Summary: Diallel crosses as mating designs are used to study the genetic properties of inbred lines in plant breeding experiments. Most of the theory of optimal diallel cross designs is based on standard linear model assumptions where the general combining ability effects are taken as fixed. In many practical situations, this assumption may not be tenable since we are studying only a sample of inbred lines from a possibly large hypothetical population. A random effects model is proposed that allows us to first estimate the variance components and then obtain the variances of the estimates.
We address the issue of optimal designs in this context by considering \(A\)-optimality criteria. We obtain designs that are \(A\)-optimal for the estimation of heritability in the sense that the designs minimize the sum of the variances of the estimates of the variance components. The approach leads to certain connections with the optimization problem under the fixed effects model. Some numerical illustrations are given.