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. 