23. Nonparametric Statistics

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23. Nonparametric Statistics

Previous: Multivariate Analysis | Next: Experimental Design

Overview

Nonparametric statistics are methods for analyzing data without assumptions about the population distribution. They are useful when normality is not satisfied or when sample size is small.

1. When Nonparametric Tests Are Needed

1.1 When to Use Nonparametric Tests?

import numpy as np
import pandas as pd
import scipy.stats as stats
import matplotlib.pyplot as plt

np.random.seed(42)

def when_to_use_nonparametric():
    """When to use nonparametric tests"""
    print("""
    ================================================
    When to Use Nonparametric Tests
    ================================================

    1. Violation of Normality
       - Data does not follow a normal distribution
       - Rejection by Shapiro-Wilk test, etc.

    2. Small Sample Size
       - When n < 30, CLT is difficult to apply
       - Normality tests also lack power

    3. Ordinal/Rank Data
       - Likert scales (1-5 points)
       - Ranked data

    4. Presence of Outliers
       - Compare medians instead of means (which are sensitive to extreme values)
       - Rank-based methods are robust to outliers

    5. Violation of Homogeneity Assumption
       - Violation of variance homogeneity (equal variance) assumption

    ================================================
    Advantages and Disadvantages of Nonparametric Tests
    ================================================

    Advantages:
    - No distributional assumptions required
    - Robust to outliers
    - Suitable for ordinal data

    Disadvantages:
    - Lower power than parametric tests when assumptions are met
    - Effect size interpretation can be difficult
    - Difficult to apply to some complex designs
    """)

when_to_use_nonparametric()

# Normality test example
def check_normality(data, alpha=0.05):
    """Perform normality test"""
    # Shapiro-Wilk
    stat_sw, p_sw = stats.shapiro(data)

    # D'Agostino-Pearson
    if len(data) >= 20:
        stat_da, p_da = stats.normaltest(data)
    else:
        stat_da, p_da = np.nan, np.nan

    print("=== Normality Tests ===")
    print(f"Sample size: {len(data)}")
    print(f"Shapiro-Wilk: W={stat_sw:.4f}, p={p_sw:.4f}")
    if not np.isnan(p_da):
        print(f"D'Agostino-Pearson: p={p_da:.4f}")

    is_normal = p_sw > alpha
    print(f"Conclusion: {'Can be considered normal' if is_normal else 'Not normal'} (ฮฑ={alpha})")

    return is_normal

# Normal data
normal_data = np.random.normal(50, 10, 30)
check_normality(normal_data)

print()

# Non-normal data (exponential distribution)
skewed_data = np.random.exponential(10, 30)
check_normality(skewed_data)

1.2 Parametric vs Nonparametric Test Correspondence

Parametric Test Nonparametric Test Situation
1-sample t-test Wilcoxon signed-rank test Single sample, median test
Independent t-test Mann-Whitney U Two independent samples
Paired t-test Wilcoxon signed-rank test Paired samples
One-way ANOVA Kruskal-Wallis H 3+ independent samples
Repeated measures ANOVA Friedman 3+ paired samples
Pearson correlation Spearman/Kendall Correlation

2. Mann-Whitney U Test

2.1 Concept

Purpose: Compare distributions of two independent samples (median or distribution location)

Hypotheses: - Hโ‚€: The two groups have identical distributions - Hโ‚: The two groups have different distributions (or one is probabilistically larger)

Test statistic: U (based on rank sum)

def mann_whitney_example():
    """Mann-Whitney U test example"""
    np.random.seed(42)

    # Scenario: Compare treatment effects of two groups (normality violation)
    # Group A: Control
    # Group B: Treatment

    group_a = np.random.exponential(20, 25)  # Non-normal
    group_b = np.random.exponential(25, 25) + 5  # Non-normal, larger values

    print("=== Mann-Whitney U Test ===")
    print(f"\nGroup A: n={len(group_a)}, median={np.median(group_a):.2f}")
    print(f"Group B: n={len(group_b)}, median={np.median(group_b):.2f}")

