ノンパラメトリック検定の全体像 — いつ使うべきか？

import numpy as np
import matplotlib.pyplot as plt
from scipy import stats

np.random.seed(42)

fig, axes = plt.subplots(1, 3, figsize=(16, 5.5))

# (a) 正規分布 vs 歪んだ分布でのt検定 vs ウィルコクソン検定
ax = axes[0]
n_sim = 5000
n = 20

# 歪んだ分布（対数正規分布）での第一種の過誤率
distributions = {
    'Normal': (lambda n: np.random.normal(0, 1, n), 0),
    'Lognormal': (lambda n: np.random.lognormal(0, 1, n) - np.exp(0.5), np.exp(0.5)),
    'Exponential': (lambda n: np.random.exponential(1, n) - 1, 1),
    't(3)': (lambda n: np.random.standard_t(3, n), 0),
}

t_errors = []
w_errors = []
dist_names = []

for name, (gen, med) in distributions.items():
    t_rej, w_rej = 0, 0
    for _ in range(n_sim):
        sample = gen(n)
        _, p_t = stats.ttest_1samp(sample, 0)
        _, p_w = stats.wilcoxon(sample - med, alternative='two-sided', zero_method='wilcox')
        if p_t < 0.05:
            t_rej += 1
        if p_w < 0.05:
            w_rej += 1
    t_errors.append(t_rej / n_sim)
    w_errors.append(w_rej / n_sim)
    dist_names.append(name)

x_pos = np.arange(len(dist_names))
width = 0.35
ax.bar(x_pos - width/2, t_errors, width, color='steelblue', alpha=0.7, label='t-test')
ax.bar(x_pos + width/2, w_errors, width, color='salmon', alpha=0.7, label='Wilcoxon')
ax.axhline(0.05, color='black', linewidth=1.5, linestyle='--', label='$\\alpha = 0.05$')
ax.set_xticks(x_pos)
ax.set_xticklabels(dist_names, fontsize=9)
ax.set_ylabel('Type I error rate', fontsize=12)
ax.set_title('Type I Error: t-test vs Wilcoxon', fontsize=13)
ax.legend(fontsize=9)
ax.grid(True, alpha=0.3, axis='y')

# (b) 検出力の比較（歪んだ分布）
ax = axes[1]
shifts = np.linspace(0, 2, 10)
n_pow = 30

power_t = []
power_w = []
power_u = []

for shift in shifts:
    t_rej, w_rej = 0, 0
    for _ in range(n_sim):
        # 対数正規分布（右に歪んだ分布）
        sample = np.random.lognormal(shift * 0.3, 0.8, n_pow)
        _, p_t = stats.ttest_1samp(sample, np.exp(0.32))  # 中央値での検定
        try:
            _, p_w = stats.wilcoxon(sample - np.exp(0.32))
        except ValueError:
            p_w = 1.0
        if p_t < 0.05:
            t_rej += 1
        if p_w < 0.05:
            w_rej += 1
    power_t.append(t_rej / n_sim)
    power_w.append(w_rej / n_sim)

ax.plot(shifts, power_t, 'bo-', linewidth=2, markersize=6, label='t-test')
ax.plot(shifts, power_w, 'rs-', linewidth=2, markersize=6, label='Wilcoxon')
ax.set_xlabel('Effect size (shift)', fontsize=12)
ax.set_ylabel('Power', fontsize=12)
ax.set_title('Power: Lognormal Distribution', fontsize=13)
ax.legend(fontsize=10)
ax.grid(True, alpha=0.3)

# (c) マン・ホイットニー U検定の例
ax = axes[2]
n1, n2 = 25, 25
group1 = np.random.exponential(2, n1)
group2 = np.random.exponential(3, n2) + 0.5

# t検定
t_stat, p_t = stats.ttest_ind(group1, group2, equal_var=False)
# U検定
u_stat, p_u = stats.mannwhitneyu(group1, group2, alternative='two-sided')

ax.hist(group1, bins=15, density=True, alpha=0.5, color='blue',
        edgecolor='gray', label=f'Group 1 (median={np.median(group1):.2f})')
ax.hist(group2, bins=15, density=True, alpha=0.5, color='red',
        edgecolor='gray', label=f'Group 2 (median={np.median(group2):.2f})')

ax.text(0.98, 0.95, f"Welch's t: p={p_t:.4f}\nMann-Whitney U: p={p_u:.4f}",
        transform=ax.transAxes, ha='right', va='top', fontsize=10,
        bbox=dict(facecolor='lightyellow', alpha=0.8))

ax.set_xlabel('Value', fontsize=12)
ax.set_ylabel('Density', fontsize=12)
ax.set_title('Mann-Whitney U vs t-test\n(Skewed data)', fontsize=12)
ax.legend(fontsize=9)
ax.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('nonparametric_tests.png', dpi=150, bbox_inches='tight')
plt.show()