Optimally weighted Z-test is a powerful method for combining probabilities in meta-analysis

The inverse normal and Fisher's methods are two common approaches for combining P-values. Whitlock demonstrated that a weighted version of the inverse normal method, or 'weighted Z-test', is superior to Fisher's method for combining P-values for one-sided T-tests. The problem with Fisher's method is that it does not take advantage of weighting and loses power to the weighted Z-test when studies are differently sized. This issue was recently revisited by Chen, who observed that Lancaster's variation of Fisher's method had higher power than the weighted Z-test. Nevertheless, the weighted Z-test has comparable power to Lancaster's method when its weights are set to square roots of sample sizes. Power can be further improved when additional information is available. Although there is no single approach that is the best in every situation, the weighted Z-test enjoys certain properties that make it an appealing choice as a combination method for meta-analysis.