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Approximate Reliability of Best Linear Unbiased Prediction in Models with and Without Relationships

Research Institute for Animal Production "Schoonoord", Driebergseweg 10 D, P.O. Box 501, 3700 AM Zeist, The Netherlands
and Processing Center for Dairy Cattle, Eusebiusbuitensingel 38, P.O. Box 454, 6800 AL Arnhem, The Netherlands

ABSTRACT

In sire summaries, attention is paid to the reliability of estimates of breeding values, often expressed as the squared correlation between estimated and true breeding values. In best linear unbiased prediction procedures, which account for relationships between sires, reliability is approximated by (n_e + ka^–1 –k)/(n_e + ka^–1), where n_e is the effective number of daughters, i.e. the diagonal of the sire coefficient matrix after absorption of fixed effects; k is the ratio of residual and sire variance; and a^–1 is the diagonal element of the inverse of the numerator relationship matrix (A^–1). In simultated data this approximation compared with the true reliability, computed from direct inversion of the sire coefficient matrix including A^–1 after absorption of fixed effects, was biased upward for sire evaluations based on many effective daughters and few direct sire comparisons or few effective daughters and relationship to its sire. In field data, the effective number of daughters and the number of direct sire comparisons were correlated .97, and the formula gave good approximation (deviation <3%) when bulls were unrelated. However, when bulls were related, bias was considerable. Another approximation derived from selection index theory reduced the bias below 4%. This approximation takes the effective number of offspring of the bulls' sire into account and reduces to n_e/(n_e + k) if a bulls' sire has no daughter records. It can be applied easily and requires only small computational effort.