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SPEARMAN'S RHO
 = 3 \bigl(\textrm{P}\bigl((X_1 - X_2)(Y_1 - Y_3) > 0\bigr) - \textrm{P}\bigl((X_1 - X_2)(Y_1 - Y_3) < 0\bigr)\bigr).\ \textbf{(1)})
If
and
have continuous marginal distributions then
 )
has the same units as Pearson's correlation. Just like Pearson's correlation it covers the whole range of [-1,1], but now -1 corresponds to a perfect negative relationship (
is any decreasing deterministic function of ) and 1 corresponds to a perfect positive relationship ( is any increasing deterministic function of ). When
or
has a discrete mass, interval [-1,1] is not covered fully. For example, if variable
takes a given value with positive probability p, then with probability of at least p2 there is a tie:
And so
falls into interval [-1 + p2, 1 - p2]
no matter what the bivariate relationship is. There are several proposals on how to adjust for ties, the most obvious one being to divide formula (1) by
(Y_1 - Y_3) = 0\bigr).)
Still, no single generalization has been widely accepted.
Note that definition (1) depends on ranks only. We only care if
is bigger than
the actual values being irrelevant. So Spearman's rho is invariant to any monotonically increasing nonlinear transformations of
and 
If we raise
to the third power Spearman's rho will stay the same. This is very important. Spearman's rho is naturally built to capture the strength of highly nonlinear relationships, where traditional linear association measures fail. The following graph illustrates the fact.
Spearman's rho has direct relation to the copula function)
generated by random variables
and
 = 12 \int\int_{[0,1]^2} u v\ dC(u,v) - 3)
The copula function does not depend on marginal distributions and captures what happens to
and
if they are transformed into random variables uniformly distributed on [0,1]. The formula above signals once again that Spearman's rho does not depend on marginal distributions of
and
and is invariant to any monotonically increasing transformations of
and
When the joint distribution of
and
is unknown Spearman's rho can be estimated from the data as the correlation of ranks.
Let
, ..., (X_n,Y_n))
denote the data and let
rank of 
in
rank of in
The estimator of Spearman's rho is given by
 (r_{yi} - \bar{r_y})}{\sqrt{\sum_{i=1,..,n} (r_{xi} - \bar{r_x})^2 \sum_{i=1,..,n} (r_{xy} - \bar{r_y})^2}}.)
Identical values are assigned the same fractional rank, which is equal to the average of their positions in the ascending order of the values. For that reason the estimator is suitable for both discrete and continuous distributions. As the sample size converges to infinity, the estimator converges to the true Spearman's rho and can be used to test if the true Spearman's rho equals 0.
The sample rank correlation coefficient is a nonparametric estimator in the sense that no assumptions are made about the joint distribution of
and
In particular, no functional form is postulated for the conditional expectation of
given
and the conditional expectation of
given
SPEARMAN'S RHO REFERENCES
Nelsen, R. B. (2006). An Introduction to Copulas (2nd ed). New York: Springer.
Salvadori, G., De Michele, C., Kottegoda, N. T., & Rosso, R. (2007). Extremes in Nature: An Approach Using Copulas. Springer.
Corder, G.W., & Foreman, D.I. (2014). Nonparametric Statistics: A Step-by-Step Approach. Wiley, Hoboken, New Jersey.
Gibbons, J. D., & Chakraborti, S. (2003). Nonparametric Statistical Inference (4th ed). New York: Marcel Dekker.
Nešlehová, J. (2007). On Rank Correlation Measures for Non-continuous Random Variables. Journal of Multivariate Analysis, Vol. 98, Issue 3, pp. 544-567.
Spearman, C. (1904). The proof and measurement of association between two things. American Journal of Psychology, Vol. 15, pp. 72–101.
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Spearman's Rho (Spearman's Rank Correlation Coefficient) is a measure of nonlinear dependence between two random variables. If random variables
and
have joint distribution
and random vectors
and
are independent realizations from that distribution, then Spearman's rho of
and
equals
If
Still, no single generalization has been widely accepted.
Note that definition (1) depends on ranks only. We only care if
Spearman's rho has direct relation to the copula function
The copula function does not depend on marginal distributions and captures what happens to
When the joint distribution of
The estimator of Spearman's rho is given by
Identical values are assigned the same fractional rank, which is equal to the average of their positions in the ascending order of the values. For that reason the estimator is suitable for both discrete and continuous distributions. As the sample size converges to infinity, the estimator converges to the true Spearman's rho and can be used to test if the true Spearman's rho equals 0.
The sample rank correlation coefficient is a nonparametric estimator in the sense that no assumptions are made about the joint distribution of
SPEARMAN'S RHO REFERENCES
Nelsen, R. B. (2006). An Introduction to Copulas (2nd ed). New York: Springer.
Salvadori, G., De Michele, C., Kottegoda, N. T., & Rosso, R. (2007). Extremes in Nature: An Approach Using Copulas. Springer.
Corder, G.W., & Foreman, D.I. (2014). Nonparametric Statistics: A Step-by-Step Approach. Wiley, Hoboken, New Jersey.
Gibbons, J. D., & Chakraborti, S. (2003). Nonparametric Statistical Inference (4th ed). New York: Marcel Dekker.
Nešlehová, J. (2007). On Rank Correlation Measures for Non-continuous Random Variables. Journal of Multivariate Analysis, Vol. 98, Issue 3, pp. 544-567.
Spearman, C. (1904). The proof and measurement of association between two things. American Journal of Psychology, Vol. 15, pp. 72–101.
BACK TO THE STATISTICAL ANALYSES DIRECTORY
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