    # Normality tests
    print("\nNormality tests:")
    _, p_a = stats.shapiro(group_a)
    _, p_b = stats.shapiro(group_b)
    print(f"  Group A: p={p_a:.4f}")
    print(f"  Group B: p={p_b:.4f}")

    # Mann-Whitney U test
    statistic, p_value = stats.mannwhitneyu(group_a, group_b, alternative='two-sided')

    print(f"\nMann-Whitney U test:")
    print(f"  U statistic: {statistic:.2f}")
    print(f"  p-value: {p_value:.4f}")

    if p_value < 0.05:
        print("  Conclusion: Significant difference between groups (p < 0.05)")
    else:
        print("  Conclusion: No significant difference between groups (p >= 0.05)")

    # Effect size: rank-biserial correlation
    n1, n2 = len(group_a), len(group_b)
    r = 1 - (2 * statistic) / (n1 * n2)
    print(f"\nEffect size (rank-biserial r): {r:.3f}")
    print(f"  |r| < 0.1: Small effect")
    print(f"  0.1 <= |r| < 0.3: Medium effect")
    print(f"  |r| >= 0.3: Large effect")

    # Visualization
    fig, axes = plt.subplots(1, 3, figsize=(14, 4))

    # Boxplot
    ax = axes[0]
    ax.boxplot([group_a, group_b], labels=['Group A', 'Group B'])
    ax.set_ylabel('Value')
    ax.set_title('Boxplot')
    ax.grid(True, alpha=0.3)

    # Histogram
    ax = axes[1]
    ax.hist(group_a, bins=15, alpha=0.5, label='Group A', density=True)
    ax.hist(group_b, bins=15, alpha=0.5, label='Group B', density=True)
    ax.axvline(np.median(group_a), color='blue', linestyle='--', label='A median')
    ax.axvline(np.median(group_b), color='orange', linestyle='--', label='B median')
    ax.set_xlabel('Value')
    ax.set_ylabel('Density')
    ax.set_title('Distribution Comparison')
    ax.legend()
    ax.grid(True, alpha=0.3)

    # Rank distribution
    ax = axes[2]
    all_data = np.concatenate([group_a, group_b])
    ranks = stats.rankdata(all_data)
    ranks_a = ranks[:len(group_a)]
    ranks_b = ranks[len(group_a):]
    ax.hist(ranks_a, bins=15, alpha=0.5, label='Group A ranks')
    ax.hist(ranks_b, bins=15, alpha=0.5, label='Group B ranks')
    ax.set_xlabel('Rank')
    ax.set_ylabel('Frequency')
    ax.set_title('Rank Distribution')
    ax.legend()
    ax.grid(True, alpha=0.3)

    plt.tight_layout()
    plt.show()

    return group_a, group_b

group_a, group_b = mann_whitney_example()

2.2 One-sided Tests

# One-sided test: Test if Group B is larger than Group A
stat_greater, p_greater = stats.mannwhitneyu(group_a, group_b, alternative='less')
print(f"One-sided test (B > A): p = {p_greater:.4f}")

stat_less, p_less = stats.mannwhitneyu(group_a, group_b, alternative='greater')
print(f"One-sided test (A > B): p = {p_less:.4f}")

3. Wilcoxon Signed-Rank Test

3.1 Paired Sample Comparison

Purpose: Test if the difference between paired measurements is zero

Use case: Before/after comparison, paired samples

def wilcoxon_signed_rank_example():
    """Wilcoxon signed-rank test example"""
    np.random.seed(42)

    # Scenario: Weight change before and after diet program
    n = 20
    before = np.random.normal(80, 10, n)
    # Mean 3kg reduction effect + non-normal error
    after = before - 3 + np.random.exponential(2, n) - np.random.exponential(2, n)

    diff = after - before

    print("=== Wilcoxon Signed-Rank Test ===")
    print(f"\nSample size: {n}")
    print(f"Before: mean={before.mean():.2f}, median={np.median(before):.2f}")
    print(f"After: mean={after.mean():.2f}, median={np.median(after):.2f}")
    print(f"Difference: mean={diff.mean():.2f}, median={np.median(diff):.2f}")

