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When using a Holm-Bonferroni adjustment method or the like, how to you treat multiple p-values of the same value when ranking?

For instance, I have the unadjusted p-values:

2.20E-16
2.20E-16
0.001387
0.03591
0.148
0.4517

Should both 2.20E-16 values be treated as rank (1) and if so, should the 0.001387 value be rank (2) or (3)? Or should one of the 2.20E-16 values be (1) and the other (2), which would make the 0.001387 value (3)?

Cheers

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  • $\begingroup$ Ignoring the impact of the higher values for the present, the third one should definitely be rank 3 and the first two shouldn't both be 1. Keeping in mind that 2.2E-16 is simply the smallest p-value R will print, they're almost certainly different and if you know the test statistics for the first two you can probably rank their p-values without knowing the exact p-values (but you probably don't need to know which is which, just that one is 1 and one is 2) $\endgroup$ – Glen_b -Reinstate Monica Oct 11 '19 at 6:22
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The only difference it makes is in terms of the adjusted p-values, not in terms of whether the tests are significant. I have never seen a proposal for exactly how to do this and it probably does not matter too much (never mind getting too obsessed about statistical significance - getting into comparing the size of "significant" p-values is certainly completely pointless).

If you are in a situation, where two p-values are exactly the same and the associated null hypotheses are getting rejected, then I would be tempted to go with the option of adjusting each one as if it had been rejected before the other. This is in a sense the "conservative" approach in that it results in higher p-values compared to, say, randomly picking one of them to have been rejected first. Doing so does not change anything else for any other tests.

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In the Holm approach, you immediately stop as soon a p value is above its treshold. So it does not matter.

Since Holm-correction is just for decision making and not for creating p values, in your situation nothing changes anyway if you flip the order of the first.

Here are two more interesting situations.

Situation 1

P values are 0.02, 0.02, 0.03

The smallest p value times 3 is above alpha 0.05. So the algo stops and all tests are non-significant. Same if you switch identical p values.

Situation 2

P values are 0.01, 0.01, 0.03

The smallest p value times 3 is below 0.05. So the algo continues with the next smallest p value times 2 (okay) etc. All tests are significant. Same if you switch identical p values.

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