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.