    # Normality test on differences
    _, p_norm = stats.shapiro(diff)
    print(f"\nNormality of differences: p={p_norm:.4f}")

    # Wilcoxon test
    statistic, p_value = stats.wilcoxon(before, after, alternative='two-sided')

    print(f"\nWilcoxon test:")
    print(f"  W statistic: {statistic:.2f}")
    print(f"  p-value: {p_value:.4f}")

    # Comparison with paired t-test
    t_stat, t_p = stats.ttest_rel(before, after)
    print(f"\nPaired t-test (for comparison):")
    print(f"  t statistic: {t_stat:.2f}")
    print(f"  p-value: {t_p:.4f}")

    # Effect size
    r = statistic / (n * (n + 1) / 2)
    print(f"\nEffect size (r): {abs(1-2*r):.3f}")

    # Visualization
    fig, axes = plt.subplots(1, 3, figsize=(14, 4))

    # Before/After comparison
    ax = axes[0]
    ax.boxplot([before, after], labels=['Before', 'After'])
    ax.set_ylabel('Weight (kg)')
    ax.set_title('Before/After Comparison')
    ax.grid(True, alpha=0.3)

    # Individual changes
    ax = axes[1]
    for i in range(n):
        ax.plot([0, 1], [before[i], after[i]], 'b-', alpha=0.5)
    ax.plot([0, 1], [before.mean(), after.mean()], 'r-', linewidth=2, label='Mean')
    ax.set_xticks([0, 1])
    ax.set_xticklabels(['Before', 'After'])
    ax.set_ylabel('Weight (kg)')
    ax.set_title('Individual Changes')
    ax.legend()
    ax.grid(True, alpha=0.3)

    # Difference distribution
    ax = axes[2]
    ax.hist(diff, bins=10, density=True, alpha=0.7, edgecolor='black')
    ax.axvline(0, color='r', linestyle='--', label='No change')
    ax.axvline(np.median(diff), color='g', linestyle='-', label=f'Median={np.median(diff):.2f}')
    ax.set_xlabel('Difference (After - Before)')
    ax.set_ylabel('Density')
    ax.set_title('Difference Distribution')
    ax.legend()
    ax.grid(True, alpha=0.3)

    plt.tight_layout()
    plt.show()

    return before, after

before, after = wilcoxon_signed_rank_example()

3.2 One-sample Test (Median Test)

# One-sample: Test if median equals a specific value
def one_sample_wilcoxon(data, hypothesized_median):
    """
    One-sample Wilcoxon test
    H0: median = hypothesized_median
    """
    diff_from_median = data - hypothesized_median

    # Exclude zero values
    diff_from_median = diff_from_median[diff_from_median != 0]

    if len(diff_from_median) == 0:
        print("All values equal the hypothesized median.")
        return

    stat, p_value = stats.wilcoxon(diff_from_median)

    print(f"=== One-sample Wilcoxon Test ===")
    print(f"H0: median = {hypothesized_median}")
    print(f"Sample median: {np.median(data):.2f}")
    print(f"W statistic: {stat:.2f}")
    print(f"p-value: {p_value:.4f}")

    if p_value < 0.05:
        print(f"Conclusion: Median is significantly different from {hypothesized_median}")
    else:
        print(f"Conclusion: Insufficient evidence that median differs from {hypothesized_median}")

# Example
sample_data = np.random.exponential(10, 30) + 5
one_sample_wilcoxon(sample_data, hypothesized_median=10)

4. Kruskal-Wallis H Test

4.1 Concept

Purpose: Compare distributions of 3 or more independent groups (nonparametric alternative to one-way ANOVA)

Hypotheses: - Hโ‚€: All groups have identical distributions - Hโ‚: At least one group differs

def kruskal_wallis_example():
    """Kruskal-Wallis H test example"""
    np.random.seed(42)

