View source for Kendall tau metric

''Kendall tau''

The non-parametric [[Correlation coefficient|correlation coefficient]] (or measure of association) known as Kendall's tau was first discussed by G.T. Fechner and others about 1900, and was rediscovered (independently) by M.G. Kendall in 1938 [[#References|[a3]]], [[#References|[a4]]]. In modern use, the term  "correlation"  refers to a measure of a linear relationship between variates (such as the [[Pearson product-moment correlation coefficient|Pearson product-moment correlation coefficient]]), while  "measure of association"  refers to a measure of a monotone relationship between variates (such as Kendall's tau and the [[Spearman rho metric|Spearman rho metric]]). For a historical review of Kendall's tau and related coefficients, see [[#References|[a5]]].

Underlying the definition of Kendall's tau is the notion of concordance. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k1300201.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k1300202.png" /> are two elements of a [[Sample|sample]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k1300203.png" /> from a bivariate population, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k1300204.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k1300205.png" /> are concordant if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k1300206.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k1300207.png" /> or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k1300208.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k1300209.png" /> (i.e., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002010.png" />); and discordant if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002012.png" /> or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002014.png" /> (i.e., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002015.png" />). There are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002016.png" /> distinct pairs of observations in the sample, and each pair (barring ties) is either concordant or discordant. Denoting by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002017.png" /> the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002018.png" /> of concordant pairs minus the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002019.png" /> of discordant pairs, Kendall's tau for the sample is defined as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002020.png" /></td> </tr></table>

When ties exist in the data, the following adjusted formula is used:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002021.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002022.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002023.png" /> the number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002024.png" /> observations that are tied at a given rank, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002025.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002026.png" /> the number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002027.png" /> observations that are tied at a given rank. For details on the use of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002028.png" /> in hypotheses testing, and for large-sample theory, see [[#References|[a2]]].

Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002029.png" /> is equal to the probability of concordance minus the probability of discordance for a pair of observations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002031.png" /> chosen randomly from the sample <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002032.png" />. The population version <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002033.png" /> of Kendall's tau is defined similarly for random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002035.png" /> (cf. also [[Random variable|Random variable]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002037.png" /> be independent random vectors with the same distribution as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002038.png" />. Then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002039.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002040.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002041.png" /></td> </tr></table>

Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002042.png" /> is the Pearson product-moment correlation coefficient of the random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002045.png" /> is sometimes called the difference sign correlation coefficient.

When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002047.png" /> are continuous,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002048.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002049.png" /> is the [[Copula|copula]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002051.png" />. Consequently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002052.png" /> is invariant under strictly increasing transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002054.png" />, a property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002055.png" /> shares with Spearman's rho, but not with the Pearson product-moment correlation coefficient. For a survey of copulas and their relationship with measures of association, see [[#References|[a6]]].

Besides Kendall's tau, there are other measures of association based on the notion of concordance, one of which is Blomqvist's coefficient [[#References|[a1]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002056.png" /> denote a sample from a continuous bivariate population, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002058.png" /> denote sample medians (cf. also [[Median (in statistics)|Median (in statistics)]]). Divide the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002059.png" />-plane into four quadrants with the lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002061.png" />; and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002062.png" /> be the number of sample points belonging to the first or third quadrants, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002063.png" /> the number of points belonging to the second or fourth quadrants. If the sample size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002064.png" /> is even, the calculation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002066.png" /> is evident. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002067.png" /> is odd, then one or two of the sample points fall on the lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002069.png" />. In the first case one ignores the point; in the second case one assigns one point to the quadrant touched by both points and ignores the other. Then Blomqvist's <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002071.png" /> is defined as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002072.png" /></td> </tr></table>

For details on the use of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002073.png" /> in hypothesis testing, and for large-sample theory, see [[#References|[a1]]].

The population parameter estimated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002074.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002075.png" />, is defined analogously to Kendall's tau (cf. Kendall tau metric). Denoting by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002076.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002077.png" /> the population medians of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002078.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002079.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002080.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002081.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002082.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002083.png" /> denotes the joint distribution function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002084.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002085.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002086.png" /> depends only on the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002087.png" /> at the point whose coordinates are the population medians of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002089.png" />, it is sometimes called the medial correlation coefficient. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002091.png" /> are continuous,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002092.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002093.png" /> again denotes the copula of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002094.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002095.png" />. Thus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002096.png" />, like <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002097.png" />, is invariant under strictly increasing transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130020/k13002099.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Blomqvist,   "On a measure of dependence between two random variables"  ''Ann. Math. Stat.'' , '''21'''  (1950)  pp. 503–600</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.D. Gibbons,   "Nonparametric methods for quantitative analysis" , Holt, Rinehart &amp; Winston  (1976)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.G. Kendall,   "A new measure of rank correlation"  ''Biometrika'' , '''30'''  (1938)  pp. 81–93</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M.G. Kendall,   "Rank correlation methods" , Charles Griffin  (1970)  (Edition: Fourth)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  W.H. Kruskal,   "Ordinal measures of association"  ''J. Amer. Statist. Assoc.'' , '''53'''  (1958)  pp. 814–861</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  R.B. Nelsen,   "An introduction to copulas" , Springer  (1999)</TD></TR></table>