
KENDALL TAU INDEPENDENCE TESTName:
\( \frac{Y_j  Y_i}{X_j  X_i} \) < 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.
A value of +1 indicates that all pairs are concordant, a value of 1 indicates that all pairs are discordant, and a value of 0 indicates no relation (i.e., independence). The Kendall tau independence test is a test of whether the Kendall tau coefficient is equal to zero. For larger n (e.g., n > 60) or the case where there are many ties, the pth upper quantile of the Kendall tau statistic can be approximated by
with z_{p} and n denoting the pth quantile of the standard normal distribution and the sample size, respectively. The lower quantile is the negative of the upper quantile. For a twosided test, the pvalue is computed as twice the minimum of the lower tailed and upper tailed quantiles. For n ≤ 60, tabulated quantiles (from Table A11 on pp. 543544 of Conover) are used. These quantiles are exact when there are no ties in the data.
<y1> <y2> <SUBSET/EXCEPT/FOR qualification> where <LOWER TAILED/UPPER TAILED> is an optional keyword that specifies either a lower tailed or an upper tailed test; <y1> is the first response variable; <y2> is the second response variable; and where the <SUBSET/EXCEPT/FOR qualification> is optional. If neither LOWER TAILED or UPPER TAILED is specified, a twotailed test is performed. Lower tailed tests are used to test for discordance (i.e., negative correlation) and upper tailed tests are used to test for concordance (i.e., positive correlation).
<y1> ... <yk> <SUBSET/EXCEPT/FOR qualification> where <LOWER TAILED/UPPER TAILED> is an optional keyword that specifies either a lower tailed or an upper tailed test; <y1> ... <yk> is a list of 1 to 30 response variables; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax will perform all the pairwise tests for the <y1> ... <yk> response variables. For example,
is equivalent to
KENDALL TAU INDEPENDENCE TEST Y1 Y3 KENDALL TAU INDEPENDENCE TEST Y1 Y4 KENDALL TAU INDEPENDENCE TEST Y2 Y3 KENDALL TAU INDEPENDENCE TEST Y2 Y4 KENDALL TAU INDEPENDENCE TEST Y3 Y4
KENDALL TAU INDEPENDENCE TEST Y1 TO Y5 LOWER TAILED KENDALL TAU INDEPENDENCE TEST Y1 Y2 UPPER TAILED KENDALL TAU INDEPENDENCE TEST Y1 Y2
LET X = SEQUENCE 1 1 N KENDALL TAU INDEPENDENCE TEST Y X According to Conover, this test is more powerful than the Cox and Stuart test. However, it is not as widely applicable as the Cox and Stuart test.
versus
or
Although the JonckheereTerpstra test is based only on the number of concordant pairs (N_{c} above), applying the Kendall tau independence test gives an equivalent result. For Dataplot, if Y is a response variable and X is a groupid variable, then to test for "less than" (positive concordance), use
and to test for "greater than" (negative concordance), use
The CORRELATION CONFIDENCE LIMITS command can be used to generate a confidence interval for the Pearson correlation coefficient. This can be used for a parametric test for independence (i.e., does the confidence interval contain zero?).
can be used to specify that they should be based on the normal approximation given above. This may be preferred if there are ties in the data. To reset the default, enter the command
LET A = KENDALL TAU CDF Y1 Y2 LET A = KENDALL TAU PVALUE Y1 Y2 LET A = KENDALL TAU LOWER TAILED PVALUE Y1 Y2 LET A = KENDALL TAU UPPER TAILED PVALUE Y1 Y2 The cdf and pvalues are based on the normal approximation given above. To see a list of commands in which these statistics can be used, enter
The paired data can also be analyzed using other techniques for comparing two response variables (e.g., ttest, bihistogram, quantilequantile plot).
