Average Group Activation

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

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

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

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

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

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

Then the adjusted second-level covariance matrix is

$$V_{\mbox{\tiny\textit{\sffamily {$\!$G2}}}}^{\mbox{}} = V_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}+Q = V_{\mbox... ...ze\textit{\sffamily {-1}}}}X\right)^{\mbox{\scriptsize\textit{\sffamily {-1}}}}$$

$$= \left[ \begin{array}{ccc} \left(X_1^{\mbox{\scriptsize\textit... ...{\sffamily {-1}}}} \end{array} \right]+\sigma_s^2{\mathbf {I}}.$$

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

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

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

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

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

has an approximate $t$-distribution.