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

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 estimating 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.

	# declared non-significant	# declared significant	Total
# true null hypotheses	U	V	m0
# non-true null hypotheses	T	S	m - m0
Total	m - R	R	m

• m₀ is the number of true null hypotheses
• m - m₀ 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
• H₁...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 and one wants to keep this value below a threshold α.

Controlling procedures

Independent tests

The Simes procedure ensures that its expected value is less than a given α (Benjamini and Hochberg 1995). This procedure is valid when the m tests are independent. Let H₁...H_m be the null hypotheses and P₁...P_m their corresponding p-values. Order these values in increasing order and denote them by P₍₁₎...P_(m). For a given α, find the largest k such that

Then reject (i.e. declare positive) all H_(i) for i = 1,...,k. ...Note, the mean α for these m tests is which could be used as a rough FDR (RFDR) or "α adjusted for m indep. tests."

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:

• 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:

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

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

References

• Benjamini, Yoav; Hochberg, Yosef (1995). "Controlling the false discovery rate: a practical and powerful approach to multiple testing". Journal of the Royal Statistical Society, Series B (Methodological) 57 (1): 289–300. MR1325392. 

• Benjamini, Yoav; Yekutieli, Daniel (2001). "The control of the false discovery rate in multiple testing under dependency". Annals of Statistics 29 (4): 1165–1188. DOI:10.1214/aos/1013699998. MR1869245. 

• Storey, John D. (2002). "A direct approach to false discovery rates". Journal of the Royal Statistical Society, Series B (Methodological) 64 (3): 479–498. DOI:10.1111/1467-9868.00346. MR1924302. 

• Storey, John D. (2003). "The positive false discovery rate: A Bayesian interpretation and the q-value". Annals of Statistics 31 (6): 2013–2035. DOI:10.1214/aos/1074290335. MR2036398.