    # Scenario: Compare effects of 3 teaching methods
    method_a = np.random.exponential(10, 25) + 60  # Traditional
    method_b = np.random.exponential(10, 25) + 65  # Online
    method_c = np.random.exponential(10, 25) + 70  # Blended

    print("=== Kruskal-Wallis H Test ===")
    print(f"\nMethod A: n={len(method_a)}, median={np.median(method_a):.2f}")
    print(f"Method B: n={len(method_b)}, median={np.median(method_b):.2f}")
    print(f"Method C: n={len(method_c)}, median={np.median(method_c):.2f}")

    # Normality tests
    print("\nNormality tests:")
    for name, data in [('A', method_a), ('B', method_b), ('C', method_c)]:
        _, p = stats.shapiro(data)
        print(f"  {name}: p={p:.4f}")

    # Kruskal-Wallis test
    H_stat, p_value = stats.kruskal(method_a, method_b, method_c)

    print(f"\nKruskal-Wallis test:")
    print(f"  H statistic: {H_stat:.2f}")
    print(f"  p-value: {p_value:.4f}")

    # Effect size: eta-squared
    N = len(method_a) + len(method_b) + len(method_c)
    k = 3
    eta_sq = (H_stat - k + 1) / (N - k)
    print(f"\nEffect size (ฮทยฒ): {eta_sq:.3f}")
    print("  0.01: Small effect")
    print("  0.06: Medium effect")
    print("  0.14: Large effect")

    # Comparison with one-way ANOVA
    F_stat, anova_p = stats.f_oneway(method_a, method_b, method_c)
    print(f"\nOne-way ANOVA (for comparison):")
    print(f"  F statistic: {F_stat:.2f}")
    print(f"  p-value: {anova_p:.4f}")

    # Visualization
    fig, axes = plt.subplots(1, 2, figsize=(12, 5))

    # Boxplot
    ax = axes[0]
    ax.boxplot([method_a, method_b, method_c],
               labels=['Method A', 'Method B', 'Method C'])
    ax.set_ylabel('Score')
    ax.set_title('Score Distribution by Teaching Method')
    ax.grid(True, alpha=0.3)

    # Violin plot
    ax = axes[1]
    parts = ax.violinplot([method_a, method_b, method_c], positions=[1, 2, 3],
                           showmeans=True, showmedians=True)
    ax.set_xticks([1, 2, 3])
    ax.set_xticklabels(['Method A', 'Method B', 'Method C'])
    ax.set_ylabel('Score')
    ax.set_title('Violin Plot')
    ax.grid(True, alpha=0.3)

    plt.tight_layout()
    plt.show()

    return method_a, method_b, method_c

method_a, method_b, method_c = kruskal_wallis_example()

4.2 Post-hoc Tests

from itertools import combinations
from scipy.stats import mannwhitneyu

def dunn_test_alternative(groups, group_names, alpha=0.05):
    """
    Alternative to Dunn test: Bonferroni-corrected Mann-Whitney U tests
    """
    n_comparisons = len(list(combinations(range(len(groups)), 2)))
    adjusted_alpha = alpha / n_comparisons

    print(f"=== Post-hoc Tests (Bonferroni correction) ===")
    print(f"Number of comparisons: {n_comparisons}")
    print(f"Adjusted ฮฑ: {adjusted_alpha:.4f}")
    print()

    results = []
    for (i, j) in combinations(range(len(groups)), 2):
        stat, p = mannwhitneyu(groups[i], groups[j], alternative='two-sided')
        significant = p < adjusted_alpha
        results.append({
            'comparison': f'{group_names[i]} vs {group_names[j]}',
            'U': stat,
            'p-value': p,
            'significant': significant
        })
        print(f"{group_names[i]} vs {group_names[j]}: U={stat:.1f}, p={p:.4f} {'*' if significant else ''}")

    return pd.DataFrame(results)

groups = [method_a, method_b, method_c]
group_names = ['A', 'B', 'C']
posthoc_results = dunn_test_alternative(groups, group_names)

5. Friedman Test

5.1 Concept

Purpose: Compare 3 or more conditions in repeated measures or blocked designs

Use case: Same subjects measured under multiple conditions

def friedman_example():
    """Friedman test example"""
    np.random.seed(42)

