$F$-tests

Assume that for a set of $N$ subjects we model their hæmodynamic response functions using the same set of $M$ basis functions for each subject and consider the case where we wish to test for the population mean activation. Thus, we implicitly assume that there exists a single hæmodynamic response function that is representative for the group activation. We will not restrict the choice of basis functions (i.e. we do not require the basis functions to be orthogonal) and therefore allow for general correlations between the individual basis funtion fits, but assume that the covariance structure is the same for each individual. That is, we model the subject-specific vector of fits as distributed according to a multivariate normal distribution

$${\mbox{\protect\boldmath$\beta$}}_k\sim{\cal N}(\mbox{\protect\boldmath$\beta$}_{\mbox{\tiny\textit{\sffamily {$\!$G}}}},V_B),$$

and let

$$V_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}=\left[\begin{array}{ccc} V_B& & 0 \\ &\ddots & \\ 0 & & V_B \end{array} \right],$$

where $V_B$ is the covariance matrix of the $M$ basis function fits. Then the group-level design matrix

$$X_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}=\left[ {\mat... ...\cdots, {\mathbf {I}}_M \right]^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}$$

combines the $M$ individual basis functions across subjects such that

$$\widehat{\beta_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}... ... {$\!$G$\mbox{}_M$}}}}}\right)^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}},$$

where the individual values are the $M$ mean basis coefficients. In order to assess the average population activation, we need to test if any of the basis function coefficients are significantly non-zero. This can be achieved by calculating

$$F=\frac{1}{M}\beta_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbo... ...d\mbox{with} \quad C_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}={\mathbf {I}}_M,$$

which approximately follows an $F$-distribution. If, instead, we wish to assess if the final $p$ basis functions contribute significantly to the mean fit, we simply set

$$C_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}=\left[0\quad{\mathbf {I}}_p\right]$$

and change $M$ to $p$.