Incorporation of Covariates in Bayesian Piecewise Growth Mixture Models

Abstract

The Bayesian Covariate Influenced Piecewise Growth Mixture Model (CI-PGMM) is an extension of the Piecewise Growth Mixture Model (PGMM, Lock et al., 2018) with the incorporation of covariates. This was done by using a piecewise nonlinear trajectory over time, meaning that the slope has a point where the trajectory changes, called a knot. Additionally, the outcome data belong to two or more latent classes with their own mean trajectories, referred to as a mixture model. Covariates were incorporated into the model in two ways. The first was influencing the outcome variable directly, explaining additional random error variance. The second is the influence of the covariates on the class membership directly with the use of multinomial logistic regression. Both uses of covariates can potentially influence the class memberships and along with that, the trajectories and locations of the knot(s). This additional explanation of class memberships and trajectories can provide information on how individuals change, who is likely to belong in certain unknown classes, and how these class memberships can affect when the rapid change of a knot will happen. The model is shown to be appropriate and effective using two steps. First, a real data application using the National Longitudinal Survey of Youth is used to show the motivation for the model. This dataset measures income over time each year for individuals following high school. Covariates of sex and dropout status were used in the class predictive logistic regression model. This resulted in a two-class solution showing effective use of the covariates with the logistic regression coefficients drastically affecting the class memberships. The second step is using a simulation after the motivating real data application. Pilot studies were used to show if the model was suitable for a full simulation using the coefficients from the real data example as a basis for the data generation. Four pilot studies were performed, and reasonable estimates were found for the full simulation. The conditions were set up with a two class model. One class containing one knot, and the second class as a linear slope. Two class predictive covariates and one outcome predictive covariate were used. A full simulation with 200 generated datasets was performed with manipulated conditions being error variance, sample size, model type, and class probability for a 3x3x3x2 model with 54 total conditions. Outcome measures of convergence, average relative bias, RMSE, and coverage rate were used to show suitability of the model. The simulation showed the use for the CI-PGMM was stable and accurate for multiple conditions. Sample size and model type were the most impactful predictors of appropriate model use. All outcome measures were worse for the small sample sizes and became more accurate when the sample sizes were larger. Also, the simpler models showed less bias and better convergence. However, these differences are smaller when the sample size is sufficiently large. These findings were supported with multi-factor ANOVA comparing simulation conditions. Use of the CI-PGMM in the real data example and the full simulation allowed for incorporation of covariates when appropriate. I show that model complexity can lead to issues of lower convergence, thus the model should only be used when appropriate and the sample size is sufficiently large. When used, however, the model can shed light on associations between covariates, class memberships, and locations of knots that were previously unavailable.