# Calculation of standard deviation of the mean changes from the p-value or z-value of the Wilcoxon test

I am conducting a meta-analysis of observational studies without a control group that compared a parameter's value before and after an operation. All included studies have reported their results as pre-and-postoperative mean ± standard deviation and statistical analysis p-value. While it is possible to determine the standard deviation of the mean changes from the p-value of the paired t-test; however, some of the included studies have used the Wilcoxon test to compare pre-and-postoperative data. How can I calculate the standard deviation of the mean changes from the p-value or z-value of the Wilcoxon test in these studies?

Without additional information, it's not possible.

Imagine a set of paired pre- and post- values with some particular $$Z_S=\frac{S-\mu_S}{\sigma_S}$$ where $$S$$ is whichever version of the signed rank statistic is your favourite.

a. Take the set of post$$-$$pre differences and double them. The signed rank statistic, $$S$$ and the $$Z$$ value will be the same.

b. Take the set of post$$-$$pre differences and cube them. The statistic and its $$Z$$ value will be unchanged.

c. Take the set of post$$-$$pre differences and take the hyperbolic arc sin of each of them ($$\operatorname{asinh}(d) = \log(d+\sqrt{d^2+1})$$). The statistic and its $$Z$$ value are again unchanged.

d. Take the set of post$$-$$pre differences and add 100 to the positive ones and subtract 100 from the negative ones. Same thing!

If this information is unfamiliar to you, start with a small set of data and actually try these and see.

Indeed apply whatever other transformation of the values of $$d$$ that leaves the ranks of $$|d|$$ and their signs unaltered that you wish to the set of pair-differences. Same $$S$$, same $$Z$$ every time. So going back from $$Z$$ to $$\bar{d}$$ or its standard error is not possible. (Similarly with p-values, whether based on the $$Z$$-value or on the exact test.)

This is because the value of $$S$$ depends only on the ranks and signs of the differences, and transformations like those above leave the signed ranks unaltered.

Since many (indeed infinitely many) distinct situations get you back to the same $$Z_S$$ value, you have no way to "go back" to the mean of post$$-$$pre. There's simply nothing to be done. It's not a reversible operation.

Example in R. First some post-pre differences and the signed rank test:

d=c(0.64, 1, -0.24, -1.34, 0.96, 0.22, -0.72, 1.24, -0.89, -0.36)

wilcox.test(d)

Wilcoxon signed rank exact test

data:  d
V = 29, p-value = 0.9219
alternative hypothesis: true location is not equal to 0

Now let's transform the differences in several ways that won't change the signed ranks (some extraneous lines of output are removed for brevity):

wilcox.test(d*2)
V = 29, p-value = 0.9219

wilcox.test(d^3)
V = 29, p-value = 0.9219

wilcox.test(asinh(d))
V = 29, p-value = 0.9219

wilcox.test(ifelse(d>0,d+100,d-100))
V = 29, p-value = 0.9219

(In that last case the possibility of a d=0 was ignored because there were no 0-differences in the data; if they occur, you'd need to make sure you left those as-is.)

All of these will have different $$\bar{d}$$ and a different standard error of the mean difference; which one should we go back to?