# What is the distribution of q-values under the null

It's well known that under the null hypothesis, the p-value has a uniform distribution. However, how can we determine the distribution for q-values (false Discovery Rate adjusted p-values)? I guess it is no longer uniform since the lowest is always increased. Thanks!

• do you mean $(1-p)$ ? Feb 15 '13 at 3:45
• @Gong-YiLiao q-value has something to do with FDR, and nothing to do with (1-p) Feb 15 '13 at 5:16
• Not only will it not be uniform, but they will not be independent, even if the p-values were independent. Out of interest, why do you want to know - curiosity (which is fine), or is there a practical purpose to this? If you feel you need the distribution for practical purposes you may be doing something wrong with FDR. Feb 15 '13 at 6:54
• @Corone: thanks for the reply! Yes, I asked just out of curiosity. The uniform fact for p-values under H0 is very useful in statistical inference, so just curious if q-values have similar properties :) Feb 15 '13 at 16:27

Under the null distribution, your true false discovery rate will always be equal to 100% (no matter what threshold you set, every hypothesis you accept will be a false discovery). Therefore, the true q-values will be nothing but a vector of 1's.

Of course, when q-values are estimated from the p-value distribution, they will not always be estimated as being exactly 1 (though they will generally be very high, perhaps .9 or above). The exact distribution you end up with will depend on the method used to estimate q-values, and the number of hypotheses tested.

• Under the null, the FDR is not 100%. It is in fact equal to the FWER. This is because the FDP is taken to be 0 when there are no rejections. Nov 10 '16 at 18:27