Intro to Business Statistics: Nonparametric Tests - KruskalWallis H Test, One-Way ANOVA Alternative

What This Is

The Kruskal-Wallis H Test is a non-parametric alternative to the one-way ANOVA test used to compare more than two independent groups. A retail chain wants to know if average daily sales exceed $10,000 across different store locations. They collect sales data from three stores and want to determine if there's a significant difference in sales among the stores.

Key Formulas & Symbols

• H = (12/n) * ?(Ri^2 - ni^3/12) where H = Kruskal-Wallis H statistic, n = sample size, Ri = rank of i-th observation, ni = number of observations in i-th group.
• df = k - 1 where df = degrees of freedom, k = number of groups.
• p-value = P(H-H) where p-value = probability of observing the data (or more extreme) if H? is true, H = observed H statistic, H = critical value from the chi-square distribution.
• ?² = (n * (H - 1)) / (k - 1) where ?² = chi-square statistic, n = sample size, H = observed H statistic, k = number of groups.
• ?²(df, ?) = ?² value from the chi-square distribution with df degrees of freedom and-= 0.05 significance level.
• H?: = = … = ?k where H? = null hypothesis of equal group means,-= population mean, k = number of groups.
• H?: Not all ?i are equal where H? = alternative hypothesis of not all group means are equal.
• ? = 0.05 where-= significance level.

Step-by-Step Procedure

1. State hypotheses: State the null and alternative hypotheses (H? and H?) as described above.
2. Choose test: Select the Kruskal-Wallis H Test as the non-parametric alternative to the one-way ANOVA test.
3. Compute test statistic: Calculate the H statistic using the formula H = (12/n) * ?(Ri^2 - ni^3/12).
4. Find p-value or critical value: Determine the p-value by comparing the observed H statistic to the critical value from the chi-square distribution, ?²(df, ?), or find the critical value from the chi-square distribution.
5. Compare to ?: Compare the p-value to the significance level? (0.05) or compare the test statistic to the critical value.
6. Conclude: Reject the null hypothesis (H?) if the p-value is less than-or the test statistic is greater than the critical value, indicating a significant difference among the groups.

Common Mistakes

• Mistake: Misinterpreting the p-value as the probability that H? is true.
• Correction: The p-value is the probability of observing the data (or more extreme) if H? is true.
• Mistake: Failing to check the assumptions of the Kruskal-Wallis H Test (independence, identical distributions, and no ties).
• Correction: Check the assumptions before conducting the test to ensure the results are valid.
• Mistake: Using the Kruskal-Wallis H Test when the data are paired or matched.
• Correction: Use a different non-parametric test, such as the Wilcoxon rank-sum test, when the data are paired or matched.

Quick Practice Problems

1. A company wants to compare the average salaries of three different job positions. They collect data from 10 employees in each position and rank the salaries from lowest to highest. The ranks are: Position A (1, 3, 5, 7, 9), Position B (2, 4, 6, 8, 10), and Position C (1, 2, 3, 4, 5). What is the p-value of the Kruskal-Wallis H Test? Answer: 0.01, The p-value is calculated using the chi-square distribution with 2 degrees of freedom and the observed H statistic.
2. A marketing firm wants to compare the average response rates of three different advertising campaigns. They collect data from 20 customers in each campaign and rank the response rates from lowest to highest. The ranks are: Campaign A (1, 3, 5, 7, 9, 11, 13, 15, 17, 19), Campaign B (2, 4, 6, 8, 10, 12, 14, 16, 18, 20), and Campaign C (1, 2, 3, 4, 5, 6, 7, 8, 9, 10). What is the critical value of the Kruskal-Wallis H Test? Answer: 5.99, The critical value is obtained from the chi-square distribution with 2 degrees of freedom and a significance level of 0.05.

Last-Minute Cram Sheet

• Kruskal-Wallis H Test: Non-parametric alternative to the one-way ANOVA test.
• H = (12/n) * ?(Ri^2 - ni^3/12): Formula for the Kruskal-Wallis H statistic.
• df = k - 1: Formula for the degrees of freedom.
• p-value = P(H-H): Formula for the p-value.
• ?² = (n * (H - 1)) / (k - 1): Formula for the chi-square statistic.
• ?²(df, ?) = ?² value from the chi-square distribution: Critical value from the chi-square distribution.
• H?: = = … = ?k: Null hypothesis of equal group means.
• H?: Not all ?i are equal: Alternative hypothesis of not all group means are equal.
• ? = 0.05: Significance level.
• p-value is NOT the probability that H? is true – it’s the probability of observing the data (or more extreme) if H? is true.
• Failing to check the assumptions of the Kruskal-Wallis H Test can lead to invalid results.
• The Kruskal-Wallis H Test is not suitable for paired or matched data.