A parametric meta-analysis.

In a meta-analysis, we assemble a sample of independent, nonidentically distributed p-values. The Fisher's combination procedure provides a chi-squared test of whether the p-values were sampled from the null uniform distribution. After rejecting the null uniform hypothesis, we are faced with the problem of how to combine the assembled p-values. We first derive a distribution for the p-values. The distribution is parameterized by the standardized mean difference (SMD) and the sample size. It includes the uniform as a special case. The maximum likelihood estimate (MLE) of the SMD can then be obtained from the independent, nonidentically distributed p-values. The MLE can be interpreted as a weighted average of the study-specific estimate of the effect size with a shrinkage. The method is broadly applicable to p-values obtained in the maximum likelihood framework. Simulation studies show that our method can effectively estimate the effect size with as few as 6 p-values in the meta-analyses. We also present a Bayes estimator for SMD and a method to account for publication bias. We demonstrate our methods on several meta-analyses that assess the potential benefits of citicoline for patients with memory disorders or patients recovering from ischemic stroke.