Variance ratio smoothing

As mentioned in the previous section, smoothing the observed variance over the whole brain would produce a better estimate of the random variance, but that assumes constant underlying variance over the brain (spatial stationarity), which does not seem to be the case (e.g. difference in white matter and grey matter[16]). But what Worsley et al.(2000)[18] supposes is that the ratio of random-effects and fixed-effects variance is locally constant, so that smoothing the ratio would produce a pooled estimate. The method consists of performing random and fixed analysis in the first place, then of spatially smoothing the ratio of variances obtained with the two methods (random/fixed), then returning to ``random-effects'' variance by multiplying the smoothed ratio with the fixed-effects variance before performing the group $t$ test. They estimate the degrees of freedom for the test as:

$$df(Wratio)=1/[1/df_{ratio}+1/df(fixed)]=df_{ratio}/(1+df_{ratio}/df(fixed))$$

where $df_{ratio}=df(random)[2(\frac{FWHM_s}{FWHM_{data}})^2+1]^{3/2}$ and $FWHM_s$ is the Gaussian smoothing parameter which enables one to move between a random analysis if set to 0 (no smoothing) and a fixed analysis if set to $\infty$ (smoothing the variance ratio to one everywhere). Sensible choices for $FWHM_s$ would be not to increase too much the degrees of freedom comparatively to its no smoothing situation ($df(random)$), an obvious limit being the degrees of freedom of a single subject experiment i.e. $df(fixed)/n$. The value recommended in [18] is 15mm for an original smoothness of 6mm ($FWHM_{data}$) which was actually reaching the upper bound for the experiment given in example. As the final degrees of freedom obtained are estimated Worsley et al. recommend to be high (they chose $100$) for the result not to be dependent on the estimation of it.