
RANK SUM TESTName:
The rank sum test is also commonly called the MannWhitney rank sum test or simply the MannWhitney test. Note that even though this test is commonly called the MannWhitney test, it was in fact developed by Wilcoxon. To form the rank sum test, rank the combined samples. Then compute the sum of the ranks for sample one, T_{1}, and the sum of the ranks for sample two, T_{2}. If the sample sizes are equal, the rank sum test statistic is the minimum of T_{1} and T_{2}. If the sample sizes are unequal, then find T_{1} equal the sum of the ranks for the smaller sample. Then compute T_{2} = n_{1}(n_{1} + n_{2} + 1)  T_{1}. T is the minimum of T_{1} and T_{2}. Sufficiently small values of T cause rejection of the null hypothesis that the sample means are equal. Significance levels have been tabulated for small values of n_{1} and n_{2}. For sufficiently large n_{1} and n_{2}, the following normal approximation is used:
where
\( \sigma = \sqrt{n_2 \mu /6} \)
where <y1> is the first response variable; <y2> is the second response variable; and where the <SUBSET/EXCEPT/FOR qualification> is optional.
RANK SUM TEST Y1 Y2 SUBSET TAG > 2
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.
LET A = MANN WHITNEY RANK SUM TEST CDF Y1 Y2 LET A = MANN WHITNEY RANK SUM TEST PVALUE Y1 Y2 LET A = MANN WHITNEY RANK SUM LOWER TAILED PVALUE Y1 Y2 LET A = MANN WHITNEY RANK SUM UPPER TAILED PVALUE Y1 Y2 In addition to the above LET command, builtin statistics are supported for about 20+ different commands (enter HELP STATISTICS for details).
MANN WHITNEY RANK SUM MANN WHITNEY TEST MANN WHITNEY RANK SUM
SKIP 25 READ NATR323.DAT Y1 Y2 RETAIN Y2 SUBSET Y2 > 90 SET WRITE DECIMALS 4 RANK SUM TEST Y1 Y2The following output is generated. Two Sample TwoSided Mann Whitney Rank Sum Test (Conover Formulation) First Response Variable: Y1 Second Response Variable: Y2 H0: F(x) = G(x) for all x Ha: F(x) <> G(x) for some x Summary Statistics: Number of Observations for Sample 1: 13 Mean for Sample 1: 80.0208 Median for Sample 1: 80.0300 Number of Observations for Sample 2: 8 Mean for Sample 2: 79.9788 Median for Sample 2: 79.9700 Number of Tied Ranks: 14 Test (Normal Approximation): Test Statistic Value (W): 2.7105 CDF Value: 0.9966 PValue (2tailed test): 0.0067 PValue (lowertailed test): 0.9966 PValue (uppertailed test): 0.0034 TwoTailed Test: Normal Approximation H0: F(x) = G(x); Ha: F(x) <> G(x) for some x  Null Significance Test Critical Hypothesis Level Statistic Value (+/) Conclusion  80.0% 2.7105 1.2816 REJECT 90.0% 2.7105 1.6449 REJECT 95.0% 2.7105 1.9600 REJECT 99.0% 2.7105 2.5758 REJECT  
Date created: 06/05/2001 Last updated: 12/11/2023 Please email comments on this WWW page to alan.heckert@nist.gov. 