In animal genetics, linear mixed models are used to deal with genetic and environmental effects. The variance and covariance terms of these models are usually estimated by restricted maximum likelihood (REML), which provides unbiased estimators. A strong hypothesis of REML estimation is the multi-normality of the response variables. However, in practice, even if the marginal distributions of each phenotype are normal, the multi-normality assumption may be violated by non-normality of the cross-sectional dependence structure, that is to say when the copula of the multivariate distribution is not Gaussian. This study uses simulations to evaluate the impact of copula miss-specification in a bivariate animal model on REML estimations of variance components. Bivariate phenotypes were simulated for populations undergoing selection, considering different copulas for the dependence structure between the error components. Two multi-trait situations were considered: two phenotypes were measured on the selection candidates, or only one phenotype was measured on the selection candidates. Three generations with random selection and five generations with truncation selection based on estimated breeding values were simulated. When selection was performed at random, no significant differences were observed between the REML estimations of variance components and the true parameters even for the non-Gaussian distributions. For the truncation selections, when two phenotypes were measured on candidates, biases were systematically observed in the variance components for high residual dependence in the case of non-Gaussian distributions, especially in the case of a heavy-tailed or asymmetric distribution when the two traits were measured. Conversely, when only one phenotype was measured on candidates, no difference was observed between the Gaussian and non-Gaussian distributions in REML estimations. This study confirms that REML can be used by geneticists to evaluate breeding values in the multivariate case even if the multivariate phenotypes deviate from normality in the situation of random selection or if one trait is not measured for the candidate under selection. Nevertheless, when the two traits are measured, the violation of the normality assumption may lead to non-negligible biases in the REML estimations of the variance-covariance components.

中文翻译：

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REML多变量遗传动物模型估计中的Copula错误规范

在动物遗传学中，线性混合模型用于处理遗传和环境影响。这些模型的方差和协方差项通常通过限制最大似然 (REML) 来估计，它提供了无偏估计量。REML 估计的一个强假设是响应变量的多正态性。然而，在实践中，即使每个表型的边缘分布是正态的，多正态假设也可能被横截面依赖结构的非正态性违反，即当多元分布的copula不是高斯分布时. 本研究使用模拟来评估双变量动物模型中 copula 未指定对方差分量的 REML 估计的影响。模拟了正在选择的种群的双变量表型，考虑误差分量之间依赖结构的不同 copula。考虑了两种多性状情况：在选择候选者上测量了两种表型，或者在选择候选者上仅测量了一种表型。模拟了基于估计育种值的随机选择三代和截断选择五代。当随机选择时，即使对于非高斯分布，方差分量的 REML 估计与真实参数之间也没有观察到显着差异。对于截断选择，当对候选者测量两种表型时，在非高斯分布的情况下，在方差分量中系统地观察到高残差依赖性的偏差，尤其是在测量两个性状时重尾或不对称分布的情况下。相反，当仅对候选者测量一种表型时，在 REML 估计中没有观察到高斯分布和非高斯分布之间的差异。这项研究证实，即使在随机选择的情况下多变量表型偏离正态性，或者如果没有为选择的候选者测量一个性状，遗传学家也可以使用 REML 来评估多变量情况下的育种值。然而，当测量这两个特征时，违反正态性假设可能会导致方差-协方差分量的 REML 估计出现不可忽略的偏差。在 REML 估计中，高斯分布和非高斯分布之间没有观察到差异。这项研究证实，即使在随机选择的情况下多变量表型偏离正态性，或者如果没有为选择的候选者测量一个性状，遗传学家也可以使用 REML 来评估多变量情况下的育种值。然而，当测量这两个特征时，违反正态性假设可能会导致方差-协方差分量的 REML 估计出现不可忽略的偏差。在 REML 估计中，高斯分布和非高斯分布之间没有观察到差异。这项研究证实，即使在随机选择的情况下多变量表型偏离正态性，或者如果没有为选择的候选者测量一个性状，遗传学家也可以使用 REML 来评估多变量情况下的育种值。然而，当测量这两个特征时，违反正态性假设可能会导致方差-协方差分量的 REML 估计出现不可忽略的偏差。这项研究证实，即使在随机选择的情况下多变量表型偏离正态性，或者如果没有为选择的候选者测量一个性状，遗传学家也可以使用 REML 来评估多变量情况下的育种值。然而，当测量这两个特征时，违反正态性假设可能会导致方差-协方差分量的 REML 估计出现不可忽略的偏差。这项研究证实，即使在随机选择的情况下多变量表型偏离正态性，或者如果没有为选择的候选者测量一个性状，遗传学家也可以使用 REML 来评估多变量情况下的育种值。然而，当测量这两个特征时，违反正态性假设可能会导致方差-协方差分量的 REML 估计出现不可忽略的偏差。