Estimating heterogeneity variances to select a random effects model

There are many collaborative studies where the reported within-study uncertainty estimates are unreliable but can be considered as lower bounds to the true uncertainties. This work is motivated by such examples; it provides a method to determine the common mean of heterogeneous observations with unknown variances which however allow for the given lower bounds. In this situation, the classical maximum likelihood estimator and the restricted maximum likelihood estimator are derived. These procedures lead to the choice of the random effects model where the unknown heterogeneity variance can depend on the individual study. The Bayes procedures against the noninformative prior restricted on the appropriate parametric subset are recommended for practical use. Published by Elsevier B.V.