Zoltan Nagy wrote:
> Dear Will, Kohkichi and other SPMers,
> In general is it okay to do 2 sample t test if one of the samples only
> contains a single measurement?
Yes.
> To me the purpose here seemed to have been
> checking if a point estimate falls within normal values (i.e. the 27 control
> subjects). Or in other words does the person fall within 95% of the mean of
> the population.
>
The two-sample t-test design option in SPM effectively allows you to do just
this.
For Np patients and Nc controls the design matrix is
X=[ones(Nc,1),zeros(Nc,1); zeros(Np,1), ones(Np,1)].
The corresponding parameter estimates are given by
beta = CX'Y
where C=inv(X'X)
For our X we therefore have C=[1/Nc 0; 0 1/Np]
and X'Y = [Sc Sp]' where Sc and Sp are the sums of the
control and patient group data. So beta = [Mc Mp]', the means of the groups.
Then the contrast c=[1 -1] gives a t value of
t=c'beta/std(c'beta)
The denominator is
std(c'beta)=sigma*sqrt(c'Cc)
where sigma is the standard deviation of model errors.
We have sqrt(c'Cc)=sqrt(1/Nc+1/Np)
Assuming sphericity (equal error variance in patient and control
groups) sigma is a pooled estimate of the SD in both groups.
So for two-sample t-tests in SPM we have
t = (Mc-Mp)/(sigma*sqrt(1/Np + 1/Nc))
For a single patient, Np=1, we get
t= (Mc-yp)/(sigma_c*sqrt(1+1/Nc))
where yp is the patient data point.
This is similar to doing a z-test with
z= (Mc-yp)/sigma_c
ie. seeing if the patient data falls within eg. a 95% confidence interval
of the control data mean - exactly as you say.
Of course, with Np=1 it is not possible to allow for non-sphericity, as we
can't measure the variance in the patient group.
Best,
Will.
--
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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