12.3 Non-parametric Tests

Non-parametric tests are crucial when data doesn't fit normal distribution patterns. They're useful for small samples, outliers, and non-continuous data. These tests are more flexible but may sacrifice some statistical power compared to parametric tests.

Researchers use non-parametric tests like Mann-Whitney U, Wilcoxon signed-rank, and Kruskal-Wallis when data violates parametric assumptions. They're robust alternatives, especially for skewed distributions or small samples, but may not provide effect sizes or pinpoint group differences.

Assumptions and Applications of Non-Parametric Tests

Assumptions of parametric tests

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• Parametric tests require data to follow a normal distribution (bell-shaped curve)
• Homogeneity of variance assumes equal variances across groups (similar spread)
• Independence requires observations to be independent of each other (no relationship between data points)
• Interval or ratio scale data is measured on a continuous scale with equal intervals (temperature in ℃, weight in kg)
• Violations of these assumptions can lead to inaccurate p-values, confidence intervals, and increased Type I (false positive) or Type II (false negative) error rates
• Reduced power to detect significant differences when assumptions are not met

Applications of non-parametric tests

• Non-parametric tests are appropriate when data violates assumptions of parametric tests
  • Non-normal distributions such as skewed (asymmetrical) or bimodal (two peaks) distributions
  • Unequal variances across groups (heteroscedasticity)
  • Dependent observations or repeated measures (multiple measurements from the same subject)
  • Ordinal (ranked) or nominal (categorical) scale data
• Suitable for small sample sizes where normality cannot be assumed (n < 30)
• Robust to the presence of outliers or extreme values (data points far from the mean)
• Non-parametric tests serve as alternatives to parametric tests (t-tests, ANOVA) when assumptions are violated

Common Non-Parametric Tests and Interpretation

Common non-parametric test methods

• Mann-Whitney U test (Wilcoxon rank-sum test) compares two independent groups
  • Null hypothesis: The two groups have the same distribution
  • Reject null hypothesis if p-value < significance level (0.05)
• Wilcoxon signed-rank test compares two related samples or repeated measures
  • Null hypothesis: The median difference between pairs is zero
  • Reject null hypothesis if p-value < significance level
• Kruskal-Wallis test compares three or more independent groups
  • Null hypothesis: All groups have the same distribution
  • Reject null hypothesis if p-value < significance level
  • Post-hoc tests like Dunn's test for pairwise comparisons between groups

Parametric vs non-parametric test comparison

• Non-parametric tests have fewer assumptions about data distribution and are applicable to ordinal or nominal scale data
• More robust to outliers and extreme values and suitable for small sample sizes
• Less powerful than parametric tests when assumptions are met and may not provide estimates of effect size or confidence intervals
• Some tests (Kruskal-Wallis) do not identify which specific groups differ
• Parametric tests are more powerful when assumptions are met and provide estimates of effect size and confidence intervals
• Wider range of parametric tests available (t-tests, ANOVA, regression)
• Parametric tests are sensitive to violations of assumptions and less robust to outliers and extreme values