read kendall.dat y1 y2 set write decimals 5 . let statval = kendall tau y1 y2 let statcdf = kendall tau cdf y1 y2 let pvalue = kendall tau pvalue y1 y2 let pvallt = kendall tau lower tailed pvalue y1 y2 let pvalut = kendall tau upper tailed pvalue y1 y2 print statval statcdf pvalue pvallt pvalut . kendall tau independence test y1 y2 upper tailed kendall tau independence test y1 y2 . set kendall tau critical values normal approximation upper tailed kendall tau independence test y1 y2The following output is generated. PARAMETERS AND CONSTANTS STATVAL  0.43548 STATCDF  0.97563 PVALUE  0.04873 PVALLT  0.97563 PVALUT  0.02437 Two Sample Kendall Tau Test for Independence First Response Variable: Y1 Second Response Variable: Y2 H0: The Two Samples are Independent Ha: Pairs of Samples Tend to be Either Concordant or Discordant Number of Observations: 12 Sample One Summary Statistics: Sample Mean: 587.08333 Sample Standard Deviation: 58.01482 Sample Minimum: 530.00000 Sample Maximum: 740.00000 Sample Two Summary Statistics: Sample Mean: 3.59999 Sample Standard Deviation: 0.28603 Sample Minimum: 3.20000 Sample Maximum: 4.00000 Test: Kendall Tau Test Statistic Value: 0.43548 CDF Value (Normal Approximation): 0.97563 TwoSided PValue (Normal Approximation): 0.04873 Conclusions (TwoTailed Test) H0: Samples are Independent  Null Significance Test Critical Hypothesis Level Statistic Region (+/) Conclusion  80.0% 0.43548 0.27270 REJECT 90.0% 0.43548 0.36360 REJECT 95.0% 0.43548 0.42420 REJECT 99.0% 0.43548 0.54550 ACCEPT Two Sample Kendall Tau Test for Independence First Response Variable: Y1 Second Response Variable: Y2 H0: The Two Samples are Independent Ha: Pairs of Samples Tend to be Concordant Number of Observations: 12 Sample One Summary Statistics: Sample Mean: 587.08333 Sample Standard Deviation: 58.01482 Sample Minimum: 530.00000 Sample Maximum: 740.00000 Sample Two Summary Statistics: Sample Mean: 3.59999 Sample Standard Deviation: 0.28603 Sample Minimum: 3.20000 Sample Maximum: 4.00000 Test: Kendall Tau Test Statistic Value: 0.43548 CDF Value (Normal Approximation): 0.97563 Upper Tailed PValue (Normal Approximation): 0.02436 Conclusions (Upper 1Tailed Test) H0: Samples are Independent  Null Significance Test Critical Hypothesis Level Statistic Region (>) Conclusion  90.0% 0.43548 0.27270 REJECT 95.0% 0.43548 0.36360 REJECT 97.5% 0.43548 0.42420 REJECT 99.0% 0.43548 0.51520 ACCEPT 99.5% 0.43548 0.54550 ACCEPT Two Sample Kendall Tau Test for Independence First Response Variable: Y1 Second Response Variable: Y2 H0: The Two Samples are Independent Ha: Pairs of Samples Tend to be Concordant Number of Observations: 12 Sample One Summary Statistics: Sample Mean: 587.08333 Sample Standard Deviation: 58.01482 Sample Minimum: 530.00000 Sample Maximum: 740.00000 Sample Two Summary Statistics: Sample Mean: 3.59999 Sample Standard Deviation: 0.28603 Sample Minimum: 3.20000 Sample Maximum: 4.00000 Test: Kendall Tau Test Statistic Value: 0.43548 CDF Value (Normal Approximation): 0.97563 Upper Tailed PValue (Normal Approximation): 0.02436 Conclusions (Upper 1Tailed Test) H0: Samples are Independent  Null Significance Test Critical Hypothesis Level Statistic Region (>) Conclusion  90.0% 0.43548 0.28316 REJECT 95.0% 0.43548 0.36344 REJECT 97.5% 0.43548 0.43306 REJECT 99.0% 0.43548 0.51402 ACCEPT 99.5% 0.43548 0.56914 ACCEPT  
Date created: 03/08/2013 Last updated: 12/11/2023 Please email comments on this WWW page to alan.heckert@nist.gov. 