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.
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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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