23. KRUSKAL–WALLIS TEST (ONE-WAY ANOVA)

The Kruskal–Wallis Test is a non-parametric alternative to One-Way ANOVA. It is used to compare the medians of three or more independent groups when the data does not meet the assumptions of parametric ANOVA, such as normality or equal variances. This test is particularly useful in biomedical and pharmaceutical research where datasets are often small, skewed, or ordinal in nature.

When to Use the Kruskal–Wallis Test?

  • When comparing three or more independent groups.
  • When data is non-normal or ordinal.
  • When sample sizes are small.
  • When variances across groups are not equal.
  • When the assumption of normality for One-Way ANOVA is violated.

Hypotheses

  • H₀: All groups come from populations with the same median.
  • H₁: At least one group differs in median.

Principle of the Test

The Kruskal–Wallis test converts all observations into ranks, then compares the distribution of ranks among the groups. If the rank sums differ significantly, it suggests that at least one group differs from the others.

Steps in Performing the Kruskal–Wallis Test

  1. Combine all data from all groups into one list.
  2. Rank all observations from lowest to highest.
  3. If ties occur, assign average ranks.
  4. Calculate the sum of ranks for each group (R₁, R₂, R₃…).
  5. Apply the Kruskal–Wallis formula to compute the test statistic (H).

Formula

H = [12 / (N(N + 1))] × Σ (Ri² / ni) − 3(N + 1)

Where:

  • N = total number of observations
  • ni = sample size of group i
  • Ri = sum of ranks of group i

Distribution

The test statistic (H) follows a Chi-square (χ²) distribution with k – 1 degrees of freedom, where k is the number of groups.

Interpreting the Results

  • If calculated H ≥ critical χ² → Reject H₀ (significant difference).
  • If calculated H < critical χ² → Fail to reject H₀ (no significant difference).

If the test is significant, post-hoc tests (e.g., Dunn’s test) are required to identify which groups differ.

Example (Illustration)

A researcher compares the effectiveness of three antihypertensive drugs. Blood pressure reduction scores are ranked and analyzed using the Kruskal–Wallis test. A significant H value indicates that at least one drug shows different performance compared to the others.

Assumptions

  • Samples are independent.
  • Observations can be ranked.
  • Data distribution shapes across groups are similar.
  • Measurements may be ordinal, interval, or ratio (non-normal).

Advantages

  • Does not require normal distribution.
  • Can compare 3+ groups unlike Mann-Whitney (which compares only two).
  • Simple and robust for non-parametric data.
  • Useful for skewed or ordinal data.

Limitations

  • Does not specify which groups differ (requires post-hoc tests).
  • Less powerful than One-Way ANOVA for normally distributed data.
  • Assumes similar distribution shapes across groups.

Applications in Research

  • Comparing different drug formulations.
  • Evaluating symptom scores among multiple treatment groups.
  • Quality control studies in pharmacy.
  • Comparing non-normal physiological measurements.

Detailed Notes:

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