    # Scenario: Same students' scores on 3 exams
    n_students = 20

    # Generate correlated data (multiple exams for same students)
    ability = np.random.normal(70, 10, n_students)  # Baseline ability
    exam1 = ability + np.random.normal(0, 5, n_students)
    exam2 = ability + np.random.normal(3, 5, n_students)  # Slightly harder
    exam3 = ability + np.random.normal(6, 5, n_students)  # Harder

    print("=== Friedman Test ===")
    print(f"\nNumber of students: {n_students}")
    print(f"Exam 1: median={np.median(exam1):.2f}")
    print(f"Exam 2: median={np.median(exam2):.2f}")
    print(f"Exam 3: median={np.median(exam3):.2f}")

    # Friedman test
    stat, p_value = stats.friedmanchisquare(exam1, exam2, exam3)

    print(f"\nFriedman test:")
    print(f"  ฯ‡ยฒ statistic: {stat:.2f}")
    print(f"  p-value: {p_value:.4f}")

    # Effect size: Kendall's W
    k = 3  # Number of conditions
    W = stat / (n_students * (k - 1))
    print(f"\nEffect size (Kendall's W): {W:.3f}")
    print("  W < 0.3: Weak agreement")
    print("  0.3 <= W < 0.5: Moderate agreement")
    print("  W >= 0.5: Strong agreement")

    # Visualization
    fig, axes = plt.subplots(1, 2, figsize=(12, 5))

    # Boxplot
    ax = axes[0]
    ax.boxplot([exam1, exam2, exam3], labels=['Exam 1', 'Exam 2', 'Exam 3'])
    ax.set_ylabel('Score')
    ax.set_title('Score Distribution by Exam')
    ax.grid(True, alpha=0.3)

    # Individual profiles
    ax = axes[1]
    for i in range(min(10, n_students)):  # First 10 students only
        ax.plot([1, 2, 3], [exam1[i], exam2[i], exam3[i]], 'o-', alpha=0.5)
    ax.plot([1, 2, 3], [exam1.mean(), exam2.mean(), exam3.mean()],
            'rs-', linewidth=2, markersize=8, label='Mean')
    ax.set_xticks([1, 2, 3])
    ax.set_xticklabels(['Exam 1', 'Exam 2', 'Exam 3'])
    ax.set_ylabel('Score')
    ax.set_title('Individual Profiles')
    ax.legend()
    ax.grid(True, alpha=0.3)

    plt.tight_layout()
    plt.show()

    return exam1, exam2, exam3

exam1, exam2, exam3 = friedman_example()

5.2 Nemenyi Post-hoc Test

def nemenyi_posthoc(groups, group_names, alpha=0.05):
    """
    Nemenyi post-hoc test (after Friedman)
    Bonferroni-corrected Wilcoxon signed-rank tests
    """
    n_comparisons = len(list(combinations(range(len(groups)), 2)))
    adjusted_alpha = alpha / n_comparisons

    print(f"=== Nemenyi Post-hoc Test ===")
    print(f"Number of comparisons: {n_comparisons}")
    print(f"Adjusted ฮฑ: {adjusted_alpha:.4f}")
    print()

    results = []
    for (i, j) in combinations(range(len(groups)), 2):
        stat, p = stats.wilcoxon(groups[i], groups[j])
        significant = p < adjusted_alpha
        results.append({
            'comparison': f'{group_names[i]} vs {group_names[j]}',
            'W': stat,
            'p-value': p,
            'significant': significant
        })
        print(f"{group_names[i]} vs {group_names[j]}: W={stat:.1f}, p={p:.4f} {'*' if significant else ''}")

    return pd.DataFrame(results)

groups = [exam1, exam2, exam3]
group_names = ['Exam1', 'Exam2', 'Exam3']
nemenyi_results = nemenyi_posthoc(groups, group_names)

6. Nonparametric Correlation (Spearman, Kendall)

6.1 Spearman Rank Correlation

Characteristics: Rank-based, measures monotonic relationships (need not be linear)

def spearman_correlation_example():
    """Spearman correlation example"""
    np.random.seed(42)

