Mixed membership latent variable model with unknown factors, factor loadings and number of extreme profiles

Mixed membership models are frequently utilized to capture complex individual heterogeneity in multivariate and longitudinal data. A key aspect of mixed membership modeling involves determining the number of extreme profiles (classes), a task traditionally managed through inefficient criterion-based methods. This task is particularly challenging when the predictors within the models are latent and derived from multiple observed variables using exploratory factor analysis. In this paper, we consider an innovative mixed membership latent variable model, which consists of an exploratory factor model to identify latent factors and a mixed membership model with latent predictors. We develop an efficient approach that integrates parameter estimation and model selection for the number of factors, extreme profiles, and the structure of the factor loading matrix. Our approach comprises a modified stochastic search item selection algorithm to automatically determine the number of latent factors and their associated manifest variables and a Bayesian penalized method to select the number of extreme profiles. We validate our methodology through extensive simulation studies, demonstrating its accuracy and efficiency in both parameter estimation and model selection. Applying this method to data from the Parkinson's Progression Markers Initiative, we identify clinically important latent traits and distinct disease profiles. The results underscore our model's enhanced ability to depict the intricate individual heterogeneity present in Parkinson's disease patients.