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Appendix

The general form of the Sherman-Morrison-Woodbury formula [8] is

$\displaystyle \left( A + B C D \right)^{\mbox{\scriptsize\textit{\sffamily {-1}...
...ptsize\textit{\sffamily {-1}}}}D A^{\mbox{\scriptsize\textit{\sffamily {-1}}}}.$ (18)

Also, the inverse for a matrix in block form is given by

$\displaystyle \left[ \begin{array}{cc} A & B \\ B^{\mbox{\scriptsize\textit{\sf...
...1}}}}B \right)^{\mbox{\scriptsize\textit{\sffamily {-1}}}}
\end{array} \right].$      


Theorem A:

Any model of the form

$\displaystyle Y = \left[ X_1 \; \; Z_1 \right] \left[ \begin{array}{c} \beta_1 \\ \alpha_1 \end{array} \right] + \epsilon
$

can be rewritten as

$\displaystyle Y = \left[ X_2 \; \; Z_2 \right] \left[ \begin{array}{c} \beta_2 \\ \alpha_2 \end{array} \right] + \epsilon,
$

where $ Z_2^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}V^{\mbox{\scriptsize\textit{\sffamily {-1}}}}X_2 = 0$ whilst being completely equivalent in terms of the estimated parameters of interest ( $ \widehat{\beta_2} =
\widehat{\beta_1} \; , \;
\textrm{Cov}(\widehat{\beta_2}) = \textrm{Cov}(\widehat{\beta_1})$) and the modelled signal space: $ \textrm{Span}(X_2) \, \cup \, \textrm{Span}(Z_2) = \textrm{Span}(X_1) \, \cup \, \textrm{Span}(Z_1)$ in the pre-whitened space. Note that $ \textrm{Cov}(\epsilon) = V$ for both models.

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 $ X_1$ with respect to $ Z_1$ gives the desired results. Let

$\displaystyle X_2 = X_1 - P_{Z_1} X_1 %% = X_1 - Z_1 ( Z_1\T V\inv Z_1)\inv Z_1\T V\inv X_1
\quad \textrm{and} \quad
Z_2 = Z_1
$

where $ P_{Z_1} = Z_1 ( Z_1^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}V^{\mbox{\sc...
...ptsize\textit{\sffamily {$\!$T}}}}V^{\mbox{\scriptsize\textit{\sffamily {-1}}}}$ is the projection matrix for $ Z_1$ in the pre-whitened space.

These equations give $ Z_2^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}V^{\mbox{\scriptsize\textit{...
...}}V^{\mbox{\scriptsize\textit{\sffamily {-1}}}}({\mathbf {I}}- P_{Z_1}) X_1 = 0$. Also, the combined span of $ X_2$ and $ Z_2$ is clearly the same as that of $ X_1$ and $ Z_1$.

Now consider the covariances

$\displaystyle \textrm{Cov}\left(\left[ \!\! \begin{array}{c} \widehat{\beta_1} ...
... D \end{array} \! \right]^{\mbox{\scriptsize\textit{\sffamily {-1}}}}\!\!\!\!.
$

Using the block matrix inverse, this gives
$\displaystyle \textrm{Cov}(\widehat{\beta_1})$ $\displaystyle =$ $\displaystyle \left(A - B D^{\mbox{\scriptsize\textit{\sffamily {-1}}}}B^{\mbox...
...textit{\sffamily {$\!$T}}}}\right)^{\mbox{\scriptsize\textit{\sffamily {-1}}}},$  

while, since the off-diagonal blocks are zero in the second case, the calculation simply gives
$\displaystyle \textrm{Cov}(\widehat{\beta_2})$ $\displaystyle =$ $\displaystyle (X_2^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}V^{\mbox{\scriptsize\textit{\sffamily {-1}}}}X_2)^{\mbox{\scriptsize\textit{\sffamily {-1}}}}$  
  $\displaystyle =$ $\displaystyle \left\{ (X_1^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}- B D^...
...xtit{\sffamily {$\!$T}}}}) \right\}^{\mbox{\scriptsize\textit{\sffamily {-1}}}}$  
  $\displaystyle =$ $\displaystyle \left( A - B D^{\mbox{\scriptsize\textit{\sffamily {-1}}}}B^{\mbo...
...\textit{\sffamily {$\!$T}}}}\right)^{\mbox{\scriptsize\textit{\sffamily {-1}}}}$  
  $\displaystyle =$ $\displaystyle \textrm{Cov}(\widehat{\beta_1}).$  

