The general form of the Sherman-Morrison-Woodbury formula [8] is
Also, the inverse for a matrix in block form is given by
Theorem A:
Any model of the form
That is, the signals of interest can be made orthogonal to the confounds without affecting the estimation of the parameters or the residuals.
Proof:
The proof is by construction, where we show that orthogonalising with respect to gives the desired results. Let
These equations give . Also, the combined span of and is clearly the same as that of and .
Now consider the covariances
For the first model, the parameter estimates, given by
equation 5, can be written using the matrix block inversion
formula, giving
Applying the Sherman-Morrison-Woodbury formula to the second term
in equation 12 gives
Substituting this into equation 12 gives
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Theorem B:
Given the standard GLM, , and a set of linearly independent contrasts specified by such that , then an equivalent model without contrasts, but with confounds, exists in the form
Proof:
The proof is, again, by construction. Firstly, let be a set of contrasts that when combined with form a complete linearly independent set of contrasts. That is, the matrix will be full rank (and hence invertible). Then let
From these definitions it is easy to see that , which represents an orthogonality condition. As before, it is straightforward to verify that the combined span of and is equal to the span of . Consequently,
The estimation equations for the model become
Thus
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