False discovery rate

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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.

1 Classification of m hypothesis tests
2 Controlling procedures
- 2.1 Independent tests
- 2.2 Dependent tests
3 References
4 External links

[edit] 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	$U$	$V$	$m 0$
# non-true null hypotheses	$T$	$S$	$m - m 0$
Total	$m - R$	$R$	$m$

$m 0$ 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 $α$ .

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

[edit] Controlling procedures

[edit] Independent tests

The Simes procedure ensures that its expected value $\mathrm{E}\!\left[ \frac{V}{V + S} \right]\,$ is less than a given $α$ (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 $α$ , 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 $α$ for these $m$ tests is $\frac{\alpha(m+1)}{2m}$ which could be used as a rough FDR (RFDR) or " $α$ adjusted for $m$ indep. tests."

[edit] 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:

Failed to parse (Cannot write to or create math output directory): 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 $α$ ) for $m$ dependent tests = RFDR / ( ln( $m$ )+ 0.57721...).

[edit] References

^ Benjamini, Y., and Hochberg Y. (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. School of Mathematical Sciences
^ Shaffer J.P. (1995) Multiple hypothesis testing, Annual Rview of Psychology 46:561-584, Annual Reviews