16. Non-parametric Tests
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 :
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|
0. Data reduction.
We need to rank the data. Ranking “in place” we have :
Using A = 1, B = 2, c = 3, the sums of the ranks for each group are
Finally note that and .
: no differences in means between the brands
: some differences exist
2. Critical statistic.
From the Chi Squared Distribution Table with , find
3. Test statistic.
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.