Investigating Stage-Sequential Growth Mixture Models with Multiphase Longitudinal Data

Abstract

This article investigates three types of stage-sequential growth mixture models in the structural equation modeling framework for the analysis of multiple-phase longitudinal data. These models can be important tools for situations in which a single-phase growth mixture model produces distorted results and can allow researchers to better understand population heterogeneity and growth over multiple phases. Through theoretical and empirical comparisons of the models, the authors discuss strategies with respect to model selection and interpreting outcomes. The unique attributes of each approach are illustrated using ecological momentary assessment data from a tobacco cessation study. Transitional discrepancy between phases as well as growth factors are examined to see whether they can give us useful information related to a distal outcome, abstinence at 6 months postquit. It is argued that these statistical models are powerful and flexible tools for the analysis of complex and detailed longitudinal data.

Keywords:

• growth mixture modeling
• latent growth modeling
• longitudinal data analysis
• multiphase longitudinal data
• stage-sequential models

Notes

¹Muthén, Khoo, Francis, and Boscardin (2003) examined an application of GMM to sequential processes, in which it is similar to the sequential process GMM in that there exist separate growth parts in each phase (separate intercepts and slopes), but it is different in that it has only one latent class variable.

²In this particular example, a growth trajectory is assumed to be linear. Four or more time points provide an opportunity to test for nonlinear trajectories in the SEM framework. The most familiar approach to nonlinear trajectories is probably the use of polynomials (Duncan, Duncan, Strycker, Li, & Alpert, 1999). The inclusion of quadratic or cubic effects is easily accomplished by including another factor or two.

³ Bauer (2007) indicated that population heterogeneity can also be captured using LGM by estimating variances around fixed effects curves.

⁴In Nagin's semiparametric approach, systematic individual differences from the mean trajectory within classes are not allowed.

⁵It is because the nonnormality of a composite density function, the aggregated density function of each component density function of classes in a mixture, is a necessary condition for obtaining a nontrivial solution for the mixture (Bauer & Curran, 2003a).

⁶ Bauer and Curran (2003a) is argued in two ways: (a) It might not be as difficult as Bauer and Curran (2003a) suggested to distinguish between a single-class model with nonnormally distributed observed variables and a true mixture (Muthén, 2003; Rindskopf, 2003); and (b) even if these two possibilities (spurious classes or a true mixture) cannot be distinguished, a growth mixture model can still provide useful insights into the data (Cudeck & Henly, 2003).

⁷In this particular example, growth trajectories within each phase are assumed to be linear as we assumed for LGM in Equation 5 or in piecewise LGM in Equation 4.

⁸The test for equal residual variance over time using piecewise LGM was rejected, χ² (20, N = 370) = 82.382, p = .000. Nevertheless, we constrained the residual variances over time in stage-sequential growth mixture models for the estimation purpose. We experienced convergence problems when estimating with different residual variances over time, perhaps because of a lot more free parameters being estimated.

⁹In Mplus, no correlations between repeated measures in a growth model are allowed by default, so we released that constraint by allowing an AR(1) structure in each phase. An autoregressive process is a stochastic process that can be described by a weighted sum of its previous values and an error. An AR(1) process is a first-order one, meaning that only the immediately previous value has a direct effect on the current value.

where c is a constant, ρ is a correlation, and ∊_t is a white noise process with a mean of zero and a variance of σ². There is another AR trend that allows autoregressive patterns in the error terms (Bollen & Curran, 2004).

¹⁰In Nylund et al. (2007), BIC performed well across all sample sizes for GMM and correctly identified a specific k number of classes close to 100% of the time for both categorical and continuous items. However, AIC does not seem to be a good indicator for identifying the k class model for any of the modeling settings. Further, when AIC is not able to identify the correct k class model, it is most likely to identify the k + 1 class model as the correct model.

¹¹ Yang (2006) performed a simulation study to explore the performance of information criteria in a set of latent class analysis (LCA) models with categorical outcomes, and the results indicated that the adjusted BIC was the best indicator of the information criteria considered. However, Nylund et al. (2007) showed that BIC has more consistent power across all sample sizes and models considered than the adjusted BIC.

¹²The equation for entropy is

where p _ik denotes the estimated posterior probability for individual i in class k. K is the total latent classes.

¹³The data used in the reported application were collected in 2001 to 2003 and made available by the University of Wisconsin Center for Tobacco Research and Intervention.

¹⁴The simplest indicator-only model and the most complicated full model indicate different numbers of latent class solutions. In our particular smoking cessation data, more latent classes were suggested by BIC in the simplest model than in the most complicated model. We used treatment-added models for the model selection procedure, not the full model, because of a tremendous amount of convergence time.

¹⁵In the initial stage, 100 random sets of starting values were generated, and an optimization was carried out for each of the sets. The ending values from the 20 optimizations with the highest log likelihoods were used as the starting values in the final stage.