Degrees of freedom and comments

Notice that $t_o(fixed) >t_o(random)$; the difference may be quite large, as the between-subject variation is normally much larger than the within-subject variation (scan-to-scan). Sample size must be considered here, as to be able to give an accurate decision about the population (as with the random-effects analysis) one needs a good estimate of the between-subject variation. There is also a difference in the degrees of freedom: $df(fixed)= n(T_A+T_B -2)$, and $df(random)=n-1$. As a rule, with more general models, the degrees of freedom for a fixed-effect approach will be the sum of the degrees of freedom from every single-subject analysis and the degrees of freedom for a random-effect approach will be the degrees of freedom of the second stage model (explained further in the GLM section). Notice the very low $df(random)$ compared with the fixed-effects approach (a typical $n$ is 10, and $T > 100$). The random-effects approach is the valid one but is difficult to be confident in (power analysis) as the sample size used in general is very small compared to what is usually needed in estimation (say at least 30). For fMRI studies it has been suggested that 12 to 15 subjects would suffice. From a large sample ($n=40$), Darrell et. al (1998)[4] presented a power analysis resulting in sufficient power observed for about 20 subjects (depending on the brain area under consideration). A possible alternative way of obtaining more degrees of freedom (and so a better estimation) would be to estimate the random variance needed using more data that is on the whole brain, for example by pooling or smoothing the variances. This is similar to the idea used by Worsley in the ``variance ratio method'' described in the next section.