Factor-analytic models for genotype x environment type problems and structured covariance matrices

Background: Analysis of data on genotypes with different expression in different environments is a classic problem in quantitative genetics. A review of models for data with genotype x environment interactions and related problems is given, linking early, analysis of variance based formulations to their modern, mixed model counterparts. Results: It is shown that models developed for the analysis of multi-environment trials in plant breeding are directly applicable in animal breeding. In particular, the 'additive main effect, multiplicative interaction' models accommodate heterogeneity of variance and are characterised by a factor-analytic covariance structure. While this can be implemented in mixed models by imposing such structure on the genetic covariance matrix in a standard, multi-trait model, an equivalent model is obtained by fitting the common and specific factors genetic separately. Properties of the mixed model equations for alternative implementations of factor-analytic models are discussed, and extensions to structured modelling of covariance matrices for multi-trait, multi-environment scenarios are described. Conclusion: Factor analytic models provide a natural framework for modelling genotype x environment interaction type problems. Mixed model analyses fitting such models are likely to see increasing use due to the parsimonious description of covariance structures available, the scope for direct interpretation of factors as well as computational advantages.