Talk:Kendall rank correlation coefficient

Suggest reformulation of kendall's tau definition
I notice there's been some debate here (more than 4 years ago!) over the definition of kendall's tau. While the current definition is technically correct, it's not the most intuitive definition. The way I've usually seen it defined is $$\tau = \frac{\text{number of concordant pairs - number discordant pairs}}{\text{number of concordant pairs + number of discordant pairs}}$$

This is the same as what's currently given because the sum of concordant and discordant pairs is just the total number of pairs, given by N choose 2 and equal to $$\frac{1}{2} n(n-1)$$. See e.g. http://ir.cis.udel.edu/~carteret/papers/sigir09.pdf

Ttodorovv (talk) 00:36, 16 February 2013 (UTC)


 * It's not equivalent because if there exists a pair $$p$$ such that $$x_i = x_j$$ or $$y_i = y_j$$ then


 * $$\frac{n(n-1)}{2} \neq \text{number of concordant pairs} + \text{number of discordant pairs}$$


 * because $$p$$ is neither concordant nor discordant. 128.30.71.236 (talk) 21:42, 10 July 2017 (UTC)

Replace the current formulation with the following...?
I found the current formulation and explanation to be wrong and misleading. I've prepared the following change:

"Kendall tau coefficient is defined


 * $$\tau = \frac{n_c-n_d}{\frac{1}{2}{n(n-1)}}$$

where $$n_c$$ is the number of concordant pairs, and $$n_d$$ is the number of discordant pairs in the data set.

The denominator in the definition of $$\tau$$ can be interpreted as the total number of pairs of items. So, a high value in the numerator means that most pairs are concordant, indicating that the two rankings are consistent. Note that a tied pair is not regarded as concordant or discordant. If there is a large number of ties, the total number of pairs (in the denominator of the expression of $$\tau$$) should be adjusted accordingly."

This is from http://www.statsdirect.com/help/nonparametric_methods/kend.htm and I've confirmed it to be correct. Understanding it simply requires understanding concordant pairs, for which there is already a rather good entry. —Preceding unsigned comment added by Squeakywaffle (talk • contribs) 23:32, 18 September 2008 (UTC)

Well if nobody has any objections I'm going to go ahead and make this change. I will have to remove the example, but this talk page is dominated by comments questioning the validity of the current formulation and explanation, and IMO having those correct is more important than having an example.

Maybe I will come back and do an example, or maybe someone else can do it. --Squeakywaffle (talk) 22:22, 23 September 2008 (UTC)

Error in equation?
According to http://www.rsscse.org.uk/TS/bts/noether/text.html and my experiments, I think the equation should be 1 - 4P/n(n-1) NOT 4P/n(n-1) - 1 —Preceding unsigned comment added by 74.95.2.89 (talk) 23:35, 11 December 2007 (UTC)

142.103.8.44 (talk) 23:16, 1 May 2008 (UTC) I'm pretty sure that $$\tau = \frac{4P}{n(n-1)} - 1$$ works.

Error in explanation
The phrasing of the actual definition needs work:

"where n is the number of items, and P is the sum, over all the items, of the number of items ranked after the given item by both rankings." —Preceding unsigned comment added by 76.191.205.197 (talk) 01:39, 3 July 2008 (UTC)

The last paragraph of the definition has a problem: it says


 * P can also be interpreted as the number of concordant pairs subtracted by the number of discordant pairs.

This can't be literally true: P (as defined above) is a positive number, while this subtraction doesn't have to be.

Instead, while tau = 2P/N-1 (when N=n*(n-1)/2)), if we write S="number of concordant pairs subtracted by the number of discorant pairs", I think that tau = S/N, so that we have S=(2P-N).

Or am I missing something?

Nyh

Example
shouldn't the example be P = 5 + 4 + 4 + 4 + 3 + 1 + 0 + 0 = 22. instead of P = 5 + 4 + 5 + 4 + 3 + 1 + 0 + 0 = 22?

Sboehringer 17:00, 5 March 2007 (UTC)

Example as described on main page appears to be correct -- —Preceding unsigned comment added by 76.31.253.197 (talk) 13:43, 14 April 2008 (UTC)

Significance tests
Article needs some discussion of how to generate p-values in order for hypothesis testing. —Preceding unsigned comment added by 128.200.138.197 (talk) 17:55, August 27, 2007 (UTC)
 * I agree. Cazort (talk) 21:17, 20 November 2008 (UTC)

It would be good if there were cited references for these equations.

I'm also a bit unclear about what terms to group in regard to z(B). For example:

v = (v0-vt-vu)/18 + v1 +v2. Strictly speaking, the denominator is 18 and does not include v1 or v2. Is that right? Some added parens would make that clearer.

v2 = sum ti(ti-1)(ti-2)*sum uj(uj-1)(uj-2)/(9n(n-1)(n-2)). Is the last term, (9n(n-1 etc., included in the sum or not? I supposed it doesn't make a difference. Nonetheless, for clarity, some delimiters for the summation terms would be appreciated. — Preceding unsigned comment added by 132.246.3.117 (talk) 19:25, 3 January 2018 (UTC)

The whole discussion on significance tests seems a bit strange; In Kendall - Rank Correlation Methods (1970) book, the variance implied by $$z_B$$ in the text is given as the variance of the asymptotic distribution of $$n_c - n_d$$ in the presence of ties - regardless of whether we are considering the A or B method. The variance implied by formula for $$z_A$$ is given as the variance of the asymptotic distribution of the same quantity in the absence of ties - again without any reference to either A or B method (since the quantity $$n_c-n_d$$ is computed in the same way in both methods). Also, what does it even mean that "the following statistic [...] is approximately distributed as a standard normal when the variables are statistically independent"? Normally distributed with what parameters? What is the random variable in that expression? Vjka (talk) 15:50, 7 October 2018 (UTC)

Incomplete definition?
The definition section says: "They are said to be discordant, if xi > xj and yi < yj or if xi yj."

Shouldn't this be stated more generally: "They are said to be discordant, if xi > xj and yi ≤ yj or if xi < xj and yi ≥ yj."?

Alopdahl (talk) 09:19, 15 March 2013 (UTC)


 * I don't think so. Any evidence?  — Arthur Rubin  (talk) 16:03, 15 March 2013 (UTC)

Empirical estimation vs formal definition
What is given in the section of the definition appears to me more like the estimator of Kendall's tau for a given sample.

The probabilistic definition should probably more look like:

$$ \tau = \operatorname{E}[ \operatorname{sign} ( ( X_1 - X_1') ( X_2 - X_2' ) ) ] $$

where $$(X_1', X_2')$$ is an independent copy of $$(X_1', X_2')$$.

77.56.29.47 (talk) 16:44, 19 January 2014 (UTC)

Definition
The definition asssumes the uniqueness of the values xi, yet makes a statement about the case xi=xj(?) 82.75.155.228 (talk) 21:20, 28 April 2014 (UTC)