Random Subject Analysis

The inadequacy of the fixed-effects approach is seen in the first assumption:⁶

$$\hat{b_i}\sim N(m_b \, , \, \hat{\sigma}^2_{b_{\epsilon_i}}) \quad i=1\cdots n$$ (6)

One must consider the estimated activation as what it is, an estimation of the activation for the given subject:

$$\hat{b_i}\sim N(b_i \, , \, \hat{\sigma}^2_{b_{\epsilon_i}}) \quad i=1\cdots n$$ (7)

and now consider the activation for a given subject as a random observation of the activation for the population: $b_i=m_b + \, \eta_i$, $b_i$ is random $\sim N(m_b \, , \,\sigma^2_{b_{\eta}})$.
It is a two levels variation (within subject and between subjects): before, one had
$\hat{b}_i=b_i \, + \, \epsilon_i$ , now:
$\hat{b}_i=m_b \, + \, \eta_i \,+ \, \epsilon_i$, i.e. two error terms.
So, $\widehat{m}_b=\bar{\hat{b}}\sim N(m_b \, , \, 1/n \,\sigma^2_{(b_\eta + b_\eps... ... b_\epsilon)}=\hat{\sigma}^2_{b_{\eta}} \, +\, \hat{\sigma}^2_{b_{\epsilon}}$ and testing $m_b=0$ is a one sample $t$-test:

$$t_o(random)=\frac{\bar{\hat{b}}}{ 1/\sqrt{n}\hat{\sigma}_{(b_\eta + b_\epsilon)}}$$ (8)

Note that only the whole variation is needed, as one is not interested in the individual variance components. The whole variation is calculated directly using the set of estimated single-subject activation levels (the $\hat{b}_i$'s) as

$$\hat{\sigma}^2_{(b_\eta + b_\epsilon)}=\sum_i (\hat{b}_i -\bar{\hat{b}}_.)^2/(n-1)$$ (9)

which usually will reflect mainly the between-subject variation. Remarks: Notice here that introducing the random effects for the subject brings a constant autocorrelation between time measures (Compound Symmetry): let $i=1\cdots n$ and $j=1\cdots T$ identifying respectively the subjects and the time of measurements; the model can be written $y_{ij}= \beta_j+ \eta_i+\epsilon_{ij}$ with $E(\eta_i)=0$, $E(\epsilon_{ij})=0$, $var(\epsilon_{ij})=\sigma_w^2$, and $var(\eta_i)=\sigma_s^2$ then $corr(y_{ij},y_{i'j'})=\frac{\sigma_s^2}{(\sigma_w^2+\sigma_s^2)}$ for $i=i'$.