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:

\( Z = \frac{|\mu - T| - 0.5} {\sigma} \)

where

\( \mu = n_1(n_1 + n_2 + 1)/2 \)

\( \sigma = \sqrt{n_2 \mu /6} \)

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 rank sum test:

STATVAL	-	The rank sum test statistic
STATCD2	-	the normal cdf value of T (only applies for sufficiently large N1 and N2)
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.

The following statistics are also supported:
LET A = MANN WHITNEY RANK SUM TEST Y1 Y2
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, built-in statistics are supported for about 20+ different commands (enter HELP STATISTICS for details).

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.

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

Confirmatory Data Analysis

1999/5

 
SKIP 25
READ NATR323.DAT Y1 Y2
RETAIN Y2 SUBSET Y2 > -90
SET WRITE DECIMALS 4
RANK SUM TEST Y1 Y2
    

The following output is generated.

            Two Sample Two-Sided 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
P-Value (2-tailed test):                 0.0067
P-Value (lower-tailed test):             0.9966
P-Value (upper-tailed test):             0.0034
 
 
            Two-Tailed 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