Dear Jason:

Just a small additional comment on top of Richard's excellent answer. It 
sounds like what you are asking about the interaction contrast is whether 
the result you obtain is independent of the results of the main effects 
(i.e., not confounded by them). The answer is yes. As Richard pointed out 
the interaction effects contrast is orthogonal to the main effects 
contrast. In this design matrix specifying interactions (properly) would 
automatically treat the main effects as confounds. However, in other 
designs, for example if you were entering a covariate  in a simple 
regression, you would have to explicitly specify the interaction and the 
main effects vectors.

Hope this helps,
Darren


At 03:02 PM 6/28/2001 +0100, you wrote:
>Dear Jason,
>
>>      I have a question concerning a factorial design. I
>>have two factors each with two levels. So a two by two design. I have
>>specified the design with four conditions, one condition per cell of
>>design. What
>>is of interest is where the two factors interact and the two main effects.
>>I have done the following tests:
>>         main effects: (-1 -1 1 1) and (-1 1 -1 1)
>>         interaction (1 -1 -1 1) as I understand it this test looks for an
>>interaction between the two factors and does not allow for confounds of
>>the main effects. (Coull et al. 2000; Orienting attention in time:...)
>
>Sorry, I am not sure what you mean here.  Whether there is an
>interaction or not is a completely independent question to whether
>there are any main effects.  To put it another way, if there is a
>main effect in your data, there is no statistical reason why there
>should be an interaction.  Thus you could have the following results
>with an interaction:
>50 100 100 50,
>
>...or with the same amplitude interaction but with a main effect in
>one direction:
>50 100 200 150
>
>...or with the same amplitude interaction, but with the opposite main effect:
>50 100 0 -50
>
>...or perhaps a main effect for the other factor
>150 100 200 50
>
>and so on.  The interaction is the same in all of these cases, and
>hasn't been influenced at all by the presence or absence of the main
>effects.
>
>>My question is whether any one has any input on a cell effect test. That
>>is the test:
>>                 (-1 -1 -1 3)
>>As I understand this should test whether one cell shows an effect greater
>>than the average of the other three cells.
>
>Agreed, but a voxel may turn up in this contrast either by virtue of
>a main effect of one of the factors, or because it shows an
>interaction between the two factors, so the result is difficult to
>interpret.
>
>>  I have done this and there is a
>>large overlap with this result and the two main effects.
>
>That's not very surprising.  Obviously the contrast -1 -1 -1 3 is not
>orthogonal to the contrast -1 -1 1 1, and voxels that show up in one
>of these contrasts will also tend to show up in the other (even if
>the underlying data just contains noise).  The same goes for -1 -1 -1
>3 and -1 1 -1 1.
>
>>         What can this test show that the main effects do not? Does this
>>test show any modulatory effects between the two factors?
>
>Yes.  The contrast -1 -1 -1 3 is not orthogonal to the contrast 1 -1
>-1 1, and voxels that show up in one of these contrasts will also
>tend to show up in the other, as before.
>
>>  How can I test
>>for modulatory effects between two factors?
>
>The whole point of a factorial design is that you can look at this
>using the interaction, and this contrast is orthogonal to the main
>effects.
>
>By the way, why did you look at the contrast -1 -1 -1 3 if you didn't
>know what the interpretation would be?  This seems like a rather
>topsy-turvy approach to functional imaging.  Surely you should come
>to your data with a specific hypothesis and then work out which is
>the right contrast to test that hypothesis?  Coming up with a new
>contrast and then trying to work out what hypothesis it would be
>testing seems rather odd, but perhaps you have your reasons.
>
>>         Thank you for any input anyone may have,
>
>Hope it's of some help!
>
>Best of luck,
>
>Richard.

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Darren R. Gitelman, M.D.
Cognitive Neurology and Alzheimer¹s Disease Center
E-mail:  [log in to unmask]
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