Factor analytic multiplicative mixed models in the analysis of multiple experiments

Analysis of groups of experiments or multi-environment trials (MET) has been traditionally based on simple models assuming error variance homogeneity between trials, independent error within trials, genotype x environment (g x e) effects as a set of independent random effects. The combined analysis of MET data through realistic models is a complex statistical problem which requires extensions to the standard linear mixed model. The relaxation of the assumption concerning the independence of g x e effects can be achieved with the use of multiplicative models. Such models have been popularised as additive main effects and multiplicative interaction effects (AMMI) and a number of applications have been found. However, AMMI analysis presents at least five great limitations: it considers the genotype and g x e effects as fixed; it is suitable only for balanced data sets; it does not consider spatial variation within trials; it does not consider the heterogeneity of variance between trials; it does not consider the different number of replications across sites. These features are not realistic in analysing field data. In a mixed model setting, Piepho (1998) presented a factor analytic multiplicative mixed (FAMM) model with random genotype and g x e effects which is conceptually and functionally better than AMMI. In the same context, Smith et al. (2001) presented a general class of FAMM models that encompass the approach of Piepho (1998) and include separate spatial errors for each environment (FAMMS). Such general class of models provides a full realistic approach for analysing MET data. The present paper deals with the application of FAMM and FAMMS models in two large unbalanced data sets (on eucalypt and tea plant) aiming at emphasising their advantages over AMMI models in terms of the assumptions of error variance homogeneity between trials and independent error within trials. Also, the ability of FAMM models in providing parsimonious models is also stressed. Parsimonious FAMM models were found for the two data sets. There were great advantages of heterogeneous variance FAMM models over homogeneous variance FAMM models. This reveals the superiority of FAMM models over AMMI models. It was noted that there was heterogeneity among the specific variances in individual environments; therefore, factor analytic models with common specific variances for all sites were not suitable. FAMM models provided estimates of the full correlation structure, facilitating practical decisions to be made. FAMM models with heterogeneous variance among traits and spatial errors within traits were advantageous over FAMM models with variance homogeneity and non-spatial error. This also shows the superiority of FAMM models over AMMI models, which do not allow for dependent or spatial errors. For analysing multi-environment data sets with longitudinal data, FAMMS models proved to be a very useful tool.