I'm in the process of running a repeated-measures ANOVA on two groups of participants from an experiment. In one group, there are 10 participants, and in the other group, there is only 1 participant. This is because the 1 participant is a patient who is being compared to the 10 other participants.

Now, my question is: is it fair/allowable to run an ANOVA comparing the patient with the other participants? I wasn't sure, so ran it in SPSS, and it seems to work, with a significant result. Is this naughty?

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#### Best Answer

Performing the ANOVA assumes that the nature and amount of variation in the hypothetical population represented by the one patient are the same as the variation in the hypothetical population represented by the ten other patients, and that all were obtained randomly and independently from their populations.

Under these assumptions a standard approach is to use data from the ten patients to compute a prediction interval for a single additional patient. The significance test only has to check whether the lone patient falls within that interval.

The prediction interval is simple to justify and compute. It depends only on the estimated mean $bar{m}$ from the ten patients and on their estimated variance $s^2$. Let $x$ be the value of the lone patient. Under the null hypothesis (that all patients are drawn independently from a single population), the variance of $x – bar{m}$ equals $s^2 + s^2/10$. *If you further assume all variation is Normal*–that's critical here because it's hard to check with just ten patients in a group–then $frac{x – bar{m}}{sqrt{s^2 + s^2/10}}$ has a Student's t distribution with 9 degrees of freedom (9 is the df used to estimate $s^2$), allowing you to erect a confidence interval in the standard way for $x$.

If this prediction interval test disagrees with SPSS's result, I would not trust the SPSS ANOVA in this case.

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