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Average Group Activation

In order to calculate the average group activation, we model the individual subject activation as being normally distributed according to

$\displaystyle \beta_k\sim{\cal N}(\beta_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}},\sigma_s^2)
$

where $ \beta_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}$ represents the average group activation and is usually estimated as

$\displaystyle \widehat{\beta_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}}=\frac{1}{N}\sum_k\widehat{\beta_k}
$

and where $ \sigma_s^2$ denotes the between-subject variance.

We will model the first-level within-subject covariances to be subject-specific and model the between-subject variances (from the group mean) as equal across the group. That is

\begin{displaymath}
V=\left[
\begin{array}{ccc}
V_1& & 0 \\
& \ddots & \\
0...
...}= (1\cdots 1)^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}.
\end{displaymath}

Then the adjusted second-level covariance matrix is

$\displaystyle V_{\mbox{\tiny\textit{\sffamily {$\!$G2}}}}^{\mbox{}}$ $\displaystyle =$ $\displaystyle V_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}+Q = V_{\mbox...
...ze\textit{\sffamily {-1}}}}X\right)^{\mbox{\scriptsize\textit{\sffamily {-1}}}}$  
  $\displaystyle =$ \begin{displaymath}\left[
\begin{array}{ccc}
\left(X_1^{\mbox{\scriptsize\textit...
...{\sffamily {-1}}}}
\end{array} \right]+\sigma_s^2{\mathbf {I}}.\end{displaymath}  

Define $ u_k=\left(X_k^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}V_k^{\mbox{\script...
...family {-1}}}}X_k\right)^{\mbox{\scriptsize\textit{\sffamily {-1}}}}+\sigma_s^2$, which is the sum of the within- and between-subject covariances. Then the estimate of the group parameter writes as

$\displaystyle \widehat{\beta_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}...
...}\cdots u_N^{\mbox{\scriptsize\textit{\sffamily {-1}}}}\right)\widehat{\beta},$ (16)

¥ where the inverse sum over $ u_k^{\mbox{\scriptsize\textit{\sffamily {-1}}}}$s is the associated variance.

Hence we see that in the general framework, the mean group activation parameter is a weighted average of the combined subject- specific activations, where the weights are inversely proportional to the subject-specific variances. This adjustment is advantageous in the case where the individual time-series model does not fit well for a particular subject, $ j$, generating an unusual value for $ \widehat{\beta_j}$ (an outlier) but also a large $ \textrm{Var}(\widehat{\beta_j})$. If no correction for the first-level variance is done, then this outlier can significantly affect the estimation of the group (between-subject) variance. If, however, this first-level correction is performed, the increased variance in this parameter will effectively de-weight the contribution of this outlier to the group variance estimate, since we use General Least Squares estimation.

In the much simpler case, where the within-subject covariances are $ V_k=\sigma_w^2{\mathbf {I}}$, and the $ X_k$ are normalised, such that $ X_k^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}X_k = 1$ for all $ k$, then

$\displaystyle u_k=\sigma^2_w+\sigma_s^2 \;$ and $\displaystyle \; \widehat{\beta_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox...
...{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}})=\frac{\sigma^2_w+\sigma_s^2}{N}.$ (17)

The test for significance is then carried out in a $ t$-test where

$\displaystyle T=\widehat{\beta_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{...
...xtrm{Var}(\widehat{\beta_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}})}
$

has an approximate $ t$-distribution.



Subsections
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Next: Numerical simulation Up: Examples Previous: Examples
Christian Beckmann 2003-07-16