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I would like to issue the following correction to users of the harmonic mean *p*-value (HMP), with apologies: The paper (Wilson 2019

*PNAS*116: 1195-1200) erroneously states that the following asymptotically exact test controls the strong-sense family-wise error rate for any subset of

*p*-values \(\mathcal{R}\):

$$\overset{\circ}{p}_\mathcal{R} \leq \alpha_{|\mathcal{R}|}\,w_\mathcal{R}$$

when it should read

$$\overset{\circ}{p}_\mathcal{R} \leq \alpha_{L}\,w_\mathcal{R}$$

where:

*L*is the total number of individual*p*-values.- \(\mathcal{R}\) represents any subset of those
*p*-values. - \(\overset{\circ}{p}_\mathcal{R} = \left(\sum_{i\in\mathcal{R}} w_i\right)/\left(\sum_{i\in\mathcal{R}} w_i/p_i\right)\) is the HMP for subset \(\mathcal{R}\).
- \(w_i\) is the weight for the
*i*th*p*-value. The weights must sum to one: \(\sum_{i=1}^L w_i=1\). For equal weights, \(w_i=1/L\). - \(w_\mathcal{R}=\sum_{i\in\mathcal{R}}w_i\) is the sum of weights for subset \(\mathcal{R}\).
- \(|\mathcal{R}|\) gives the number of
*p*-values in subset \(\mathcal{R}\). - \(\alpha_{|\mathcal{R}|}\) and \(\alpha_{L}\) are significance thresholds provided by the Landau distribution (Table 1).

*p*-values. If argument L is omitted, a warning is issued and L is assumed to equal the length of the first argument, p, preserving previous behaviour. Please update the R package.

An updated tutorial is available as a vignette in the R package and online here: http://www.danielwilson.me.uk/harmonicmeanp/hmpTutorial.html

### Why does this matter?

The family-wise error rate (FWER) controls the probability of falsely rejecting any null hypotheses, or groups of null hypotheses, when they are true. The strong-sense FWER maintains control even when some null hypotheses are false, thereby offering control across much broader and more relevant scenarios.

Using the more lenient threshold \(\alpha_{|\mathcal{R}|}\) rather than the corrected threshold \(\alpha_L\), both derived via Table 1 of the paper from the desired ssFWER \(\alpha\), means the ssFWER is not controlled at the expected rate.

Tests with small numbers of

*p*-values are far more likely to be affected in practice. In particular, individual*p*-values should be assessed against the threshold \(\alpha_{L}/L\) when the HMP is used, not the more lenient \(\alpha_{1}/L\) nor the still more lenient \(\alpha/L\) (assuming equal weights). This shows that there is a cost to using the HMP compared to Bonferroni correction in the evaluation of individual*p*-values. For one billion tests \(\left(L=10^9\right)\) and a desired ssFWER of \(\alpha=0.01\), the fold difference in thresholds from Table 1 would be \(\alpha/\alpha_L=0.01/0.008=1.25\).
However, it remains the case that HMP is much more powerful than Bonferroni for assessing the significance of

*groups*of hypotheses. This is the motivation for using the HMP, and combined tests in general, because the power to find significant*groups*of hypotheses will be much higher than the power to detect significant*individual*hypotheses when the total number of tests (*L*) is large and the aim is to control the ssFWER.### How does it affect the paper?

I have submitted a request to correct the paper to

*PNAS*. It is up to the editors whether to agree to this request. A copy of the published paper, annotated with the requested corrections, is available here: http://www.danielwilson.me.uk/files/wilson_2019_annotated_corrections.pdf. Please use Adobe Reader to properly view the annotations and the embedded corrections to Figures 1 and 2.### Where did the error come from?

Page 11 of the supplementary information gave a correct version of the full closed testing procedure that controls the ssFWER (Equation 37). However, it went on to erroneously claim that "one can apply weighted Bonferroni correction to make a simple adjustment to Equation 6 by substituting \(\alpha_{|\mathcal{R}|}\) for \(\alpha\)." This reasoning would only be valid if the subsets of

*p*-values to be combined were pre-selected and did not overlap. However, this would no longer constitute a flexible multilevel test in which every combination of*p*-values can be tested while controlling the ssFWER. The examples in Figures 1 and 2 pursued multilevel testing, in which the same*p*-values were assessed multiple times in subsets of different sizes, and in partially overlapping subsets of equal sizes. For the multilevel test, a formal shortcut to Equation 37, which makes it computationally practicable to control the ssFWER, is required. The simplest such shortcut procedure is the corrected test
$$\overset{\circ}{p}_\mathcal{R} \leq \alpha_{L}\,w_\mathcal{R}$$

One can show this is a valid multilevel test because if $$\overset{\circ}{p}_\mathcal{R}\leq\alpha_L\,w_\mathcal{R}$$ then $$\overset{\circ}{p}=\left(w_\mathcal{R}\,\overset{\circ}{p}^{-1}_\mathcal{R}+w_{\mathcal{R}^\prime}\,\overset{\circ}{p}^{-1}_{\mathcal{R}^\prime}\right)^{-1} \leq w^{-1}_\mathcal{R}\,\overset{\circ}{p}_\mathcal{R}\leq\alpha_L$$an argument that mirrors the logic of Equation 7 for direct interpretation of the HMP, which is not affected by this correction.

### More information

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