
KENDALLS TAUName:
\( \frac{Y_j  Y_i}{X_j  X_i} \) < 0  pair is discordant \( \frac{Y_j  Y_i}{X_j  X_i} \) = 0  pair is considered a tie X_{i} = X_{j}  pair is not compared Kendall's tau is computed as
with N_{c} and N_{d} denoting the number of concordant pairs and the number of discordant pairs, respectively, in the sample. Ties add 0.5 to both the concordant and discordant counts. There are \( \left( \begin{array}{c} n \\ 2 \end{array} \right) \) possible pairs in the bivariate sample. The above definition of Kendall's tau is from Conover. This is equivalent to the Goodman and Kruskal gamma coefficient. There are several alternative definitions of Kendall's tau in the literature. In particular, Kendall's original definition, referred to as taua, is
where the \( N_c \) and \( N_d \) do not add 0.5 for tied values. The Conover formulation accounts for ties while the Kendall taua statistic does not. Kendall's taub is defined as
with \( T_x \) denoting the number of pairs tied for the first response variable only and \( T_y \) denoting the number of pairs tied for the second variable only. As with Kendall's taua, the \( N_c \) and \( N_d \) do not add 0.5 for tied values. Kendall's taub is equal to Kendall's taua when there are no ties but is preferred to Kendall's taua when there are ties. Kendall's tauc is used when the two response variables can only take a discrete number of values, but the scales for the response variables are different. For example, X can take integer values from 1 to 10 while Y can take integer values from 1 to 20. The formula for Kendall's tauc is
where m is the minimun of \( X_d \) and \( Y_d \) where \( X_d \) is the number of distinct values of X and \( Y_d \) is the number of distinct values for Y. Kendall's tau is an alternative to the Spearman's rho rank correlation. Kendall's tau or the rank correlation may be preferred to the standard correlation coefficient in the following cases:
<SUBSET/EXCEPT/FOR qualification> where <y1> is the first response variable; <y2> is the second response variable; <par> is a parameter where the computed Kendall's tau is stored; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax computes Kendall's tau as formulated by Conover.
<SUBSET/EXCEPT/FOR qualification> where <y1> is the first response variable; <y2> is the second response variable; <par> is a parameter where the computed Kendall's tau is stored; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax computes Kendall's taua.
<SUBSET/EXCEPT/FOR qualification> where <y1> is the first response variable; <y2> is the second response variable; <par> is a parameter where the computed Kendall's tau is stored; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax computes Kendall's taub.
<SUBSET/EXCEPT/FOR qualification> where <y1> is the first response variable; <y2> is the second response variable; <par> is a parameter where the computed Kendall's tau is stored; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax computes Kendall's tauc.
LET A = KENDALLS TAU Y1 Y2 SUBSET TAG > 2 LET A = KENDALLS TAU A Y1 Y2 LET A = KENDALLS TAU B Y1 Y2 LET A = KENDALLS TAU C Y1 Y2
2019/08: Added KENDALL TAU A 2019/08: Added KENDALL TAU B 2019/08: Added KENDALL TAU C . Following data from page 320 of Conover, "Practical . Nonparametric Statistics", Third Edition, 1999, Wiley. LET Y1 = DATA 7 8 4 5.5 4.5 4 5 3 2 0.5 1 LET Y2 = DATA 4 2 5 0.5 1.5 2 0 1 0 1.5 0 LET A1 = KENDALLS TAU Y1 Y2 The computed value of Kendall's tau is 0.4355.  
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Date created: 12/22/2004 