How can I convert a q-value distribution to a p-value distribution?

Question

Let's say I have a vector of q-values, which allow for handling multiple hypothesis testing by controlling the false discovery rate. Usually, these q-values will be generated from a distribution of p-values.

However, what if I don't have the original p-value distribution, only the vector of q-values? Is there a way to convert from the q-values to the p-values?

Answer 1

You can convert from a q-value distribution to a p-value distribution rather simply (indeed, it's easier than the other way around!).

The way to do this in R is (explanation is in the comments):

convert.qval.pval = function(qvalues) {
    # you need to know the estimate of pi0 used to create the q-value
    # that's the maximum q-value (or very, very close to it)
    pi0 = max(qvalues)
    # compute m0, the estimated number of true nulls
    m0 = length(qvalues) * pi0
    # then you multiply each q-value by the proportion of true nulls
    # expected to be under it (the inverse of how you get there from
    # the p-value):
    return(qvalues * rank(qvalues) / m0)
}

It can be done in one line as

qvalues * rank(qvalues) / (max(qvalues) * length(qvalues))

As a demonstration, using the package qvalue:

library(qvalue)
pvals = replicate(1000, t.test(rnorm(100, .1))$p.value)
qvals = qvalue(pvals)$qvalue
plot(pvals, convert.qval.pval(qvals))