Test Questions for Kruskal-Wallis and Friedman Tests with Solutions

📚 Quick Study Guide

• 📊 Kruskal-Wallis Test: A non-parametric test used to compare two or more independent samples of equal or different sample sizes. It's an alternative to one-way ANOVA when the assumption of normality is not met. The test statistic, $H$, is calculated as: $H = \frac{12}{N(N+1)} \sum_{i=1}^{k} \frac{R_i^2}{n_i} - 3(N+1)$, where $N$ is the total number of observations, $k$ is the number of groups, $R_i$ is the sum of ranks for group $i$, and $n_i$ is the sample size for group $i$.
• 📝 Friedman Test: A non-parametric test used to compare two or more related samples. It's an alternative to repeated measures ANOVA when the assumption of normality is not met. It is essentially the Kruskal-Wallis test applied to repeated measures. The test statistic, $Q$, is calculated as: $Q = \frac{12}{nk(k+1)} \sum_{j=1}^{k} R_j^2 - 3n(k+1)$, where $n$ is the number of blocks (subjects), $k$ is the number of treatments (conditions), and $R_j$ is the sum of ranks for treatment $j$.
• 💡 Assumptions:
  • Kruskal-Wallis: Independent samples, ordinal or continuous data.
  • Friedman: Related samples (repeated measures), ordinal or continuous data.
• 🧭 Null Hypothesis:
  • Kruskal-Wallis: The population medians of all groups are equal.
  • Friedman: The population medians of all related groups are equal.
• 🧮 Post-hoc Tests: If the null hypothesis is rejected, post-hoc tests (e.g., Dunn's test for Kruskal-Wallis, Wilcoxon signed-rank test with Bonferroni correction for Friedman) can be used to determine which specific groups differ significantly.

Practice Quiz

1. Which of the following is a key assumption of the Kruskal-Wallis test?
   1. A. Data must be normally distributed.
   2. B. Samples must be independent.
   3. C. Samples must be paired.
   4. D. Data must be interval or ratio.
2. The Friedman test is a non-parametric alternative to which parametric test?
   1. A. Independent samples t-test.
   2. B. Paired samples t-test.
   3. C. One-way ANOVA.
   4. D. Repeated measures ANOVA.
3. What type of data is most appropriate for the Kruskal-Wallis test?
   1. A. Nominal data.
   2. B. Ordinal or continuous data.
   3. C. Interval data only.
   4. D. Ratio data only.
4. In the Kruskal-Wallis test, what does $R_i$ represent in the formula $H = \frac{12}{N(N+1)} \sum_{i=1}^{k} \frac{R_i^2}{n_i} - 3(N+1)$?
   1. A. The range of group i.
   2. B. The sum of ranks for group i.
   3. C. The mean rank of group i.
   4. D. The median of group i.
5. Which post-hoc test is commonly used after a significant Kruskal-Wallis test to determine which groups differ significantly?
   1. A. Tukey's HSD.
   2. B. Bonferroni correction.
   3. C. Dunn's test.
   4. D. Fisher's LSD.
6. In the Friedman test, what does 'n' represent in the formula $Q = \frac{12}{nk(k+1)} \sum_{j=1}^{k} R_j^2 - 3n(k+1)$?
   1. A. The number of treatments.
   2. B. The number of blocks (subjects).
   3. C. The total number of observations.
   4. D. The number of groups.
7. What is the null hypothesis for the Friedman test?
   1. A. The population means of all groups are equal.
   2. B. The population medians of all related groups are equal.
   3. C. The population variances of all groups are equal.
   4. D. There is no difference between any of the groups.