On the Analysis of Variance of a Half Diallel Table

Abstract

Hayman's (1954) extension of Yates' (1947) analysis of the diallel cross to yield estimates of the effects of various forms of dominance has been widely criticised (Gilbert 1958, Griffing 1956, Kempthorne 1956, and Topham 1966). In part this controversy arises from equating genetical additive and interaction effects with statistical main and interaction effects. Such equating is particularly problematic where the model used by Hayman is innately nonorthogonal. Hayman dealt with the non-orthogonality by redefining the mean of origin of the model, so that by progressive fitting of the parameters the sums of squares in the analysis were rendered additive. This device distorts the parameters from their normal genetic meaning, without satisfying statistical criticisms based on the non-orthogonality. For where gene interaction which cannot be removed by scalar transformation persists, estimates of even the distorted genetic main effects are of arguable statistical validity. The extension of this technique to the half diallel table by Morley-Jones (1965) is naturally open to the same objection. A method to remove the effects of the non-orthogonality from the full diallel table was recently proposed by Walters and Gale (1977). The present note extends this method to the half-diallel table and explains its advantages over the earlier formulations.