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Methods to account for selection in estimation of variance components are computationally difficult and require inclusion of records on which selection was based. The last criterion often cannot be met. Within a time records of daughters in the small sample set for a bull should be relatively free of effects of selection. If only such records are used, many herd-year- season subclasses, however, contain only one record, so that those records are eliminated when herd-year-season effects are absorbed. Including records of daughters of few but heavily used and selected bulls would provide more comparisons within herd-year-seasons, but treating effects of such sires as random and unselected would bias estimates of variance components. Effects for proved sires can be treated as fixed and for sampling sires as random for estimation of sire and residual variances. For example, Method 3 estimates for an operational model including fixed herd-year-season effects (h), fixed effects of proved sires (s1), and random effects of sampling sires (s2) are:
ôe2 = [ýy - R(h,s1,s2)]/(N - c)
ôs2 = [R(s2|h,s1) - (r - 1) ôe2]/tr(Z’WZ)
where ýy is total sum of squares, R( ) indicates a least squares reduction in sum of squares, N is number of records, c is rank of full coefficient matrix, r is rank, and tr(Z'WZ) is trace of coefficient matrix after absorption of effects of herd-year- seasons and proved sires.