I think Alexa was correct: I don't think its matter of df's per se, but whether the underlying linear model is correct (and which terms in that model are used to construct the F-contrast). If in a multifactor design, each of the effect-by-subject interaction terms is drawn from the same, single IID error distribution (the "pooling" assumption), such that their SSs are proportional to their df's, then pooling provides a statistically more efficient estimate of that error distribution, so would be preferred. If the effect-by-subject terms are drawn from different distributions, however, only the partitioned error approach will be strictly valid (a better model). One might argue that the partitioning of the error terms is therefore the safer option, but that does not mean the pooling approach is invalid: if there are a priori reasons to expect a single error term, pooling could be valid and statistically more powerful. Fortunately, in my experience, the results of the two approaches are usually similar, but pooling also offers an additional (orthogonal) advantage of a greater number of df's with which to estimate the spatial smoothness of the residuals, which reduces the conservativeness of RFT (for low numbers of subjects). But I would be interested to hear the opinions of a proper statistician!

Best wishes
Rik
________________________________________
From: SPM (Statistical Parametric Mapping) [[log in to unmask]] on behalf of Stephen J. Fromm [[log in to unmask]]
Sent: 06 January 2011 15:25
To: [log in to unmask]
Subject: Re: [SPM] Design matrix for each or all subject(s)?

"In probably rather simple terms - in a pooled error test, if done correctly, the df are greater but are valid. They are not 'inflated'. The error is also greater - being pooled from what would be different partitioned error cells."

The df are not inflated, but only in the sense that they're appropriate for that F-test.  The crux here is that the F-test is the wrong one:  that is, the F-test is inappropriate for the inference being undertaken.  The correct F-ratio has both a smaller error term and fewer dof.  In that sense, the dof of the "pooled" test is inflated.

"If sphericity holds (or is corrected for), this should be just as valid as a partitioned error test. I understand it to involve a pooled *estimate* of an error variance that may (or may not) be the same for all cells - and this estimate is more powerful, with more df, if assumptions hold."

I disagree.  The classical texts state that the assumption that has to hold is that certain interaction terms vanish (in the sense of being statistically negligible), which is not true in general.  I'm not an expert and haven't had time to fully explore the issue, but I think sphericity is a different issue.

"If many contrasts per subject are taken to the 2nd level as in the 3 subjects/ 12 levels example, and subject effects are not modelled (discounted from the error term) as the Flexible Factorial allows..."

It's more complicated than modeling pure subject effects.  Rather, the variance components at issue are the _interaction_ with the effect of subject.  This cannot be done correctly using the "pooled" method.

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One citation:  David C. Howell, _Statistical Methods for Psychology_.  It's one of the references cited by Henson and Penny ("ANOVAs and SPM"), at
     http://www.fil.ion.ucl.ac.uk/~wpenny/publications/rik_anova.pdf
(Unfortunately, the bibliography/endnote list was cut out accidentally in the revised version.)  I have the 4th ed (1997).

One can try reading the Henson/Penny monograph itself, though it's relatively heavy going.