How to check permutation testing exchangeability assumption when using a General Linear Model I have a question on the assumption of exchangeability in permutation tests. Although I read a lot about this topic, I am still confused.
For $N$ subjects, I have the value of a clinical measure $Y$ (a numerical quantity such as the volume of a ventricle) together with a set of other clinical parameters (such as age, gender, height...) and a categorical variable which represents the presence of a genetic mutation. I would like to use permutation testing with the following multiple regression model:
$Y = \beta_0 + \beta_1 \; age + \beta_2 \; gender \; + \; ... \; + \;\beta_n \; mutation + \epsilon$
to see if $\beta_n \neq 0$. 
The only assumption required by the technique I would like to employ 
(Freedman and Lane, 1983) is the one required by all the permutation tests: exchangeability.
If my understanding is correct I need to check if I can shuffle the values of $Y$ across the $N$ subjects under the null hypothesis (no effect of the genetic mutation) and this doesn't affect the error ($\epsilon$) distribution.
I believe that this is not true in this setting as Y depends also on the other parameters in the model (age, gender etc.), but I am not sure about that. I was wondering what you think about that and what I should check to correctly apply permutation testing in this setting.
 A: Assuming you mean the paper I mentioned in comments, Freedman and Lane [1] explain it clearly.
The last sentence of the abstract for their paper* says:

This development parallels that of randomization tests, but there is a crucial technical difference: our approach involves permuting observed residuals; the classical randomization approach involves permuting unobservable, or perhaps nonexistent, stochastic disturbance terms.

*(which you can read without access to the journal, but I assume you do have access to the paper if you're trying to do what it does) 
That says it all, pretty much. 
What is exchangeable in regression is the error term, $\epsilon$. Note that under the null hypothesis the error term of interest is that for the model where Y is regressed on all the predictors but the one being tested (the reduced model).
You can't get at $\epsilon$. So they use the best available estimate of them, the residuals. 
Those aren't really exchangeable (they don't have the same distribution and they're not equicorrelated). So that's then not quite a permutation(/randomization) test, and they clearly say so right there in that abstract -- they say it "parallels" the classical randomization test, which it does in a particular sense.
In essence, they fit the reduced model, 
 then permuted residuals from that are added to the fitted values for the reduced model to produce a new "resampled" Y* which is then regressed on all the predictors to obtain a "randomization-like" distribution for the test statistic of interest.
You might like to compare this with another resampling technique, the bootstrap which involves resampling with replacement (while randomization tests resample without replacement), one version of that for the regression situation, the residual bootstrap operates in a similar fashion. (If you seek a reference, I'd suggest a recent printing of the book by Davison and Hinkley [2])
[1] Freedman D. and Lane, D. (1983),
A Nonstochastic Interpretation of Reported Significance Levels,
Journal of Business & Economic Statistics, Vol. 1, No. 4 (Oct.), pp. 292-298
[2] A. C. Davison. & D. V. Hinkley (1997),
Bootstrap Methods and their Application,
Cambridge Series in Statistical and Probabilistic Mathematics,
Cambridge University Press
