The harmonic mean p-value (HMP) is a statistical technique for addressing the multiple comparisons problem that controls the strong-sense family-wise error rate (this claim has been disputed). It improves on the power of Bonferroni correction by performing combined tests, i.e. by testing whether groups of p-values are statistically significant, like Fisher's method. However, it avoids the restrictive assumption that the p-values are independent, unlike Fisher's method. Consequently, it controls the false positive rate when tests are dependent, at the expense of less power (i.e. a higher false negative rate) when tests are independent. Besides providing an alternative to approaches such as Bonferroni correction that controls the stringent family-wise error rate, it also provides an alternative to the widely-used Benjamini-Hochberg procedure (BH) for controlling the less-stringent false discovery rate. This is because the power of the HMP to detect significant groups of hypotheses is greater than the power of BH to detect significant individual hypotheses. There are two versions of the technique: (i) direct interpretation of the HMP as an approximate p-value and (ii) a procedure for transforming the HMP into an asymptotically exact p-value. The approach provides a multilevel test procedure in which the smallest groups of p-values that are statistically significant may be sought. Direct interpretation of the harmonic mean p-value The weighted harmonic mean of p-values is defined as where are weights that must sum to one, i.e. . Equal weights may be chosen, in which case . In general, interpreting the HMP directly as a p-value is anti-conservative, meaning that the false positive rate is higher than expected. However, as the HMP becomes smaller, under certain assumptions, the discrepancy decreases, so that direct interpretation of significance achieves a false positive rate close to that implied for sufficiently small values (e.g. ). The HMP is never anti-conservative by more than a factor of for small , or for large . However, these bounds represent worst case scenarios under arbitrary dependence that are likely to be conservative in practice. Rather than applying these bounds, asymptotically exact p-values can be produced by transforming the HMP. Asymptotically exact harmonic mean p-value procedure Generalized central limit theorem shows that an asymptotically exact p-value, , can be computed from the HMP, , using the formula Subject to the assumptions of generalized central limit theorem, this transformed p-value becomes exact as the number of tests, , becomes large. The computation uses the Landau distribution,