Probabilities of Non-Positive Definite between-Group or Genetic Covariance Matrices

The probability (Q) that the estimated between-group covariance matrix is not positive deJinite is computed for the balanced single classlfication multivariate analysis of variance with random effects. It is shown that Q depends only on the roots of the matrix product of the inverse of the true within-group and the true between-group covariance matrices which, for independent variables, reduces to expressions in intra-class correlations. Values of Q are computedfor ranges of size of experiment, intra-class correlation and number of variables. Even for large experiments, Q can approach 100% if there are many variables, for example with 160 groups of size 10 and either 8 independent variables each with intra-class 0.025 or 14 variables each with intra-class correlation 0.0625. Some rationalization of the results is given in terms of the bias in the roots of the sample between-group covariance matrix. In genetic applications, the between-group covariance matrix is proportional to the genetic covariance matrix, if non-positive definite, heritabilities and ordinary or partial genetic correlations are outside their valid limits, and the effiect on selection index construction is discussed.