    # Nonlinear relationship data
    n = 50
    x = np.random.uniform(0, 10, n)
    y = np.log(x + 1) + np.random.normal(0, 0.3, n)  # Log relationship + noise

    # Pearson correlation
    pearson_r, pearson_p = stats.pearsonr(x, y)

    # Spearman correlation
    spearman_r, spearman_p = stats.spearmanr(x, y)

    print("=== Correlation Analysis ===")
    print(f"\nPearson correlation: r={pearson_r:.4f}, p={pearson_p:.4f}")
    print(f"Spearman correlation: ฯ={spearman_r:.4f}, p={spearman_p:.4f}")
    print("\nSpearman is more appropriate for nonlinear relationships")

    # Visualization
    fig, axes = plt.subplots(1, 3, figsize=(15, 4))

    # Original data
    ax = axes[0]
    ax.scatter(x, y, alpha=0.7)
    ax.set_xlabel('X')
    ax.set_ylabel('Y')
    ax.set_title(f'Original Data\nPearson r={pearson_r:.3f}, Spearman ฯ={spearman_r:.3f}')
    ax.grid(True, alpha=0.3)

    # Rank transformation
    ax = axes[1]
    x_ranks = stats.rankdata(x)
    y_ranks = stats.rankdata(y)
    ax.scatter(x_ranks, y_ranks, alpha=0.7)
    ax.set_xlabel('X Rank')
    ax.set_ylabel('Y Rank')
    ax.set_title('After Rank Transformation')
    ax.grid(True, alpha=0.3)

    # With outlier
    ax = axes[2]
    x_outlier = np.append(x, [50])  # Extreme outlier
    y_outlier = np.append(y, [y.mean()])

    pearson_out, _ = stats.pearsonr(x_outlier, y_outlier)
    spearman_out, _ = stats.spearmanr(x_outlier, y_outlier)

    ax.scatter(x_outlier[:-1], y_outlier[:-1], alpha=0.7, label='Normal data')
    ax.scatter(x_outlier[-1], y_outlier[-1], color='r', s=100, label='Outlier', marker='x')
    ax.set_xlabel('X')
    ax.set_ylabel('Y')
    ax.set_title(f'With Outlier\nPearson={pearson_out:.3f}, Spearman={spearman_out:.3f}')
    ax.legend()
    ax.grid(True, alpha=0.3)

    plt.tight_layout()
    plt.show()

    return x, y

x, y = spearman_correlation_example()

6.2 Kendall Rank Correlation

Characteristics: Based on pairwise comparisons, more robust than Spearman

def kendall_correlation_example():
    """Kendall tau correlation example"""
    np.random.seed(42)

    # Ordinal data (Likert scale, etc.)
    n = 30
    rater1 = np.random.randint(1, 6, n)  # 1-5 points
    # Somewhat consistent but not identical
    rater2 = np.clip(rater1 + np.random.randint(-1, 2, n), 1, 5)

    # Pearson (inappropriate)
    pearson_r, _ = stats.pearsonr(rater1, rater2)

    # Spearman
    spearman_r, _ = stats.spearmanr(rater1, rater2)

    # Kendall
    kendall_tau, kendall_p = stats.kendalltau(rater1, rater2)

    print("=== Ordinal Data Correlation Analysis ===")
    print(f"\nRater 1 distribution: {np.bincount(rater1)[1:]}")
    print(f"Rater 2 distribution: {np.bincount(rater2)[1:]}")
    print(f"\nPearson r: {pearson_r:.4f} (inappropriate for ordinal data)")
    print(f"Spearman ฯ: {spearman_r:.4f}")
    print(f"Kendall ฯ„: {kendall_tau:.4f}, p={kendall_p:.4f}")

    print("\nInterpretation:")
    print("  ฯ„ = 0: Equal concordant/discordant pairs")
    print("  ฯ„ = 1: Perfect rank agreement")
    print("  ฯ„ = -1: Perfect rank disagreement")

