Spearman rho metric
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 [a4] as a coefficient of correlation on ranks (cf. also Correlation coefficient; 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), while "measure of association" refers to a measure of a monotone relationship between variates (such as the Kendall tau metric and Spearman's rho). For an historical review of Spearman's rho and related coefficients, see [a2].
Spearman's rho, denoted , is computed by applying the Pearson product-moment correlation coefficient procedure to the ranks associated with a sample
. Let
and
; then computing the sample (Pearson) correlation coefficient
for
yields
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where . When ties exist in the data, the following adjusted formula for
is used:
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where for
the number of
observations that are tied at a given rank, and
for
the number of
observations that are tied at a given rank. For details on the use of
in hypothesis testing, and for large-sample theory, see [a1].
If and
are random variables (cf. Random variable) with respective distribution functions
and
, then the population parameter estimated by
, usually denoted
, is defined to be the Pearson product-moment correlation coefficient of the random variables
and
:
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Spearman's is occasionally referred to as the grade correlation coefficient, since
and
are sometimes called the "grades" of
and
.
Like Kendall's tau, is a measure of association based on the notion of concordance. One says that two pairs
and
of real numbers are concordant if
and
or if
and
(i.e., if
); and discordant if
and
or if
and
(i.e., if
). Now, let
,
and
be independent random vectors with the same distribution as
. Then
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that is, is proportional to the difference between the probabilities of concordance and discordance between the random vectors
and
(clearly,
can be replaced by
).
When and
are continuous,
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where is the copula of
and
. Consequently,
is invariant under strictly increasing transformations of
and
, a property
shares with Kendall's tau but not with the Pearson product-moment correlation coefficient. Note that
is proportional to the signed volume between the graphs of the copula
and the "product" copula
, the copula of independent random variables. For a survey of copulas and their relationship with measures of association, see [a3].
Spearman [a5] also proposed an version of
, known as Spearman's footrule, based on absolute differences
in ranks rather than squared differences:
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The population parameter estimated by
is given by
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References
[a1] | J.D. Gibbons, "Nonparametric methods for quantitative analysis" , Holt, Rinehart & Winston (1976) |
[a2] | W.H. Kruskal, "Ordinal measures of association" J. Amer. Statist. Assoc. , 53 (1958) pp. 814–861 |
[a3] | R.B. Nelsen, "An introduction to copulas" , Springer (1999) |
[a4] | C. Spearman, "The proof and measurement of association between two things" Amer. J. Psychol. , 15 (1904) pp. 72–101 |
[a5] | C. Spearman, "A footrule for measuring correlation" Brit. J. Psychol. , 2 (1906) pp. 89–108 |
Spearman rho metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spearman_rho_metric&oldid=15466