A method for combined analysis of the diallel crosses repeated in several environments

Abstract

This paper presents a metodology derived from the model proposed by Gardner & Eberhart For computing combined analysis of variance of diallel crosses tested in several environments, aiming to obtain estimators for the parameters and sums of squares formulae. The following mathematical model was adapted for a combined analysis of the complete diallel crosses:
y_ijj’ = m + ½ (v_j + v_j’) + e_i + ½ (ev_ij + ev_ij’) + θ(h + eh_i + h_j + eh_ij + h_j’ + eh_ij’ + s_jj’ + es_ijj’) + ε_ijj’,
where: Y_ijj' is the variety mean if j = j' or the cross mean if j≠j' in the i^th environment; e_i is the environment effects; ev_jj and ev_ij, are the effects of the interaction environment x variety, eh_i is the effect of the interaction environment x average heterosis; eh_ij and eh_ij' are the effects of the interaction environment x variety heterosis and es_ijj' is the interaction environment x specific heterosis. The other parameters of this model are similarly defined in Gardner & Eberhart work. When j = j' then θ=0 and for j≠j' then θ=1. The estimators for the parameters and the sums of squares formulas were determined through the least squared method. Variances of the parameters estimators and the analysis of variance were also determined. An example is presented with its correspondent analysis.