  The DerSimonian-Laird (DL) weighted average method has been widely used for
estimation of a pooled effect size from an aggregated data meta-analysis study.
It is mainly criticized for its underestimation of the standard error of the
pooled effect size in the presence of heterogeneous study effect sizes. The
uncertainty in the estimation of the between-study variance is not accounted
for in the calculation of this standard error. Due to this negative property,
many alternative estimation approaches have been proposed in literature. One
approach was developed by Hardy and Thompson (HT), who implemented a profile
likelihood approach instead of the moment-based approach of DL. Others have
further extended the likelihood approach and proposed higher-order likelihood
inferences (e.g., Bartlett-type corrections). Likelihood-based methods better
address the uncertainty in estimating the between-study variance than the DL
method, but all these methods assume that the within-study standard deviations
are known and equal to the observed standard error of the study effect sizes.
Here we treat the observed standard errors as estimators for the within-study
variability and we propose a bivariate likelihood approach that jointly
estimates the pooled effect size, the between-study variance, and the
potentially heteroscedastic within-study variances. We study the performance of
the proposed method by means of simulation, and compare it to DL, HT, and the
higher-order likelihood methods. Our proposed approach appears to be less
sensitive to the number of studies, and less biased in case of
heteroscedasticity.
