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False discovery rate is a Statistical method used to correct for Multiple Comparisons . It is different in the sense that instead of using Significance Levels , it allows for a certain fraction (defined by a q-value) of the tests declared positive by the statistics to be truly negative.


USUAL DEFINITION


  • m_0 represents the number of time the Null Hypothesis is true

  • m - m_0 represents the number time the null hypothesis is false

  • V represents a False Positive

  • T represents a False Negative

  • H_i the null hypothesis being tested

  • Random Variable s are capitalized and numbers, observable or not, are in small caps


The false discovery rate is given by Q = rac{V}{V + S} and one wants to keep this value below a threshold q.


CONTROLLING PROCEDURES


Independent tests

Since Q is a random variable that cannot be computed, the ''Benjamini and Hochberg'' procedure ensures that its Expected Value E \left Q ight is less than a given q. This procedure is only 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 q-value, find the largest k such that

: orall i \leq k\ P_{(i)} \leq rac{i}{m} q.

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


Dependent tests

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

: orall i \leq k\ P_{(i)} \leq rac{i}{m . c(m)} q

  • 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 nagative correlation, c(m) can be by approximated using the Euler-Mascheroni Constant

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


EXTERNAL LINKS

  • 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'' 57 (1), 289–300. {Link without Title}


  • Benjamini, Y., and Yekutieli, D. (2001). "The Control of the False Discovery Rate in Multiple Testing under Dependency,". ''The Annals of Statistics'' 29 (4), 1165–1188. {Link without Title}


  • Genovese, C., Lazar, N., and Nichols, T. (2002). "Thresholding of Statistical Maps in Functional Neuroimaging Using the False Discovery Rate". ''NeuroImage'' 15, 870–878. {Link without Title}