The assumption of symmetricity for signed rank test (and its relevance) is becoming extremely confusing for me. I am hypothesizing that sub-population A (before treatment) and sub-population B (after treatment) come from same population (no effect of treatment). Does my paired difference need to conform to the assumption of symmetricity?
jbowman noted in his response here that "Note that under the typical null hypothesis, if you assume sub-populations A and B are drawn from the same distribution, symmetry of the paired differences between A and B is assured regardless of the lack of symmetry of the underlying distribution." Whereas texts elsewhere say "If you are testing the null hypothesis that the mean (= median) of the paired differences is zero, then the paired differences must all come from a continuous symmetrical distribution. "
Best Answer
Although on the surface the two statements above may appear contradictory, they aren't. The Wilcoxon Signed Rank test does require that the paired differences come from a continuous symmetric distribution (under the null hypothesis, as Michael Chernick points out in comments.) In the special case when the two sub-populations $A$ and $B$ from which the paired samples will be drawn (one each from $A$ and $B$) have the same (continuous) distribution, it is guaranteed that the pairwise differences between the samples $a_i,b_i$ will come from a continuous symmetric distribution.
You can see this by observing that if the two samples come from the same distribution, $p(a_i = x, b_i = y) = p(a_i = y, b_i = x)$. In the former case, the paired difference $\delta_{i,1} = x-y$, and in the latter case the paired difference $\delta_{i,2} = y-x = -\delta_{i,1}$. Since the probabilities of the two cases are equal, it follows that $p(\delta_i) = p(-\delta_i)$, i.e., that the distribution is symmetric around 0.
Therefore, if you can make the assumption that the two sub-populations have the same continuous distribution under the null hypothesis, you've satisfied the Wilcoxon Signed Rank assumption requirements. Often it is easier to see why this assumption might be true than to see why the more general "pairwise differences come from a continuous symmetric distribution" might be true, hence its occasional use.