Hi all,
Tom Nichols posted the following very helpful reply to the FSL list, which I
thought I'd forward here for the sake of completeness of the SPM archive.
Also, for anyone who's interested, I've put up a little m-file demonstration
of this here:
http://www.cs.ucl.ac.uk/staff/gridgway/stats/fdr_query.html
Best,
Ged.
[Forwarded text follows]
-------- Original Message --------
Subject: Re: [FSL] FDR adjustment (cf. correction)
Date: Fri, 20 Apr 2007 12:58:01 +0100
From: Thomas Nichols <[log in to unmask]>
Reply-To: FSL - FMRIB's Software Library <[log in to unmask]>
To: [log in to unmask]
References: <[log in to unmask]>
Ged,
I've noticed that in SPM's function for converting uncorrected
p-values to FDR p-values (spm_P_FDR), there is a distinction drawn
between "corrected" and "adjusted" p-values, with reference made to:
Yekutieli & Benjamini (1999) (eqn 3)
http://dx.doi.org/10.1016/S0378-3758(99)00041-5
<http://dx.doi.org/10.1016/S0378-3758(99)00041-5>
It is the *adjusted* p-values which spm_P_FDR actually returns, and so
I think these are the ones users report in their tables of voxel-wise
pFDR.
For the record, these terms were introduced in that reference and I've
haven't seen the distinction widely adopted; most authors just write
'corrected FDR P-values'. But, for the purposes of clarity, I'll use
the Yekutieli & Benjamini (YB) terms in quotes for this email.
Best way to distinguish 'corrected' from 'adjusted' FDR P-values is to
look at the FDR inequality from Benjamini-Hochberg original paper:
p(i) <= alpha i / v
where p(i) are the ordered P-values p(1)<=p(2)<=...<=p(v), alpha is the
desired rate at which to control FDR, and v is the number of voxels.
If you just change the inequality to an equality and solve for alpha,
you get the YB definition of 'corrected' P-values
Pc(i) = p(i) v / i
The problem is that there is no guarantee that these 'corrected'
P-values are monotonic; it may well be that
p(1) = 0.00001 Pc(1) = 0.06
but
p(10) = 0.00005 Pc(10) = 0.03
Hence, if you just used these 'corrected' P-value you'd reject the 10th
smallest P-value as FDR-0.05 significant, but not the very smallest P-value!
So 'adjusted' P-values are just 'corrected' P-values with montonicity
enforced, i.e.
Pa(i) = min(Pc(i),Pc(i+1),Pc(i+2),...,Pc(v))
So my question is, are the "adjusted p-values" returned by SPM
preferable to the corrected ones / FSL's q-rate? Or should they be
differently interpreted? Incidentally, SPM's "Qs -> P" step can be
very slow for large images...
The generic definition corrected P-value for multiple false positive
measure BLAH is: Smallest alpha BLAH false positive rate for which a
test is significant. This is what is meant by 'FDR corrected' and 'FWE
corrected' P-values in SPM.
It is YB's 'adjusted' P-values that satistify this definition of
corrected FDR P-values and hence the ones I'd recomend for use.
-Tom
____________________________________________
Thomas Nichols, PhD
Director, Modelling & Genetics
GlaxoSmithKline Clinical Imaging Centre
Senior Research Fellow
Oxford University FMRIB Centre
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