Exactly what does the Wilcoxon signed rank test compare?

Question

The Wilcoxon signed rank test is a nonparametric version of the t-test. That is, it converts parametric values, which may be non-normally distributed into EDIT for Glen_B more normally distributed signed ranks; e.g., from Harrell and Slaughter "Biostatistics for Biomedical Research" (BBR) 7.2 We look up |z| against the normal distribution. A complication arises when ties occur. Now the problem is, that when the test is usually described it is presented as a "test of median values," which is no more true than it is a test of mean values, as the mean and median are the same for a normal distribution arrived at by nonparametric transformation. This is complicated by the fact that there is an actual median test, which is a Chi-squared test for different median values, and which is certainly not the same thing as signed rank test which combines the rankings of paired samples of the data.

So the question is, "Rather than say that the Wilcoxon test is a test for "median values" which if true applies to the nonparametric transformed random variates, what words would best describe what the Wilcoxon test actually is?"

A further complication is that in the ranked data, the transformed medians correspond position-wise to the data medians (if odd), but the values being compared are different, i.e., ranks, and not data medians.

Edit In the one-sample case, we would be asking how plausibly the signed ranks sums would include a given point value, but there is no comparison single value for the signed rank sums, but rather a region or range in which the same probability would apply to slightly different tested points, thus statements suggesting that the signed rank sum test is a test of median or mean values do not make sense. Think about the sign test for a moment, the same probability occurs irrespective of what the mean or median is, as long as the relative number of sample values greater or less than a comparison value is maintained. The Wilcoxon test is similar to the sign test but with the additional information of signed rank sums providing better power.

Answer 1

I can think of a few useful ways to think of the signed ranks test. The following notes may not be technically rigorous.

• One way to think of the hypothesis tested is described by Frank Harrell in a comment here: "As stated earlier it tests precisely whether the probability that a randomly chosen pair of values sums to a positive number is equal to 0.5 or not." stats.stackexchange.com/questions/569439/how-to-interprete-‌​and-describe-the-res‌​ults-of-a-wilcoxon-s‌​igned-rank-test. @statmerkur mentioned in the chat the Wikipedia description of the hypotheses (en.wikipedia.org/wiki/Wilcoxon_signed-rank_test#Null_and_alternative_hypotheses.

• With the understanding that it first takes the differences of the paired values, and then uses the ranks, the test looks to see if the [signed rank values of the differences] values are symmetrically distributed around zero. That is, there are two ways you could get an e.g. significant result. a) If the values are symmetrical around a value other than zero, or b) If the values are not symmetric even if centered (in some sense) around zero.

• It could be considered a test of location of the differences, where location is understood through the lens of signed ranks. This "location" may be related to the Hodges–Lehmann estimator. ( I don't understand this relationship well enough to say anything more definitive.)

As a parenthetical comment, personally, I think it's not helpful to describe tests like Wilcoxon-Mann-Whitney or Wilcoxon signed rank tests as tests of median, even if we add caveats about assumptions of the distributions of data or populations for this interpretation. As far as I can tell, this avenue just causes confusion in people trying to understand these tests. We have different methods to test medians if that's what we really want.