Some statistical properties of an index of multiple traits

Summary Hazel (1943) defined a selection index that maximizes the correlation, R_IH, between the index, I=b_iX_i and its aggregate genetic value, H=a_ig_i, where the b_i's are the derived coefficients of the observed traits, X_i, the a_i's are their relative economic values and the g_i's are their respective breeding values. This is called an optimum index. The expected value of the index is not H but rather K=b_ig_i. The ratio of R_HI/R_KI 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_I²=(K)/H. This is not the same as R_IH², 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.Journal Paper No. J-8176 of the Iowa Agriculture and Home Economics Experiment Station, Project 1711.Editor's Note: This paper and one closely similar (Lin and Allaire T.A.G. 51, 1–3 (1977) were independently derived and reviewed by different corresponding editors. The earlier paper, however, lacks numerical examples and a possible application as discussed in the present paper.