Some statistical properties of an index of multiple traits

Hazel (1943) defined a selection index that maximizes the correlation, RIH, between the index, I=ΣbiXi and its aggregate genetic value, H=Σaigi, where the bi's are the derived coefficients of the observed traits, Xi, the ai's are their relative economic values and the gi's are their respective breeding values. This is called an optimum index. The expected value of the index is not H but rather K=Σbigi. The ratio of RHI/RKI is always less than or equal to 1.0. With selection on I, the ratio of the change in K to the change in H is the true heritability of the indez, that is, h I (2) =Δ(K)/ΔH. This is not the same as R IH (2) , which serves only as the predictor of the change in H with selection on I. If the index, I, itself is considered as a unit trait, studies can then be made of correlated response in I when selection is based only on a single trait in the index. I is then called a performance index. This approach provides for additional insight into the question of the failure of selection to make gains in total performance over many generations.