statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

, a concordant pair is a pair of observations, each on two variables, (X_''1'',Y_''1'') and (X_''2'',Y_''2''), having the property that :

\sgn (X_2 - X_1)\ = \sgn (Y_2 - Y_1),

where "sgn" refers to whether a number is positive, zero, or negative (its sign). Specifically, the

signum function In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is often represented as . To avo ...

, often represented as sgn, is defined as: :

\sgn x = \begin 
-1,  &  x < 0 \\
0 ,  &  x = 0 \\
1 ,  &  x > 0
\end

That is, in a concordant pair, both elements of one pair are either greater than, equal to, or less than the corresponding elements of the other pair. In contrast, a discordant pair is a pair of two-variable observations such that :

\sgn (X_2 - X_1)\ = - \sgn (Y_2 - Y_1).

That is, if one pair contains a higher value of ''X'' then the other pair contains a higher value of ''Y''.

Uses

The

Kendall tau distance The Kendall tau rank distance is a metric (distance function) that counts the number of pairwise disagreements between two ranking lists. The larger the distance, the more dissimilar the two lists are. Kendall tau distance is also called bubble-sor ...

between two series is the total number of discordant pairs. The

Kendall tau rank correlation coefficient In statistics, the Kendall rank correlation coefficient, commonly referred to as Kendall's τ coefficient (after the Greek letter τ, tau), is a statistic used to measure the ordinal association between two measured quantities. A τ test is a n ...

, which measures how closely related two series of numbers are, is proportional to the difference between the number of concordant pairs and the number of discordant pairs. An estimate of

Goodman and Kruskal's gamma 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. It measures the strength of association of the cross tabulated data when both va ...

, another measure of

rank correlation 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 o ...

, is given by the ratio of the difference to the sum of the numbers of concordant and discordant pairs. Somers' ''D'' is another similar but asymmetric measure given by the ratio of the difference in the number of concordant and discordant pairs to the number of pairs with unequal values for one of the two variables.

References

*Abdi, Hervé (2007)
"The Kendall Rank Correlation Coefficient".
In: Neil Salkind (Ed.), ''Encyclopedia of Measurement and Statistics''. Thousand Oaks (CA): Sage. * Kendall, M. (1948) ''Rank Correlation Methods'', Charles Griffin & Company Limited * Kendall, M. (1938) "A New Measure of Rank Correlation", ''

Biometrika ''Biometrika'' is a peer-reviewed scientific journal published by Oxford University Press for thBiometrika Trust The editor-in-chief is Paul Fearnhead (Lancaster University). The principal focus of this journal is theoretical statistics. It was es ...

'', 30:81-89. * {{cite journal, last1=Newson, first1=Roger, title=Parameters behind "nonparametric" statistics: Kendall's tau, Somers' ''D'' and median differences, journal=Stata Journal, date=2002, volume=2, issue=1, pages=45–64, url=http://www.stata-journal.com/article.html?article=st0007

External links

MacTutor: David George Kendall

Janus: The Papers of Professor David Kendall
Nonparametric statistics

Uses

See also

References

External links