Dear Rik and Will,
Thank you very much for your kind reply.
The modified t test described by Sokal and Rohlf (1995) is new to me.
I had modified SPM and compared the results of the one sample T test
and the two sample T test. I attached the results.
The results of both tests are similar but the statistically active areas
are resticted in modified t test approach due to 1/sqrt(Nc+1) factor, in this case 1/3.
I agree to Will's suggestion that patient is a random variable.
But I think that we can also use the 1 sample T test to examine
whether this patient fall within e.g. 95% of the mean of the control population or not.
Anyway, I will get the textbook "Biometry" and learn about the modified t test.
Best regards,
Takanori Kochiyama.
In message "Re: [SPM] How to use contrast?",
>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.
In message "Re: [SPM] How to use contrast?",
Rik Henson FIL wrote...
>
>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]
>> -------------------------------------------------------------
-------------------------------------------------------------
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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