Approximating Prediction Error Variances for Multiple Trait Sire Evaluations

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Abstract
The coefficient matrix for multiple trait (milk, fat, and protein) mixed model equations may be too large to obtain prediction error variances from inverse elements. The commonly used reciprocals of diagonal elements may not be accurate approximations when sire relationships or multiple traits are included since much information is contained in offdiagnonal elements. Approximations incorporating increased information from coefficient matrix were compared with actual prediction error variances for multiple trait evaluations for milk, fat, protein, and dollar value (relationships included) of 229 Ayrshire and 248 Brown Swiss bulls. Six approximations were selection index using number of daughter records, inverses of individual sire diagonal blocks, inverses of group and individual sire blocks, and inverses of all diagonal blocks and offdiagonal blocks associated with individual sires. All approximations underestimated actual prediciton error variances, but most, except selection index, were highly correlated (.90 to .99) with actual prediction error variances of sire evaluations for milk yield and product value for contemporary bulls. The approximation incorporating most information from the coefficient matrix is recommended for use on basis of high correlation with and closeness to actual prediction error variances.