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''Spearman rho''

The non-parametric [[correlation coefficient]] (or measure of association) known as Spearman's rho was first discussed by the psychologist C. Spearman in 1904 [[#References|[a4]]] as a coefficient of correlation on ranks (cf. also [[Correlation coefficient|Correlation coefficient]]; [[Rank statistic|Rank statistic]]). 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 the [[Kendall tau metric|Kendall tau metric]] and Spearman's rho). For an historical review of Spearman's rho and related coefficients, see [[#References|[a2]]].

Spearman's rho, denoted $r _ { S }$, is computed by applying the Pearson product-moment correlation coefficient procedure to the ranks associated with a sample $\{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { n }$. Let $R _ { i } = \operatorname { rank } ( x _ { i } )$ and $S _ { i } = \operatorname { rank } ( y _ { i } )$; then computing the sample (Pearson) correlation coefficient $r$ for $\{ ( R _ { i } , S _ { i } ) \} _ { i = 1 } ^ { n }$ yields

\begin{equation*} r _{S} = \frac { \sum _ { i = 1 } ^ { n } ( R _ { i } - \overline { R } ) ( S _ { i } - \overline{S} ) } { \sqrt { \sum _ { i = 1 } ^ { n } ( R _ { i } - \overline { R } ) ^ { 2 }\cdot \sum _ { i = 1 } ^ { n } ( S _ { i } - \overline { S } ) ^ { 2 } } } = \end{equation*}

\begin{equation*} = 1 - \frac { 6 \sum _ { i = 1 } ^ { n } ( R _ { i } - S _ { i } ) ^ { 2 } } { n ( n ^ { 2 } - 1 ) }, \end{equation*}

where $\overline { R } = \sum _ { i = 1 } ^ { n } R _ { i } / n = ( n + 1 ) / 2 = \sum _ { i = 1 } ^ { n } S _ { i } / n = \overline { S }$. When ties exist in the data, the following adjusted formula for $r _ { S }$ is used:

\begin{equation*} r_{S} = \frac { n ( n ^ { 2 } - 1 ) - 6 \sum _ { i = 1 } ^ { n } ( R _ { i } - S _ { i } ) ^ { 2 } - 6 ( T + U ) } { \sqrt { n ( n ^ { 2 } - 1 ) - 12 T } \sqrt { n ( n ^ { 2 } - 1 ) - 12 U } }, \end{equation*}

where $T = \sum _ { t } t ( t ^ { 2 } - 1 ) / 12$ for $t$ the number of $X$ observations that are tied at a given rank, and $U = \sum _ { u } u ( u ^ { 2 } - 1 ) / 12$ for $u$ the number of $Y$ observations that are tied at a given rank. For details on the use of $r _ { S }$ in hypothesis testing, and for large-sample theory, see [[#References|[a1]]].

If $X$ and $Y$ are random variables (cf. [[Random variable|Random variable]]) with respective distribution functions $F _ { X }$ and $F _{Y}$, then the population parameter estimated by $r _ { S }$, usually denoted $\rho_{ S}$, is defined to be the Pearson product-moment correlation coefficient of the random variables $F _ { X } ( X )$ and $F _ { Y } ( Y )$:

\begin{equation*} \rho _ { S } = \operatorname { corr } [ F _ { X } ( X ) , F _ { Y } ( Y ) ] = \end{equation*}

\begin{equation*} = 12 \mathsf{E} [ F_{ X} ( X ) F _ { Y } ( Y ) ] - 3. \end{equation*}

Spearman's $\rho_{ S}$ is occasionally referred to as the grade correlation coefficient, since $F _ { X } ( X )$ and $F _ { Y } ( Y )$ are sometimes called the  "grades"  of $X$ and $Y$.

