I performed a one-sided Wilcoxon signed-rank (WSR) test. I had a matched pair of continuous random variables, say $X_1, X_2, ..., X_n$ and $Y_1, Y_2, ..., Y_n$. For my system, it is safe to assume that under the null hypothesis, $X_i$ and $Y_i$ are identically distributed for all $i$. Hence, the difference, say $U_i = X_i - Y_i$ should have a symmetrical distribution around 0, i.e., under the null hypothesis. Also, $U_i$'s are independent. Hence, I believe that assumptions of Wilcoxon signed-rank test are satisfied. So, I performed a one-sided test to check if $U_i$ tend to be greater than 0.
I am writing a report on my analysis but I am a bit confused about how exactly to phrase the test. Following are a couple statements that I have considered.
I performed Wilcoxon signed-rank test to test null hypothesis against one-sided alternative that population mean of $U_i>0$.
I performed Wilcoxon signed-rank test to test null hypothesis against one-sided alternative that population median of $U_i>0$.
I have noticed that WSR null is generally reported in terms of median. For one, I am not clear on why that is. Second, my understanding is that since $U_i$'s are symmetrically distributed about 0 under the null hypothesis, mean and median should be the same. So perhaps, it should be ok to describe the test in terms of mean. If it is so, I'd prefer to report in terms of mean because a lot of my readers might not know WSR. If they read a statement in terms of mean, they might find it easy to relate to as simply an alternative to two sample t-test.
If you have a suggestion for a better way of reporting the test, please let me know.
I have read several answers over Stackexchange but could not find exactly what I was looking for. Also, I read a paper (Li and Johnson, 2014, Pharmaceutical Statistics; DOI: 10.1002/pst.1628) that describes two different null hypothesis - one for "test of median" and another for "test of symmetry" with the name of Wilcoxon signed-rank test.
I simply used
wilcox.test function in R as
wilcox.test(X, Y, alternative = "greater", paired = T). How do I know if my test was a "test of median" or a "test of symmetry"?