Fixed Subject Analysis

Following the assumption that each subject's estimated activation parameters are taken from the distributions:

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

the population mean activation by $m_b$ is estimated $\widehat{m}_b=\bar{\hat{b}}$ with the following distribution $\bar{\hat{b}}\sim N(m_b \, , \, 1/n \,\hat{\sigma}^2_{b_\epsilon}) \quad , \quad \hat{\sigma}^2_{b_\epsilon}=1/n \sum_i \hat{\sigma}^2_{b_{\epsilon_i}}$.
Now, testing $m_b=0$ (null hypothesis) is a one sample $t$-test:

$$t_o(fixed)=\frac{\bar{\hat{b}}}{ 1/\sqrt{n}\hat{\sigma}_{b_\epsilon}}$$ (4)

Note that deciding that the pooled variance is a ``better'' estimate of single-subject variance, that is, for every $i$, $\hat{\sigma}=\hat{\sigma}_{b_\epsilon}$ then:

$$t_o(fixed)=\frac{1/n \sum_i \hat{b_i}} {1/ \sqrt{n} \hat{\sigma}}= 1/\sqrt{n} \sum_i t_o(i)$$ (5)

This simplified fixed-effects analysis is therefore virtually identical to the ``Group Z'' method: $Z_{group}=1/\sqrt{n} \sum_i Z(i)$. The main shortcoming of the fixed-effects approach is that it considers the errors of measurement estimated for the subjects as the only source of variation when estimating the population mean. That is to say, only within-subject variation is accounted for. No consideration of between-subject variation is considered; therefore it is valid only for the subjects chosen in this experiment (no sampling variation). Put another way, when looking at variation of an activation, it has to account for variation during the experiment on a subject (between measures or within-subject) and the choice of subjects (between subject).