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Dataplot Vol 1 Auxiliary Chapter

KRUSKAL WALLIS

Name:
    KRUSKAL WALLIS
Type:
    Analysis Command
Purpose:
    Perform a Kruskal Wallis test that k population means are equal.
Description:
    Analysis of Variance (ANOVA) is a data analysis technique for examining the significance of the factors (= independent variables) in a multi-factor model. The one factor model can be thought of as a generalization of the two sample t-test. That is, the two sample t-test is a test of the hypothesis that two population means are equal. The one factor ANOVA tests the hypothesis that k population means are equal.

    The Kruskal Wallis test can be applied in the one factor ANOVA case. It is a non-parametric test for the situation where the ANOVA normality assumptions may not apply.

    Let ni (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 Ri = the sum of the ranks for group i. Then the Kruskal Wallis test statistic is:

      H = [12/(N(N+1)] SUM[Ri**2/Ni] - 3(N+1)

    This statistic approximates a chi-square distribution with k-1 degrees of freedom if the null hypothesis of equal populations is true. Each of the ni 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,K-1) where CHIPPF is the chi-square percent point function

    More formally,

    H0: U<sub>1</sub>=U<sub>2</sub>= ... = U<sub>k</sub>
    HA: Ui not equal Uj
    for at least one set of i and j.
    Test Statistic: H = [12/(N(N+1)] SUM[Ri**2/Ni] - 3(N+1)
    Significance Level: ALPHA, typically set to 0.05.
    Critical Region: H > CHIPPF( alpha,k-1) where CHIPPF is the chi-square percent point function.
    Conclusion: Reject the null hypothesis if the test statistic is in the critical region.

Syntax:
    KRUSKAL WALLIS <y> <x>             <SUBSET/EXCEPT/FOR qualification>
    where <y> is the response (= dependent) variable;
                <x> is the factor (= independent) variable;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
    KRUSKAL WALLIS Y X
    KRUSKAL WALLIS Y X SUBSET X = 1 TO 4
Default:
    None
Synonyms:
    The following are synonyms for KRUSKAL WALLIS:

      KRUSKAL WALLIS TEST
      KRUSKAL TEST
Related Commands:
    ANOVA = Perform an analysis of variance.
    MEDIAN POLISH = Carries out a robust ANOVA.
    YATES ANALYSIS = Analyze a Yate's design.
    BLOCK PLOT = Generate a block plot.
    T TEST = Carries out a t test.
    RANK SUM TEST = Perform a rank sum test.
    SIGNED RANK TEST = Perform a signed rank test.
    PLOT = Plots (e.g., residuals and GANOVA ).
Reference:
    "Probability and Statistics for Engineers and Scientists", 2nd ed., Walpole and Myers, MacMillian, 1978.
Applications:
    Analysis of Variance
Implementation Date:
    1999/8
Program:
    SKIP 25
    READ SPLETT2.DAT Y MACHINE
    KRUSKAL WALLIS Y MACHINE

    The following output is generated. 

                  KRUSKALL-WALLIS TEST FOR ONE-WAY ANOVA
 
1. STATISTICS
      NUMBER OF OBSERVATIONS    =       99
      NUMBER OF GROUPS          =        4
      KRUSKALL-WALLIS TEST STATISTIC   =    41.06223
 
2. PERCENT POINTS OF THE CHI-SQUARE REFERENCE DISTRIBUTION
   FOR KRUSKALL-WALLIS TEST STATISTIC
      0          % POINT    =    .0000000
      50         % POINT    =    2.365974
      75         % POINT    =    4.108345
      90         % POINT    =    6.251388
      95         % POINT    =    7.814727
      99         % POINT    =    11.34487
      99.9       % POINT    =    16.26626
 
 
         100.0000       % Point:     41.06223
 
3. CONCLUSION (AT THE 5% LEVEL):
      THE        4 SAMPLES DO NOT COME FROM IDENTICAL POPULATIONS.

Date created: 6/5/2001
Last updated: 4/4/2003
Please email comments on this WWW page to alan.heckert@nist.gov.