View source for Spearman coefficient of rank correlation

A measure of the dependence of two random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s0862501.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s0862502.png" />, based on the rankings of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s0862503.png" />'s and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s0862504.png" />'s in independent pairs of observations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s0862505.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s0862506.png" /> is the [[Rank|rank]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s0862507.png" /> corresponding to that pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s0862508.png" /> for which the rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s0862509.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625010.png" />, then the Spearman coefficient of rank correlation is defined by the formula

<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/s/s086/s086250/s08625011.png" /></td> </tr></table>

or, equivalently, by

<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/s/s086/s086250/s08625012.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625013.png" /> is the difference between the ranks of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625015.png" />. The value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625016.png" /> lies between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625018.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625019.png" /> when the rank sequences completely coincide, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625021.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625022.png" /> when the rank sequences are completely opposite, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625024.png" />. This coefficient, like any other [[Rank statistic|rank statistic]], is applied to test the hypothesis of independence of two variables. If the variables are independent, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625025.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625026.png" />. Thus, the amount of deviation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625027.png" /> from zero gives information about the dependence or independence of the variables. To construct the corresponding test one computes the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625028.png" /> for independent variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625030.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625031.png" /> one can use tables of the exact distribution (see [[#References|[2]]], [[#References|[4]]]), and when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625032.png" /> one can take advantage, for example, of the fact that as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625033.png" /> the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625034.png" /> is asymptotically distributed as a standard normal distribution. In the latter case the hypothesis of independence is rejected if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625035.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625036.png" /> is the root of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625038.png" /> is the standard [[Normal distribution|normal distribution]] function.

Under the assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625040.png" /> have a joint normal distribution with (ordinary) [[Correlation coefficient|correlation coefficient]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625041.png" />,

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as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625043.png" />, and therefore the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625044.png" /> can be used as an estimator for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625045.png" />.

The Spearman coefficient of rank correlation was named in honour of the psychologist C. Spearman (1904), who used it in research on psychology in place of the ordinary correlation coefficient. The tests based on the Spearman coefficient of rank correlation and on the [[Kendall coefficient of rank correlation|Kendall coefficient of rank correlation]] are asymptotically equivalent (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086250/s08625046.png" />, the corresponding rank statistics coincide).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Spearman,   "The proof and measurement of association between two rings"  ''Amer. J. Psychol.'' , '''15'''  (1904)  pp. 72–101</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.G. Kendall,   "Rank correlation methods" , Griffin  (1962)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.L. van der Waerden,   "Mathematische Statistik" , Springer  (1957)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L.N. Bol'shev,   N.V. Smirnov,   "Tables of mathematical statistics" , ''Libr. math. tables'' , '''46''' , Nauka  (1983)  (In Russian)  (Processed by L.S. Bark and E.S. Kedrova)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Hájek,   Z. Sidák,   "Theory of rank tests" , Acad. Press  (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Hollander,   D.A. Wolfe,   "Nonparametric statistical methods" , Wiley  (1973)</TD></TR></table>