Estimation of Singular Covariance Matrices of Random Effects

Abstract The variance-covariance matrix of random effects in a mixed linear model can be singular because identical twins are used or because a base population has been selected. As a consequence, the usual mixed model equations cannot be used for estimation and prediction. This paper presents a modification of these equations, the solution to which yields best linear unbiased estimators of fixed effects and best linear unbiased predictors of random effects. General formulae are presented for estimation of variances and covariances by minimum variance quadratic unbiased estimation and by restricted maximum likelihood estimation when the variance-covariance matrix of random effects is singular. These methods are illustrated by a numerical example in which identical twins have made one record each.