# Large p-values have lower FDRs than small p-values

This a conceptual question.

I ran 10 tests and sorted the p-values in an increasing manner.

I calculate the FDR as p-value * (10 / rank)

Then, in this case, the larger the p-value, the lower the FDR:

p-value rank    FDR
0.16    1   1.60
0.17    2   0.85
0.18    3   0.60
0.19    4   0.48
0.2 5   0.40
0.21    6   0.35
0.22    7   0.31
0.23    8   0.29
0.24    9   0.27
0.25    10  0.25


If I set a significance threshold of p-value=<0.25, and an FDR threshold of =< 0.25, should I consider that the test with a p-value = 0.25 has only a probability of 25% of being a false positive, while the other tests, although having lower p-values, have higher probabilities of being false positives, just because they are higher in the ranking?

How should this be interpreted?

Thanks!

It is common practice to keep original p-values and p-values resulting from FDR (or any other correction) in the same order (possibly with ties).

So, if for original p-values, you have $$p_1 < p_2$$, and after FDR correction, you get $$FDR(p_1) > FDR(p_2)$$, you can simply take $$FDR(p_1) = FDR(p_2) = min\{FDR(p_1) , FDR(p_2)\}$$.

This is what, for example, R function p.adjust does:

p.adjust((16:25)/100, method='fdr')
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25

• Thanks for your answer. Would you consider equally significant the p-value=0.16 and the p-value=0.25, given that both of them have the same FDR? May 14 '20 at 10:46
• Yes, that would be mine interpretation of this result. May 14 '20 at 10:49