16. Non-parametric Tests

# 16.6 Kruskal-Wallis Test (H Test)

The Kruskal-Wallis Test is a non-parametric one-way ANOVA. It detects differences in means between groups. The distribution behind the test is a new discrete distribution called the distribution that assumes the group samples come from populations with identically shaped distributions. We will use a approximation of for computing the critical statistic so, for that approximation, we need for , where is the number of groups. The hypothesis tested is :

: means of groups all equal

: means of groups not all equal

As mentioned, the *critical statistic* is with degrees of freedom which we can find using the Chi Squared Distribution Table.

The *test statistic* is :

where

The test is always right-tailed.

**Example 16.8** : With the following data on ml of potassium/quart in brands of drink, determine if there is a significant difference in the potassium content between brands.

Brand A | Brand B | Brand C |

4.7 | 5.3 | 6.3 |

3.2 | 6.4 | 8.2 |

5.1 | 7.3 | 6.2 |

5.2 | 6.8 | 7.1 |

5.0 | 7.2 | 6.6 |

0. Data reduction.

We need to rank the data. Ranking “in place” we have :

Brand (IV) | DV | Rank |

A | 4.7 | 2 |

A | 3.2 | 1 |

A | 5.1 | 4 |

A | 5.2 | 5 |

A | 5.0 | 3 |

B | 5.3 | 6 |

B | 6.4 | 9 |

B | 7.3 | 14 |

B | 6.8 | 11 |

B | 7.2 | 13 |

C | 6.3 | 8 |

C | 8.2 | 15 |

C | 6.2 | 7 |

C | 7.1 | 12 |

C | 6.6 | 10 |

Using A = 1, B = 2, c = 3, the sums of the ranks for each group are

Finally note that and .

1. Hypothesis.

: no differences in means between the brands

: some differences exist

2. Critical statistic.

From the Chi Squared Distribution Table with , find

3. Test statistic.

4. Decision.

Reject .

5. Interpretation.

At least one of the brands is different. Since is far less than the rank sums of the other two brands, we know that Brand A is different before we do any kind of post hoc testing.

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