Reformulating the meta-analytical random effects model of the standardized mean difference as a mixture model.

The classical meta-analytical random effects model (REM) has some weaknesses when applied to the standardized mean difference, g. Essentially, the variance of the studies involved is taken as the conditional variance, given a δ value, instead of the unconditional variance. As a consequence, the estimators of the variances involve a dependency between the g values and their variances that distorts the estimates. The classical REM is expressed as a linear model and the variance of g is obtained through a framework of components of variance. Although the weaknesses of the REM are negligible in practical terms in a wide range of realistic scenarios, all together, they make up an approximate, simplified version of the meta-analytical random effects model. We present an alternative formulation, as a mixture model, and provide formulas for the expected value, variance and skewness of the marginal distribution of g. A Monte Carlo simulation supports the accuracy of the formulas. Then, unbiased estimators of both the mean and the variance of the true effects are proposed, and assessed through Monte Carlo simulations. The advantages of the mixture model formulation over the "classical" formulation are discussed.