Multivariate generalized linear mixed models for continuous bounded outcomes: Analyzing the body fat percentage data

Abstract

We propose a multivariate regression model to handle multiple continuous bounded outcomes. We adopted the maximum likelihood approach for parameter estimation and inference. The model is specified by the product of univariate probability distributions and the correlation between the response variables is obtained through the correlation matrix of the random intercepts. For modeling continuous bounded variables on the interval $(0,1)$ we considered the beta and unit gamma distributions. The main advantage of the proposed model is that we can easily combine different marginal distributions for the response variable vector. The computational implementation is performed using Template Model Builder, which combines the Laplace approximation with automatic differentiation. Therefore, the proposed approach allows us to estimate the model parameters quickly and efficiently. We conducted a simulation study to evaluate the computational implementation and the properties of the maximum likelihood estimators under different scenarios. Moreover, we investigate the impact of distribution misspecification in the proposed model. Our model was motivated by a data set with multiple continuous bounded outcomes, which refer to the body fat percentage measured at five regions of the body. Simulation studies and data analysis showed that the proposed model provides a general and rich framework to deal with multiple continuous bounded outcomes.