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Introduction
Growth mixture modeling combines the conventional Laird and Ware [1] random effects modeling with latent trajectory classes as in finite mixture modeling; see, for example, [2]. Growth mixture modeling was introduced in Verbeke and LeSaffre [3] and Muthén and Shedden [4] with related development in Nagin and Land [5] and Roeder et al. [6]. Following this, many extensions and applications have been presented such as Lin et al. [7] considering prostate-specific antigen (PSA) biomarker trajectories with irregularly scheduled observations, Lin et al. [8] adding joint estimation of survival with prostate cancer, Muthén and Brown [9] considering causal inference in randomized trials of antidepressants with placebo effects, Muthén and Asparouhov [10] adding general multilevel growth mixture modeling, and Muthén et al. [11] modeling non-ignorable dropout in antidepressant trials. For overviews of methods with illustrations by a variety of applications, see [10, 12]. A limiting feature of the aforementioned approaches is the assumption of normally distributed variables within each latent class. With strongly non-normal outcomes, this means that several latent classes are required to capture the observed variable distributions. Consider a typical example involving body mass index (BMI) development over age. BMI is defined as kg/m2, where the normal range is 18 < BMI> 30. The distribution of BMI at age 15 years for men is given in Figure 1 using data from the National Longitudinal Survey of Youth (NLSY) with n = 3194 showing skewness of 1.5 and kurtosis of 3.1. The figure also shows the fitting of a mixture of normal distributions. The left part of Table I shows the loglikelihood and BIC values for 1–4 classes using a normal distribution. Although the four-class solution has a smaller (better) BIC than three classes, one class has less than 1%, and a three-class solution is therefore chosen.
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