Abstract
BackgroundAn important issue in genetic evaluation is the comparability of random effects (breeding values), particularly between pairs of animals in different contemporary groups. This is usually referred to as genetic connectedness. While various measures of connectedness have been proposed in the literature, there is general agreement that the most appropriate measure is some function of the prediction error variance–covariance matrix. However, obtaining the prediction error variance–covariance matrix is computationally demanding for large-scale genetic evaluations. Many alternative statistics have been proposed that avoid the computational cost of obtaining the prediction error variance–covariance matrix, such as counts of genetic links between contemporary groups, gene flow matrices, and functions of the variance–covariance matrix of estimated contemporary group fixed effects.ResultsIn this paper, we show that a correction to the variance–covariance matrix of estimated contemporary group fixed effects will produce the exact prediction error variance–covariance matrix averaged by contemporary group for univariate models in the presence of single or multiple fixed effects and one random effect. We demonstrate the correction for a series of models and show that approximations to the prediction error matrix based solely on the variance–covariance matrix of estimated contemporary group fixed effects are inappropriate in certain circumstances.ConclusionsOur method allows for the calculation of a connectedness measure based on the prediction error variance–covariance matrix by calculating only the variance–covariance matrix of estimated fixed effects. Since the number of fixed effects in genetic evaluation is usually orders of magnitudes smaller than the number of random effect levels, the computational requirements for our method should be reduced.
Highlights
An important issue in genetic evaluation is the comparability of random effects, between pairs of animals in different contemporary groups
Connectedness was defined as the loss of information due to a lack of orthogonality [4] measured by using the Kullback–Leibler divergence. It was shown in Laloë [5] that for a linear mixed model, the expected information is a function of the ratio of the posterior and prior variance for u, alternatively known as the prediction error variance–covariance matrix (PEV) and the relationship matrix, respectively
The first of the measures that we investigated was the PEV of contemporary group differences (PEVDij) [12]
Summary
An important issue in genetic evaluation is the comparability of random effects (breeding values), between pairs of animals in different contemporary groups. Connectedness was defined as the loss of information due to a lack of orthogonality [4] measured by using the Kullback–Leibler divergence It was shown in Laloë [5] that for a linear mixed model, the expected information is a function of the ratio of the posterior and prior variance for u, alternatively known as the prediction error variance–covariance matrix (PEV) and the relationship matrix, respectively. They showed that the expected information could be re-arranged to give a co-efficient of determination (CD) statistic [5, 6]. To reduce the computational cost of this measure, simulation and the repeated use of iterative solvers were proposed [11]
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