Guillaume -
I would agree with the general sentiment, though would reiterate your
final point that there would not seem anything wrong with multifactorial
designs when the contrast images do represent commensurate quantities
(ie all apples), and I would have thought this would be the case in the
majority of studies (i.e, I have not come across many studies in print,
or from colleagues, that have mixed parameter estimates from
incommmensurate regressors). One obvious exception to this is multiple
temporal basis functions, which I agree should only be compared with
F-tests in such designs (ie, F-contrasts in which the weight for each
basis function occurs in a different row).
Furthermore, I agree that one should be wary of contrasts that do not
include all conditions in a multifactorial design: ie conventional main
effect and interaction contrasts that span all conditions (cells) in the
design are fine, but if one is only ever testing contrasts across a
subset of conditions (eg simple effects), then one should be wary of
including other conditions in the design matrix.
Finally, and perhaps most importantly, one advantage with assuming a
pooled error (with appropriate corrections for nonsphericity) in
multifactorial designs, relative to partitioning the error in a series
of one-sample T-tests for example, is that there will be more df's with
which to estimate the spatial smoothness, which can become important (as
I understand it) to avoid RFT becoming too conservative (unless one
smooths more to compensate), as often happens when there are relatively
few (eg <~18) participants (as Tom Nichols has shown). But please let me
know if this is not the case!
Rik
PS Could SPM help the user avoid comparing apples and oranges by
associating each contrast image with the type of regressor from which it
was derived, at least those "types" distinguished by SPM during the
creation of design matrices, eg epochs vs events, temporal basis
functions, parametric modulations, user-specified regressors...?
Guillaume Flandin wrote:
> 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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Dr Richard Henson
MRC Cognition & Brain Sciences Unit
15 Chaucer Road
Cambridge
CB2 7EF, UK
Office: +44 (0)1223 355 294 x522
Mob: +44 (0)794 1377 345
Fax: +44 (0)1223 359 062
http://www.mrc-cbu.cam.ac.uk/people/rik.henson/personal
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