Michael Wrote:
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> I'm thinking we want the 1st case, where we want the average effect of
> both stimuli, not just either one.

You may think again about your choice here. "Either one" is not really an accurate description about the second test. More specifically the second one compares the following two models:

Full model: y = Effect_Of_Stimulus1 + Effect_Of_Stimulus2 + other_effects + residuals

Reduced model: y = other_effects + residuals (or beta1 = beta2 = 0)

So if the F-statistic for the second null hypothesis is significant, it indicates that either Effect_Of_Stimulus1 or Effect_Of_Stimulus2, or both are significant. As you can see, this test does tell you about the "composite" effects of the two stimuli.

My reservation about the first test in my previous response is that it weighs the effects of the two stimuli equally and one effect has the opposite sign to the other, which is rarely the case in the reality of FMRI, except for some special scenarios. More specifically the first test compares the following two models:

Full model: y = Effect_Of_Stimulus1 + Effect_Of_Stimulus2 + other_effects + residuals

Reduced model: y = Effect_Of_Stimulus1 + Effect_Of_Stimulus2 + other_effects + residuals, with the constraint of beta1 + beta2 = 0

> Could there be a case where the 1st null hypothesis fails and the
> 2nd null hypothesis above does not?

Certainly. These are two different tests, and they have both overlapping and exclusive scenarios. One obvious fact is that the first test puts a constraint on the two effects (they are presumed to have equal weight with opposite sign) but the second one doesn't.

Gang Chen
[afni.nimh.nih.gov]



Edited 9 time(s). Last edit at 07/21/2012 08:31AM by Gang.