Variance and efficiency of the combined estimator in incomplete block designs of use in forest genetics: a numerical study

The efficiency of combined interblock-intrablock and intrablock analysis fo r the estimation of treatment contrasts in alpha designs is compared using Monte-Carlo simulation. The combined estimator considers treatments and rep lications as fixed effects and blocks as random effects, whereas the intrab lock estimator considers treatments, replications, and blocks as fixed effe cts. The variances of the estimators are used as the criterion for comparis on. The combined estimator yields more accurate estimates than the intrablo ck estimator when the ratio of the block to the error variance is small, es pecially for designs with the fewest degrees of freedom. The accuracy of bo th estimators is similar when the ratio of variances is large. The variance of the combined estimator is very close to that of the best linear unbiase d estimator except for designs with small number of replicates and families or provenances. Approximations commonly used for the variance of the combi ned estimator when variances of the random effects are unknown are studied. The downward or negative bias in the estimates of the variance given by th e standard approximation used in statistical packages is largest under the conditions in which the combined estimator is more efficient than the intra block estimator. Estimates of the relative efficiency of combined estimator s have an upward bias that can exceed 10% of the true value in small- and m iddle-sized designs with two or three replicates. In designs with four or m ore replicates, often used in forest genetics, the bias is negligible.