Nonlinear Change Models in Populations with Unobserved Heterogeneity

Abstract

Abstract. When unobserved heterogeneity exists in populations where the phenomenon of interest is governed by a functional form of change linear in its parameters, the growth mixture model (GMM) is useful for modeling change conditional on latent class. However, when the functional form of interest is nonlinear in its parameters, the GMM is not very useful because it is based on a system of equations linear in its parameters. The nonlinear change mixture model (NCMM) is proposed, which explicitly addresses unobserved heterogeneity in situations where change follows a nonlinear functional form. Due to the integration of nonlinear multilevel models and finite mixture models, neither of which generally have closed form solutions, analytic solutions do not generally exist for the NCMM. Five methods of parameter estimation are developed and evaluated with a comprehensive Monte Carlo simulation study. The simulation showed that the parameters of the NCMM can be accurately estimated with several of the proposed methods, and that the method of choice depends on the precise question of interest.Keywords: longitudinal data analysis, analysis of change, growth modeling, growth mixture modeling, heterogeneous population, heterogeneous change, nonlinear growth models, nonlinear change models, functional form of growth, functional form of change DOI 10.1027/1614-2241. 4.3.97 In the behavioral, education, and social sciences, idiographic conceptualizations of change tend to focus on the individual, whereas nomothetic conceptualizations tend to focus on the group (e.g., see Allport, 1937). Muthén and Muthén (2000) discuss that, at least in general, statistical methods designed to help researchers answer questions about behavior are either variable-centered or pattern-centered. Statistical methods that are variable-centered focus more on relationships among variables (e.g., structural equation models and its variants), whereas statistical methods that are pattern-centered focus more on the relationships among the individuals (e.g., finite mixture models and its variants). Variable-centered methods are consistent with the "strong concept of growth," because this view states that "a single developmental function can adequately describe the change of all individuals from some population" (Burchinal & Appelbaum, 1991, p. 25), whereas pattern-centered approaches are consistent with the "weak concept of development," because this view seeks to identify intraindividual patterns of change and interindividual differences in those patterns (Burchinal & Appelbaum, 1991;Nesselroade & Baltes, 1979). Idiographic and nomothetic conceptualizations of behavior are traditionally considered antithetical (Dunn, 1994, p. 377) and as such potential benefits of combining the approaches in an integrated fashion often go unrecognized. Some research questions, however, demand an integration of variable-centered and pattern-centered statistical techniques (e.g., Burchinal & Appelbaum, 1991;Dumenci & Windle, 2001;Magnusson & Bergmen, 1988;Muthén & Muthé...

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