Running head : Approximate REML sampling variances Approximation of Sampling Variances and Confidence Intervals for Maximum Likelihood Estimates of Variance Components

After reviewing pertinent literature on the estimation of sampling variances and confidence intervals in the maximum likelihood framework, a method to approximate these for individual parameters in a multi-parameter analysis is described. It is based on the profile likelihood, defined as the likelihood for a subset of parameter(s) of interest with the remaining parameters equal to their maximum likelihood estimates given the former. The formation (= inverse of information) matrix for the parameters of the profile likelihood is equal to the corresponding submatrix from the full likelihood. The likelihood ratio test for composite hypotheses effectively compares points on the profile likelihood surface for the parameters tested. Hence sampling errors and confidence intervals for each parameter can be estimated considering its profile likelihood. This can be approximated fitting a one-dimensional quadratic or higher order polynomial function. Numerical examples for a balanced hierarchical full-sib design are given. Address for correspondence : Animal Genetics and Breeding Unit, University of New England, Armidale NSW 2351, Australia