Korean, Edit

Chapter 14-3. Kruskal-Wallis H test

Recommended Article: 【Statistics】 Chapter 14. Statistical Tests


1. Overview

2. Kruskal-Wallis H Test

3. Friedman test



1. Overview

⑴ Definition

① A test method for comparing distributions of three or more groups

② Used for the same purpose as one-way ANOVA in parametric methods

③ Tests whether the group median is the same, not the mean

④ Sample sizes may vary across groups

⑵ (Reference) Choice of Test Method

① Single sample

○ Parametric test: Single sample T-test

○ Non-parametric test: Sign test, Wilcoxon signed rank test

② Two samples (paired samples): Essentially the same as a single sample

○ Parametric test: Paired sample T-test

○ Non-parametric test: Sign test, Wilcoxon signed rank test

③ Two samples (independent samples)

○ Parametric test: Independent sample T-test

○ Non-parametric test: Wilcoxon rank sum test

④ Analysis of variance

○ Parametric test: ANOVA

○ Non-parametric test: Kruskal-Wallis test

⑤ Randomness

○ Non-parametric test: Run test

⑥ Correlation analysis

○ Pearson correlation coefficient

○ Spearman rank correlation coefficient



2. Kruskal-Wallis H Test

⑴ Example sample


스크린샷 2025-03-31 오후 2 29 33

Figure 1. Example sample


Step 1. Set up hypotheses

① Null hypothesis H0: Medians of each group are the same

② Alternative hypothesis H1: At least one group’s median is different

Step 2. Assign ranks to 16 data points (four data points from each of four sample groups)

Step 3. Define the test statistic H as follows: It has a closed form.


스크린샷 2025-04-01 오전 8 44 59


① N: Total number of samples

② Rij: Overall rank of Yij

③ R.j: Sum of ranks of group j

④ R..: Sum of all ranks

⑤ The total sum of squares is used in the denominator as opposed to the error sum of squares as in regular one-way ANOVA.

Step 4. Apply H to the chi-squared test

① H is distribution-free like other rank-based tests, but it asymptotically follows a chi-squared distribution.

② Degrees of freedom: the number of groups (J) − 1. That is, in the example above, the degrees of freedom are 4 − 1 = 3.

③ Each group follows χ2(1), and due to the constraint imposed by the total sum, the degrees of freedom for the chi-squared test become J−1.

⑹ Rejection region for significance level α

① If H ≥ h(α, k, (1, 2, ∙∙∙, m)), then reject H0

② h(α, k, (1, 2, ∙∙∙, m)) is the upper 100α percentile of H satisfying P(H ≥ h(α, k, (1, 2, ∙∙∙, m)))

RStudio

kruskal.test(y ~ x, data = my_data)



**3. Friedman test

⑴ A rank-based procedure for repeated measure designs, to test

① Null hypothesis H0: For all i = 1, …, I, (Yi,1, …, Yi,J) are exchangeable

② Alternative hypothesis H1: The negation on H0

⑵ Formula


스크린샷 2025-04-01 오전 9 02 53


① The treatment reponses are compared within each subject.

② Rij: The rank of Yi,j among (Yi,1, …, Yi,J).

③ I: The number of repetitive experiments.

④ Under the null, G has asymptotically (as I → ∞) the chi-squared distribution with J-1 degrees of freedom.



Posted: 2019.08.24 00:58

results matching ""

    No results matching ""