Dear Pierre, Richard, Joe and everyone,
while certainly not being an expert, I thought I should add my five cents to
this very interesting discussion.
Joe gave a nice explanation of deMorgans inequality, which states that if you
have a set of statements then the "set" is true only if all the statements
are true. Or conversly the "set" is false if one or more of the statements
are false.
I think this is in fact precisely what Pierre and Richard (and I) have a
problem with. That means that we (might) reject the null hypothesis if a
single one of the individual hypotheses is wrong. This is certainly not in
accordance with my intuitive interpretation of "conjunctions" which I have
thought of as testing if they are all wrong (i.e. if we have "activations" in
all of the constituent contrasts).
When does one use a "minimum" statistic? Well, say that we have 5 independent
samples of THE SAME effect, for example we have tested treatment A in 5
groups of 10 subjects. The null hypothesis is that A doesn't work, and we
might proceed to test this with the minimum t-statistic of these 5 different
t values.
Now, say instead we have 5 independent samples testing DIFFERENT effects, for
example treatments with compounds A to E in groups of five subjects. The null
hypothesis is that none of the compounds work, and again we use the minimum
t-statistic to test it. I suggest that if we reject the null hypothesis we
conclude that at least one of the compunds work. But I suspect we cannot say
that all compounds work, which is what we would say in "neuroimaging
conjunctions".
So, isn't perhaps the problem that an assumption (one which we never test) in
the use of the minimum t-statistic is that we test THE SAME effect in all
tests? Whereas this in fact ought to be the hypothesis that we test.
I hope I haven't added to the confusion too much with this.
Jesper
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