Statistical Methods IIStatistical Methods II

Session 10Non Parametric Testing – The Kruskal-Wallis Test

Kruskal-Wallis Test

Recall what we know about ANOVA:Used when we want to test if the means of 3 or more groups are different.

Hypotheses are written as:H0: The k level means are the same

H1: At least one of means is different

Assumptions include:1.All the groups are normally distributed.2.All the populations sampled have approximately equal variance (you can check this by generating side-by-side boxplots).3.The samples of the groups are independent of each other and subjects within the groups were randomly selected.

Kruskal-Wallis Test

From these assumptions, what deviations should concern us the most?

If the normality assumptions are not met AND we do not have a balanced design, ANOVA is not an appropriate tool.

If these assumptions are violated, Plan B is the non parametric alternative – The Kruskal-Wallis Test.

Kruskal-Wallis Test

Things to know about the K-W Test:

It is a non-parametric alternative and therefore should never be “Plan A”.Like all non-parametric tests, the focus is on ranks, counting and the medians. The hypotheses statements are written as:

H0: All k populations have the same median.

H1: Not all of the k population medians are the same.

As the ANOVA is a conceptual extension of the two sample ttest, so the K-W test is a conceptual extension of the Wilcoxon Rank Sum (Mann Whitney) test.

Kruskal-Wallis Test

Steps for the K-W Test:

1.Rank the entire dataset from highest to lowest. Handle ties by assigning the average of the tied ranks to all involved in the tie.2.Compute Ti as the sum of the ranks for group i, for each of the k groups.

3.The test statistic is: H={12/N(N+1)}Σni {Ri – ((1+N)/2)2}Where, N = the total number of obs in the sample

ni = the number of obs in group i

Ri = the average of the ranks in group i

Kruskal-Wallis Test

Consider the following example –

To assess the effects of expectation on the aesthetic quality, an investigator randomly sorts 24 amateur wine aficionados into three groups – A, B and C of 8 subjects each. Each subject is scheduled for an individual interview to taste and rate wines. The only difference among the three groups is the expectations set regarding the quality of the wine (it’s the same wine being tested across the groups). Group A is told that the wine is of VERY fine quality, Group B is told that the wine is medium quality and Group C is told that the wine is of low quality. The wine will be ranked from 1 (low) to 10 (high).

Kruskal-Wallis Test

Unfortunately, on the day of the interviews, some people don’t show up. By the end of the day, the researcher has the following data:

GROUP A (n=8)

GROUP B (n=7)

GROUP C (n=6)

6.4 2.5 1.3

6.8 3.7 4.1

7.2 4.9 4.9

8.3 5.4 5.2

8.4 5.9 5.5

9.1 8.1 8.2

9.4 8.2

9.7

Lets execute this test using SAS…

Kruskal-Wallis Test