Dear James,
> can you confirm that for random effects analysis...
>
> 1) temporal smoothing is not necessary at the first level, unless you also
> wish to infer the significance at the first level ?
This is correct. The motivation for temproral smoothing is to ensure a
r0bust estimate for the standard error of the parameter estimates. As
this error estimate does not enter into the second level there is no
need to smooth. Generally smoothing decreases the efficiency of
parameter estimation. High-pass filtering however should be used at
the first level because this generally renders the parameter estimation
more efficient.
> 2) spatial smooting is also unnecessary at the first level, or at least
> minimal smoothing. for similar reasons (GRF theory rather than
> autocorrelations etc). All spatial smoothing can be done on the contrast
> images before entering the second level. One may wish to use minimal
> smoothing for subject by subject analysis at the first level, say 6 mm
> kernel, for optimal spatial resolution and localisation, but larger
> smoothing at the second level to accomodate inter subject variability, say
> 10 mm kernel.
Absolutely. Spatial convolution is a linear operation and so is
parameter estimation. They right and left-hand operators respectively
therefore the order of smoothing and estimation does not matter:
Beta_hat = [pinv(Kt*X)*Kt*Y] * Ks = pinv(Kt*X) * [Kt*Y*Ks]
estimate then smooth = smooth then estimate
where Kt and Ks are temporal and spatial filter matrices respectively
(as you note you can do some smoothing before and the rest after). Note
this is not the case for temporal filtering:
pinv(Kt*X)*Kt*Y*Ks ~= Kt*pinv(X)*Y*Ks
With very best wishes - Karl
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