Somas de quadrados e hipóteses associadas ao modelo dialélico de Gardner e Eberhart

Abstract

The diallel analysis model proposed by Gardner and Eberhart (1966) has been used to evaluate and select superior genotypes in plant breeding programs and to study several characters of genetic inheritance. However, the diallel model is non orthogonal as pointed out by the authors. Dependencies can also be generated by losses of genotypes' means, and that can lead to erroneous interpretation of the tested hypotheses because when the model is non orthogonal or unbalanced there are different approaches to the proposed hypotheses that generate different values for the sums of squares. This work intends to determine how the sums of squares of the diallel model effect and their associated hypotheses are affected, including the situations of losses of genotype means. Examples of complete and partial diallels were used and the type I sums of squares were determined by means of successive reductions of the unrestricted model according to the procedure used by the SAS and they were compared with the type III sums of squares, obtained through the inverse-of-part-of-inverse algorithm. The associated hypotheses were also compared. The type I was obtained through the estimable functions reported in the GLM of the SAS system and the type III was obtained by means of the procedure described by Searle (1987), that are correspondent to obtain the solution of Wrβ = Wβ. The analysis of the complete and partial diallel models, with or without losses of means, was affected in a way similar to the sums of squares and associated hypotheses. It was verified that the losses of crossing means affected in a different way the type I and III sums of squares and the associated hypotheses of most of the effects and that heterosis was not affected by losses of parents' means. The complete and partial diallel, without losses of means, only presented differences between the sums of squares and associated type I and III hypotheses for parents confirming the natural non-orthogonality of the adopted model.