Abstract
Growth mixture modeling has become a prominent tool for studying the heterogeneity of developmental trajectories within a population. In this article we develop graphical diagnostics to detect misspecification in growth mixture models regarding the number of growth classes, growth trajectory means, and covariance structures. For each model misspecification, we propose a different type of empirical Bayes residual to quantify the departure. Our procedure begins by imputing multiple independent sets of growth classes for the sample. Then, from these so-called "pseudoclass" draws, we form diagnostic plots to examine the averaged empirical distributions of residuals in each such class. Our proposals draw on the property that each single set of pseudoclass adjusted residuals is asymptotically normal with known mean and (co)variance when the underlying model is correct. These methods are justified in simulation studies involving two classes of linear growth curves that also differ by their covariance structures. These are then applied to longitudinal data from a randomized field trial that tests whether children's trajectories of aggressive behavior could be modified during elementary and middle school. Our diagnostics lead to a solution involving a mixture of three growth classes. When comparing the diagnostics obtained from multiple pseudoclasses with those from multiple imputations, we show the computational advantage of the former and obtain a criterion for determining the minimum number of pseudoclass draws.
Original language | English (US) |
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Pages (from-to) | 1054-1076 |
Number of pages | 23 |
Journal | Journal of the American Statistical Association |
Volume | 100 |
Issue number | 471 |
DOIs | |
State | Published - Sep 2005 |
Funding
Chen-Pin Wang is Assistant Professor, Department of Medicine, University of Texas Health Science Center, San Antonio, TX 78230 (E-mail: [email protected]). C. Hendricks Brown is Professor, Department of Epidemiology and Biostatistics, University of South Florida, Tampa, FL 33647 (E-mail: [email protected]). Karen Bandeen-Roche is Professor, Department of Biostatistics, Johns Hopkins University, Baltimore, MD 21205 (E-mail: [email protected]). An earlier version of this article was presented at the 2001 Society for Prevention Research meeting and the 2002 Joint Statistical Meeting. Work on this article was supported by facilities within the Veterans Evidence-Based Research, Dissemination, and Implementation Center; National Institute of Mental Health grants MH40859, MH01259, MH38725, MH42968, and MH-56639-01 A1; National Institute on Drug Abuse grant MH40859; National Institute of Child Health and Human Development grant HD40051; National Science Foundation grant BCS-9978453, and the William T. Grant Foundation grant 2045. The work has benefitted from many helpful discussions within the Prevention Science and Methodology Group, especially those with Bengt Muthén, Sheppard Kellam, and Nick Ialongo. The authors also acknowledge insightful comments from the two referees and the associate editor.
Keywords
- Empirical Bayes
- Growth mixture modeling
- Latent variables
- Marginal maximum likelihood
- Preventive intervention
- Pseudoclass
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty