Dear Takanori,
I think you could make sensible arguments for both approaches
eg.
1. The two-sample t-test; because we really do have two
different populations. Just because we have N=1 for one
population, this does'nt change.
2. The one-sample t-test; because we want to compare
the control mean to a particular value.
However, the one-sample t-test is usually used to
compare eg. a control mean to a *fixed known value*
(eg zero) - not to a *random variable* (eg. patient
scan). So we need to adjust the test to account for this.
Perhaps others would like to comment (Tom ?)
Anyway, in practice the different approaches will
produce very similar results.
For a 12 subject study the difference in t-values
is only 4% (sqrt(13)/sqrt(12)). For 24 subjects its 2%.
Best,
Will.
Takanori Kochiyama wrote:
> 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]
> -------------------------------------------------------------
>
>
>
--
William D. Penny
Wellcome Department of Imaging Neuroscience
University College London
12 Queen Square
London WC1N 3BG
Tel: 020 7833 7475
FAX: 020 7813 1420
Email: [log in to unmask]
URL: http://www.fil.ion.ucl.ac.uk/~wpenny/
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