Nonparametric Methods
Overview
Nonparametric methods replace distributional assumptions with rank-based or permutation-based approaches. The Mann-Whitney U test is the nonparametric alternative to the two-sample t-test, comparing distributions using ranks. The Wilcoxon signed-rank test replaces the paired t-test for matched data. The Kruskal-Wallis test extends to multiple groups (nonparametric ANOVA). The Friedman test handles repeated measures. Permutation tests build exact null distributions by shuffling labels, providing p-values for any test statistic. Bootstrap methods approximate sampling distributions without assumptions. These tests are slightly less powerful than parametric tests when assumptions hold, but far more robust when they don't.
Key Concepts
Nonparametric Test Equivalents
| Nonparametric Test | Parametric Equivalent | Use Case | What It Tests |
|---|---|---|---|
| Mann-Whitney U | Two-sample t-test | Independent samples | Distributions differ |
| Wilcoxon signed-rank | Paired t-test | Paired samples | Median difference β 0 |
| Kruskal-Wallis | One-way ANOVA | Multiple independent groups | At least one group differs |
| Friedman | Repeated-measures ANOVA | Multiple related groups | Treatment effects |
| Permutation test | Any parametric test | Any test statistic | Any null hypothesis |
When to Use Each Test
| Data Characteristic | Recommended Test |
|---|---|
| Skewed, 2 independent groups | Mann-Whitney U |
| Skewed, paired data | Wilcoxon signed-rank |
| Skewed, 3+ independent groups | Kruskal-Wallis |
| Ordinal data | Any rank-based test |
| Very small sample () | Permutation test |
| Complex test statistic | Bootstrap |
Quick Example
Key Takeaways
Deep Dive
For detailed explanations, worked examples, and Python implementations, explore the dedicated statistics lessons:
Permutation Tests
- Permutation Tests β Exact p-values by shuffling labels, no distributional assumptions, and when to use
Paired Tests
- Wilcoxon Signed-Rank Test β Nonparametric paired test for medians with full derivation
Independent Group Tests
- Mann-Whitney U Test β Nonparametric two-sample test using ranks with examples
Multiple Group Tests
- Kruskal-Wallis Test β Nonparametric one-way ANOVA using ranks
Repeated Measures
- Friedman Test β Nonparametric repeated-measures ANOVA for related groups
Related Topics
- t-Tests β The parametric alternatives these tests replace
- One-Way ANOVA β The parametric counterpart to Kruskal-Wallis
- Bootstrap Confidence Intervals β Distribution-free confidence intervals
- Effect Size β Measuring practical significance alongside nonparametric tests