Clarification and Reevaluation of Population‐Based Diallel Analyses

Abstract

Plant breeders and geneticists often use diallel mating designs to obtain genetic information about a trait of interest from a fixed or randomly chosen set of parental lines. Diallel analyses of broad‐based populations have frequently been conducted by means of three analyses presented by Gardner and Eberhart in 1966. The original paper of Gardner and Eberhart used sequential model fitting to obtain estimates of effects and corresponding sums of squares. This approach, although having a long history, suffers from shortcomings which have led to confusion about what hypotheses the analyses actually test. The objectives of this paper were to delineate clearly all models implicitly required to perform Gardner and Eberhart Analyses II and III, and to present explicit formulas for effects in terms of the population means which are fundamental and unambiguous. While developing formulas of effects, we discovered a typographic error associated with variety effects in the original example of Analysis II. Our results also indicate that Analyses II and III effect formulas are rather nonintuitive both biologically and genetically, and incorporate multipliers that are functions of the number of parents. Another specific result shows that the varietal effects obtained in Gardner and Eberhart's Analysis III are “unconstrained” estimates, while those from the Analysis II are estimates constrained by the assumption of “no heterosis.” These results have implications for the use and interpretation of such effects. A SAS computer program for analyzing diallels among broad‐based populations according to Gardner and Eberhart's Analyses II and III is also reported.