### Abstract

For the comparison of more than two independent samples the Kruskal-Wallis H test is a preferred procedure in many situations. However, the exact null and alternative hypotheses, as well as the assumptions of this test, do not seem to be very clear among behavioral scientists. This article attempts to bring some order to the inconsistent, sometimes controversial treatments of the Kruskal-Wallis test. First we clarify that the H test cannot detect with consistently increasing power any alternative hypothesis other than exceptions to stochastic homogeneity. It is then shown by a mathematical derivation that stochastic homogeneity is equivalent to the equality of the expected values of the rank sample means. This finding implies that the null hypothesis of stochastic homogeneity can be tested by an ANOVA performed on the rank transforms, which is essentially equivalent to doing a Kruskal-Wallis H test. If the variance homogeneity condition does not hold then it is suggested that robust ANOVA alternatives performed on ranks be used for testing stochastic homogeneity. Generalizations are also made with respect to Friedman's G test.

Original language | English |
---|---|

Pages (from-to) | 170-192 |

Number of pages | 23 |

Journal | Journal of Educational and Behavioral Statistics |

Volume | 23 |

Issue number | 2 |

Publication status | Published - 1998 |

### Fingerprint

### Keywords

- ANOVA
- Friedman's g test
- Kruskal-wallis h test
- Mann-whitney test
- Measure of stochastic superiority
- Nonparametric anova
- Stochastic equality
- Stochastichomogeneity

### ASJC Scopus subject areas

- Social Sciences (miscellaneous)
- Education

### Cite this

*Journal of Educational and Behavioral Statistics*,

*23*(2), 170-192.

**The Kruskal-Wallis Test and Stochastic Homogeneity.** / Vargha, A.; Delaney, Harold D.

Research output: Contribution to journal › Article

*Journal of Educational and Behavioral Statistics*, vol. 23, no. 2, pp. 170-192.

}

TY - JOUR

T1 - The Kruskal-Wallis Test and Stochastic Homogeneity

AU - Vargha, A.

AU - Delaney, Harold D.

PY - 1998

Y1 - 1998

N2 - For the comparison of more than two independent samples the Kruskal-Wallis H test is a preferred procedure in many situations. However, the exact null and alternative hypotheses, as well as the assumptions of this test, do not seem to be very clear among behavioral scientists. This article attempts to bring some order to the inconsistent, sometimes controversial treatments of the Kruskal-Wallis test. First we clarify that the H test cannot detect with consistently increasing power any alternative hypothesis other than exceptions to stochastic homogeneity. It is then shown by a mathematical derivation that stochastic homogeneity is equivalent to the equality of the expected values of the rank sample means. This finding implies that the null hypothesis of stochastic homogeneity can be tested by an ANOVA performed on the rank transforms, which is essentially equivalent to doing a Kruskal-Wallis H test. If the variance homogeneity condition does not hold then it is suggested that robust ANOVA alternatives performed on ranks be used for testing stochastic homogeneity. Generalizations are also made with respect to Friedman's G test.

AB - For the comparison of more than two independent samples the Kruskal-Wallis H test is a preferred procedure in many situations. However, the exact null and alternative hypotheses, as well as the assumptions of this test, do not seem to be very clear among behavioral scientists. This article attempts to bring some order to the inconsistent, sometimes controversial treatments of the Kruskal-Wallis test. First we clarify that the H test cannot detect with consistently increasing power any alternative hypothesis other than exceptions to stochastic homogeneity. It is then shown by a mathematical derivation that stochastic homogeneity is equivalent to the equality of the expected values of the rank sample means. This finding implies that the null hypothesis of stochastic homogeneity can be tested by an ANOVA performed on the rank transforms, which is essentially equivalent to doing a Kruskal-Wallis H test. If the variance homogeneity condition does not hold then it is suggested that robust ANOVA alternatives performed on ranks be used for testing stochastic homogeneity. Generalizations are also made with respect to Friedman's G test.

KW - ANOVA

KW - Friedman's g test

KW - Kruskal-wallis h test

KW - Mann-whitney test

KW - Measure of stochastic superiority

KW - Nonparametric anova

KW - Stochastic equality

KW - Stochastichomogeneity

UR - http://www.scopus.com/inward/record.url?scp=0032343867&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032343867&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0032343867

VL - 23

SP - 170

EP - 192

JO - Journal of Educational and Behavioral Statistics

JF - Journal of Educational and Behavioral Statistics

SN - 1076-9986

IS - 2

ER -