Measurement and Structural Model Class Separation in Mixture CFA: ML/EM Versus MCMC

Parameter recovery was assessed within mixture confirmatory factor analysis across multiple estimator conditions under different simulated levels of mixture class separation. Mixture class separation was defined in the measurement model (through factor loadings) and the structural model (through factor variances). Maximum likelihood (ML) via the EM algorithm was compared to a Markov chain Monte Carlo (MCMC) estimator condition using weak priors and a condition using tight priors. Results indicated that the MCMC weak condition produced the highest bias, particularly with a weak Dirichlet prior for the mixture class proportions. Specifically, the weak Dirichlet prior affected parameter estimates under all mixture class separation conditions, even with moderate and large sample sizes. With little knowledge about parameters, ML/EM should be used over MCMC weak. However, MCMC tight produced the lowest bias under all mixture class separation conditions and should be used if tight and accurate priors can be placed on parameters. Keywords: Bayesian estimationconfirmatory factory analysisclass separationparameter recoveryfinite mixture models Notes 1Although the multivariate Bernoulli density is discussed here, note also that many of the mixture distributions encountered in practice are Gaussian (univariate or multivariate). There are also other mixture distributions, such as exponential and Weibull, that are used in biostatistics and related fields to model the distribution of survival time. For a discussion of some of the common finite mixture distributions used, see CitationEveritt (1996). 2Note that this diagnostic can also be used to assess convergence in a single MCMC chain by comparing the first portion of the post burn-in iterations to the last portion of the chain. See Muth=n and Muth=n (1998–2010) for more details of how this is implemented in Mplus. 3Although it is true that nonconjugate priors can be specified for any model, this is typically not encouraged for mixture models (see, e.g., CitationDiebolt & Robert, 1994; CitationLee, 2007). The use of fully noninformative priors (e.g., uniform) can lead to improper posterior distributions. As a result, it is common for mixture models to be specified with conjugate priors to avoid this problem altogether. 4 Hyperparameters are the parameters of a prior distribution. For example, the hyperparameters for the normal distribution are the mean and variance terms. 5Although this article deals with priors in terms of the variance–covariance matrix by using an IW distribution, there are some software programs (e.g., WinBUGS) that can only place prior distributions on the precisions (inverse variance–covariance matrix). In this case, the prior distribution on the precision matrix would be the Wishart distribution. 6For the ML/EM estimator, the number of random starts for the models specified in this study was 100. The number of final stage optimization steps was 10. These settings were used to ensure that estimates were not the result of problems with local maxima. Likewise, there was no evidence of any class label switching, a potential problem in mixture simulation studies, in any of the estimator conditions presented here. 7For the ML/EM condition and both MCMC estimator conditions, the number of replications that converged properly under the poor and moderate class separation conditions were as low as 58%. These results are similar to those obtained in CitationTueller and Lubke (2010), which indicated poor convergence under lower class separation conditions. ML/EM and MCMC appeared to have equal difficulty in obtaining convergence under these separation conditions. 8Bias is computed by using the following equation: 100*([estimate − true value]/true value). 9Note, however, that structural model class separation could have also been defined through the factor means or factor covariances or correlations. Nevertheless, it was deemed sufficient, for the purposes of this study, to define structural class separation in terms of the factor variances. 10Each condition took less than 2 minutes to run for the ML/EM and MCMC estimators. aBias decreased below 10.00% when sample size increased to 300. bBias decreased below 10.00% when sample size increased to 800. aBias decreased below 10.00% when sample size increased to 300. bBias decreased below 10.00% when sample size increased to 800. aBias decreased below 10.00% when sample size increased to 300. bBias decreased below 10.00% when sample size increased to 800. aBias decreased below 10.00% when sample size increased to 300. bBias decreased below 10.00% when sample size increased to 800. aBias decreased below 10.00% when sample size increased to 300. bBias decreased below 10.00% when sample size increased to 800. aBias decreased below 10.00% when sample size increased to 300. bBias decreased below 10.00% when sample size increased to 800.