[I just saw Archana Singh's reply... hope this isn't redundant.]
Livio,
The FDR method implemented in SPM is that of Benjamini & Hochberg. It is
not a step-down method, but rather a step-up method. The definitions of
step-up and step-down make sense in terms of statistic values; a step-up
starts from small--least significant--statistic values, and successively
accepts nulls until the first significant test is found. A step-down
test starts from large statistic values and rejects successively until
the first non-signiificant test is found.
> having a look on your FDR.m function I see:
>
> pID = p(max(find(p<=I/V*q/cVID)));
>
> Does it mean, we are performing an FDR procedure in a step-down
> fashion? I mean, in this way we are starting from the less significant
> p-value, then going down to the most significant, then rejecting all
> the remaining hypotheses after the first hypothesis is rejected (with
> a fdr correction).
>
> indeed we do not impose monotonicity. is it right?
No, it's a step up test. And an artifact of this is that the inequality
may not be satisfied for all rejected tests; it doesn't need to be.
It simply must be the case that the most significant test rejected
satisfies the inequality for the desired q.
The "max" in the code snippet above grabs the best threshold,
and hence enforces monotonicity.
> As an example, consider the vector:
>
> p=[.01 .01 .03 .05 .3 .31 .32 .4 .5 .6];
> and q=.5;
>
> pID (p-value threshold based on independence or positive dependence) = .4
>
>
> the ans to p<=I/V*q/cVID, is
> 1 1 1 1 0 0 1 1 0 0.
>
> this mean that the values .3 and .31 do not respect the criteria,
> altought thei are rejected.
That's right... it makes sense, doesn't? Would you want
to reject a P-value of 0.4 but *not* reject a P-value of 0.31?
> there exist some work proving that this step-down procedure control
> the FDR?
Yes, the original paper proves that this step-down method controls
FDR
Benjamini & Hochberg (1995), "Controlling the False Discovery Rate: A
Practical and Powerful Approach to Multiple Testing". J Royal Stat Soc,
Ser B. 57:289-300.
and this paper shows that the original method is valid under positive
dependence
Benjamini & Yekutieli (2001), "The Control of the false discovery rate
in multiple testing under dependency". Annals of Statistics,
29(4):1165--1188.
Hope this helps.
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