Dear Joe,

Thank you for your extremely clear e mail.  However, I am still not
happy with the interpretation of the conjoint p value (0.001 in my
example), so I will try to explain why.

>I think the issue of interpreting conjunction analyses may be easier than the
>previous emails indicate.

I hope so.

>Essentially, because conjunctions require orthogonal contrasts, the
>probabilities
>associated with each contrast are independent.  Thus their joint
>probabilities are
>simply the product of the individual probabilities.  In simple terms, if
>you have
>three contrasts each present at p<0.1 uncorrected (within a voxel) then you
>would be likely to reject each of these individually because of the low
>significance
>level (as Richard stated).  However, since they are independent measures,
>the likelihood
>of all three activations being false positives is:
>
>p(A-B) * p(C-D) * p(E-F)
>
>where p(X-Y) is the probability that condition X is significantly more
>active than Y.  If each
>of these contrasts has a p<0.1, then the conjoint probability is:
>
>p(A-B, C-D, E-F) = 0.1 * 0.1 * 0.1 = 0.001.
>
>Whether you accept this as a *significant* activation is still a matter of
>interpretation, of course.

So far, so good.  But to rephrase the meaning of this p value ('the
likelihood of all three activations being false positives'), it is
the probability that the voxel exceeds the threshold p<0.1 in all
three contrasts under the null hypothesis that there is no 'real'
activation in any of the three contrasts.

This just doesn't appear to be the right null hypothesis to be using,
as outlined in the previous e mails (unless one is happy to assume a
priori that the result, one way or the other, will be the same in all
three contrasts).

>Leaving that aside, by computing the min{t} map, SPM looks for the least
>active condition and uses
>that t-value to compute the probability.  Because higher t-values will have
>lower associated p-values,
>the limit is imposed by the minimum t.
>
>For example, let's say that the three conditions above have the following
>t-values:
>       Contrast        t-statistic at a particular voxel
>       p-value associated (uncorrected)
>       A-B             4.2                             0.0005
>       C-D             2.9                             0.009
>       E-F             3.9                             0.0009
>(PS. I made these values up for illustration only)
>
>In a typical PET study none of these individual contrasts reach a corrected
>significance level.  Taken
>together, however, they might.  Their uncorrected joint p-value would be
>0.0005 * 0.009 * 0.0009 = 4.05 * 10^-10
>(i.e something really small which is likely to reach significance after
>Gaussian random field correction).
>However, the limit is clearly the second contrast (C-D) as the other two
>are less likely to be false
>positives.  Thus it is sufficient to find the minimum t-value and compute
>the probability of that contrast
>given the number of independent (orthogonal, actually) contrasts.

So in this example, we calculated the probability, under the null
hypothesis, that in all three contrasts the t statistic would exceed
that level which is in fact observed (in this data set) for the
second contrast, i.e. t = 2.9.  All well and good.

But the null hypothesis in this case is still that there is no
activation in any of these contrasts.  This still leaves the
possibility of activation in only one contrast within the realm of
the alternative hypothesis, which isn't where we want it to be.

>If you choose an uncorrected p-value as your cutoff (say 0.001) then the
>minimum contrast needs to
>have an associated p-value of 0.001 ^ -n (that is, 0.001 to the negative
>nth power where n is the number of orthogonal
>contrasts).

I think that you must mean 0.001^(1/n) here.  Otherwise, agreed!
>
>The interpretation is fairly straightforward.  If the results of your
>conjunction analysis reach significance,
>however you define that, then you can confidently state that the effect was
>present in each and every
>one of the contrasts which contributed to the conjunction analysis.

Yes, but only at the level of the individual lowest p value (0.009 in
your example).

We can't calculate the p value which expresses the probability of the
voxel exceeding this p value in all three contrasts given a null
hypothesis of real activation in fewer than all three contrasts (or
at least, I can't) for the reason outlined in my previous e mail.

>Note
>that this is a very conservative
>statement.  If it is done over subjects in a study then each and every
>subject activated a given voxel for
>a particular task.  If it is not present, then at least one subject
>(contrast) did NOT activate that voxel.

As Pierre originally pointed out, this might not be so much of a
problem when using conjunctions of subject-specific contrasts.  Here
you might be OK in adopting the simplifying assumption that whatever
the 'real' activity pattern is found one brain is the same in all of
the others.  In this case, with a null hypothesis of no activation in
any of the subjects, the alternative hypothesis would be activation
in all of the brains (any other possibility being given a probability
of zero).  The conjoint probability would therefore be appropriate in
testing for a conjunction.

>A random effects analysis, in contrast, would allow some variability in the
>sample when making
>inferences.  Consider the case where each subject's contrast was present at
>p<0.001 uncorrected except
>one whose contrast was only p<0.9.  If you had sufficient numbers of
>subjects (say n>12) then the
>RFX would be likely to identify this region as active in the population
>based on your sample.  In a
>conjunction analysis, however, you would not see this region because it was
>not present in ALL subjects.

Agreed.

>I've attached some code which I find useful for looking at
>conjunction analyses.

Thanks.  I have never used a conjunction analysis, which is why I
find myself tackling this problem for the first time now, in response
to Pierre's original message.  So I probably won't make use of this
code at the moment, although I am sure that others will.

I have to admit, although you have run through the thinking behind
conjunctions very clearly, I remain unhappy about the idea that, in
the case of several contrasts in one subject or group of subjects,
the 'conjoint' probability of activation in all three contrasts
(under a null hypothesis of no activation in any of the contrasts) is
appropriate when testing for a conjunction.

  Best wishes,

Richard.
--
from: Dr Richard Perry,
Clinical Lecturer, Wellcome Department of Cognitive Neurology,
Institute of Neurology, Darwin Building, University College London,
Gower Street, London WC1E 6BT.
Tel: 0207 679 2187;  e mail: [log in to unmask]