I am working on some stats coursework, and have non parametric bivariate data. n=19, so small sample. There are a number of tied ranks, so I'm planning to use Kendall's tau rather than Spearman's rho, as I have found some literature supporting this. My SPSS output has calculated tau = -.982, p <.001. What I have absolutely no idea of is how to calculate the confidence intervals. Am I correct in thinking I can't use the Fisher Z transformation with tau, and if so why not? Do I need to do something with bootstrapping (which I have never done!)? If anyone could help, in as basic terms as possible (very much a beginner!) I would really appreciate it.
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1$\begingroup$ I should have mentioned that I have a basic understanding of how bootstrapping works (thank you youtube!) - I am just unsure if that is an appropriate way of calculating CIs for Kendall's tau-b $\endgroup$– MillieJul 21, 2015 at 19:56
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This paper discusses the contexts where you can and can't use a normal approximation for Tau. According to Wikipedia, it also looks like the validity of normal/Z depends on how your version of Tau handles ties. My sense is that it's probably safer not to assume that it's Gaussian, especially with relatively low sample sizes.
I couldn't think of a reason why Kendall's Tau wouldn't be compatible with the bootstrap, but I wasn't 100% sure. So I looked it up:
Here's a paper by Brad Efron, the inventor of the bootstrap, that uses it for Tau (Section 5).
Here's a paper that spends some time discussing the bootstrap in the context of Tau (mostly Section 4).
Looks like you shouldn't have any serious problems using the bootstrap for tau.
answered Mar 13, 2017 at 17:53
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$\begingroup$ Very nice. Any idea of whether there's a better way than bootstrapping? $\endgroup$ Mar 13, 2017 at 18:20
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$\begingroup$ If we don't have tau's sampling distribution, the only alternatives I'm aware of are to approximate it (e.g. with a Gaussian) or to do some kind of resampling (like bootstrapping). This paper claims that the Gaussian approximation isn't too bad, and gives a formula for a Z-test (section 4). Keep in mind that the different options for handling ties will make a difference here. utdallas.edu/~herve/Abdi-KendallCorrelation2007-pretty.pdf $\endgroup$ Mar 13, 2017 at 21:41
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$\begingroup$ (or, in a few words: bootstrapping is probably best, but it's not the only option) $\endgroup$ Mar 14, 2017 at 2:29