An Approximate Procedure for Determining Prediction Error Variances of Sire Evaluations

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Abstract
Prediction errors of sire evaluations can be obtained directly from the inverse of the appropriate coefficient matrix. Considerably more effort is required to obtain the inverse in practical situations than can be justified for publication of a confidence figure. An approximate pre- diction error variance, k/(n + 20), is used currently in the Northeast Artificial Insemination Sire Comparison where n is the number of daughters and k is an appropriate breed constant corresponding to the residual variance. This procedure, however, does not account for distribu- tion of sires across herds nor several lactations per daughter. Thus, the diago- nal elements of the sire equations after absorption of cow, sire-by-herd, natural service sire, and herd-year-season equa- tions were chosen as likely indicators of the prediction error variance for this more complicated model. Simple regres- sion was used to relate prediction error variance obtained from the inverse to the diagonal after absorption. The coefficient of determination was .995 or greater in all cases. A single approximate prediction error variance of sire evaluation (group plus sire solution) could be used for Ayrshire, Guernsey, Jersey, and Brown Swiss bulls (and probably for Holsteins, which were not studied). The approxi- mate prediction error variance is (-.0014 + 1.08/diagonal) times the appropriate residual variance. An approximation comparable to repeatability for herdmate comparisons also was derived as (1.01 - 9/diagonal).

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