Estimators of random effects variance components in meta-analysis

In meta-analyses, groups of study effect sizes often do not fit the model of a single population with only sampling, or estimation, variance differentiating the estimates. If the effect sizes in a group of studies are not homogeneous, a random effects model should be calculated, and a variance component for the random effect estimated. This estimate can be made in several ways, but two closed form estimators are in common use. The comparative efficiency of the two is the focus of this report. We show here that these estimators vary in relative efficiency with the actual size of the random effects model variance component. The latter depends on the study effect sizes. The closed form estimators are linear functions of quadratic forms whose moments can be calculated according to a well-known theorem in linear models. We use this theorem to derive the variances of the estimators, and show that one of them is smaller when the random effects model variance is near zero; however; the valiance of the other is smaller when the model variance is larger This leads to conclusions about their relative efficiency.