# Solved – Why are the q-values equal to 1 after adjusting for multiple testing via the FDR

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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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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