In animal genetics, linear mixed models are used to address genetic and environmental effects (model G+E). Variance components are often estimated using the restricted maximum likelihood estimator (REML), which relies on normality assumptions. However, when dealing with multiple traits, the assumption of multivariate normality for the phenotypes may be violated, particularly due to the non-normal dependence structure between the phenotypes. This is notably caused by a non-Gaussian copula in the environmental part, i.e., the residuals. We demonstrated in recent paper that using a Gaussian multi-trait animal model for traits sampled through a non-Gaussian copula for the residuals can bias the estimated genetic parameters, such as heritability and genetic correlations, especially in populations undergoing non-random selection of reproducers. To address this issue, we propose a non-Gaussian inference model that considers not only the genetic and environmental parts but also a copula structure on the residuals, which can be non-Gaussian. We develop stochastic gradient strategies to maximize the considered log-likelihoods, allowing us to jointly estimate the genetic variance components, residual variances, and copula parameters. We will compare the estimations of the genetic and residual parameters obtained from the non-Gaussian inference model with those from Gaussian multi-trait models estimated by AI-REML. This comparison will use simulated data generated from the copula inference model with various copulas, including the Gaussian one and heavy-tailed distributions. Additionally, we will present illustrations using real data from animal farming traits, where multivariate Gaussian models appear inadequate.
