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Estimation of Variance of Prediction Error for Best Linear Unbiased Prediction Models with Relationships Included¹

Department of Animal Sciences, University of Maryland, College Park 20742

H. D. Norman and G. R. Wiggans

Animal Improvement Programs Laboratory, Agricultural Research Service, United States Department of Agriculture, Beltsville, MD 20705

ABSTRACT

For mixed models, variance of prediction error of sire evaluations is required for computing Repeatabilities or confidence intervals. Although variance of prediction error can be computed by inverting the coefficient matrix, this is not practical because of prohibitive computational cost. Therefore, variance of prediction error is estimated. Three sire evaluation models were analyzed: 1) a multiparity model for single traits (i.e., multiple lactations of a cow were repeat samples of a single trait) without relationships included; 2) a multiparity model for single traits with relationships included; and 3) a multitrait model for the first three parities as correlated traits with relationships included. Data sets were constructed small enough that variances of prediction error could be computed by inversion and compared with estimates from various functions. Reciprocal of the diagonal element was an accurate estimate of variance of prediction error only for the multiparity model for single traits without relationships. However, variance of prediction error was estimated best by a cubic function of the reciprocal. For the multiparity model for single traits with relationships included, variance of prediction error was estimated best by a function with four terms: the reciprocal of the diagonal element of the coefficient matrix minus contributions of relatives, the square and cube of this term, and the reciprocal of the diagonal element of the inverse of the matrix of sire variance. For the multitrait model, all functions investigated resulted in less accuracy than desired for estimating variance of prediction error.

¹ This research was sponsored in part by the US-Israel Binational Agricultural Research and Development Fund, Project No. US-334-81.

² Institute of Animal Sciences, Volcani Center, Bet Dagan, Israel.