For the first model, the parameter estimates, given by equation 5, can be written using the matrix block inversion formula, giving

    $\displaystyle \widehat{\beta_1} =\left( A - B D^{\mbox{\scriptsize\textit{\sffa...
...tsize\textit{\sffamily {$\!$T}}}}V^{\mbox{\scriptsize\textit{\sffamily {-1}}}}Y$ (19)

while for the second model, the block diagonal form yields the familiar form
$\displaystyle \widehat{\beta_2}$ $\displaystyle =$ $\displaystyle (X_2^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}V^{\mbox{\scri...
...tsize\textit{\sffamily {$\!$T}}}}V^{\mbox{\scriptsize\textit{\sffamily {-1}}}}Y$  
  $\displaystyle =$ $\displaystyle \textrm{Cov}(\widehat{\beta_2}) \left( X_1^{\mbox{\scriptsize\tex...
...tit{\sffamily {$\!$T}}}}V^{\mbox{\scriptsize\textit{\sffamily {-1}}}}\right) Y.$  

Applying the Sherman-Morrison-Woodbury formula to the second term in equation 12 gives

$\displaystyle A^{\mbox{\scriptsize\textit{\sffamily {-1}}}}B \left( D - B^{\mbo...
...e\textit{\sffamily {-1}}}}B \right)^{\mbox{\scriptsize\textit{\sffamily {-1}}}}$ $\displaystyle =$ $\displaystyle A^{\mbox{\scriptsize\textit{\sffamily {-1}}}}\left( {\mathbf {I}}...
...textit{\sffamily {-1}}}}\right) B D^{\mbox{\scriptsize\textit{\sffamily {-1}}}}$  
  $\displaystyle =$ $\displaystyle (A - B D^{\mbox{\scriptsize\textit{\sffamily {-1}}}}B^{\mbox{\scr...
...iptsize\textit{\sffamily {-1}}}}B D^{\mbox{\scriptsize\textit{\sffamily {-1}}}}$  
  $\displaystyle =$ $\displaystyle \textrm{Cov}(\widehat{\beta_1}) B D^{\mbox{\scriptsize\textit{\sffamily {-1}}}}.$  

Substituting this into equation 12 gives


$\displaystyle \widehat{\beta_1}$ $\displaystyle =$ $\displaystyle \textrm{Cov}(\widehat{\beta_1}) \left( X_1^{\mbox{\scriptsize\tex...
...xtit{\sffamily {$\!$T}}}}V^{\mbox{\scriptsize\textit{\sffamily {-1}}}}Y \right)$  
  $\displaystyle =$ $\displaystyle \widehat{\beta_2}$  

$ \Box$


Theorem B:

Given the standard GLM, $ Y = X\beta + \epsilon$, and a set of linearly independent contrasts specified by $ C_1$ such that $ \widehat{b_1} =
C_1^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}\widehat{\beta}$, then an equivalent model without contrasts, but with confounds, exists in the form

$\displaystyle Y = \left[ X_2 \; \; Z_2 \right] \left[ \begin{array}{c} b \\ \alpha \end{array}\right] + \epsilon.
$

That is, $ \widehat{b} = C_1^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}\widehat{\beta}$, $ \textrm{Cov}(\widehat{b}) = C_1^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}\textrm{Cov}(\widehat{\beta}) C_1$ and the modelled signal space: $ \textrm{Span}(X_2) \, \cup \, \textrm{Span}(Z_2) = \textrm{Span}(X)$ in the pre-whitened space. Note that $ \textrm{Cov}(\epsilon) = V$ for both models.

