# How is q value defined?

1. From Applied Statistical Genetics with R by Foulkes (pFDR = positive false discovery rate):

The q-value is defined similarly as the greatest lower bound of the $pFDR$ that can result from rejecting a test statistic $T$ and a sample $x$ with $T(x)=t$ based on a rejection region $Γ$ in the set of nested rejection regions. Formally, the q-value is written $$q(t) =\inf_{Γ :t∈Γ } [pFDR(Γ )]$$

Since $pFDR$ is involved here, there must be several null hypotheses, and different null may have different testing rule. But here does it assume that there is one common testing rule $(T, Γ)$ for all the nulls?

2. From Wikipedia

The q-value is defined to be the FDR analogue of the p-value. The q-value of an individual hypothesis test is the maximum FDR at which the test may be called significant. One approach is to directly estimate q-values rather than fixing a level at which to control the FDR.

• Why are we talking about the q-value of an individual hypothesis test? Isn't the q-value for multiple nulls and their test rules?

• Should "maximum FDR" be "minimum FDR" instead?

• What does "One approach is to directly estimate q-values rather than fixing a level at which to control the FDR" try to mean? What is "one approach" trying to solve?

3. Am I right that pFDR depends on and changes with the sample's distribution, so does q-value?

Thanks and regards!

1. Indeed. It is assumed that all test statistics that fall in $\Gamma$ are rejected and others are not. Also note that in a pure (i.e., subjective) Bayesian framework, there need not be multiple hypotheses to define the pFDR. This is, however, now the view Storey has been promoting.
3. The estimated $pFDR(\Gamma)$ depends on the sample. It's value does not depend on the sampling distribution of each test statistic (it is simple arithmetics), but it's sampling properties will naturally depend on the sampling distribution. Particularly, the test statistics need to be independent for Storey's estimator to be unbiased to the true $pFDR(\Gamma)$.
• Also in the first definition, is $T$ the test statistic of just one of the multiple comparisons? If yes, does the definition of q-value not require other comparisons's test statistics to be rejected? – Tim Jul 9 '13 at 1:01