How to Analyze Kruskal Wallis Test - Non Parametric

The Kruskal-Wallis test is a non-parametric analysis method used to test differences among three or more independent groups on ordinal data. This test can be considered an extension of the Mann-Whitney U test, which is specific for two independent groups. If only two groups are being tested, then the result of the Kruskal-Wallis test will be identical to the Mann-Whitney U test. The Kruskal-Wallis test serves as an appropriate alternative for One-Way Analysis of Variance (ANOVA) or Completely Randomized Design (CRD) when the assumptions of normality and homogeneity of variance for ANOVA are not met, and data transformation is either not possible or ineffective. If the results of the Kruskal-Wallis test indicate a significant difference among groups, it is recommended to proceed with Post hoc Tests to determine which groups differ. In SmartstatXL, several Post hoc Test options available include Dunn, Nemenyi, and Sach.

Case Example

Sourced from: Handbook of Parametric and Nonparametric Statistical Procedures, Fifth Edition by David J. Sheskin

Example 22.1

A psychologist conducts research to determine whether noise can inhibit learning. Each of the 15 subjects is randomly assigned to one of three groups. Each subject is given 20 minutes to memorize a list of 10 nonsense syllables, being informed that they will be tested the following day. Five subjects assigned to Group 1, the no-noise condition, learn the list of nonsense syllables while in a quiet room. Five subjects assigned to Group 2, the moderate noise condition, learn the list of nonsense syllables while listening to classical music. Five subjects assigned to Group 3, the extreme noise condition, learn the list of nonsense syllables while listening to rock music. The number of nonsense syllables correctly remembered by the 15 subjects is as follows: Group 1: 8, 10, 9, 10, 9; Group 2: 7, 8, 5, 8, 5; Group 3: 4, 8, 7, 5, 7. Does the data indicate that noise affects the subjects' performance?

Analysis Steps

The following are the steps for conducting the Kruskal-Wallis test and Post hoc Tests using SmartstatXL, Excel Add-in:

1. Activate the worksheet (Sheet) that will be analyzed.
2. Place the cursor on the dataset (to create a dataset, see the Data Preparation method). The dataset can be arranged in two layouts:
3. Grouped based on level (comparison between levels)
4. Grouped based on variables (comparison between variables)
   
5. If the active cell is not on the dataset, SmartstatXL will automatically try to determine the dataset.
6. Activate the SmartstatXL Tab
7. Click the Non-Parametric Menu. SmartstatXL will display a dialog box to ensure whether the dataset is correct or not (usually the cell address of the dataset is automatically selected correctly).
   
8. If correct, Click the Next Button
9. Next, the Non-Parametric Test Dialog Box will appear:
   
10. If the data layout used is based on comparison between variables, the following dialog box will appear:
    
11. Finally, press the "OK" button

Analysis Results

Below is the Output Analysis of the Kruskal-Wallis (One Way ANOVA):

Statistical Summary:

Based on the analysis results:

1. Proposed Hypothesis:
   • H0 (Null Hypothesis): All samples come from populations with the same median. This means there is no difference between groups based on noise level.
   • H1 (Alternative Hypothesis): Samples come from populations with not all the same medians. This means there is a difference between groups based on noise level.
2. Statistical Results:
   • The H-value (test statistic) is 8.435, which is adjusted to 8.747.
   • The critical value for this test (with df = 2) is 5.991.
   • The p-value (probability) is 0.013.
3. Interpretation:
   • Since the H-value (8.747) is greater than the critical value (5.991), and the p-value (0.013) is smaller than 0.05, we reject the null hypothesis (H0). This means there is a significant difference between the groups based on noise level in terms of their performance in recalling nonsense syllables.

Conclusion:

Based on the Kruskal-Wallis analysis, there is sufficient evidence to state that noise affects the subjects' performance in recalling nonsense syllables. In other words, the noise level has an effect on someone's ability to remember information.

Average Comparison

Table of average values, Mean Rank, Sum Rank, and Post hoc Tests

From this table, we can see the comparison between groups based on average values, mean ranks, and sum ranks. Additionally, there are also results from various Post hoc Tests (Dunn, Nemenyi, Sach) used to determine which differences are significant after the Kruskal-Wallis Test indicated there were differences.

Let's interpret these comparison results:

1. Data Description:
   • Group 1: With an average of 9.20, mean rank of 12.70, and sum rank of 63.50.
   • Group 2: With an average of 6.60, mean rank of 6.20, and sum rank of 31.00.
   • Group 3: With an average of 6.20, mean rank of 5.10, and sum rank of 25.50.
2. Post hoc Test Interpretation:
   • Dunn: Group 1 (b) significantly differs from Group 3 (a), but does not significantly differ from Group 2 (ab). Groups 2 and 3 do not significantly differ from each other as they both carry the label "a".
   • Nemenyi: The result is the same as the Dunn test.
   • Sach: The result is also the same as the Dunn and Nemenyi tests.
3. Critical Distance (CD-KW):
   • CD-KW provides the critical distance between the mean ranks of two groups to determine if their difference is significant. If the difference between the mean ranks of two groups is greater than CD-KW, then the difference is significant.

Conclusion:

Based on the Post hoc Tests (Dunn, Nemenyi, Sach), Group 1 performs significantly better than Group 3 in recalling nonsense syllables. However, there is no significant difference between Groups 2 and 3, as well as between Groups 1 and 2. This indicates that extreme noise (rock music) may have a greater impact on recall ability compared to no-noise or moderate noise (classical music) conditions.