Key Takeaways: Nonparametric Methods

One-Sentence Summary

Nonparametric methods — including the sign test, Wilcoxon rank-sum (Mann-Whitney U), Wilcoxon signed-rank, and Kruskal-Wallis tests — replace raw data values with ranks to compare groups without requiring normality, making them ideal for ordinal data, small samples, and datasets with heavy outliers, at a cost of only about 5% efficiency when normality actually holds.

Core Concepts at a Glance

The Nonparametric Toolkit

When to Use Each Test

Information Used by Each Test

More information used = more power (when assumptions hold). Less information used = more robustness (when assumptions fail).

The Ranking Procedure

1. Combine all observations from all groups into a single list
2. Sort from smallest to largest
3. Assign ranks 1, 2, 3, ... from smallest to largest
4. Handle ties with average ranks (midranks)
5. Return each rank to its original group

Quick check: Sum of all ranks = $N(N+1)/2$

Test Procedures

Sign Test (Paired Data)

1. Compute differences $d_i$
2. Drop zeros
3. Count positives ($n^+$) and negatives ($n^-$)
4. Under $H_0$: $n^+ \sim \text{Binomial}(n, 0.5)$
5. p-value from binomial distribution

Mann-Whitney U (Two Independent Groups)

1. Combine and rank all $N = n_1 + n_2$ observations
2. Compute rank sums: $W_1$, $W_2$
3. Compute: $U_1 = W_1 - n_1(n_1+1)/2$, $U_2 = W_2 - n_2(n_2+1)/2$
4. Check: $U_1 + U_2 = n_1 \times n_2$
5. p-value from Mann-Whitney distribution or normal approximation

Wilcoxon Signed-Rank (Paired Data)

1. Compute differences $d_i$
2. Drop zeros
3. Rank absolute differences $|d_i|$
4. Apply signs to ranks
5. Compute $W^+ = \sum$ positive ranks, $W^- = \sum$ negative ranks
6. Check: $W^+ + W^- = n(n+1)/2$
7. Test statistic: $W = \min(W^+, W^-)$

Kruskal-Wallis (Three or More Groups)

1. Combine and rank all $N$ observations
2. Compute mean rank $\bar{R}_i$ for each group
3. Test statistic: $H = \frac{12}{N(N+1)} \sum n_i (\bar{R}_i - \bar{R})^2$
4. Under $H_0$: $H \sim \chi^2(k-1)$
5. If significant: pairwise Mann-Whitney U with Bonferroni correction

Key Python Code

from scipy import stats

# Sign test (via binomial test)
p_value = stats.binomtest(n_positive, n_total, 0.5,
                          alternative='two-sided').pvalue

# Mann-Whitney U test
stat, p_value = stats.mannwhitneyu(group1, group2,
                                    alternative='two-sided')

# Wilcoxon signed-rank test
stat, p_value = stats.wilcoxon(x, y,
                                alternative='two-sided')

# Kruskal-Wallis test
H_stat, p_value = stats.kruskal(group1, group2, group3)

Excel: Limited Support

Excel's Data Analysis ToolPak does not include nonparametric tests. Options: - Manual calculation using RANK.AVG() - Free add-in: Real Statistics Resource Pack - Recommended: Use Python for nonparametric analysis

The Power Tradeoff

Asymptotic Relative Efficiency (normal data): ~0.955 for all three rank-based tests vs. their parametric counterparts. That's a 4.5% efficiency loss — roughly equivalent to needing one extra observation per group of 20.

Common Mistakes

Decision Flowchart Summary

Is your data ordinal (1-5 scale, rankings)?
  → YES: Use nonparametric
  → NO: Continue

Is n < 15 per group AND data clearly non-normal?
  → YES: Use nonparametric
  → NO: Continue

Are there heavy outliers that distort means?
  → YES: Use nonparametric
  → NO: Parametric tests are likely fine

Still unsure?
  → Run BOTH. If they agree, report the one matching
    your data type. If they disagree, investigate why.

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