Dear Uta,
I believe Karl is referring to the fact that in a fixed-effects (1st level)
analysis, you cannot generalise your findings beyond your (subject) sample. A
fixed-effect, first-level SPM analysis treats 'subject' as a factor. However, it
does not assume that the levels of the factor (i.e. your group of subjects) are
selected at random from a distribution of such levels (the subject population).
The implicit assumption is that the subjects are being studied because you're
interested in *them*, and not them as a sample from a larger group.
So, if you have x subjects and perform a first-level analysis, you are limited
to discussing the results of your experiment as they occur in your subject group
- not the group/population that the subjects were sampled from. This is because
your first level-analysis models only scan/scan variance, not subject/subject
variance, which is necessary for the second level of inference.
The mean effect of an experimental manipulation in a fixed effects analysis is a
way of summarising the data across that group, and only that group. Assessing
the mean effect of an experimental manipulation in a random effects analysis
entails weighting the size of the average effect with your estimation of the
population error, which you get from your sample. In other words, here's how
variable the effect is in this group - what does this tell us about the
prevalance of the effect in the population. Thus, you gain inference in the
second case, but lose sensitivity (due to the df reduction).
However, I don't think that this means that fixed effects analyses are a poor
cousin of random effects. It just means, as Karl said, that you have to qualify
your 'sphere of influence' i.e. that the effects you see can only be said to
occur in your group. Most statistical tests will suffer from low dfs, and so
there is no point in obscuring potentially interesting results just because they
fail to make the 2nd-level grade. Your results can be qualified by doing
fixed-effects replication studies, or by performing a second experiment in which
you use the results of the first to constrain the search space (a la small
volume correction) in your random FX analysis.
A more thorough treatment of this, with added equations, can be found in the
paper by Friston, Holmes and Worsley,
http://a1700.g.akamaitech.net/v/1700/1861/5s/extra.idealibrary.com/production/nimg/1999/10/1/nimg.1999.0439/0439a.pdf?sessionID=MMRYQSAAAAETACQAABYAAAA
Best,
Dave.
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