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My question is if data are in paired groups (before and after treatment), where the pre-treatment group is normally distributed and the post-treatment group is not normally distributed (i.e. it is skewed), then which test should be applied? Is the Wilcoxon sign rank test appropriate?

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The Wilcoxon sign rank test is appropriate. However you should be careful in your interpretation of the test.

In its most general form, the sign rank test is a test for stochastic dominance. That is, strong enough evidence will lead to a conclusion that a random observation from the pre-treatment group has a greater than (or less than) 50/50 chance of being larger than a random observation from the post-treatment group. The null hypothesis is that 50/50 chance, or: H$_{0}\text{: P}\left(X_{\text{pre}} > X_{\text{post}}\right) = 0.5$, and H$_{\text{A}}\text{: P}\left(X_{\text{pre}} > X_{\text{post}}\right) \ne 0.5$.

Under conditions where (1) the distributions in both pre- and post-treatment are the same shape, and (2) differ only in the central location of that shape (i.e. they are merely shifted with respect to one another, not 'stretched out' with different variances), then the Wilcoxon sign rank test can be interpreted as a test for median difference, with the H$_{0}\text{: }\tilde{\mu}_{\text{pre}} - \tilde{\mu}_{\text{post}} = 0$. However, these more rigid assumptions clearly do not obtain in your data, so you would interpret the results of the sign rank test solely in terms of stochastic dominance.

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  • $\begingroup$ Indeed, if they have the same shape, the signed rank test can as legitimately be interpreted as a test of means (as long as means exist), just as it can be a test of medians, or 10th percentiles, or any number of other location-measures. $\endgroup$ – Glen_b Jul 10 '14 at 1:56

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