INFERENCES ON HOMOGENEITY OF BETWEEN-FAMILY COMPONENTS OF VARIANCE AND COVARIANCE AMONG ENVIRONMENTS IN BALANCED CROSS-CLASSIFIED DESIGNS

Estimation and testing of homogeneity of between-family components of variance and covariance among environments are investigated for balanc ed cross-classified designs. The variance-covariance structure of the residuals is assumed to be diagonal and heteroskedastic. The testing p rocedure for homogeneity of family components is based on the ratio of maximized log-restricted likelihoods for the reduced (hypothesis of h omogeneity) and saturated models. An expectation-maximization (EM) alg orithm is proposed for calculating restricted maximum likelihood (REML ) estimates of the residual and between-family components of variance and covariance. The EM formulae to implement this are iterative and us e the classical analysis of variance (ANOVA) statistics, ie the betwee n- and within-family sums of squares and cross-products. They can be a pplied both to the saturated and reduced models and guarantee the solu tions to be in the Parameter space. Procedures presented in this paper are illustrated with the analysis of 5 vegetative and reproductive tr aits recorded in an experiment on 20 full-sib families of black medic (Medicago lupulina L) tested in 3 environments. Application to pure ma ximum likelihood procedures, extension to unbalanced designs and compa rison with approaches relying on alternative models are also discussed .