False discovery rate (FDR) control is a statistical method used in multiple hypothesis testing to correct for multiple comparisons. In a list of rejected hypotheses, FDR controls the expected proportion of incorrectly rejected null hypotheses (type I errors).^[1] It is a less conservative procedure for comparison, with greater power than familywise error rate (FWER) control, at a cost of increasing the likelihood of obtaining type I errors.^[2]

The q value is defined to be the FDR analogue of the p-value. The q-value of an individual hypothesis test is the minimum 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.

Classification of m hypothesis tests

The following table defines some random variables related to the m hypothesis tests.

• m_0 is the number of true null hypotheses
• m - m_0 is the number of false null hypotheses
• U is the number of true negatives
• V is the number of false positives
• T is the number of false negatives
• S is the number of true positives
• H_1 ... H_m the null hypotheses being tested
• In m hypothesis tests of which m₀ are true null hypotheses, R is an observable random variable, and S, T, U, and V are unobservable random variables.

The false discovery rate is given by \mathrm{E}\!\left [\frac{V}{V+S}\right ] = \mathrm{E}\!\left [\frac{V}{R}\right ] and one wants to keep this value below a threshold \alpha.

( \frac{V}{R} is defined to be 0 when R = 0 )

Controlling procedures

Independent tests

The Simes procedure ensures that its expected value \mathrm{E}\!\left[ \frac{V}{V + S} \right]\, is less than a given \alpha (Benjamini and Hochberg 1995). This procedure is valid when the m tests are independent. Let H_1 \ldots H_m be the null hypotheses and P_1 \ldots P_m their corresponding p-values. Order these values in increasing order and denote them by P_{(1)} \ldots P_{(m)}. For a given \alpha, find the largest k such that P_{(k)} \leq \frac{k}{m} \alpha.

Then reject (i.e. declare positive) all H_{(i)} for i = 1, \ldots, k.

...Note, the mean \alpha for these m tests is \frac{\alpha(m+1)}{2m} which could be used as a rough FDR (RFDR) or "\alpha adjusted for m indep. tests." The RFDR calculation shown here provides a useful approximation and is not part of the Benjamini and Hochberg method. See AFDR below.

Dependent tests

The Benjamini and Yekutieli procedure controls the false discovery rate under dependence assumptions. This refinement modifies the threshold and finds the largest k such that:

P_{(k)} \leq \frac{k}{m \cdot c(m)} \alpha

• If the tests are independent: c(m) = 1 (same as above)
• If the tests are positively correlated: c(m) = 1
• If the tests are negatively correlated: c(m) = \sum _{i=1} ^m \frac{1}{i}

In the case of negative correlation, c(m) can be approximated by using the Euler-Mascheroni constant

\sum _{i=1} ^m \frac{1}{i} \approx \ln(m) + \gamma.

Using RFDR above, an approximate FDR (AFDR) is the min(mean \alpha) for m tests = RFDR / ( ln(m)+ 0.57721...).

References

External links

• False Discovery Rate Analysis in R - Lists links with popular R packages

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