Factor Analysis of correlated random effects
Abstract: Multivariate mixed linear models with correlated random effects are of increasing importance in many fields
(e.g., quantitative genetics). When the number of response variables is large, the vector of random effects and the corresponding covariance matrix are of high dimension. This may bring computational problems (due to numerically singular (co)variance matrices) and makes interpretation of the result
difficult. Here we develop a hierarchical model where in the first level data is modeled as a linear function of fixed and random effects, and in the second level the vector of random effects is modeled using a factor analysis structure. The model is not only more parsimonious than the standard mixed model, but could also be useful in finding patterns of co-variation at the level of the random effects. Analytical results derived shown that implementation does not pose special difficulties neither in a maximum likelihood nor in a Bayesian context, and minor
modifications of existing software would allow estimation of the model. With purpose of exploratory analysis, a simple two-step methodology applied to two cases, each involving 12-diseases traits in dairy cattle. Preliminary results show that an important reduction in the number of traits can be attained. Based on information contained in data, a few number of factors with substantial biological meaning were identified.