Rank correlation

ordinal associationrank correlation coefficientrank regression
In statistics, a rank correlation is any of several statistics that measure an ordinal association—the relationship between rankings of different ordinal variables or different rankings of the same variable, where a "ranking" is the assignment of the ordering labels "first", "second", "third", etc. to different observations of a particular variable.wikipedia
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Ranking

rankrankedrankings
In statistics, a rank correlation is any of several statistics that measure an ordinal association—the relationship between rankings of different ordinal variables or different rankings of the same variable, where a "ranking" is the assignment of the ordering labels "first", "second", "third", etc. to different observations of a particular variable.
A rank correlation can be used to compare two rankings for the same set of objects.

Correlation and dependence

correlationcorrelatedcorrelations
Rank correlation coefficients, such as Spearman's rank correlation coefficient and Kendall's rank correlation coefficient measure the extent to which, as one variable increases, the other variable tends to increase, without requiring that increase to be represented by a linear relationship.

Mann–Whitney U test

Mann–Whitney UWilcoxon rank-sum testMann-Whitney U test
For example, two common nonparametric methods of significance that use rank correlation are the Mann–Whitney U test and the Wilcoxon signed-rank test.
A method of reporting the effect size for the Mann–Whitney U test is with a measure of rank correlation known as the rank-biserial correlation.

Spearman's rank correlation coefficient

Spearman's rank correlationrank correlation coefficientSpearman
which is exactly Spearman's rank correlation coefficient \rho.
In statistics, Spearman's rank correlation coefficient or Spearman's rho, named after Charles Spearman and often denoted by the Greek letter \rho (rho) or as r_s, is a nonparametric measure of rank correlation (statistical dependence between the rankings of two variables).

Goodman and Kruskal's gamma

Gamma test (statistics)Yule's ''Qgamma
In statistics, Goodman and Kruskal's gamma is a measure of rank correlation, i.e., the similarity of the orderings of the data when ranked by each of the quantities.

Somers' D

d yx /d xy Somers' ''D
In statistics, Somers’ D, sometimes incorrectly referred to as Somer’s D, is a measure of ordinal association between two possibly dependent random variables X and Y. Somers’ D takes values between -1 when all pairs of the variables disagree and 1 when all pairs of the variables agree.

Kendall rank correlation coefficient

Kendall tau rank correlation coefficientKendall's tauKendall's τ
The sum is the number of concordant pairs minus the number of discordant pairs (see Kendall tau rank correlation coefficient).
In statistics, the Kendall rank correlation coefficient, commonly referred to as Kendall's tau coefficient (after the Greek letter τ), is a statistic used to measure the ordinal association between two measured quantities.

Statistics

statisticalstatistical analysisstatistician
In statistics, a rank correlation is any of several statistics that measure an ordinal association—the relationship between rankings of different ordinal variables or different rankings of the same variable, where a "ranking" is the assignment of the ordering labels "first", "second", "third", etc. to different observations of a particular variable.

Ordinal data

ordinalordinal variableordered categorical data
In statistics, a rank correlation is any of several statistics that measure an ordinal association—the relationship between rankings of different ordinal variables or different rankings of the same variable, where a "ranking" is the assignment of the ordering labels "first", "second", "third", etc. to different observations of a particular variable.

Statistical significance

statistically significantsignificantsignificance level
A rank correlation coefficient measures the degree of similarity between two rankings, and can be used to assess the significance of the relation between them.

Nonparametric statistics

non-parametricnon-parametric statisticsnonparametric
For example, two common nonparametric methods of significance that use rank correlation are the Mann–Whitney U test and the Wilcoxon signed-rank test.

Wilcoxon signed-rank test

Wilcoxon Signed Rank Testsigned-ranksWilcoxon
For example, two common nonparametric methods of significance that use rank correlation are the Mann–Whitney U test and the Wilcoxon signed-rank test.

Contingency table

cross tabulationcontingency tablescrosstab
As another example, in a contingency table with low income, medium income, and high income in the row variable and educational level—no high school, high school, university—in the column variable), a rank correlation measures the relationship between income and educational level.

Coefficient

coefficientsleading coefficientfactor
An increasing rank correlation coefficient implies increasing agreement between rankings.

Permutation

permutationscycle notationpermuted
Following, a ranking can be seen as a permutation of a set of objects.

Set (mathematics)

setsetsmathematical set
Following, a ranking can be seen as a permutation of a set of objects.

Symmetric group

symmetricinfinite symmetric groupS 4
Thus we can look at observed rankings as data obtained when the sample space is (identified with) a symmetric group.

Metric (mathematics)

metricdistance functionmetrics
We can then introduce a metric, making the symmetric group into a metric space.

Metric space

metricmetric spacesmetric geometry
We can then introduce a metric, making the symmetric group into a metric space.

Frobenius inner product

FrobeniusFrobenius inner productsmatrix inner product
where is the Frobenius inner product and the Frobenius norm.

Matrix norm

Frobenius normnormspectral norm
where is the Frobenius inner product and the Frobenius norm.

Stochastic empirical loading and dilution model

In SELDM, these three treatment variables are modeled by using the trapezoidal distribution and the rank correlation with the associated highway-runoff variables.

Maurice Kendall

M. G. KendallMaurice George KendallM.G. Kendall
During this period he also began work on the rank correlation coefficient which currently bears his name (Kendall's tau), which eventually led to a monograph on Rank Correlation in 1948.

Bivariate analysis

bivariate
If both variables are ordinal, meaning they are ranked in a sequence as first, second, etc., then a rank correlation coefficient can be computed.

Jonckheere's trend test

JonckheereJonckheere test for ordered alternatives
The test can be seen as a special case of Maurice Kendall’s more general method of rank correlation and makes use of the Kendall’s S statistic.