Thanks for the explanation, and the book reference.
I worked through the problem in MatLab and discovered what was confusing. The
procedure works if only the covariate associated with the second (activation)
scan is modified.
If this is correct then:
1) Would one use exactly the same procedure for a multi-subject design where
the condition and score covariates are correlated ?
and
2) How does your procedure relate to Andrew Holmes procedure deriving a
correlation between the difference in performance scores and the difference in
images across conditions. Specifically, does one have to orthogonalize the
performance score covariate before computing the change scores ?
What follows is a detailed description of how I came to this conclusion.
The initial description of the modification procedure given on pg 484 of the
Andrade et al. paper is:
C2* = C2 - correlation (C1,C2)*square_root(SSQ_C2/SSQ_C1)
since the subtracted amount is a scalar, the equation implies that this scalar
is subtracted from _each_ covariate value (both the 0 for resting scans and the
non-zero values for activation scans)
But in the appendix, the operation is given in matrix notation as:
C2* = C2 - C1*inv(C1'*C1)*(C1'C2)
Since C1 is the condition matrix that alternates between 0 and 1 and (C1'*C1)
and (C1'*C2) result in scalars, the amount that is subtracted is also a matrix
that alternates between a 0 and a non-zero amount. Therefore, the covariate
value associated with the rest scans (condition = 0) are unchanged since zero is
subtracted from them, whereas the covariates associated with the activation
scans have non-zero amounts subtracted from them.
When I only change the values of the covariate associated with the activation
condition, I get the results you describe in the paper. Specifically,
1) Although condition(C1) and covariate (C2)are highly correlated, there is no
correlation between modified covariate (C2*) and C1, and C1 and C2* are now, by
definition, orthogonalized.
2) The overall fit of a GLM is not altered by substituting C2* for C2, and the
fits for C2* and C2 are identical. However, the fit of C1 is radically altered
when C2* is used in the model instead of C2. With C2 in the model, C1 shows a
marginal fit with a negative beta. Using C2*, C1 shows a significant fit with a
positive beta.
So am I correct that the adjustment is only made to the covariates associated
with activation conditions, i.e. conditions with non-zero entries in the design
matrix ?
Here are the specific values I used in my test
C1 C2 Pixel Value
0 0 1
1 1 4
0 0 3
1 2 5
0 0 4
1 3 10
0 0 3
1 4 14
0 0 5
1 5 15
0 0 6
1 6 20
Note that the formula using correlations and sums of squares gives a slightly
different subtractitve factor compared to the matrix formula:
Correlation of C1 and C2 = 0.823
Scaling factor = square root(SSQ_C1/SSQ_C1) = sqrt(91/6) = 3.90
Proportional factor = 0.823 * 3.90 = 3.20
inv(C1'*C1)= 1/6
(C1'*C2) = 21
C1 * 21/6= C1 * 3.5 = [0,3.5,0,3,5...0,3.5]
Therefore Correlation Matrix
Proportional Factor 3.2 3.5
sg
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