Reduced rank factor analytic models for capturing genotype by environment interactions in livestock

Genotype by environment (GxE) interactions occur when the genetic correlation between a trait measured in different environments is less than one, or when the genetic variance of a trait changes between environments. This is often captured using a multi-trait model with an unstructured genetic covariance matrix, where performance in a different environment is considered as a separate but correlated trait. Such an approach becomes computationally infeasible with a large number of environments; an analysis with n=30 environments would require the estimation of n[n+1]/2 or 465 genetic parameters. Hence, we need methods that enable the estimation of GxE interactions with fewer parameters. Factor analytic models approximate the multi-trait model by assuming the pattern of GxE across environments can be described by the regression of genetic effects on latent common factors. The latent common factors are estimated from the data such that they explain the maximum amount of covariance between environments. These models are potentially more flexible and less prescriptive compared to other methods such as reaction norms commonly used in livestock genetics. This study analysed post-weaning body weights from 15,908 lambs across 31 flock-years. The flocks were linked via common sires artificial insemination, while the years were linked via dams used across years. Each flock-year had at least 350 lambs. A reduced rank factor analytic model with two latent common factors for the additive genetic effects and genetic group effects, respectively, provided the best fit to the data based on a log-likelihood ratio test (LRT) and the AIC. The 465 pairwise genetic correlations between environments that were derived from the factor analysis ranged between -0.69 and 1.00, with an average of 0.68. Of these, 22% were significantly less than 1, while 12% were significantly less than 0.80. An alternative approach using a reaction norm model that regressed over the mean performance was also investigated. It was unclear which model was preferred; the reaction norm was significantly poorer than the reduced-rank factor analytic models based on the LRT and AIC but were preferred based on the BIC. However, when the underlying GxE interactions are multi-dimensional, factor analytic models present appealing formulation.