False Discovery Rate

False discovery rate (FDR) control is a Shaffer J.P. (1995) Multiple hypothesis testing, Annual Rview of Psychology 46:561-584 http://dx.doi.org/10.1146/annurev.ps.46.020195.003021
(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 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 Variable s.

The false discovery rate is given by

\mathrm{E}\!\left [rac{V}{V+S}
ight ] = \mathrm{E}\!\left [rac{V}{R}
ight ]

and one wants to keep this value below a threshold

\alpha

.

CONTROLLING PROCEDURES

Independent tests

The ''Simes'' procedure ensures that its Expected Value

\mathrm{E}\!\left[ rac{V}{V + S} 
ight]\,

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-value s. 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 rac{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

rac{\alpha(m+1)}{2m}

which could be used as a rough FDR (RFDR) or "

\alpha

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:

:

P_{(k)} \leq rac{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: $c(m) = \sum _{i=1} ^m rac{1}{i}$

In the case of negative correlation,

c(m)

can be approximated by using the Euler-Mascheroni Constant

:

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

Using RFDR above, an approximate FDR (AFDR) is the min(mean

\alpha

) for

m

dependent tests = RFDR / ( ln(

m

)+ 0.57721...).

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	CATEGORIES ABOUT FALSE DISCOVERY RATE

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Information About

False Discovery Rate