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This email is probably aimed at Jeanette Mumford and Tom Nichols, but I'd be very happy to hear input from the wider FSL community.

I really appreciate the excellent examples at http://fsl.fmrib.ox.ac.uk/fsl/fslwiki/GLM and I find them really useful. I do however have a question about comparing a t-test with covariate, to an anova. I apologise in advance for the rather long email but I hope it makes the question clearly!

The question is basically, how are t-tests with a dichotomous covariate, or t-tests with dichotomous covariate interaction different to a 2x2 anova?

To create a toy example, lets say I have a group of people who have either brown or red hair, and either blue or green eyes. There are therefore four groups of people:

If I have a hypothesis that there is a main effect of hair color on FA in a TBSS analysis (or whatever the dependent variable may be) "correcting for eye color" I could set up a t-test with the following design matrix (assuming there are 2 people in each group):

1   0   0.5
1   0   0.5
0   1   0.5
0   1   0.5
1   0  -0.5
1   0  -0.5
0   1  -0.5
0   1  -0.5

Contrasts:
1  -1   0
-1  1   0

I would interpret significant regions from this test as those in which there was a "main effect of hair color correcting for eye color". The test would be directional (ie: the two contrasts would not show the same regions).

If I wanted to test the hypothesis that there is a main effect of eye color on FA "correcting for hair color" then I could set up the following design:

1   0   0.5
1   0   0.5
1   0  -0.5
1   0  -0.5
0   1   0.5
0   1   0.5
0   1  -0.5
0   1  -0.5

Contrasts:
1  -1   0
-1  1   0

If I wanted to test an interaction between hair color and eye color I could setup the following design file:

 1   0   0.5   0
 1   0   0.5   0
 0   1   0.5   0
 0   1   0.5   0
 1   0   0     0.5
 1   0   0     0.5
 0   1   0     0.5
 0   1   0     0.5

Contrasts:
0    0    1    -1
0    0   -1     1


These all look very different to the 2 way between subjects anova example at http://fsl.fmrib.ox.ac.uk/fsl/fslwiki/GLM#ANOVA:_2-factors_2-levels_.282-way_between-subjects_ANOVA.29

An example for that design file for my test would be:

 1  -1   1   1
 1  -1   1   1
-1   1   1   1
-1   1   1   1
 1  -1  -1   1
 1  -1  -1   1
-1   1  -1   1
-1   1  -1   1

Contrasts:
 1   0   0   0
 0   1   0   0
 0   0   1   0

Ftests:
1  0  0
0  1  0
0  0  1

I would interpret the Ftests as "a main effect of hair color", "a main effect of eye color" and an "interaction between hair color and eye color" respectively.

So, here's my question again: how are t-tests with a dichotomous covariate, or t-tests with dichotomous covariate interaction different to a 2x2 anova? Are the t-tests that I outlined valid? Do they converge with the anova? I do appreciate that the anova is conducting an F test while the others are pulling from a t-distribution. But is that the only difference?

Any thoughts would be greatly appreciated.

Thank you very much, and apologies again for the long email.

Kx

--
Kirstie Whitaker, PhD
Research Associate

Department of Psychiatry
University of Cambridge

Mailing Address
Douglas House
18b Trumpington Road
Cambridge, CB2 8AH

Phone: +44 7583 535 307
Website: www.kirstiewhitaker.com