
RANK CORRELATION INDEPENDENCE TESTName:
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 pth upper quantile of the rank correlation 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 ≤ 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.
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 twotailed test is performed. Lower tailed tests are used to test for negative correlation and upper tailed tests are used to test for positive 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 pairwise tests for the <y1> ... <yk> response variables. For example,
is equivalent to
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 TO Y5 LOWER TAILED RANK CORRELATION INDEPENDENCE TEST Y1 Y2 UPPER TAILED RANK CORRELATION INDEPENDENCE TEST Y1 Y2
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 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 = 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 pvalues are based on the normal approximation given above.
The paired data can also be analyzed using other techniques for comparing two response variables (e.g., ttest, bihistogram, quantilequantile plot).
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 y2The 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 TwoSided PValue (Normal Approximation): 0.05036 Conclusions (TwoTailed 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 PValue (Normal Approximation): 0.02518 Conclusions (Upper 1Tailed 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 PValue (Normal Approximation): 0.02518 Conclusions (Upper 1Tailed 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  
Date created: 03/08/2013 Last updated: 12/11/2023 Please email comments on this WWW page to alan.heckert@nist.gov. 