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1、 A Stata Plugin for Estimating Group-Based Trajectory Models Bobby L. Jones University of Pittsburgh Medical Center Daniel S. Nagin Carnegie Mellon University May 21, 2012 This work was generously supported by National Science Foundation Grants SES-102459 and SES- 0647576 Abstract Group-based trajec
2、tory models are used to investigate population differences in the developmental courses of behaviors or outcomes . This article demonstrates a new Stata command, traj, for fitting to longitudinal data finite (discrete) mixture models designed to identify clusters of individuals following similar pro
3、gressions of some behavior or outcome over age or time. Censored normal, Poisson, zero-inflated Poisson, and Bernoulli distributions are supported. Applications to psychometric scale data, count data, and a dichotomous prevalence measure are illustrated. Introduction A developmental trajectory measu
4、res the course of an outcome over age or time. The study of developmental trajectories is a central theme of developmental and abnormal psychology and psychiatry, of life course studies in sociology and criminology, of physical and biological outcomes in medicine and gerontology. A wide variety of s
5、tatistical methods are used to study these phenomena. This article demonstrates a Stata plugin for estimating group-based trajectory models. The Stata program we demonstrate adapts a well-established SAS-based procedure for estimating group-based trajectory model (Jones, Nagin, and Roeder, 2001; Jon
6、es and Nagin, 2007) to the Stata platform. Group-based trajectory modeling is a specialized form of finite mixture modeling. The method is designed identify groups of individuals following similarly developmental trajectories. For a recent review of applications of group-based trajectory modeling se
7、e Nagin and Odgers (2010) and for an extended discussion of the method, including technical details, see Nagin (2005). A Brief Overview of Group-Based Trajectory Modeling Using finite mixtures of suitably defined probability distributions, the group-based approach for modeling developmental trajecto
8、ries is intended to provide a flexible and easily applied method for identifying distinctive clusters of individual trajectories within the population and for profiling the characteristics of individuals within the clusters. Thus, whereas the hierarchical and latent curve methodologies model populat
9、ion variability in growth with multivariate continuous distribution functions, the group-based approach utilizes a multinomial modeling strategy. Technically, the group-based trajectory model is an example of a finite mixture model. Maximum likelihood is used for the estimation of the model paramete
10、rs. The maximization is performed using a general quasi-Newton procedure (Dennis, Gay, and Welsch 1981; Dennis and Mei 1979). The fundamental concept of interest is the distribution of outcomes conditional on age (or time); that is, the distribution of outcome trajectories denoted by ),|(iiAgeYP whe
11、re the random vector Yi represents individual is longitudinal sequence of behavioral outcomes and the vector Agei represents individual is age when each of those measurements is recorded.1 The group-based trajectory model assumes that the population distribution of trajectories arises from a finite
12、mixture of unknown order J. The likelihood for each individual i, conditional on the number of groups J, may be written as 1 Trajectories can also be defined by time (e.g., time from treatment). 1(|)(|, ;)(1),J jj iiii jP Y AgeP Y Age jwhere j is the probability of membership in group j, and the con
13、ditional distribution of Yi given membership in j is indexed by the unknown parameter vector j which among other things determines the shape of the group-specific trajectory. The trajectory is modeled with up to a 5th order polynomial function of age (or time). For given j, conditional independence
14、is assumed for the sequential realizations of the elements of Yi , yit, over the T periods of measurement. Thus, we may write Titj ititj iijageypjAgeYP),2();,|();,|( where p(.) is the distribution of yit conditional on membership in group j and the age of individual i at time t. 2 The software provi
15、des three alternative specifications of p(.): the censored normal distribution also known as the Tobit model, the zero-inflated Poisson distribution, and the binary logit distribution. The censored normal distribution is designed for the analysis of repeatedly measured, (approximately) continuous sc
16、ales which may be censored by either a scale minimum or maximum or both (e.g., longitudinal data on a scale of depression symptoms). A special case is a scale or other outcome variable with no minimum or maximum. The zero-inflated Poisson distribution is designed for the analysis of longitudinal count data (e.g., arrests by age) and binary logit distribution for the analysis of longitudinal data on a dichotomous outcome variable
