Hi Karl,

Indeed - an even simpler way to view this is that if you view this as
a 2-group t-test the question comes up of whether to estimate
different variances for the two "groups" or not, which is clearly a
problem if you do try to. By running the analysis through GLM instead
you are implicitly making the decision to pool variance across all
subjects, hence the "problem" does not arise.

Cheers, Steve.



On 19 Apr 2007, at 13:27, Karl Friston wrote:

> Dear Christian,
>
>> I'm sorry to bother you directly with what might seem a trivial
>> question, but we are stuck with a paper using SPM, with one of the
>> reviewer claiming that we cannot do something that I'm quite
>> convinced we can, and I would be very glad about your input:
>> we have conducted a study comparing 2 rare patients against 16
>> controls. We have the same contrast (lets call it C) in all 18
>> subjects at the first level. We want to examine if the average
>> activation in the patients differs from the average in the
>> normals. At the second level I therefore performed a 2-sample t-
>> test (voxelwise) with 2 subjects in group 1 and 16 subjects in
>> group 2. One of the reviewers states:
>>
>> "Regarding the 2 sample t-test. I wonder if SPM2 is able to
>> compare 2 subjects versus 16. I believe this is not the case and
>> that one must use a different methodology to make this
>> comparison. I would like to see a citation to a study that uses a
>> similar approach that might alleviate this concern.".
>>
>> Do you have any comments we could quote?
>
> Yes; " it is perfectly valid to compare groups of unequal size,
> using the
> general linear model under parametric assumptions; although generally
> referred to as a t-test, it is formally identical to a regression
> analysis, where the
> regressor has indicator variables denoting group membership (i.e.,
> 1, 1, ... ,-1, -1, ...)"
>
> This argument extends to comparing a single subject with a group
> (i.e., using a
> regressor (1, -1, -1, ...). I am afraid I do not know of any
> references that would
> alleviate the reviewers concern because there is no mathematical
> reason write
> such a paper.
>
> Clearly, there are other people, apart from your reviewer, who do
> not appreciate the
> generality of parametric inference under the general linear model.
> In fact, some authors
> have even written papers about comparing single cases with a group,
> as if it was a
> special problem; e.g.,
>
> Crawford JR, Garthwaite PH. Statistical methods for single-case
> studies in neuropsychology:
> comparing the slope of a patient's regression line with those of a
> control sample.
> Cortex. 2004 Jun;40(3):533-48.
>
> All these tests are simple instances of the GLM as implemented in SPM.
>
>
>> Second, the reviewer asks for a power analysis. I guess that the
>> SPM approach using the GLM looks at the difference in means
>> compared to the overall error, with the power influenced more by
>> the total group size (18) than the number of subjects in each
>> group... Is that reasonable?
>
> It is certainly possible to do post-hoc power analyses using the
> estimated variance of
> the noise (i.e., the values in ResMS.ing, which are the sum of
> squared residuals divided by
> the d.f. or trace(RV)). We do not generally encourage this because
> this analysis should,
> strictly speaking, be repeated for every voxel and every contrast.
> Furthermore, the power
> will change arbitrarily with different adjustments to the p-value
> to control family-wise error.
> Finally, the power is a function of some alternate hypothesis,
> which is not specified
> in any general sense.
>
> Power analyses are usually only performed before the experiment to
> see how many subjects
> are necessary for a particular effect size. They are not necessary
> after the experiment because
> they can only be used to quantify type II error (false negative
> rate). This information is usually
> useless in imaging because this rate depends on a precise
> specification of the alternate hypothesis,
> which will not generalize to all brain regions.
>
> I hope this helps - Karl


------------------------------------------------------------------------
---
Stephen M. Smith, Professor of Biomedical Engineering
Associate Director, Oxford University FMRIB Centre

FMRIB, JR Hospital, Headington, Oxford OX3 9DU, UK
+44 (0) 1865 222726 (fax 222717)
[log in to unmask] http://www.fmrib.ox.ac.uk/~steve
------------------------------------------------------------------------
---