Hi Eric,
You are right about Random Effects (which we call Mixed Effects in FSL as it is a mixture of the fixed effects variance and the random effects variance - as no one uses pure random effects, and in the paper they are effectively talking about Mixed Effects). That is, it cannot extrapolate to the entire population if your sample is of a certain subset. For example, if you only scan young healthy controls, then your Mixed Effects analysis will not extrapolate to elderly or pathological subjects. What it will do is to extrapolate to the entire population of young healthy controls (in this example). If you just did a Fixed Effects analysis then your inference would only be about that particular set of individuals that you scanned. It would not extrapolate beyond those specific individuals that were scanned. Thus it is rare that this is of interest for the between-subject case, as you always want to extrapolate to the wider group and not restrict your analysis to just the people you specifically recruited. If you were dealing with a very rare disease and had scanned every single subject with that disease then maybe Fixed Effects would be sensible, but in general you want Mixed Effects for groups of subjects. It just extrapolates to the population of "similar" individuals (e.g. young healthy controls in my example before) and therefore gives you results that would be consistent if you scanned any other group (of the same size) selected with the same characteristics (e.g. young healthy controls). I hope this clarifies things for you.
All the best,
Mark
On 21 Jun 2012, at 04:31, Eric Walden wrote:
> I am trying to understand the issue of “inference” vis-à-vis fixed and random effects. Penny and Holms (2003) say, “It is not possible to make formal inferences about population effects using FFX. Random-Effects (RFX) analysis, however, takes into account both sources of variation and makes it possible to make formal inferences about the population from which the subjects are drawn.”
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> I get how to calculate them and that random effects take into account a type of variance that probably exists. I get that fixed effects probably mis-estimates variance so that p-values are not reported correctly.
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> What I don’t get is what is meant by making inferences about a population.
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> Does this statement simply mean that p-values of fixed effects are incorrect, or does it mean something more?
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> The problem that I have is that if the statement “It is not possible to make formal inferences about population effects using FFX.” Simply means that if a model does not take into account some type of variance that exists, the p-values are wrong, then random effects analysis also probably fails to capture some variance that also exists. The simplest example is that of variance between subgroups, such as males vs. females, or young vs. old.
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> Does the statement “It is not possible to make formal inferences about population effects using FFX. Random-Effects (RFX) analysis, however, takes into account both sources of variation and makes it possible to make formal inferences about the population from which the subjects are drawn.” Simply mean that “if the assumptions of the random effects model are true, then the fixed effects model produces incorrect p-values?”
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> This is important because the statement ““It is not possible to make formal inferences about population effects using FFX. Random-Effects (RFX) analysis, however, takes into account both sources of variation and makes it possible to make formal inferences about the population from which the subjects are drawn.” Really seems to ascribe some almost mystical power to random effects analysis.
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> I agree that random effects analysis is probably closer to the truth (and hence better) than fixed effects analysis for fMRI data. I am just concerned that if all the assumptions of random effects analysis are not met, that it is also not accurate and the statement above, which seems to be repeated with no explanation or discussion, makes people think that random effects analysis is true.
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