalburgi, Y. (1999). Problematic formulations of SAS PROC. IXED models for repeated measurements. Journal of Biopharmaceutical Statistics, 9, 189– 216Overall, J.E. , Doyle, S. (1994). Estimating samples sizes for repeated measurement designs. Controlled Clinical Trials, 15, 100– 123Overall, J.E. , Tonidandel, S. (2002). Measuring change in controlled longitudinal studies. British Journal of Mathematical and Statistical Psychology, 55, 109– 124Overall, J.E. , Tonidandel, S. (2004). Robustness of generalized estimating equation (GEE) tests of significance against misspecification of the error structure model. Biometrical Statistics, 46, 203– 213Overall, J.E. , Tonidandel, S. (2006a). Rule-of-thumb adjustment of sample sizes to accommodate dropouts in a two-stage analysis of repeated measurements. International Journal of Methods in Psychiatric Research, 15, 1– 11Overall, J.E. , Tonidandel, S. (2006b). A two-stage analysis of repeated measurements with dropouts and/or intermittent missing data. Journal of Clinical Psychology, 62, 285– 292Palta, M. , Cook, T. (1987). Some considerations in the analysis of rates of change in longitudinal studies. Statistics in Medicine, 6, 599– 611Rubin, D.B. (1976). Inference and missing data. Biometrika, 63, 581– 592Talwalker, S. (1996). Analysis of repeated measurements with dropouts among Alzheimer patients using summary measures and meta-analysis. Journal of Biopharmaceutical Statistics, 6, 49– 58Weinberg, J.M. , Lagakos, S.W. (2001). Efficiency comparisons of rank and permutation tests based on summary statistics computed from repeated measures data. Statistics in Medicine, 20, 705– 731Winer, B.J. , Brown, D.R. , Michels, K.M. (1991). Statistical principles in experimental design . New York: McGraw-HillWu, M.C. , Bailey, K.R. (1989). Estimation and comparison of changes in the presence of informative right censoring: Conditional linear model. Biometrics, 45, 939– 955Wu, M.C. , Carroll, R.J. (1988). Estimation and comparison of changes in the presence of informative right censoring by modeling the censoring process. Biometrics, 4, 175– 188