In univariate linear mixed models, it is usually assumed that both residual and random effects have homogeneous variance-covariance components. This assumption may, however, not be completely realistic in practice. This presentation addresses the problem of taking into account such a phenomenon. We first discuss how to model such components of variance.
A natural approach lies in postulating a log link function relating the variance components to explanatory variables, the nature and influence of which can be tested. The general approach involves a structural linear model on logvariances involving both fixed and random effects, as originally proposed by Foulley et al (1992). We can also accommodate relationships between the random components of variance and the residual variance, as well as semiparametric techniques. This procedure can also be extended to heterogeneity of covariances. It provides great flexibility in modelling potential sources of variation in the variance-covariance components with the key idea of parsimony thanks to the introduction of random effects or the use of hierarchical models.
Inference procedures can be based either on ML-REML theory via the EM algorithms and extensions, or on Bayesian theory with MCMC implementation. Finally, the procedure is illustrated with several data examples from different fields of application (clinical trials, growth and calving data, differentially expressed genes) analyzed with linear, generalized linear and non linear mixed models.
Foulley, J.-L., San Cristobal, M., Gianola, D., Im, S.(1992) Marginal likelihood and Bayesian approaches to the analysis of heterogeneous residual variances in mixed linear Gaussian models. Computational Statistics and Data Analysis, 13, 291-305.
Jaffrézic F., Marot G., Degrelle S., Hue I., Foulley J.-L. (2007) A structural mixed model for variances in differential gene expression studies. Genetical Research, 89, 19-25.
Foulley J.-L., Jaffrézic F. (2009) Modelling and estimating heterogeneous variances in threshold models for ordinal discrete data via Winbugs/Openbugs. Computer Methods and Programs in Biomedicine, 97,19-27