A related issue has been debated in behavioural classical statistics, see eg:
Crawford, J. R., Howell, D. C., & Garthwaite, P. H. (1998). Payne and Jones revisited: Estimating
the abnormality of test score differences using a modified paired samples t test. Journal of
Clinical and Experimental Neuropsychology, 20, 898-905.
http://www.abdn.ac.uk/~psy086/dept/SingleCaseMethodology.htm
I think that SPM's two-sample t-test is equivalent to the (unpaired) t-test
described in the reference above, ie denominator Vc*sqrt(1/Nc+1).
Rik
----- Original Message -----
From: "Takanori Kochiyama" <[log in to unmask]>
To: <[log in to unmask]>
Sent: Monday, January 17, 2005 11:30 AM
Subject: Re: [SPM] How to use contrast?
> Dear Will and SPMers,
>
> I would like to ask you about the 2 sample t test if one of the samples
> contains a single measurement because I face the similar problem.
>
> We agree on the point that GLM can deal with the two-sample t-test design
> including a single measurement in one of the samples. However,
> I think that conventinal one-sample t-test design is suitable in this case.
> (although we need some modification in spm.)
>
> Based on your previous Email, we think about the following:
>
> The formula of T statistics is
>
> t = (Mc-Mp)/SE.
>
> Here,
> Mc(p): mean of the contol (or patient) groups, and
> SE: standard error.
> and also,
> Vc(p): variance of the contol (or patient) groups, and
> Nc(p): Number of sub. in the contol (or patient) groups.
>
> The denominator for one-sample t-test is
> SE = sigma*sqrt(c'*inv(X'X)*c)
> where
> sigma = Vc
> sqrt(c'*inv(X'X)*c) = sqrt(1/Nc)
>
> The denominator for two-sample t-test is
> SE = sigma*sqrt(c'*inv(X'X)*c)
> where
> sigma = {(Nc-1)Vc + (Np-1)Vp}/{Nc+Np-2}
> = Vc for Np = 1
> sqrt(c'*inv(X'X)*c) = sqrt(1/Nc+1/Np)
> = sqrt(1/Nc+1) for Np = 1
>
> The numerator {Mc-Mp} and df {Nc-1} of both tests are same.
>
> As a result, we have the following relationship in T value
> between 1sample and 2sample T-test:
> T_(2sample) = {1/sqrt(Nc+1)}*T_(1sample)
>
> i.e. T_(2sample) is smaller than T_(1sample).
>
> This seems to affect the confidence interval.
> If we want to check e.g. a 95% confidence interval of the "control data mean",
> I think, 1sample T is preferable.
>
> And I am worried about the equal variance assumption
> between control and patient group which is required by 2 sample T test,
> because we never can measure the variance in the patient group with single subject.
> In this point, I think, 1sample T is safe approach
> Please correct me if I am wrong.
>
> Thanks in advance for any clarification.
>
> -------------------------------------------------------------
> Takanori Kochiyama
> Faculty of Engineering
> Kagawa Univ., Hayashi-cho 2217-20,Takamatsu, JAPAN
> Phone: +81-87-864-2337,Fax: +81-87-864-2369
> e-mail: [log in to unmask]
> -------------------------------------------------------------
>
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