
SIGNED RANK TESTName:
The signed rank test is also commonly called the Wilcoxon signed rank test or simply the Wilcoxon test. To form the signed rank test, compute d_{i} = X_{i}  Y_{i} where X and Y are the two samples. Rank the d_{i} without regard to sign. Tied values are not included in the Wilcoxon test. After ranking, restore the sign (plus or minus) to the ranks. Then compute W+ and W as the sums of the positive and negative ranks respectively. If the two population means are in fact equal, then the sums of the ranks should also be nearly equal. If the difference between the sum of the ranks is too great, we reject the null hypothesis that the population means are equal. Significance levels are based on the fact that if there is no difference in the population means, then there are 2^{n} equally likely ways for the n ranks to recieve signs. More formally, the hypothesis test is defined as follows.
Although the above discussion was in terms of a paired two sample test, it can easily be adapted to the following additional cases:
where <y1> is a response variable; <mu> is a number or parameter that is the hypothesized mean value; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax implements the one sample signed rank test.
where <y1> is the first response variable; <y2> is the second response variable; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax implements the two sample paired signed rank test where the hypothesized difference between the population means for the two samples is zero.
<SUBSET/EXCEPT/FOR qualification> where <y1> is the first response variable; <y2> is the second response variable; <mu> is a number or parameter that is the hypothesized difference between the means of the two samples; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax implements the two sample paired signed rank test where the hypothesized difference between the population means for the two samples is equal to a nonzero value.
SIGNED RANK TEST Y1 Y2 SIGNED RANK TEST Y1 Y2 MU SIGNED RANK TEST Y1 Y2 SUBSET TAG > 2
STATCD2 = the normal cdf value of W CUTLOW90 = 0.05 critical value CUTUPP90 = 0.95 critical value CUTLOW95 = 0.025 critical value CUTUPP95 = 0.975 critical value CUTLOW99 = 0.005 critical value CUTUPP99 = 0.995 critical value Note that the above critical values are the lower and upper tails for two sided tests (i.e., each tail is alpha/2. For example, CUTLOW90 is the lower 5% of the normal percent point function (adjusted for the mean and standard deviation). This is the critical regions for alpha = 0.10, so there is 0.05 in each tail.
So the signed rank test weakens the assumption of normality of the paired ttest to an assumption of symmetry. The signed rank test is more powerful than a sign test (it takes the magnitude of the differences into account as well as the sign), but it has stronger assumptions than the sign test. So if your data is at approximately symmetric, then the signed rank test is preferred to the sign test. However, if the symmetry assumption is not reasonable, the sign test is preferred. Also according to Conover, the null hypothesis can be stated either in terms of the mean or the median. This is due to the assumption of symmetry.
LET A = ONE SAMPLE WILCOXON SIGNED RANK TEST CDF Y LET A = ONE SAMPLE WILCOXON SIGNED RANK TEST PVALUE Y LET A = ONE SAMPLE WILCOXON SIGNED RANK TEST LOWER ... TAIL PVALUE Y LET A = ONE SAMPLE WILCOXON SIGNED RANK TEST UPPER ... TAIL PVALUE Y
LET A = TWO SAMPLE WILCOXON SIGNED RANK TEST
Y1 Y2 In addition to the above LET command, builtin statistics are supported for about 20+ different commands (enter HELP STATISTICS for details).
WILCOXON SIGNED RANK WILCOXON SIGN TEST WILCOXON TEST SIGNED RANK
Snedecor and Cochran (1989), "Statistical Methods," Eigth Edition, Iowa State University Press, pp. 140142.
2011/05: Switch algorithm to that given in Conover. SKIP 25 READ NATR332.DAT Y1 Y2 SET WRITE DECIMALS 4 SIGNED RANK TEST Y1 Y2The following output is generated. Two Sample TwoSided Wilcoxon Signed Rank Test (Conover Formulation) First Response Variable: Y1 Second Response Variable: Y2 H0: Mu1  Mu2 Equal 0.0000 Ha: Mu1  Mu2 Not Equal 0.0000 Summary Statistics: Number of Observations: 10 Number of Zero Differences (Omitted): 3 Number of Positive Differences: 3 Number of Negative Differences: 4 Number of Tied Ranks: 2 Sum of Positive Ranks: 13.5000 Sum of Negative Ranks: 14.5000 Test (Large Sample Approximation): Test Statistic Value: 0.0853 CDF Value: 0.4660 PValue (2tailed test): 1.0000 PValue (lowertailed test): 0.5000 PValue (uppertailed test): 0.5677 TwoTailed Test: Normal Approximation H0: u1  u2 = d0; Ha: u1  u2 <> d0  Null Significance Test Critical Hypothesis Level Statistic Value (+/) Conclusion  60.0% 0.0853 0.8416 ACCEPT 80.0% 0.0853 1.2816 ACCEPT 90.0% 0.0853 1.6449 ACCEPT 95.0% 0.0853 1.9600 ACCEPT 99.0% 0.0853 2.5758 ACCEPT  
Date created: 06/05/2001 Last updated: 12/11/2023 Please email comments on this WWW page to alan.heckert@nist.gov. 