Both Spearman and Kendall tau correlations measure monotonicity relationships and have nice interpretation. The paper listed below compares Spearman and Kendall correlations in terms of robustness and statistical efficiency. In the normal case, the Kendall correlation is preferred than the Spearman correlation because of a smaller gross error sensitivity (GES) (more robust) and a smaller asymptotic variance (AV) (more efficient). If you are interested in other cases, you may compute their GES and AV by yourself. Make sure that two correlations measure the same quantity such that their GES and AV are comparable. 

However, in terms of computation, Kendall correlation has a O(n^2) computation complexity comparing with O(n logn) of Spearman correlation, where n is the sample size. 

Croux, C. and Dehon, C. (2010). Influence functions of the Spearman and Kendall correlation measures. Statistical Methods and Applications, 19, 497-515. 

Thanks Professor Iosif Pinelis for pointing out the reference https://www.jstor.org/stable/2282833?seq=1#page_scan_tab_contents, which states that Kendall's correlation statistic can be computed in O(n log n) steps.

See if the links below may help

http://stats.stackexchange.com/questions/3943/kendall-tau-or-spearmans-rho

http://www.unesco.org/webworld/idams/advguide/Chapt4_2.htm

The link below provides a more recent study of the two

http://www.eee.hku.hk/~yhhou/pdf/1-s2.0-S0165168412002721-main.pdf

Spearman's rho is usually used for ordinal scale data: Pagano, R. R. (2009). Understanding statistics in the behavioral sciences (9th Edition). Belmont, CA: Wadsworth Publishing.

Kendall's tau is used in small samples or when we have many values with the same score: Field, A. (2009). Discovering statistics using SPSS (3rd Edition). London: Sage Publications.

I have another suggestion here: http://www.sciencedirect.com/science/article/pii/S0378375806002588

http://link.springer.com/chapter/10.1007/978-3-319-13254-9_71

http://arxiv.org/pdf/1011.2009.pdf

Hi Ronán.  In Statistical Methods for Psychology, Dave Howell concludes that Kendall's tau is generally preferred over Spearman's rho, "because it is a better estimate of the corresponding population parameter, and its standard error is known."  See the attached screen-capture from Google Books.

HTH.

Bruce : good point.

Tau is also interpretable as the percentage of pairs of data points that show a positive correlation. The actual percentage is tau+(half of(1-tau)). So a tau of 0·6 means that there is a positive correlation between 80% of all possible pairs of data points. And, of course, a tau of –1 means that no data points show a positive correlation.

By contrast, there is no precise interpretation of rho. 

While I find tau intrinsically appealing I think that Spearman's rho can also have an intuitive interpretation in terms of capturing a monotonically increasing or decreasing trend (in which case rho is 1 or -1.

Thom: what I was referring to was an actual interpretation for a measured value of rho. While r is interpretable (as r-squared) and tau is interpretable as above, I can think of no interpretation for a rho of 0·8 – can anyone?

Paul: That's a very important point: it's a defined property, first and foremost, of the data.

If you don't have computer/software resources handy, Spearman's rho is pretty easy to calculate by hand. Which of course has a lot to do with why Spearman designed it the way he did.

Stephen … tau might be even easier as I remember doing it by hand as part my Maths O level. You start by drawing lines between values of equal ranks when one set of values is sorted in ascending order and counting the crossed lines … (which is also visually quite powerful in conveying what information is being used).

Dear Ronán,

  I have experiences about that. I want to describe that Spearman's rho is useful for only to describe about linear relationship between  two variables that are nonparametric data -- the nonparametric data is a data that can not exactly measure in numeric scale, represent  only the levels of the data, but  Kendall's tau is useful describe about the linear relationship of  the data that are both of parametric data or parametric data and nonparamertric data or both of of them are nonparametric data.

I think this answer may useful for you.  If my answer is not correct, please tell me.

Thank you very much.

Mathee: I would disagree with your use of parametric and nonparametric to describe data. A parameter is a property of the distribution of data. Some statistical methods calculate such parameters (including, for example, the Wilcoxon Mann-Whitney test and Kendall's tau). Other do not. 

Ronán: Thank you very much. I agree with you. Could I still share with you?

I think it also depends upon the context upon which you are implementing the statistical analyses, as well as what kind of sensitivities you are looking to report in your analysis.  For this reason, tau is the least sensitive of the two and is more robust to outliers, so would be better for analyzing smaller sample sizes.  Spearman is usually a go-to because it is more popular, but is more sensitive to error and discrepancies in data.  Because they are similar measures, it can be suggested that they might lead to similar results.  Still, I would perform both calculations and see which best fits your data and the conclusion you are trying to reach.  Personally a fan of Spearman, but my internet searches have suggested that Kendall is the better test.

Austin – I'm curious to know why you are a fan of Spearman? It doesn't seem to present any compelling advantages. 

It was the test that I learned how to use first and I had read of it for looking at ecological sampling.  I agree with you that there are no real advantages of using it over Kendall.  I'm currently doing gene coexpression research and both tests yielded the same results.  Regardless, I think Kendall is probably better, but it depends upon how you want to portray the sensitivities of your data (I.e. individual differences versus discordance across a dataset).

Both Spearman and Kendall tau correlations measure monotonicity relationships and have nice interpretation. The paper listed below compares Spearman and Kendall correlations in terms of robustness and statistical efficiency. In the normal case, the Kendall correlation is preferred than the Spearman correlation because of a smaller gross error sensitivity (GES) (more robust) and a smaller asymptotic variance (AV) (more efficient). If you are interested in other cases, you may compute their GES and AV by yourself. Make sure that two correlations measure the same quantity such that their GES and AV are comparable. 

However, in terms of computation, Kendall correlation has a O(n^2) computation complexity comparing with O(n logn) of Spearman correlation, where n is the sample size. 

Croux, C. and Dehon, C. (2010). Influence functions of the Spearman and Kendall correlation measures. Statistical Methods and Applications, 19, 497-515. 

Thanks Professor Iosif Pinelis for pointing out the reference https://www.jstor.org/stable/2282833?seq=1#page_scan_tab_contents, which states that Kendall's correlation statistic can be computed in O(n log n) steps.

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Thanks, Xin Dang, for the reference. Certainly, when you were working by hand, or with early computers, Kendall's tau was computationally expensive. Nowadays, though, it's rarely a problem. It astonishes me what you can do on a computer now, compared with the days of punched card…