Mixed and mixture regression models for continuous bounded responses using the beta distribution

Doubly bounded continuous data are common in the social and behavioral sciences. Examples include judged probabilities, confidence ratings, derived proportions such as percent time on task, and bounded scale scores. Dependent variables of this kind are often difficult to analyze using normal theory models because their distributions may be quite poorly modeled by the normal distribution. The authors extend the beta-distributed generalized linear model (GLM) proposed in Smithson and Verkuilen (2006) to discrete and continuous mixtures of beta distributions, which enables modeling dependent data structures commonly found in real settings. The authors discuss estimation using both deterministic marginal maximum likelihood and stochastic Markov chain Monte Carlo (MCMC) methods. The results are illustrated using three data sets from cognitive psychology experiments.