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What does the symmetry requirement for the Conover rank squared tests entail? In that Mathematica link it is stated that the ConoverTest assumes the data is symmetric about a common median. We have no idea what the ConoverTest symmetry test is about.

It is unclear how this symmetry test relates to the literature, and we have looked everywhere, and now turn to this site for an explanation. In specific, according to Glen_b of the CrossValidated site,

"Wolfram is remiss in not offering a reference for the test. References on this do not appear to list symmetry. You can read about what is listed in Conover (1980)[1] (which is widely available... it also appears to be in 3rd Ed.), but the test predates the book (on p247 Conover refers to Conover & Iman, 1978 [2] and a related test by Talwar & Gentle, 1977). Looking at the abstract of Conover & Iman, if this test is to be named after someone it should be named for Taha (who wrote about it in 1964 [3]). It is curious that Taha explicitly says it's for asymmetrical distributions in the title, but my guess is Taha is ranking the values themselves* rather than the deviations from the mean, as Conover describes.

Conover doesn't list symmetry among the assumptions; instead he explicitly assumes identical distributions under the null. In either case the assumption would refer to the original distribution of values. In the case of the assumption Conover gives, this would be in order that the permutations of the ranks be equally probable under the null (edit: indeed, see the "Theory" section starting from the end of p243, where he makes that explicit).

It may be that the assumption of symmetry (+ equal variance under the null) also leads to a suitable permutation argument, but we don't find it discussed in Conover and your link curiously doesn't seem to offer any references at all.

* Turns out that guess was correct. By the look of it the idea to subtract the individual population means and rank the deviations from it, then apply the squared ranks test to that was made by several different people, and in Conover&Iman they say it seems to have been first proposed by Shorack in 1965 (though it doesn't sound like they were able to read the paper in question). So maybe it should really be named Taha-Shorack. Or we could stick to the name Conover uses and just call it the squared ranks test.

[1] Conover, W.J. (1980),
Practical Nonparametric Statistics, 2ed., pp. 239-248.
John Wiley and Sons, New York

[2] Conover, W. J. and R. L. Iman (1978),
"Some exact tables for the squared ranks test"
Communications in Statistics - Simulation and Computation 7:5, pp 491-513

[3] Taha, M.A.H. (1964),
"Rank test for scale parameter for asymmetrical one-sided distributions".
In Publications de L'Institute de Statistiques, Vol. 13, 169–180.
Paris: de L'Université de."

So where can we learn about this symmetry requirement?

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