Proof:

The proof is, again, by construction. Firstly, let $ C_2$ be a set of contrasts that when combined with $ C_1$ form a complete linearly independent set of contrasts. That is, the matrix $ C = [ C_1 \; \; C_2 ]$ will be full rank (and hence invertible). Then let

$\displaystyle X_2 = X Q C_1 F_1 \quad \textrm{and} \quad Z_2 = X Q C_3 F_3
$

where

$\displaystyle Q = (X^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}V^{\mbox{\sc...
...textit{\sffamily {$\!$T}}}}Q C_3)^{\mbox{\scriptsize\textit{\sffamily {-1}}}}.
$

From these definitions it is easy to see that $ C_1^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}Q C_3 = 0$, which represents an orthogonality condition. As before, it is straightforward to verify that the combined span of $ X_2$ and $ Z_2$ is equal to the span of $ X$. Consequently,

$\displaystyle Z_2^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}V^{\mbox{\scrip...
...C_1 F_1 = F_3 C_3^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}Q C_1 F_1 = 0.
$

Therefore, $ Z_2$ and $ X_2$ are orthogonal as well.

The estimation equations for the model become

$\displaystyle \textrm{Cov}\left(\left[ \begin{array}{c} \widehat{b} \\ \widehat{\alpha} \end{array} \right]\right)$ $\displaystyle =$ $\displaystyle \left[ \begin{array}{cc} (X_2^{\mbox{\scriptsize\textit{\sffamily...
...mily {-1}}}}Z_2)^{\mbox{\scriptsize\textit{\sffamily {-1}}}}\end{array}\right],$  
$\displaystyle \left[ \begin{array}{c} \widehat{b} \\ \widehat{\alpha} \end{array} \right]$ $\displaystyle =$ $\displaystyle \left[ \begin{array}{c} (X_2^{\mbox{\scriptsize\textit{\sffamily ...
...y {$\!$T}}}}V^{\mbox{\scriptsize\textit{\sffamily {-1}}}}\end{array} \right] Y.$  

Thus

$\displaystyle \textrm{Cov}(\widehat{b})$ $\displaystyle =$ $\displaystyle \left(F_1 C_1^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}Q X^{...
...\sffamily {-1}}}}X Q C_1 F_1\right)^{\mbox{\scriptsize\textit{\sffamily {-1}}}}$  
  $\displaystyle =$ $\displaystyle \left(F_1 C_1^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}Q C_1 F_1 \right)^{\mbox{\scriptsize\textit{\sffamily {-1}}}}$  
  $\displaystyle =$ $\displaystyle F_1^{\mbox{\scriptsize\textit{\sffamily {-1}}}}= C_1^{\mbox{\scri...
...tsize\textit{\sffamily {-1}}}}X)^{\mbox{\scriptsize\textit{\sffamily {-1}}}}C_1$  
  $\displaystyle =$ $\displaystyle C_1^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}\textrm{Cov}(\widehat{\beta}) C_1$  

and


$\displaystyle \widehat{b}$ $\displaystyle =$ $\displaystyle \textrm{Cov}(\widehat{b}) (F_1 C_1^{\mbox{\scriptsize\textit{\sff...
...ize\textit{\sffamily {$\!$T}}}}) V^{\mbox{\scriptsize\textit{\sffamily {-1}}}}Y$  
  $\displaystyle =$ $\displaystyle C_1^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}(Q X^{\mbox{\sc...
...ize\textit{\sffamily {$\!$T}}}}V^{\mbox{\scriptsize\textit{\sffamily {-1}}}}) Y$  
  $\displaystyle =$ $\displaystyle C_1^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}\widehat{\beta}$  

$ \Box$


next up previous
Next: Bibliography Up: tr01cb1 Previous: Acknowledgements
Christian Beckmann 2003-07-16