    # Visualization
    fig, ax = plt.subplots(figsize=(8, 6))

    # Add jitter (to prevent overlapping same scores)
    jitter1 = rater1 + np.random.uniform(-0.1, 0.1, n)
    jitter2 = rater2 + np.random.uniform(-0.1, 0.1, n)

    ax.scatter(jitter1, jitter2, alpha=0.7, s=50)
    ax.plot([0, 6], [0, 6], 'r--', label='Perfect agreement line')
    ax.set_xlabel('Rater 1')
    ax.set_ylabel('Rater 2')
    ax.set_title(f'Inter-rater Agreement\nKendall ฯ„ = {kendall_tau:.3f}')
    ax.set_xlim(0.5, 5.5)
    ax.set_ylim(0.5, 5.5)
    ax.set_xticks(range(1, 6))
    ax.set_yticks(range(1, 6))
    ax.legend()
    ax.grid(True, alpha=0.3)

    plt.show()

    return rater1, rater2

rater1, rater2 = kendall_correlation_example()

6.3 Correlation Coefficient Comparison

def compare_correlations():
    """Compare three correlation coefficients"""
    print("""
    =================================================
    Correlation Coefficient Comparison
    =================================================

    | Characteristic | Pearson | Spearman | Kendall |
    |----------------|---------|----------|---------|
    | Relationship | Linear | Monotonic | Monotonic |
    | Data type | Continuous | Ordinal/Continuous | Ordinal |
    | Outliers | Sensitive | Robust | Very robust |
    | Ties handling | N/A | Average rank | Correctable |
    | Computation | O(n) | O(n log n) | O(nยฒ) |
    | Range | [-1, 1] | [-1, 1] | [-1, 1] |

    Selection criteria:
    - Continuous & linear relationship โ†’ Pearson
    - Non-normal/outliers/nonlinear โ†’ Spearman
    - Ordinal/ranked data โ†’ Kendall
    - Small sample size โ†’ Kendall
    """)

compare_correlations()

7. Practice Examples

7.1 Comprehensive Nonparametric Analysis

def comprehensive_nonparametric_analysis(data_dict, alpha=0.05):
    """
    Perform comprehensive nonparametric analysis

    Parameters:
    -----------
    data_dict : dict
        Format: {group_name: data}
    """
    print("="*60)
    print("Comprehensive Nonparametric Analysis")
    print("="*60)

    groups = list(data_dict.values())
    group_names = list(data_dict.keys())
    n_groups = len(groups)

    # 1. Descriptive statistics
    print("\n[1] Descriptive Statistics")
    for name, data in data_dict.items():
        print(f"  {name}: n={len(data)}, median={np.median(data):.2f}, "
              f"IQR={np.percentile(data, 75)-np.percentile(data, 25):.2f}")

    # 2. Normality tests
    print("\n[2] Normality Tests (Shapiro-Wilk)")
    all_normal = True
    for name, data in data_dict.items():
        _, p = stats.shapiro(data)
        is_normal = p > alpha
        all_normal = all_normal and is_normal
        print(f"  {name}: p={p:.4f} {'(normal)' if is_normal else '(non-normal)'}")

    # 3. Select and perform appropriate test
    print(f"\n[3] Group Comparison Test (number of groups: {n_groups})")

    if n_groups == 2:
        # Two groups: Mann-Whitney U
        stat, p = stats.mannwhitneyu(groups[0], groups[1], alternative='two-sided')
        print(f"  Mann-Whitney U: U={stat:.2f}, p={p:.4f}")
        test_name = "Mann-Whitney U"
    else:
        # Three or more groups: Kruskal-Wallis
        stat, p = stats.kruskal(*groups)
        print(f"  Kruskal-Wallis H: H={stat:.2f}, p={p:.4f}")
        test_name = "Kruskal-Wallis"

