Bayesian Inference for Growth Mixture Models with Latent Class Dependent Missing Data

Abstract

Growth mixture models (GMMs) with nonignorable missing data have drawn increasing attention in research communities but have not been fully studied. The goal of this article is to propose and to evaluate a Bayesian method to estimate the GMMs with latent class dependent missing data. An extended GMM is first presented in which class probabilities depend on some observed explanatory variables and data missingness depends on both the explanatory variables and a latent class variable. A full Bayesian method is then proposed to estimate the model. Through the data augmentation method, conditional posterior distributions for all model parameters and missing data are obtained. A Gibbs sampling procedure is then used to generate Markov chains of model parameters for statistical inference. The application of the model and the method is first demonstrated through the analysis of mathematical ability growth data from the National Longitudinal Survey of Youth 1997 (Bureau of Labor Statistics, U.S. Department of Labor, 1997). A simulation study considering 3 main factors (the sample size, the class probability, and the missing data mechanism) is then conducted and the results show that the proposed Bayesian estimation approach performs very well under the studied conditions. Finally, some implications of this study, including the misspecified missingness mechanism, the sample size, the sensitivity of the model, the number of latent classes, the model comparison, and the future directions of the approach, are discussed.

Notes

¹Throughout the article, MN_n denotes an n-dimensional multivariate normal distribution.

²Here we have two probabilities that need to be distinguished. The class probability π _k is a class-specific population parameter in the model, whereas the posthoc posterior probability is an individual variable that is computed for each individual once model parameters have been estimated.

³Note that this is only one way to specify a regression model for categorical variables.

⁴Specifically, suppose for each k (k = 1, 2, …, K) there exists an underlying continuous random variable c_ik *, which follows a normal distribution with mean ϕ _k0 + x _i ′ ϕ _k1 and variance 1,

where e_i ~ N(0, 1). The outcome y_i comes from the first k classes when c_ik * is positive. In other words,

⁵With 10,000 burn-ins, the Markov chains for all parameters converged.

⁶This method tests the convergence of Markov chain by comparing the means of two subsets of the chain.

⁷With 70,000 iterations, the ratio of MCse/sd is less than 0.05 for all parameters, which indicates that the estimates are accurate. An example of inaccurate estimates obtained with 2,000 burn-ins and 5,000 iterations can be found on our website: (http://nd.psychstat.org/research/luzhanglubke2010) for comparison.

^a The significance of parameter estimates can be judged based on the confidence intervals. If zero is included in the interval, then the parameter estimate is not significantly different from zero.

^b The growth curve parameters for Class 1. Specifically, β[1]: initial level; β[2]: slope; Ψ[11]: variance of initial level; Ψ[22]: variance of slope; Ψ[12]: covariance of initial level and slope; φ: variance of error.

^c The growth curve parameters for Class 2.

^d The probit parameters of class proportion as in Equation (6).

^e The probit parameters of missing data rate. Note that although the γ_0t * and γ_1t * here are different with the γ _{zt 1} and γ _{zt 2} in Equation (9), they are equivalent after reparameterizing γ_0t * = γ _{zt 1} and γ_1t * = γ _{zt 2} – γ _{zt 1} .

⁸To be consistent with the real data analysis, γ _{zt 1} and γ _{zt 2} are reparameterized as γ_0t * and γ_1t * with γ_0t * = γ _{zt 1} and γ_1t * = γ _{zt 2} – γ _{zt 1} .

^a The average absolute relative bias across all model parameters, defined by |Bias.rel| = Σ _j=1 ^p |Bias.rel _j |p.

^b The average absolute difference between the empirical SDs and the average Bayesian SDs across all model parameters, defined by |SD.diff| = Σ _j=1 ^p |SD.emp _j – SD.avg _j |p.

^c The average coverage probability across all model parameters, defined by HPD.cvr = Σ _j=1 ^p HPD.cvr _j /p.

^d With a sample size of 500, the convergence rate under unequal classes and MNAR missingness is 100/147 ≈ 67%. MNAR = missing not at random; MCAR = missing completely at random.

^a Modeling GMM and latent class dependent missingness.

^b Modeling GMM only, ignore the missingness mechanism.

^c GMM with latent class dependent missing data.