Variance of prediction error with mixed model equations when relationships are ignored

Summary

Formulas are presented to illustrate the calculation of correct variances of prediction error (PEV) and the correlation between true and predicted values (r_TI) when the incorrect variance-covariance matrix for the random effects is used in mixed-model equations (MME). The example with progeny records of highly related and inbred sires showed that PEV were underestimated from the diagonals of the inverse of the coefficient matrix of the MME when sires were assumed unrelated and not inbred and were overestimated when relationships among sires were calculated with Henderson's simple rules for the inverse of the numerator relationship matrix, A^-1, which do not consider inbreeding. When Quaas' rules for A^-1, which do consider inbreeding, are used, the correct PEV are obtained. In the example, calculations of r_TI from the diagonals of the inverse of the coefficient matrix were too large when relationships and inbreeding were ignored and were obviously wrong when the approximation to the numerator relationship matrix, A, was based on the simple rules for calculating A^-1. If the correct A is used in the MME, the calculation of r_TI may be incorrect if inbreeding of the evaluated individual is not considered. If inbreeding is known, adjustment for inbreeding is easy for calculation of r_TI.