Dear Guillaume,
Many thanks for your e-mails, I think I now sort of understand where this came from.
If I am correct, the formula p <= i/V*q in Genovese or the original Benjamini & Hochberg (1995) becomes p*V/I to get the FDR adjusted q-value. In my case, with 25 clusters, I would have, for instance, for p-values 4-5-6 (.00007, .0991 and .1113) the FDR adjusted q-values .0004, .4955, and .4635 (rounded).
However, given equation 3 in Yekutieli & Benjamini (1999), if a higher uncorrected p-value gets a lower FDR adjusted q-value, all q-values between i and that low one (at least, in the paper, the ones below the threshold, which is extended to those above the threshold in SPM) get this q-value. Thus, as the adjusted q .4955 (for the 5th p-value) is higher than .4635 (for the 6th p-value), both adjusted q-values 5 and 6 are set to .4635.
I was able to plot all cluster-level uncorrected p-values (see attached, if images are being sent); I plotted them along with their 'true' FDR adjusted q-value to which I also applied the 'min' argument. It becomes obvious that in case of very close p-values, having the step (i) in the equation makes the FDR-adjusted lower and lower, so that many resulting FDR are set to the same adjusted p-values, and that this is done, according to T. Nichols, to enforce monotonicity, which may result in "many voxels with different statistic values that share the same p-value".
I do not know if I am all correct, but I was able to reproduce these results and understand what is happening know.
Again, thank you for your time, and happy wishes for the end-of-2016/start-of-2017 to you and all SPMers.
Kind Regards,
--
Maxime Résibois
PhD Student (KU Leuven)
Quantitative Psychology and Individual Differences
Tiensestraat 102 bus 3713
3000 LEUVEN
tel. +32 16 37 30 98
-----Original Message-----
From: SPM (Statistical Parametric Mapping) [mailto:[log in to unmask]] On Behalf Of Guillaume Flandin
Sent: Friday 16 December 2016 12:16
To: [log in to unmask]
Subject: Re: [SPM] Boundary of voxel-wise FDR correction
Dear Maxime,
As mentioned in the help text of spm_P_FDR.m, the FDR q-values are computed using the definition of equation 3 from:
Yekutieli & Benjamini (1999). "Resampling-based false discovery rate controlling multiple test procedures for correlated test statistics". J of Statistical Planning and Inference, 82:171-196.
http://dx.doi.org/10.1016/S0378-3758(99)00041-5
Otherwise, draw a line passing through the origin and the uncorrected p-values of the statistic you want to compute the q-value for, and read the ordinate of the intersection of this line with the right axis (at x=1). Depending of the profile of sorted uncorrected p-values, it might be clearer, at least intuitively, why the ordinate described above might not increase monotonically with increasing uncorrected p-values. This is the pendant of the use of find(Ps<=Fi, 1, 'last') in spm_uc_FDR.m to find the FDR critical threshold.
Maybe it will be simpler if you share the list of uncorrected p-values (for peaks or clusters so that you don't have too many) that give rise to the behaviour you describe.
Best regards,
Guillaume.
|