# False Discovery Rate

False Discovery Rate (FDR) is a new approach to the multiple comparisons problem. Instead of controlling the chance of *any* false positives (as Bonferroni or random field methods do), FDR controls the expected *proportion* of false positives among suprathreshold voxels. A FDR threshold is determined from the observed p-value distribution, and hence is adaptive to the amount of signal in your data.

FDR is more sensitive than traditional methods simply because it is using a more lenient metric for false positives. However, if there is truly no signal anywhere in the brain, a FDR-controlling method has the same control as standard methods. That is, if the null hypothesis is true everywhere, a FDR procedure will control the chance of a false positive anywhere in the brain at the specified level. (In technical terms, FDR methods have weak control of Familywise Type I error; for more on the standard Familywise approach, see Nichols & Hayasaka, Controlling the Familywise Error Rate in Functional Neuroimaging: A Comparative Review, Stat. Meth. in Med. Research, 12(5): 419-446, 2003.)

Below is software, in the form of an extension to SPM, and an article introducing FDR to the neuroimaging community. Please feel free to contact me with any questions.

## Software: FDR inference in Matlab

The following function will supply p-value thresholds which control the expected FDR at a specified rate.

## Software: FDR for multiple images New!

Use this Matlab function, `multFDR.m`, to find a common FDR threshold for a set of images. It will return a single common t threshold as well as a set of individual t thresholds for comparison.

It requires SPM99 or SPM2, but can be used with any Analyze-format *T* image.

## Software: Illustrating the FDR threshold

Use this Matlab function, `FDRill.m`, to evaluate an image's t distribution and observe the PP-plot graph that determines the FDR threshold.

It requires SPM99 or SPM2, but can be used with any Analyze-format *T* image.

## Software: Extending SPM99 for FDR inference

*(Note: SPM2 already includes FDR, so no patches are needed.) *

SPM provides corrected p-values based on Familywise (Type I) Error control; a Familywise Error (FWE) is one or more false positive anywhere in the brain. The following new and modified SPM m-files will add a column of FDR-corrected p-values to SPM output and allow for the specification of FDR-corrected thresholds. Otherwise it does not alter SPM's behavior^{*}.

The compressed tar file below contains the 11 m-files needed to implement FDR within SPM. The files will expand into a separate `spm_fdr` directory. Include this directory before your spm99 directory, e.g.

path('/my/path/spm_fdr',path)

Please note changes to `spm_VOI.m` as of September 30, 2002. See this email. Thanks to Ahmed Toosy for reporting this problem.

Files:

See this note on a bug fix to Matlab core function`betainc`/

`betacore.`The symptom is that SPM hangs when it's determing FDR p-values.

^{*}This tar file includes new modifications to SPM that were released June 28, 2001, in the form of the last six functions listed above. These modifications modify the (FWE) corrected p-value procedures to always take the minimum of random field and Bonferroni results. As released, SPM99 only gives random field p-values and thresholds even if Bonferroni results are more powerful. This is a minor change will, at most, reduce corrected p-values and thresholds in certain situations (typically, low degrees of freedom analyses).

Changes to `spm_P.m` as of October 14, 2001. See this email.

Changes as of August 7, 2001. See this email.

## Article

Thresholding of Statistical Maps in Functional Neuroimaging Using the False Discovery Rate.

Christopher R. Genovese, Nicole A. Lazar, Thomas E. Nichols (2002).

*NeuroImage* **15**:870-878.

Download: Published Version | Preprint

Note on the preprint: The preprint reflected the current results at the time of submission. Since then, the independence FDR result of 1995 (BH 1995; in the above paper, "c(V)=1" on pg 9) has been shown to be more general (BY 2001). In particular, a technical condition known as *positive regression dependency* is sufficient to use the 1995 result. Examples of this condition include multivariate normal data where the covariance matrix has all positive elements; this seems a reasonable assumption for imaging, and hence the code above uses the c(V)=1 result.

BH 1995: Y. Benjamini and Y Hochberg. Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing. J. Roy Stat Soc, Ser B. 57:289-300.

BY 2001: Y. Benjamini and D. Yekutieli. The Control of the false discovery rate in multiple testing under dependency. The Annals of Statistics, 29(4):1165-1188, 2001. Preprint verison available at http://www.math.tau.ac.il/~ybenja/ under "Papers"

## Presentation Slides

These PowerPoint slides are from a talk presented Feburary 6, 2004, at the Department of Statistical Science at Southern Methodist University, Dallas, TX. They illustrate how the FDR threshold will be lower with greater extent of activation.

## FDR Citations in Neuroimaging

If you know of other papers which have used FDR thresholds, please send me the reference!

** Neuroimaging
Statistics**

**Contact Info**

Room D0.03

Deptment of Statistics

University of Warwick

Coventry

CV4 7AL

United Kingdom

Tel: +44(0)24 761 51086

Email: t.e.nichols 'at' warwick.ac.uk

Web: http://nisox.org

Blog: NISOx blog

*Handbook of fMRI Data Analysis* by Russ Poldrack, Thomas Nichols and Jeanette Mumford