
KRUSKAL WALLISName:
The Kruskal Wallis test can be applied in the one factor ANOVA case. It is a nonparametric test for the situation where the ANOVA normality assumptions may not apply. Although this test is for identical populations, it is designed to be sensitive to unequal means. Let n_{i} (i = 1, 2, ..., k) represent the sample sizes for each of the k groups (i.e., samples) in the data. Next, rank the combined sample. Then compute R_{i} = the sum of the ranks for group i. Then the Kruskal Wallis test statistic is:
This statistic approximates a chisquare distribution with k1 degrees of freedom if the null hypothesis of equal populations is true. Each of the n_{i} should be at least 5 for the approximation to be valid. We reject the null hypothesis of equal population means if the test statistic H is greater than CHIPPF(ALPHA,K1) where CHIPPF is the chisquare percent point function More formally,
where <y> is the response (= dependent) variable; <x> is the factor (= independent) variable; and where the <SUBSET/EXCEPT/FOR qualification> is optional.
<SUBSET/EXCEPT/FOR qualification> where <y1> ... <yk> is a list of 1 to 30 response variables; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax is used for the case when the data for each group is stored in a separate variable. This syntax accepts matrix arguments.
KRUSKAL WALLIS Y X SUBSET X = 1 TO 4 MULTIPLE KRUSKAL WALLIS Y1 Y2 Y3 Y4 MULTIPLE KRUSKAL WALLIS Y1 TO Y4
The populations i and j seem to be different if the following inequality is satisfied:
with TPPF and T denoting the t percent point function with N  k degrees of freedom and the KruskalWallis test statistic, respectively.
LET A = KRUSKAL WALLIS TEST CDF Y X LET A = KRUSKAL WALLIS TEST PVALUE Y X with Y denoting the response variable, X denoting the groupid variable, and ALPHA denoting the significance level for the critical value. In addition to the above LET command, builtin statistics are supported for about 20+ different commands (enter HELP STATISTICS for details).
KRUSKAL TEST
Walpole and Myers (1978), "Probability and Statistics for Engineers and Scientists," Second Edition, MacMillian.
2004/10: Modified test to use Conover formulation rather than the Walpole Meyers formulation 2011/06: Reformatted the output, support for SET WRITE DECIMALS 2011/06: Support for the MULTIPLE option SKIP 25 READ SPLETT2.DAT Y MACHINE SET WRITE DECIMALS 5 KRUSKAL WALLIS Y MACHINEThe following output is generated. KruskalWallis One Factor Test Response Variable: Y GroupID Variable: MACHINE H0: Samples Come From Identical Populations Ha: Samples Do Not Come From Identical Populations Summary Statistics: Total Number of Observations: 99 Number of Groups: 4 KruskalWallis Test Statistic Value: 41.10239 CDF of Test Statistic: 0.99999 PValue: 0.00000 Percent Points of the ChiSquare Reference Distribution  Percent Point Value  0.0 = 0.000 50.0 = 2.366 75.0 = 4.107 90.0 = 6.251 95.0 = 7.815 97.5 = 9.348 99.0 = 11.345 99.9 = 16.265 Conclusions (Upper 1Tailed Test)  Alpha CDF Critical Value Conclusion  10% 90% 6.251 Reject H0 5% 95% 7.815 Reject H0 2.5% 97.5% 9.348 Reject H0 1% 99% 11.345 Reject H0 Multiple Comparisons Table  I J Ri/Ni  Rj/Nj 90% CV 95% CV 99% CV  1 2 18.82083 10.54643 12.60485 16.68947 1 3 47.56083 10.54643 12.60485 16.68947 1 4 4.98083 10.54643 12.60485 16.68947 2 3 28.74000 10.43825 12.47556 16.51830 2 4 13.83999 10.43825 12.47556 16.51830 3 4 42.58000 10.43825 12.47556 16.51830  
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Date created: 6/5/2001 