Fisher's Method - Interpretation

Interpretation

Fisher's method is typically applied to a collection of independent test statistics, usually from separate studies having the same null hypothesis. The meta-analysis null hypothesis is that all of the separate null hypotheses are true. The meta-analysis alternative hypothesis is that at least one of the separate alternative hypotheses is true.

In some settings, it makes sense to consider the possibility of "heterogeneity," in which the null hypothesis holds in some studies but not in others, or where different alternative hypotheses may hold in different studies. A common reason for the latter form of heterogeneity is that effect sizes may differ among populations. For example, consider a collection of medical studies looking at the risk of a high glucose diet for developing type II diabetes. Due to genetic or environmental factors, the true risk associated with a given level of glucose consumption may be greater in some human populations than in others.

In other settings, the alternative hypothesis is either universally false, or universally true – there is no possibility of it holding in some settings but not in others. For example, consider several experiments designed to test a particular physical law. Any discrepancies among the results from separate studies or experiments must be due to chance, possibly driven by differences in power.

In the case of a meta-analysis using two-sided tests, it is possible to reject the meta-analysis null hypothesis even when the individual studies show strong effects in differing directions. In this case, we are rejecting the hypothesis that the null hypothesis is true in every study, but this does not imply that there is a uniform alternative hypothesis that holds across all studies. Thus, two-sided meta-analysis is particularly sensitive to heterogeneity in the alternative hypotheses. One sided meta-analysis can detect heterogeneity in the effect magnitudes, but focuses on a single, pre-specified effect direction.