Restricted likelihood representation for multivariate heterogeneous random effects models

In the random effects model of meta-analysis for heterogeneous multidimensional data a canonical representation of the restricted likelihood function is obtained. This representation is related to a linear data transform which is based on the algebraic characteristics of error covariance matrices which are supposed to commute. The relationship between the heterogeneity covariance matrix estimators and the mean effect estimators is explored. It is noted that the sample mean exhibits the Stein-type phenomenon being an inadmissible estimator of the effect size under the quadratic loss when the number of studies exceeds three.