Like Kendall's tau, $\rho_{ S}$ is a measure of association based on the notion of concordance. One says that two pairs $( x _ { 1 } , y _ { 1 } )$ and $( x _ { 2 } , y _ { 2 } )$ of real numbers are concordant if $x _ { 1 } < x _ { 2 }$ and $y _ { 1 } < y _ { 2 }$ or if $x _ { 1 } &gt; x _ { 2 }$ and $y _ { 1 } &gt; y _ { 2 }$ (i.e., if $( x _ { 1 } - x _ { 2 } ) ( y _ { 1 } - y _ { 2 } ) &gt; 0$); and discordant if $x _ { 1 } < x _ { 2 }$ and $y _ { 1 } &gt; y _ { 2 }$ or if $x _ { 1 } &gt; x _ { 2 }$ and $y _ { 1 } < y _ { 2 }$ (i.e., if $( x _ { 1 } - x _ { 2 } ) ( y _ { 1 } - y _ { 2 } ) < 0$). Now, let $( X _ { 1 } , Y _ { 1 } )$, $( X _ { 2 } , Y _ { 2 } )$ and $( X _ { 3 } , Y _ { 3 } )$ be independent random vectors with the same distribution as $( X , Y )$. Then

\begin{equation*} \rho _ { S } = 3 \mathsf{P} [ ( X _ { 1 } - X _ { 2 } ) ( Y _ { 1 } - Y _ { 3 } ) &gt; 0 ] + \end{equation*}

\begin{equation*} - 3 \mathsf{P} [ ( X _ { 1 } - X _ { 2 } ) ( Y _ { 1 } - Y _ { 3 } ) < 0 ], \end{equation*}

that is, $\rho_{ S}$ is proportional to the difference between the probabilities of concordance and discordance between the random vectors $( X _ { 1 } , Y _ { 1 } )$ and $( X _ { 2 } , Y _ { 3 } )$ (clearly, $( X _ { 2 } , Y _ { 3 } )$ can be replaced by $( X _ { 3 } , Y _ { 2 } )$).

When $X$ and $Y$ are continuous,

\begin{equation*} \rho _ { S } = 12 \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } u v d C _ { X , Y } ( u , v ) - 3 = \end{equation*}

\begin{equation*} = 12 \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } [ C _ { X , Y } ( u , v ) - u v ] d u d v, \end{equation*}

where $C _ { X , Y }$ is the [[Copula|copula]] of $X$ and $Y$. Consequently, $\rho_{ S}$ is invariant under strictly increasing transformations of $X$ and $Y$, a property $\rho_{ S}$ shares with Kendall's tau but not with the Pearson product-moment correlation coefficient. Note that $\rho_{ S}$ is proportional to the signed volume between the graphs of the copula $C _ { X , Y } ( u , v )$ and the  "product"  copula $\Pi ( u , v ) = u v$, the copula of independent random variables. For a survey of copulas and their relationship with measures of association, see [[#References|[a3]]].

Spearman [[#References|[a5]]] also proposed an $L_1$ version of $r _ { S }$, known as Spearman's footrule, based on absolute differences $| R _ { i } - S _ { i } |$ in ranks rather than squared differences:

\begin{equation*} f _ { S } = 1 - \frac { 3 \sum _ { i = 1 } ^ { n } | R _ { i } - S _ { i } | } { n ^ {2} - 1 }. \end{equation*}

The population parameter $\phi_S$ estimated by $f _ { S }$ is given by

\begin{equation*} \phi _ { S } = 1 - 3 \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } | u - v | d C _ { X , Y } \gamma ( u , v ) = \end{equation*}

\begin{equation*} = 6 \int _ { 0 } ^ { 1 } C _ { X , Y } ( u , u ) d u - 2. \end{equation*}

====References====
<table>
<tr><td valign="top">[a1]</td> <td valign="top">  J.D. Gibbons,   "Nonparametric methods for quantitative analysis" , Holt, Rinehart &amp; Winston  (1976)</td></tr><tr><td valign="top">[a2]</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">[a3]</td> <td valign="top">  R.B. Nelsen,   "An introduction to copulas" , Springer  (1999)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  C. Spearman, "The proof and measurement of association between two things"  ''Amer. J. Psychol.'' , '''15'''  (1904)  pp. 72–101</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  C. Spearman,   "A footrule for measuring correlation"  ''Brit. J. Psychol.'' , '''2'''  (1906)  pp. 89–108</td></tr>
</table>