I am testing for differential expression of micro RNAs (miRNA), and looking for changes in the mean of a continuous variable under a categorical variable that takes values 0 and 1. There are many different miRNA features, and I test each one in turn for a change in its mean. I performed a false-discovery rate (FDR) adjustment to account for these multiple comparisons. My q-values (reported by the false discovery rate procedure) turn out to be all equal to 1.

Can someone explain why this happened, and what it means?

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#### Best Answer

**Two points**

**Point the First**

If you are using something like R's `p.adjust()`

to calculate $q$ values, then the 1 values simply indicate not rejected at any level of FDR. $q$-values are actually a little problematic to interpret directly, since they have a subtle mathematical artifice and because they do not communicate the step-wise nature of the FDR adjustment process (and *one cannot make FDR rejection decisions based on $q$-values alone*). Backing up to a single two-sided hypothesis can help illustrate why:

Reject $H_{0}$ if $p le alpha/2$. So for $alpha = 0.05$, we would reject $H_{0}$ if $p le 0.025$. **Alternately**, we could express this same rejection criterion as reject $H_{0}$ if $2p le alpha$. The first expression perhaps emphasizes the meaning of $p$, and the second emphasizes the meaning of $alpha$.

If we think about the Bonferroni method (FWER, not FDR), we can see that we have two way to express the rejection criterion given $m$ number of comparisons:

Reject $H_{0}$ if $p le frac{alpha/2}{m}$, or

Reject $H_{0}$ if $2mp le alpha$.

That $2mp$ is an 'adjusted $p$-value', sometimes called a '$q$-value'.

(I suppose there's also a third way: reject $H_{0}$ if $mp le alpha/2$.)

But look: $2mp$ is $>1$ when $p>.5/m$, which is quite possible. Unfortunately $p$ (or $q$) is supposed to be a *probability* which means that it's value is strictly bounded by zero and one inclusive. So many folks, and many statistical software authors will take an expression like $q = mp$, and replace it with $q=max(1,mp)$. The same applies to FDR (whether using the Benjamini-Hochberg or Benjamini-Yekutieli method)… the adjustments are more complicated than the Bonferroni, but they cap the $q$-value results at 1.

In a way, I suspect that this implies that expressing such adjustments as adjustments of rejection levels, rather than adjustments to $p$-values is a little more coherent, because the artifice of $max(1,f(p,i))$ does not apply.

**Point the Second**

We can't tell for sure because you have not provided, for example, your vector of $p$-values, but the likelihood is that your $p$-values are all too high, and that you are not achieving significance.

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