Abstract
In an extension of Robertson''s (1957) work (Biometrics 13, 442-450) the optimum test design and maximum improvement were studied when selection comprises (i) non-tested members of selected families, (ii) all members of selected families, and (iii) only tested animals, and considering the breeding requirement as a total of individuals rather than of families. Of the three cases the maximum gains are likely to be obtained with all members, tested and untested, of the selected families are used. The total improvement is largely determined by the ratio to the total number tested to the number required for breeding and by the heritability. The efficiency of the test is further affected by the size and nature of the family and by the test group size. These details can be varied to some extent to fit the requirements of efficiency. Consideration was also given of how best to use the testing facilities and selected individuals, so as to produce the maximum improvement in the breed or population as a whole. Maximum improvement will be obtained if testing facilities are restricted to a nucleus group of breeders, who in turn supply the rest of the breeders with breeding stock, so that full opportunity for testing and selection is confined to the nucleus group.

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