Dear SPM'ers
I carefully read the numerous mails concerning the use of
conjunction. For me, it seems clear that when we use conjunction , we
have to know that a priori the contrasts share a common cognitive
component . If my interpretation of the different mails is correct,
Jesper, Joe , Mly and Richard have the same opinion (Am I right?)
and Mly expressed that in a very clear form:
>> My impression on the matter is that it is true
>> that we usually use 5 different treatments A-E,
>> but all of these have something in common.
>> Say A-E are different tasks, all of which share
>> a common cognitive component "X".
>>
>> For a brain area that respond to X,
>> we are effectively testing 5 time the same effect (X).
>>
>> In this case it seeems to me
>> that the min-t-stats would be approprate!
>>
However, reporting such result, I think we have always to specify (in a
non ambiguous manner) that the test validity requires that "we admit
(or know or suppose or are sure, ..) that the used contrasts share a
common cognitive component".
Now, say that we wish to search for voxels activated during two tasks (we
do not know if these two tasks share the same cognitive component and the
aim of the experiment is to prove this). I consider we can not use
conjunction because conjunction takes as null hypothesis that: no effect
of task 1 AND no effect of task 2 AND ....) . I agree with the mails
which propose to test the null hypothesis:" no effect of task 1 OR
no effect of task 2 OR ...") the negation of which is, according to
the deMorgan law " effect of task1 AND effect of task 2 AND
...". How can we calculate the risk when rejecting such hypothesis?
Richard propose a solution but I think that this is too conservative.
Take H0: "no effect of task 1 OR no effect of task 2"
If H0 is true, we calculate P{contrast1>t1 OR contrast2>t2} t1 and
t2 being the observed values of the contrasts. This probability is
(because P{A OR B}=P{A}+P{B}-P{A AND B}):
P{contrast1>t1}+P{contrast2>t2}-P{ contrast1>t1 AND
contrast2>t2};
If the events "contrast1>t1" and
"contrast2>t2" are independent then
P{ contrast1>t1 AND contrast2>t2}=
P{contrast1>t1}*P{contrast2>t2}
which can be neglected . If the events are not independent this value is
always between 0 and min(P{contrast1>t1},P{contrast2>t2}).
Thus, for me, the risk when rejecting H0 is between the
max(P{contrast1>t1},P{contrast2>t2}) and
P{contrast1>t1}+P{contrast2>t2}.
Some other comments/questions
1) A more naive approach of conjunction analysis is perhaps to consider
that we test for the first contrast, then for the second, .... We take
the risk 1, the risk 2, ... So the global risk is the sum of the risk.
What is wrong ?
2) Alternatively, some different statistical approaches can be used, at
least when we have factorial plans. Say 4 tasks: seeing words, seeing
pictures, hearing words, hearing sounds.
The plan is two factors, two levels each:
visual auditory
words
task1 task3
non words
task2 task4
This plan can be analysed as two main effects and one interaction
term. This could give a piece of information (which are not exactly the
same as conjunction) that has a meaning close to conjunction.
3) If two contrast are orthogonal, can we consider that events
"contrast1>t1" and "contrast2>t2" are
independent? I am not sure. Consider the following example:
say two tasks (A and B) and the control conditions (A0 B0) and the
contrasts:
A B A0 B0
1 -1 0 0
1 1 -1 -1
They are orthogonal but are the values of theses contrasts
independent?
Could somebody hint me towards any reference of a book or publication
demonstrating this?
Cordially
Pierre
Pierre Fonlupt
NSERM - Unité 280
Processus Mentaux et Activation Cérébrale
151 Cours Albert Thomas
69424 LYON CEDEX 03
Tél : 06 60 54 68 29