Spearman coefficient of rank correlation
A measure of the dependence of two random variables 
and 
, based on the rankings of the 
's and 
's in independent pairs of observations 
. If 
is the rank of 
corresponding to that pair 
for which the rank of 
is equal to 
, then the Spearman coefficient of rank correlation is defined by the formula
 |
or, equivalently, by
 |
where 
is the difference between the ranks of 
and 
. The value of 
lies between 
and 
; 
when the rank sequences completely coincide, i.e. 
, 
; and 
when the rank sequences are completely opposite, i.e. 
, 
. This coefficient, like any other rank statistic, is applied to test the hypothesis of independence of two variables. If the variables are independent, then 
, and 
. Thus, the amount of deviation of 
from zero gives information about the dependence or independence of the variables. To construct the corresponding test one computes the distribution of 
for independent variables 
and 
. When 
one can use tables of the exact distribution (see [2], [4]), and when 
one can take advantage, for example, of the fact that as 
the random variable 
is asymptotically distributed as a standard normal distribution. In the latter case the hypothesis of independence is rejected if 
, where 
is the root of the equation 
and 
is the standard normal distribution function.
Under the assumption that 
and 
have a joint normal distribution with (ordinary) correlation coefficient 
,
 |
as 
, and therefore the variable 
can be used as an estimator for 
.
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 are asymptotically equivalent (when 
, the corresponding rank statistics coincide).
References
| [1] | C. Spearman, "The proof and measurement of association between two rings" Amer. J. Psychol. , 15 (1904) pp. 72–101 |
| [2] | M.G. Kendall, "Rank correlation methods" , Griffin (1962) |
| [3] | B.L. van der Waerden, "Mathematische Statistik" , Springer (1957) |
| [4] | 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) |
Comments
References
| [a1] | J. Hájek, Z. Sidák, "Theory of rank tests" , Acad. Press (1967) |
| [a2] | M. Hollander, D.A. Wolfe, "Nonparametric statistical methods" , Wiley (1973) |
Spearman coefficient of rank correlation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spearman_coefficient_of_rank_correlation&oldid=15078