Simplex Mixed-Effects Models for Longitudinal Proportional Data

Continuous proportional outcomes are collected from many practical studies, where responses are confined within the unit interval (0,1). Utilizing Barndorff-Nielsen and Jorgensen's simplex distribution, we propose a new type of generalized linear mixed-effects model for longitudinal proportional data, where the expected value of proportion is directly modelled through a logit function of fixed and random effects. We establish statistical inference along the lines of Breslow and Clayton's penalized quasi-likelihood (PQL) and restricted maximum likelihood (REML) in the proposed model. We derive the PQL/REML using the high-order multivariate Laplace approximation, which gives satisfactory estimation of the model parameters. The proposed model and inference are illustrated by simulation studies and a data example. The simulation studies conclude that the fourth order approximate PQL/REML performs satisfactorily. The data example shows that Aitchison's technique of the normal linear mixed model for logit-transformed proportional outcomes is not robust against outliers.