Kruskal Wallis Test

The Kruskal-Wallis test (H test) is a nonparametric statistical test used to compare three or more independent groups to determine if there are statistically significant differences between them. It is an extension of the Mann-Whitney U test, which is used for comparing two groups.

Unlike the one-way ANOVA, which assumes normality in data distribution, the Kruskal-Wallis test does not require the data to be normally distributed. Instead, it ranks the data, making it suitable for ordinal or non-normally distributed continuous data.

Key Characteristics

• Nonparametric: Does not assume a normal distribution, making it useful for skewed or ordinal data.
• Rank-based: Converts data to ranks instead of using raw values.
• Independent groups: Assumes that observations in different groups are independent of each other.

When to Use the Kruskal-Wallis Test

This test is applicable in the following scenarios:

• When comparing three or more independent groups.
• When the data is ordinal or not normally distributed.
• When sample sizes are small or unequal across groups.

The Kruskal-Wallis test is useful when the assumptions for one-way ANOVA are not met. Some common examples include: Medical Research, Social Sciences, Psychology Studies.

Hypotheses in the Kruskal-Wallis Test

• Null Hypothesis (H₀): All groups have the same central tendency and come from the same population.
• Alternative Hypothesis (H₁): At least one group has a different central tendency from the others.

How the Kruskal-Wallis Test Works

Instead of analyzing means, the Kruskal-Wallis test examines differences in the rank sums of the groups. The procedure involves the following steps:

Step 1: Rank the Data

• Combine all data from different groups.
• Assign ranks to the data, with tied values receiving the average of their ranks.

Step 2: Compute the Rank Sum

• Sum up the ranks for each group.

Step 3: Calculate the Test Statistic (H)

The Kruskal-Wallis test statistic H is given by:

H = \left[ \frac{12}{n(n+1)} \sum \frac{R_i^2}{n_i} \right] - 3(n+1)

Where:

• n = Total number of observations.
• n_i​ = Number of observations in group iii.
• R_i​ = Sum of ranks for group iii.
• k = Number of groups.

Step 4: Determine the Critical Value

• Compare H to the Chi-square distribution with k − 1 degrees of freedom.
• If H exceeds the critical value, reject the null hypothesis.

Interpretation of Results

• If the p-value is less than the significance level (e.g., 0.05), at least one group differs significantly.
• If the p-value is greater than the significance level, we fail to reject the null hypothesis, meaning there is no strong evidence of differences among the groups._.

Example: Evaluating Training Methods

A university wants to evaluate the effectiveness of three different training methods on student performance. A sample of 20 students was divided into three groups based on their training method:

1. Video Lectures
2. Books and Articles
3. Classroom Training

The students' examination scores are as follows:

Step 1: Identify Variables

• Independent Variable: Training method (3 levels)
• Dependent Variable: Examination scores

Step 2: State Hypotheses

• H₀: The mean exam scores across all training methods are equal (μ₁ = μ₂ = μ₃).
• H₁: At least one training method leads to different exam scores.

Step 3: Rank the Data

The scores from all groups are combined, sorted in ascending order and assigned ranks. If there are ties, the average rank is assigned.

Step 4: Calculate Sum of Ranks

Each group’s ranks are summed:

Step 5: Calculate H-Statistic

Using the formula:

H = \frac{12}{20(21)} \left[\frac{66^2}{6}+\frac{70^2}{7}+\frac{74^2}{7}\right] - 3(21)

H = 0.0938

Step 6: Determine Critical Value

• Degrees of freedom (df) = K - 1 = 3 - 1 = 2
• Significance Level (α) = 0.10
• From the chi-square table: X^2_{0.10,2} = 4.605

Step 7: Compare H with Critical Value

• If H_{calc} < X^2, fail to reject H₀ (no significant difference).
• If H_{calc} > X^2, reject H₀ (significant difference exists).

Since 0.0938 < 4.605, we fail to reject H₀, meaning there is no significant difference in students' performance across the three training methods.