Kruskal-Wallis Test Ben Rindlisbacher Brigham Young University Marriott School of Management

1. Kruskal-Wallis TestBen RindlisbacherBrigham Young UniversityMarriott School of Management

2. Outline/Agenda • Who are Kruskal and Wallis? • What is the Kruskal-Wallis Test? • Brainstorming activity: Organizational Application • Nuts and Bolts • How does the Kruskal-Wallis Test work? • Why should I care? A Real-World Example • Practice Exercise • Summary • Readings list

3. Who were Kruskal and Wallis? • W. Allen Wallis: • American Economist • Statistician • President – University of Rochester William H. Kruskal: • Accomplished Mathematician • Statistician • Ph.D. from Columbia University Together they developed the Kruskal-Wallis Test

4. What is the Kruskal-Wallis Test? • Kruskal-Wallis Test (one-way analysis of variance): • A statistical test to see whether samples originate from the same distribution (ex: post-production inspection) • Primarily used when comparing more than two samples that come from different sources or when the samples are of unequal size

5. Brainstorming Activity:Organizational Application • Within your organization (or a supplier’s organization) identify a process that currently displays an unacceptable amount of variance. (ex: manufacturing, shipping, customer service, etc.) With this process in mind, identify three reasons why being able to pin-point the source of the variance from multiple, un-evenly distributed samples could be helpful.

6. Nuts and Bolts • The Kruskal-Wallis Test is comprised of the following seven steps: • Define the Null and Alternative Hypothesis • State your Alpha • Calculate the Degrees of Freedom • State your Decision Rule • Calculate your Test Statistic • State your Results • State your Conclusion

7. How does the Kruskal-Wallis Test work? • Example: The following scores (data) have been gathered and ranked from a process within your company: • Using the ordinal data from the table above determine if there is a difference between the three samples, using an alpha of .05

8. How does the Kruskal-Wallis Test work? • Step #1: Define the Null and Alternative Hypothesis • = Null Hypothesis • = Alternative Hypothesis • So…

9. How does the Kruskal-Wallis Test work? • Step #2: State your Alpha Alpha = = 0.05

10. How does the Kruskal-Wallis Test work? • Step #3: Calculate the Degrees of Freedom • DF = K - 1 • DF = 3 - 1 = 2

11. How does the Kruskal-Wallis Test work? • Step #4: State our Decision Rule • Using a Chi-squared table we need to find our critical value • With an alpha of 0.05 and DF of 2 our critical value = 5.99 • So if our (Chi – squared) is greater than 5.99 we will reject our null hypothesis

12. How does the Kruskal-Wallis Test work? • Step #5: Calculate the Test Statistic • Rank the 18 scores in the example from smallest to largest (right) • Once the data has been ranked, replace the original scores with the rankings (as shown below)

13. How does the Kruskal-Wallis Test work? • Step #5: Calculate the Test Statistic (continued…) • Calculate T or the sum of each group/sample rankings • Calculate n or the sample size for each group/sample • Sample 1; T = 39 and n = 6 • Sample 2; T = 65 and n = 6 • Sample 3; T = 67 and n = 6

14. How does the Kruskal-Wallis Test work? • Step #5: Calculate the Test Statistic (continued…) • Using the following formula and the previously solved variables, calculate the test statistic

15. How does the Kruskal-Wallis Test work? • Step #6: State your Result • Our Decision Rule: If > 5.99, reject the null hypothesis AND • H = 2.854 SO • Because 2.854 < 5.99 we will NOT reject the null hypothesis

16. How does the Kruskal-Wallis Test work? • Step #7: State your Conclusion • Because we do not reject the null hypothesis, we can conclude that there is no significant difference among the three samples compared in this example

17. Why should I care?A Real World Example • A commercial plumbing co. in the Midwest • Wanted to improve customer satisfaction scores • Customer surveys  customer service attributes • Kruskal-Wallis Test  compare the different survey samples to calculate if there was difference between the ratings of each attribute • Most important attributes: Courtesy, Convenience, Reliability, Quality, and Competitive Pricing

18. Practice Exercise • Imagine that your company performs specialty treatments for customers. The following data has been recorded from the treatment process: • Using the data from the table above, utilize the seven steps of the Kruskal-Wallis test in order to see if there is a difference between the four samples, using an alpha of 0.05

19. Summary • William H. Kruskal and W. Allen Wallis developed the Kruskal-Wallis one-way analysis of variance test • A statistical test to see whether samples originate from the same distribution • Useful when comparing three or more samples, or when samples are of different sizes • The Seven Steps of the Kruskal-Wallis Test • Can be useful in real-world applications to help improve quality

20. Reading list • Koppes, Steve. “William Kruskal, Statistician, 1919-2005” The University of Chicago News. University of Chicago News Office. 5 May 2005. • Elliott, Alan C.; Hynan, Linda S. “A SAS® macro implementation of a multiple comparison post hoc test for a Kruskal-Wallis analysis” Computer Methods and Programs in Biomedicine. Vol. 102 pp. 75-80. 2011. • Pace, Eric. “W. Allen Wallis, Economist and President of the U. of Rochester” The New York Times. 14 October 1998 pp. 5-9. • Kruskal, Willliam, and Judith Tanner. International Encyclopedia of Statistics. New York Free Press, 1978. Print. • Statistics Lectures. “The Kruskal-Wallis Test.” <www.statisticslectures.com /topics/kruskalwallis/>. 2010-2012. • Goldstein, Sheldon D. “Using Kruskal-Wallis to Improve Customer Satisfaction” The Steele Group. 2011. pp. 1-10.