Dear All,
Following some recent posts on this list regarding second level
factorial designs, I had an interesting discussion with Karl, that I
thought some of you might be interested in.
> Guillaume and I were discussing the problems that a few people have
> had, when estimating non sphericity in factorial designs at the
> second (between-subject) level. Almost universally, these problems
> arise from the second-level modelling of different sorts of summary
> statistic contrasts. This can become particularly problematic when
> the contrasts reflect parametric effects at the first level. The
> reason for this is that the scaling of these contrasts is arbitrary
> and depends upon the scaling of the parameters used at the first
> level: For example, the scaling of a contrast for a regression of an
> event related response on reaction time will change by three orders
> of magnitude, depending upon whether the reaction time was measured
> in milliseconds or seconds. This means that one cannot meaningfully
> compare activations due to the presentation of, say, faces with the
> effects due to reaction time. This is likely comparing apples and
> oranges. For example, a T-contrast weight factor of +1 and -1 will
> be dominated by the contrast with the larger scaling and will not
> reflect the difference in activation due to seeing faces and those
> due to increases in reaction time. The only way that these
> pathological T-contrasts can arise is when the second level design is
> multifactorial and both sorts of contrasts have been modelled
> together. To prevent this happening good (i.e., conservative)
> practice would be to always perform a one sample T-test at the second
> level. In other words, if you want to test a hypothesis with a
> T-test, summarize the effect with a single contrast per subject and
> perform a single sample T-test at the second level. This will give
> (almost) identical results to conventional software packages and
> ensure the degrees of freedom are appropriate for the effect tested.
> So what is the motivation for a multifactorial model at the second
> level?
>
> The motivation is when one wants to test a multifactorial hypothesis
> with (and only with) an F-test. In other words one wants to search
> for regions in which the effect could have been in one contrast type
> or another or both. In this instance each row of the F-contrast
> matrix pertains a single sort of contrast and the difficulties of
> interpretation above disappear. However, when it comes to performing
> post hoc T-tests to test for specific main effects and interactions,
> these would generally use single-sample T-tests as above. This
> ensures that any differences in scaling that have been introduced
> inadvertently through the use of parametric regressors do not
> confound the interpretation of the second-level T-tests. The only
> situation in which second-level T-contrasts should be specified in
> the context of a multifactorial design is when one can be fairly
> confident that the first-level contrasts (i.e., summary statistics)
> are measuring the same sort of thing and have the same scaling. In
> this instance, SPM will use a pooled variance assumption and use the
> degrees of freedom from all summary statistic contrasts to estimate
> the standard error for any second level T-test; even if these test
> for effects in a small subset of the summary statistics. This can be
> used to increase the power of the inference but should be used
> carefully. Note that the pooled variance assumption over levels of
> second level factors is always a feature of post hoc T-tests within a
> multifactorial model; irrespective of whether one has made IID
> assumptions or has estimated the non-sphericity. The degrees of
> freedom will be the same because SPM uses maximum likelihood
> estimators after accounting for unequal variance or non-independence
> between levels of a factor. We hope that this helps.
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