### What is Kruskal–Wallis One-Way Analysis of Variance?

The Kruskal Wallis one-way analysis of variance is a statistical hypothesis test to compare the medians among more than two groups.

- Null Hypothesis (H
_{0}):*η*_{1}= η_{2}= … = η_{k} - Alternative Hypothesis
*(H*_{a}): at least one of the medians is different from others.

Where: *n*is the median of population_{i}*i**k*is the number of groups of our interest

It is an extension of Mann–Whitney test. While the Mann–Whitney test allows us to compare the samples of two populations, the Kruskal Wallis test allows us to compare the samples of more than two populations.

One key difference between this test and the Mann–Whitney test is the robustness of the test when the populations are not identically shaped. If this is the case, there is a different test, called Mood’s median, which is more appropriate.

### Kruskal Wallis One-Way Analysis of Variance: Assumptions

- The sample data drawn from the populations of interest are unbiased and representative
- The data of
*k*populations are continuous or ordinal when the spacing between adjacent values is not constant - The
*k*populations are independent to each other - The Kruskal–Wallis test is robust for the non-normally distributed population

### How Kruskal Wallis One-Way ANOVA Works

The Kruskal Wallis test works very similarly to the Mann–Whitney Test.

**Step 1:** Group the* k* samples from *k* populations (sample* i* is from population *i*) into one single data set and then sort the data in ascending order ranked from 1* to N*, where *N* is the total number of observations across* k* groups.

**Step 2:** Add up the ranks for all the observations from sample *i* and call it *r _{i}*, where i can be any integer between

*1 and k.*

**Step 3:** Calculate the test statistic

Where:

*k*is the number of groups*n*is the sample size of sample_{i}*i**N*is the total number of all the observations across*k*groups*r*is the rank (among all the observations) of observation_{ij}*j*from group*i*

**Step 4:** Make a decision of whether to reject the null hypothesis.

- Null Hypothesis
*(H*_{0}):*η*_{1}= η_{2}= … = η_{k} - Alternative Hypothesis
*(H*_{a}):*at least one of the medians is different from others*

The test statistic follows chi-square distribution when the null hypothesis is true. If T is greater than(the critical chi-square statistic), we reject the null and claim there is at least one median statistically different from other medians. If T is smaller than, we fail to reject the null and claim the medians of k groups are equal.

### Use JMP to Run a Kruskal–Wallis One-Way ANOVA

Case study: We are interested in comparing customer satisfaction among three types of customers using a nonparametric (i.e., distribution-free) hypothesis test: Kruskal Wallis one-way ANOVA.

Data File: “Kruskal–Wallis.jmp”

- Null Hypothesis
*(H*_{0}):*η*_{1}= η_{2}= … = η_{k} - Alternative Hypothesis (H
_{a}): at least one of the customer types has different overall satisfaction levels from the others

### Steps to run a Kruskal Wallis One-Way ANOVA in JMP:

- Click Analyze -> Fit Y by X
- Select “Overall Satisfaction” as “Y, Response”
- Select “Customer Type” as “X, Factor”

- Click “OK”
- Click on the red triangle button next to “One-Way Analysis of Overall Satisfaction by Customer Type”
- Click Nonparametric -> Wilcoxon Test

Model summary: The p-value of the test is lower than alpha level (0.05), and we reject the null hypothesis and conclude that at least the overall satisfaction median of one customer type is statistically different from the others. The result of the test is boxed in: The p-value is lower than the alpha value of 0.05; therefore, we must reject the null hypothesis and claim that at least one of the customer types has a different level of satisfaction than the others.

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