RANK CORRELATION INDEPENDENCE TEST

RANK CORRELATION INDEPENDENCE TEST

Analysis Command

Perform a Spearman rho rank correlation test for whether two samples are independent (i.e., not correlated).

If the measurements in the two samples are replaced with their ranks (and average ranks in the case of ties) and the Pearson correlation coefficient is computed, the result is the Spearman rho correlation coefficient.

A value of +1 indicates perfect positive correlation, a value of -1 indicates perfect negative correlation, and a value of 0 indicates no relation (i.e., independence).

The rank correlation independence test is a test whether the rank correlation coefficient is equal to zero.

For larger n (e.g., n > 30) or the case where there are many ties, the p-th upper quantile of the rank correlation statistic can be approximated by

\( w_{p} = \frac{z_{p}}{\sqrt{n-1}} \)

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 ≤ 30, tabulated quantiles (from Table A10 on p. 542 of Conover) are used. These quantiles are exact when there are no ties in the data.

<LOWER TAILED/UPPER TAILED> RANK CORRELATION
                        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 negative correlation and upper tailed tests are used to test for positive correlation).

<LOWER TAILED/UPPER TAILED> RANK CORRELATION
                        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,

RANK CORRELATION INDEPENDENCE TEST Y1 TO Y4

is equivalent to

RANK CORRELATION INDEPENDENCE TEST Y1 Y2
RANK CORRELATION INDEPENDENCE TEST Y1 Y3
RANK CORRELATION INDEPENDENCE TEST Y1 Y4
RANK CORRELATION INDEPENDENCE TEST Y2 Y3
RANK CORRELATION INDEPENDENCE TEST Y2 Y4
RANK CORRELATION INDEPENDENCE TEST Y3 Y4

RANK CORRELATION INDEPENDENCE TEST Y1 Y2
RANK CORRELATION INDEPENDENCE TEST Y1 TO Y5
LOWER TAILED RANK CORRELATION INDEPENDENCE TEST Y1 Y2
UPPER TAILED RANK CORRELATION 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
RANK CORRELATION 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 for trend is referred to as the Daniels test for trend.

The KENDALL TAU INDEPENDENCE TEST can be used to perform a test for independence based on the Kendall tau statistic.

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 ≤ 30. The command
SET RANK CORRELATION 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 RANK CORRELATION CRITICAL VALUES TABLE

The RANK CORRELATION 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 rank correlation 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
CTLOW999	=	the 99.9% lower tailed critical value
CTUPP999	=	the 99.9% upper tailed critical value

The following statistics can also be computed
LET A = RANK CORRELATION Y1 Y2
LET A = RANK CORRELATION CDF Y1 Y2
LET A = RANK CORRELATION PVALUE Y1 Y2
LET A = RANK CORRELATION LOWER TAILED PVALUE Y1 Y2
LET A = RANK CORRELATION UPPER TAILED PVALUE Y1 Y2


The cdf and p-values are based on the normal approximation given above.

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

KENDALL TAU INDEPENDENCE TEST	= Compute an independence test based on the Kendall tau 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

 
skip 25
read kendall.dat y1 y2
set write decimals 5
.
let statval  = rank correlation y1 y2
let statcdf  = rank correlation cdf y1 y2
let pvalue   = rank correlation pvalue y1 y2
let pvallt   = rank correlation lower tailed pvalue y1 y2
let pvalut   = rank correlation upper tailed pvalue y1 y2
print statval statcdf pvalue pvallt pvalut
.
rank correlation independence test y1 y2
.
upper tailed rank correlation independence test y1 y2
.
set rank correlation critical values normal approximation
upper tailed rank correlation independence test y1 y2
    

The following output is generated.

 
 PARAMETERS AND CONSTANTS--

    STATVAL --        0.59002
    STATCDF --        0.97482
    PVALUE  --        0.05036
    PVALLT  --        0.97482
    PVALUT  --        0.02518
 

            Two Sample Rank Correlation Test for Independence
 
First Response Variable:  Y1
Second Response Variable: Y2
 
H0: The Two Samples are Independent
Ha: The Two Samples Are Not Independent
 
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:
Spearman Rho Rank Correlation Value:             0.59001
CDF Value (Normal Approximation):                0.97481
Two-Sided P-Value (Normal Approximation):        0.05036
 
 
            Conclusions (Two-Tailed Test)
 
H0: Samples are Independent
------------------------------------------------------------
                                                        Null
   Significance           Test       Critical     Hypothesis
          Level      Statistic   Region (+/-)     Conclusion
------------------------------------------------------------
          80.0%        0.59001        0.39860         REJECT
          90.0%        0.59001        0.49650         REJECT
          95.0%        0.59001        0.58040         REJECT
          99.0%        0.59001        0.72030         ACCEPT
 
 
            Two Sample Rank Correlation Test for Independence
 
First Response Variable:  Y1
Second Response Variable: Y2
 
H0: The Two Samples are Independent
Ha: The Two Samples Are Positively Correlated
 
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:
Spearman Rho Rank Correlation Value:                 0.59001
CDF Value (Normal Approximation):                    0.97481
Upper Tailed P-Value (Normal Approximation):         0.02518
 
 
            Conclusions (Upper 1-Tailed Test)
 
H0: Samples are Independent
------------------------------------------------------------
                                                        Null
   Significance           Test       Critical     Hypothesis
          Level      Statistic     Region (>)     Conclusion
------------------------------------------------------------
          90.0%        0.59001        0.39860         REJECT
          95.0%        0.59001        0.49650         REJECT
          97.5%        0.59001        0.58040         REJECT
          99.0%        0.59001        0.67130         ACCEPT
          99.5%        0.59001        0.72030         ACCEPT
          99.9%        0.59001        0.81120         ACCEPT
 
 
            Two Sample Rank Correlation Test for Independence
 
First Response Variable:  Y1
Second Response Variable: Y2
 
H0: The Two Samples are Independent
Ha: The Two Samples Are Positively Correlated
 
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:
Spearman Rho Rank Correlation Value:                 0.59001
CDF Value (Normal Approximation):                    0.97481
Upper Tailed P-Value (Normal Approximation):         0.02518
 
 
            Conclusions (Upper 1-Tailed Test)
 
H0: Samples are Independent
------------------------------------------------------------
                                                        Null
   Significance           Test       Critical     Hypothesis
          Level      Statistic     Region (>)     Conclusion
------------------------------------------------------------
          90.0%        0.59001        0.38640         REJECT
          95.0%        0.59001        0.49594         REJECT
          97.5%        0.59001        0.59095         ACCEPT
          99.0%        0.59001        0.70142         ACCEPT
          99.5%        0.59001        0.77664         ACCEPT
          99.9%        0.59001        0.93174         ACCEPT