Marginal likelihood and Bayesian approaches to the analysis of heterogeneous residual variances in mixed linear Gaussian models

Abstract

A statistical procedure for assessing the extend and sources of heterogeneity of residual variances in Gaussian mixed linear models is presented. The method assumes a structural linear model for logvariances. Hypothesis testing about the effects of factors potentially involved in the variation of dispersion parameters is based on a likelihood ratio test using marginal likelihood functions where location parameters are integrated out. This method allows us to identify meaningful sources of variation in heterogeneous variances and to assess how these factors act (additivity, interactions). Dispersion parameters are estimated either by maximum marginal likelihood or by the mode of their posterior distribution using a log inverted gamma distribution as prior. Algorithms so obtained have remarkable similarities with the mixed model methodology of Henderson for location parameters. The analysis of a small example is used to illustrate the theory