        # Post-hoc if significant
        if p < alpha:
            print("\n  Post-hoc tests (Bonferroni correction):")
            n_comp = n_groups * (n_groups - 1) // 2
            adj_alpha = alpha / n_comp
            for (i, j) in combinations(range(n_groups), 2):
                _, ph_p = stats.mannwhitneyu(groups[i], groups[j])
                sig = '*' if ph_p < adj_alpha else ''
                print(f"    {group_names[i]} vs {group_names[j]}: p={ph_p:.4f} {sig}")

    # 4. Visualization
    fig, axes = plt.subplots(1, 2, figsize=(12, 5))

    # Boxplot
    ax = axes[0]
    ax.boxplot(groups, labels=group_names)
    ax.set_ylabel('Value')
    ax.set_title(f'{test_name} Test\np = {p:.4f}')
    ax.grid(True, alpha=0.3)

    # Histogram
    ax = axes[1]
    for name, data in data_dict.items():
        ax.hist(data, bins=15, alpha=0.5, label=name, density=True)
    ax.set_xlabel('Value')
    ax.set_ylabel('Density')
    ax.set_title('Distribution Comparison')
    ax.legend()
    ax.grid(True, alpha=0.3)

    plt.tight_layout()
    plt.show()

# Usage example
np.random.seed(42)
data = {
    'Control': np.random.exponential(10, 30) + 50,
    'Treatment A': np.random.exponential(10, 30) + 55,
    'Treatment B': np.random.exponential(10, 30) + 60
}
comprehensive_nonparametric_analysis(data)

8. Practice Problems

Problem 1: Test Selection

Select an appropriate nonparametric test for each situation: 1. Comparing test scores between male and female students (skewed score distribution) 2. Comparing waiting times at 3 hospitals 3. Comparing pain scores before and after treatment for the same patients

Problem 2: Mann-Whitney U

Given data for two groups: - Group A: [23, 28, 31, 35, 39, 42] - Group B: [18, 22, 25, 29, 33]

Calculate the U statistic manually and compare with scipy results.

Problem 3: Correlation Analysis

For 10 pairs of ranked data: 1. Calculate Spearman correlation coefficient 2. Calculate Kendall tau 3. Interpret the difference between the two coefficients

Problem 4: Kruskal-Wallis Post-hoc

When Kruskal-Wallis is significant in comparing 4 groups: 1. Number of post-hoc comparisons needed 2. Bonferroni-corrected significance level 3. Determine which pairs differ significantly

9. Key Summary

Test Selection Flowchart

Check data type
    โ”‚
    โ”œโ”€โ”€ Normality satisfied? โ”€โ”€โ”€โ”ฌโ”€โ”€ Yes โ†’ Parametric test
    โ”‚                           โ””โ”€โ”€ No โ†’ Nonparametric test
    โ”‚
Nonparametric test selection:
    โ”‚
    โ”œโ”€โ”€ 2 independent groups โ†’ Mann-Whitney U
    โ”‚
    โ”œโ”€โ”€ 2 paired groups โ†’ Wilcoxon signed-rank
    โ”‚
    โ”œโ”€โ”€ 3+ independent groups โ†’ Kruskal-Wallis H โ†’ Post-hoc: Dunn
    โ”‚
    โ”œโ”€โ”€ 3+ paired groups โ†’ Friedman โ†’ Post-hoc: Nemenyi
    โ”‚
    โ””โ”€โ”€ Correlation โ†’ Spearman (continuous) / Kendall (ordinal)

scipy.stats Functions

Test Function
Mann-Whitney U mannwhitneyu(x, y)
Wilcoxon wilcoxon(x, y)
Kruskal-Wallis kruskal(g1, g2, g3, ...)
Friedman friedmanchisquare(g1, g2, g3, ...)
Spearman spearmanr(x, y)
Kendall kendalltau(x, y)

Effect Size Interpretation

Test Effect Size Small Medium Large
Mann-Whitney rank-biserial r 0.1 0.3 0.5
Kruskal-Wallis ฮทยฒ 0.01 0.06 0.14
Friedman Kendall's W 0.1 0.3 0.5

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In Chapter 14, Experimental Design, we'll cover: - Basic principles of experimental design - A/B testing - Sample size determination (power analysis) - Sequential testing

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