RANK SUM TEST

RANK SUM TEST

Analysis Command

Perform a two sample rank sum test.

The t-test is the standard test for testing that the difference between population means for two non-paired samples are equal. If the populations are non-normal, particularly for small samples, then the t-test may not be valid. The rank sum test is an alternative that can be applied when distributional assumptions are suspect. However, it is not as powerful as the t-test when the distributional assumptions are in fact valid.

The rank sum test is also commonly called the Mann-Whitney rank sum test or simply the Mann-Whitney test. Note that even though this test is commonly called the Mann-Whitney 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, T₁, and the sum of the ranks for sample two, T₂. If the sample sizes are equal, the rank sum test statistic is the minimum of T₁ and T₂. If the sample sizes are unequal, then find T₁ equal the sum of the ranks for the smaller sample. Then compute T₂ = n₁(n₁ + n₂ + 1) - T₁. T is the minimum of T₁ and T₂. 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 n₁ and n₂. For sufficiently large n₁ and n₂, the following normal approximation is used:

where

RANK SUM TEST <y1> <y2>             <SUBSET/EXCEPT/FOR qualification>
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
RANK SUM TEST Y1 Y2 SUBSET TAG > 2

Dataplot saves the following internal parameters after a sign test:
STATVAL = The rank sum test statistic

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.

None

The following are synonyms for RANK SUM TEST:
MANN WHITNEY RANK SUM TEST
MANN WHITNEY RANK SUM
MANN WHITNEY TEST
MANN WHITNEY
RANK SUM

T-TEST	= Compute a t-test.
SIGN TEST	= Compute a sign test.
SIGNED RANK TEST	= Compute a signed rank test.
CHI-SQUARED 2 SAMPLE TEST	= Compute a two sample chi-square test.
BIHISTOGRAM	= Generates a bihistogram.
QUANTILE-QUANTILE PLOT	= Generate a quantile-quantile plot.
BOX PLOT	= Generates a box plot.

"Statistical Methods", Eighth Edition, Snedecor and Cochran, 1989, Iowa State University Press, pp. 142-144.

Confirmatory Data Analysis

1999/5

SKIP 25
READ NATR323.DAT Y1 Y2
RETAIN Y2 SUBSET Y2 > -90
RANK SUM TEST Y1 Y2


The following output is generated.

                 MANN WHITNEY RANK SUM TEST
                    (2-SAMPLE)
HYPOTHESIS BEING TESTING--POPULATION MEANS MU1 = MU2
SAMPLE SIZE FOR VARIABLE 1              =       13
SAMPLE SIZE FOR VARIABLE 2              =        8
RANK SUM FOR VARIABLE 1                 =    180.0000
RANK SUM FOR VARIABLE 2                 =    51.00000
RANK SUM TEST STATITIC (U)              =    51.00000
 
HYPOTHESIS     ACCEPTANCE INTERVAL      CONCLUSION
MU1 = MU2      U  >    60.00000         REJECT