nonparametric-tests-guide

Nonparametric Tests Guide

A skill for selecting and applying nonparametric statistical tests when data violate parametric assumptions. Covers rank-based tests for group comparisons, correlation, and paired data, with implementation examples and guidance on reporting.

When to Use Nonparametric Tests

Decision Criteria

Use nonparametric tests when:
  - Data are ordinal (Likert scales, rankings)
  - Distribution is clearly non-normal (heavy skew, outliers)
  - Sample size is very small (n < 15-20 per group)
  - Homogeneity of variance is violated
  - You are analyzing ranks or medians rather than means

Use parametric tests when:
  - Data are approximately normal (or n > 30 by CLT)
  - Variance is homogeneous across groups
  - You need greater statistical power
  - The parametric assumptions are reasonably met

Test Selection Guide

Parametric Test	Nonparametric Alternative	Use Case
Independent t-test	Mann-Whitney U	Compare 2 independent groups
Paired t-test	Wilcoxon signed-rank	Compare 2 related samples
One-way ANOVA	Kruskal-Wallis H	Compare 3+ independent groups
Repeated measures ANOVA	Friedman test	Compare 3+ related samples
Pearson correlation	Spearman rank correlation	Measure association
Chi-square test	Fisher's exact test	Compare proportions (small n)

Mann-Whitney U Test

Two Independent Groups

from scipy import stats
import numpy as np


def mann_whitney_test(group_a: list, group_b: list) -> dict:
    """
    Perform Mann-Whitney U test for two independent groups.

    Args:
        group_a: Observations from group A
        group_b: Observations from group B
    """
    statistic, p_value = stats.mannwhitneyu(
        group_a, group_b, alternative="two-sided"
    )

    n_a, n_b = len(group_a), len(group_b)

    # Rank-biserial correlation as effect size
    r = 1 - (2 * statistic) / (n_a * n_b)

    return {
        "U_statistic": statistic,
        "p_value": p_value,
        "n_a": n_a,
        "n_b": n_b,
        "median_a": np.median(group_a),
        "median_b": np.median(group_b),
        "effect_size_r": abs(r),
        "effect_interpretation": (
            "small" if abs(r) < 0.3
            else "medium" if abs(r) < 0.5
            else "large"
        )
    }


# Example usage
control = [12, 15, 14, 10, 13, 11, 16, 9, 14, 12]
treatment = [18, 22, 19, 17, 20, 21, 16, 23, 19, 20]
result = mann_whitney_test(control, treatment)
print(f"U = {result['U_statistic']}, p = {result['p_value']:.4f}")
print(f"Effect size r = {result['effect_size_r']:.3f} ({result['effect_interpretation']})")