Estimating nonlinear change models in heterogeneous populations when class membership is unknown: Defining and developing the latent classification differential change model

Standard methods for analyzing change generally assume that the population of interest is homogeneous or that heterogeneity is known. When a population consists of unknown subpopulations, the parameters within each of the latent classes may be unique to that particular class. In such a situation the results of standard techniques for analyzing change are misleading, because such methods ignore unobserved heterogeneity and treat the population as if it were homogeneous. The growth mixture model (GMM; Muthen, 2001a; Muthen, 2001b; Muthen, 2002) partly addresses the problem of unknown heterogeneity because the parameters of the GMM are conditional on latent class membership. However, the GMM is necessarily restricted to models of change linear in their parameters (such as polynomial change models). The latent classification differential change (LCDC) model allows change to follow a nonlinear functional form and parameter estimates to be conditional on latent class membership. Due to the integration of nonlinear multilevel models and finite mixture models, neither or which have closed form solutions, analytic solutions do not generally exist for the LCDC model. Five methods of parameter estimation are developed and evaluated with a Monte Carlo simulation study. The simulation showed that the parameters of the LCDC model can be accurately estimated with each of the proposed methods, and that the method of choice depends on the particular question of interest. In situations where a different functional form of change or a different error structure is desired across the latent classes, Method 3.1 is recommended. In situations where the functional form of change and the error structure are constant across class, and an interest exists in cross-class hypothesis testing and/or cross-class constraints, Method 3.2 is recommended. In situations where the functional form of change and fixed effect model specification is constant, with or without holding the error structure constant across class, Method 4 is recommended. Method 4 was shown to be the most accurate method in recovering the population fixed effect values in straightforward applications of the LCDC model. The LCDC model provides a novel method of modeling and understanding change when change is governed by a nonlinear functional form in populations where latent classes exist.