KENDALLS TAU

KENDALLS TAU (LET)

Let Subcommand

Compute Kendall's tau coefficient between two paired variables.

Kendall's tau coefficient is a measure of concordance between two paired variables. Given the pairs (X_i,Y_i) and (X_j,Y_j), then
\( \frac{Y_j - Y_i}{X_j - X_i} \) > 0 - pair is concordant

\( \frac{Y_j - Y_i}{X_j - X_i} \) < 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

\( \tau = \frac{N_c - N_d}{N_c + N_d} \)

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 tau-a, is

\( \tau_a = \frac{N_c - N_d}{N(N-1)/2} \)

where the \( N_c \) and \( N_d \) do not add 0.5 for tied values.

The Conover formulation accounts for ties while the Kendall tau-a statistic does not.

Kendall's tau-b is defined as

\( \tau_b = \frac{N_c - N_d} {\sqrt{(N_c + N_d + T_x) (N_c + N_d + T_y)}} \)

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 tau-a, the \( N_c \) and \( N_d \) do not add 0.5 for tied values. Kendall's tau-b is equal to Kendall's tau-a when there are no ties but is preferred to Kendall's tau-a when there are ties.

Kendall's tau-c 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 tau-c is

\( \tau_c = \frac{2(N_c - N_d)} {N^2 (m - 1)/m} \)

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:

1. When the underlying data does not have a meaningful numerical measure, but it can be ranked;

2. When the relationship between the two variables is not linear;

3. When the normality assumption for two variables is not valid.

LET <par> = KENDALLS TAU <y1> <y2>
                        <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.

LET <par> = KENDALLS TAU A <y1> <y2>
                        <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-a.

Note that since Kendall's tau-a is equivalent to Kendall's tau-b when there are no ties and it assumes there are no ties, the Kendall's tau-a will simply return the Kendall's tau-b value.

LET <par> = KENDALLS TAU B <y1> <y2>
                        <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-b.

LET <par> = KENDALLS TAU C <y1> <y2>
                        <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-c.

LET A = KENDALLS TAU Y1 Y2
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

The two variables must have the same number of elements.

Dataplot implements Kendall's tau by computing all the pairwise slopes. Although straightforward, this is not a particularly efficient computational algorithm.

The 2026/04 version of Dataplot added the capability to compute Kendall's tau-b with a more efficient algorithm. Specifically, it adapts the ktau.f routine from the R mcr package (some minor code modifications were made for incorporation into Dataplot). For larger sample sizes, this can dramatically reduce the computational time required. Note that the computational time for smaller data sets is typically not an issue. To specify the sample size at which the more efficient algorithm will be used (i.e., any sample size larger than or equal to this will use the more efficient algorithm), enter the command

SET KENDALL TAU CUTOFF <value>

The default value is 100.

Dataplot statistics can be used in a number of commands. For details, enter
HELP STATISTICS

None

None

CORRELATION (LET)	= Compute the Pearson correlation coefficient between two variables.
RANK CORRELATION (LET)	= Compute the Spearman's rho correlation coefficient between two variables.
STATISTICS PLOT	= Generate a statistic versus subset plot.
BOOTSTRAP PLOT	= Generate a bootstrap plot.
TABULATE	= Perform a tabulation for a specified statistic.

Conover (1999), "Practical Nonparametric Statistics," Third Edition, Wiley, pp. 319-323.

Exploratory Data Analysis

2004/10
2019/08: Added KENDALL TAU A
2019/08: Added KENDALL TAU B
2019/08: Added KENDALL TAU C
2026/04: More efficient algorithm for Kendall's Tau-B

 
. 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.