KENDALL TAU INDEPENDENCE TEST

KENDALL TAU INDEPENDENCE TEST

Analysis Command

Perform a Kendall tau test for whether two samples are independent (i.e., not correlated).

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.

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 p-th upper quantile of the Kendall tau statistic can be approximated by

\( w_{p} = z_{p} \frac{\sqrt{2(2n + 5)}}{3\sqrt{n(n-1)}} \)

with z_p and n denoting the p-th quantile of the standard normal distribution and the sample size, respectively. The lower quantile is the negative of the upper quantile.

For a two-sided test, the p-value is computed as twice the minimum of the lower tailed and upper tailed quantiles.

For n ≤ 60, tabulated quantiles (from Table A11 on pp. 543-544 of Conover) are used. These quantiles are exact when there are no ties in the data.

<LOWER TAILED/UPPER TAILED> KENDALL TAU INDEPENDENCE TEST
                        <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 two-tailed 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).

<LOWER TAILED/UPPER TAILED> KENDALL TAU INDEPENDENCE TEST
                        <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 pair-wise tests for the <y1> ... <yk> response variables. For example,

KENDALL TAU INDEPENDENCE TEST Y1 TO Y4

is equivalent to

KENDALL TAU INDEPENDENCE TEST Y1 Y2
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 Y2
KENDALL TAU INDEPENDENCE TEST Y1 TO Y5
LOWER TAILED KENDALL TAU INDEPENDENCE TEST Y1 Y2
UPPER TAILED KENDALL TAU INDEPENDENCE TEST Y1 Y2

This command can be used to test for trend in a univariate variable. For example
LET N = SIZE Y
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.

This test can be used to perform the Jonckheere-Terpstra test. Given two or more independent samples, the Jonckheere-Terpstra is used to test the null hypothesis that all the samples came from the same distribution against the ordered alternative that the distributions differ in a specified direction. That is
H0: F1(x) = F2(x) = .... = Fk(x)

versus

Ha: F1(x) <= F2(x) <= .... <= Fk(x)

or

Ha: F1(x) >= F2(x) >= .... >= Fk(x)

Although the Jonckheere-Terpstra 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 group-id variable, then to test for "less than" (positive concordance), use

UPPER TAILED KENDALL TAU INDEPENDENCE TEST Y X

and to test for "greater than" (negative concordance), use

LOWER TAILED KENDALL TAU INDEPENDENCE TEST Y X

The RANK CORRELATION INDEPENDENCE TEST can be used to perform a test for independence based on the Spearman rho rank correlation.

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?).

By default, critical values are based on tabulated values for n <= 60. The command
SET KENDALL TAU CRITICAL VALUES NORMAL APPROXIMATION

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

SET KENDALL TAU CRITICAL VALUES TABLE

The KENDALL TAU INDEPENDENCE TEST will accept matrix arguments. If a matrix is given, the data elements in the matrix will be collected in column order to form a vector before performing the test.

Dataplot saves the following internal parameters after a Kendall tau independence test:

STATVAL	=	the value of the test statistic
STATCDF	=	the CDF of the test statistic
PVALUE	=	the p-value for the two-sided test
PVALUELT	=	the p-value for the lower tailed test
PVALUEUT	=	the p-value for the upper tailed test
CUTLOW90	=	the 90% lower tailed critical value
CUTUPP90	=	the 90% upper tailed critical value
CUTLOW95	=	the 95% lower tailed critical value
CUTUPP95	=	the 95% upper tailed critical value
CTLOW975	=	the 97.5% lower tailed critical value
CTUPP975	=	the 97.5% upper tailed critical value
CUTLOW99	=	the 99% lower tailed critical value
CUTUPP99	=	the 99% upper tailed critical value
CTLOW995	=	the 99.5% lower tailed critical value
CTUPP995	=	the 99.5% upper tailed critical value

The following statistics can also be computed
LET A = KENDALL TAU Y1 Y2
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 p-values are based on the normal approximation given above.

To see a list of commands in which these statistics can be used, enter

HELP STATISTICS

The run sequence plot can be used to graphically assess whether or not there is trend in the data. The 4-plot can be used to assess the more general assumption of "independent, identically distributed" data.

The paired data can also be analyzed using other techniques for comparing two response variables (e.g., t-test, bihistogram, quantile-quantile plot).

None

None

RANK CORRELATION INDEPENDENCE TEST	= Compute an independence test based on the Spearman rho rank correlation statistic.
CORRELATION CONFIDENCE LIMITS	= Generate confidence limits for the Pearson correlation coefficient.
COX AND STUART TEST	= Compute a Cox and Stuart trend test.
T-TEST	= Compute a t-test.
4-PLOT	= Generate a 4-plot.
RUN SEQUENCE PLOT	= Generate a run sequence plot.
BIHISTOGRAM	= Generates a bihistogram.
QUANTILE-QUANTILE PLOT	= Generate a quantile-quantile plot.

Conover (1999), "Practical Nonparametric Statistics", Third Edition, Wiley, pp. 319-327.

Confirmatory Data Analysis

2013/3

 
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 y2
    

The 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
Two-Sided P-Value (Normal Approximation):        0.04873
 
 
            Conclusions (Two-Tailed 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 P-Value (Normal Approximation):        0.02436
 
 
            Conclusions (Upper 1-Tailed 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 P-Value (Normal Approximation):        0.02436
 
 
            Conclusions (Upper 1-Tailed 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