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I know that rank tests are useful because we they are more or less performed without distribution assumptions of the unranked data; they are particular cases of permutation tests...

But what are the real advantages and inconveniences of rank based tests, and in which cases are they especially indicated?

Moreover, which textbooks about rank tests would you advise (I have the Hajek and Sivak: Theory of Rank Tests)?

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    $\begingroup$ Possible duplicate of When to use the Wilcoxon rank-sum test instead of the unpaired t-test? $\endgroup$ – Tim Nov 15 '15 at 18:31
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    $\begingroup$ @Tim I checked this question. I do not think it is a duplicate since I'm talking about Rank Tests in general, while the Wilcoxon one is very particular. $\endgroup$ – MoebiusCorzer Nov 15 '15 at 19:09
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    $\begingroup$ @Tim I agree with MoebiusCorzer's point: rank-based tests are a broad class of tests of which the rank-sum test is but one example. See for example, Conover, W. J. (1999). Practical Nonparametric Statistics. Wiley, Hoboken, NJ, 3rd edition. $\endgroup$ – Alexis Nov 15 '15 at 19:11
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    $\begingroup$ @FrankHarrell Why not make an answer? :) $\endgroup$ – Alexis Nov 15 '15 at 19:18
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    $\begingroup$ I'd second Alexis' suggestion of Conover. One thing to keep in mind is that often almost all the location information is conveyed by the relative positions (the ranks) rather than the actual numerical values; once you use the information in the ranks the numerical values typically add little additional information. $\endgroup$ – Glen_b -Reinstate Monica Nov 15 '15 at 22:24
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This is an awfully general question. Rank tests are not necessarily special cases of permutation tests. Huge advantages of rank tests and their generalizations (semiparametric regression models) include not being dependent on the correct transformation of $Y$, being robust, and being more powerful than parametric tests on the average (and only losing a tiny bit of power if e.g. normality holds).

Rank tests and semiparametric models do not make any assumptions about the distribution of $Y$ given a particular set of values for the covariates $X$ but do assume how the distribution of $Y|X=x_{1}$ is connected to the distribution of $Y|X=x_{2}$. For example the Wilcoxon test (a special case of the proportional odds ordinal logistic model) assumes that the two distributions are in proportional odds for the Wilcoxon test to be optimally powerful. Contrast that with the $t$-test which assumes that both distributions are Gaussian.

There are many excellent references, Agresti being one of them. Note that all continuous $Y$ are ordinal. See also my course notes which includes a detailed case study of ordinal regression for continuous $Y$.

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  • $\begingroup$ Thank you for your answer. I understand that my question is very general, but I was wondering why Rank Tests were so important and I have a partial answer with yours. Would you advise particular reference works? $\endgroup$ – MoebiusCorzer Nov 15 '15 at 19:33
  • $\begingroup$ "…all continuous Y are ordinal…" Boy howdy, I have been delighted and in wonder about this for about a decade now: ordinality is a foundational property of quantity, and not the other way around. :) I think I cottoned to it when reading a history of calculus, and marveling at the 2,000 years it took to develop limits as a way out of the infinitely divisible/infinitesimal quandary. $\endgroup$ – Alexis Nov 16 '15 at 19:38

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