A comparison via simulation of least squares Lehmann-Scheffé estimators of two variances and heritability with those of restricted maximum likelihood.

The objective was to compare the performance of a recently derived, new method of estimating variances and covariances with any mixed linear model and any pattern of missing data with that of restricted maximum likelihood. For each of 96 combinations of six three-herd x four-sire unbalanced designs of 39 offspring each, four heritability values, two ratios of sire variance to interaction variance, and two distributions (multivariate normal and multivariate chi2, 3 df), 15,000 vectors (n = 39) were generated. Least squares Lehmann-Scheffé (LSLS) estimators of sire variance, interaction variance, and heritability were compared to those of REML with the performance measures of percentage of estimates (of the 15,000) that were positive, mean square error, variance, percentage of estimates within +/- 50% of the parameter, bias, maximum value, skewness, and kurtosis. The LSLS method vastly outperformed REML in almost all 96 combinations. Averaged over the 48 combinations with multivariate normal data, the average percentage that REML estimators of heritability performed relative to those of LSLS for the first five of the above listed eight performance measures was -100%. The number of times LSLS was better than REML was 235 out of 240. The analogous values for the 48 combinations with multivariate chi2, 3-df data were -90% and 230 out of 240. The REML maximum values were always larger than the LSLS values. The LSLS skewness and kurtosis values were about the same as those for REML, with the exception of LSLS heritability kurtosis values, which were notably less than those for REML. The explicit expectations of the LSLS estimators showed that the LSLS estimators were surprisingly unbiased given the paucity of data. Explicit coefficients for calculating mean square errors, variances, and biases squared of the LSLS estimators of the three variances were obtained for each design. The LSLS advantage was not quite so large with the multivariate chi2, 3-df data as with the multivariate normal data. Results with a symmetric multinomial distribution were the same as with the multivariate normal. The overall result was that the LSLS estimators produced substantially more non-zero estimates than REML estimators and these more abundant positive estimates were substantially grouped closer to their respective parameters. Results justify efforts to make the LSLS procedure computationally available.