统计假设检验Python实现指南：T检验、ANOVA、卡方检验等常用方法详解
Statistical Hypothesis Testing by aj-geddes/useful-ai-prompts
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统计假设检验
概述
假设检验提供了一个通过检验观察到的差异是否具有统计显著性或仅是偶然结果，来做出数据驱动决策的框架。
检验框架
- 零假设 (H0) : 不存在效应或差异
- 备择假设 (H1) : 存在效应或差异
- 显著性水平 (α) : 拒绝 H0 的阈值（通常为 0.05）
- P值 : 在 H0 为真的情况下观察到数据的概率
常用检验
- T检验 : 比较两组间的均值
- 方差分析 (ANOVA) : 比较多组间的均值
- 卡方检验 : 检验分类变量的独立性
- 曼-惠特尼U检验 : T检验的非参数替代方法
- 克鲁斯卡尔-沃利斯检验 : 方差分析的非参数替代方法
使用 Python 实现
import pandas as pd
import numpy as np
from scipy import stats
import matplotlib.pyplot as plt
# 样本数据
group_a = np.random.normal(100, 15, 50) # Mean=100, SD=15
group_b = np.random.normal(105, 15, 50) # Mean=105, SD=15
# 检验 1: 独立样本 t 检验
t_stat, p_value = stats.ttest_ind(group_a, group_b)
print(f"T-test: t={t_stat:.4f}, p-value={p_value:.4f}")
if p_value < 0.05:
print("Reject null hypothesis: Groups are significantly different")
else:
print("Fail to reject null hypothesis: No significant difference")
# 检验 2: 配对 t 检验（相同受试者，两种条件）
before = np.array([85, 90, 88, 92, 87, 89, 91, 86, 88, 90])
after = np.array([92, 95, 91, 98, 94, 96, 99, 93, 95, 97])
t_stat, p_value = stats.ttest_rel(before, after)
print(f"\nPaired t-test: t={t_stat:.4f}, p-value={p_value:.4f}")
# 检验 3: 单因素方差分析（多组）
group1 = np.random.normal(100, 10, 30)
group2 = np.random.normal(105, 10, 30)
group3 = np.random.normal(102, 10, 30)
f_stat, p_value = stats.f_oneway(group1, group2, group3)
print(f"\nANOVA: F={f_stat:.4f}, p-value={p_value:.4f}")
# 检验 4: 卡方检验（分类变量）
# 创建列联表
contingency = np.array([
[50, 30], # Control: success, failure
[45, 35] # Treatment: success, failure
])
chi2, p_value, dof, expected = stats.chi2_contingency(contingency)
print(f"\nChi-square: χ²={chi2:.4f}, p-value={p_value:.4f}")
# 检验 5: 曼-惠特尼 U 检验（非参数）
u_stat, p_value = stats.mannwhitneyu(group_a, group_b)
print(f"\nMann-Whitney U: U={u_stat:.4f}, p-value={p_value:.4f}")
# 可视化
fig, axes = plt.subplots(2, 2, figsize=(12, 10))
# 分布比较
axes[0, 0].hist(group_a, alpha=0.5, label='Group A', bins=20)
axes[0, 0].hist(group_b, alpha=0.5, label='Group B', bins=20)
axes[0, 0].set_title('Group Distributions')
axes[0, 0].legend()
# 正态性 Q-Q 图
stats.probplot(group_a, dist="norm", plot=axes[0, 1])
axes[0, 1].set_title('Q-Q Plot (Group A)')
# 前后对比
axes[1, 0].plot(before, 'o-', label='Before', alpha=0.7)
axes[1, 0].plot(after, 's-', label='After', alpha=0.7)
axes[1, 0].set_title('Paired Comparison')
axes[1, 0].legend()
# 效应量（Cohen's d）
cohens_d = (np.mean(group_a) - np.mean(group_b)) / np.sqrt(
((len(group_a)-1)*np.var(group_a, ddof=1) +
(len(group_b)-1)*np.var(group_b, ddof=1)) /
(len(group_a) + len(group_b) - 2)
)
axes[1, 1].text(0.5, 0.5, f"Cohen's d = {cohens_d:.4f}",
ha='center', va='center', fontsize=14)
axes[1, 1].axis('off')
plt.tight_layout()
plt.show()
# 正态性检验（Shapiro-Wilk）
stat, p = stats.shapiro(group_a)
print(f"\nShapiro-Wilk normality test: W={stat:.4f}, p-value={p:.4f}")
# 效应量计算
def calculate_effect_size(group1, group2):
n1, n2 = len(group1), len(group2)
var1, var2 = np.var(group1, ddof=1), np.var(group2, ddof=1)
pooled_std = np.sqrt(((n1-1)*var1 + (n2-1)*var2) / (n1+n2-2))
cohens_d = (np.mean(group1) - np.mean(group2)) / pooled_std
return cohens_d
effect_size = calculate_effect_size(group_a, group_b)
print(f"Effect size (Cohen's d): {effect_size:.4f}")
# 置信区间
from scipy.stats import t as t_dist
def calculate_ci(data, confidence=0.95):
n = len(data)
mean = np.mean(data)
se = np.std(data, ddof=1) / np.sqrt(n)
margin = t_dist.ppf((1 + confidence) / 2, n - 1) * se
return mean - margin, mean + margin
ci = calculate_ci(group_a)
print(f"95% CI for Group A: ({ci[0]:.2f}, {ci[1]:.2f})")
# 附加检验和可视化
# 检验 6: Levene 方差齐性检验
stat_levene, p_levene = stats.levene(group_a, group_b)
print(f"\nLevene's Test for Equal Variance:")
print(f"Statistic: {stat_levene:.4f}, P-value: {p_levene:.4f}")
# 检验 7: Welch t 检验（不假设方差齐性）
t_stat_welch, p_welch = stats.ttest_ind(group_a, group_b, equal_var=False)
print(f"\nWelch's t-test (unequal variance):")
print(f"t-stat: {t_stat_welch:.4f}, p-value: {p_welch:.4f}")
# 功效分析
from scipy.stats import nct
def calculate_power(effect_size, sample_size, alpha=0.05):
t_critical = stats.t.ppf(1 - alpha/2, 2*sample_size - 2)
ncp = effect_size * np.sqrt(sample_size / 2)
power = 1 - stats.nct.cdf(t_critical, 2*sample_size - 2, ncp)
return power
power = calculate_power(abs(effect_size), len(group_a))
print(f"\nStatistical Power: {power:.2%}")
# Bootstrap 置信区间
def bootstrap_ci(data, n_bootstrap=10000, ci=95):
bootstrap_means = []
for _ in range(n_bootstrap):
sample = np.random.choice(data, size=len(data), replace=True)
bootstrap_means.append(np.mean(sample))
lower = np.percentile(bootstrap_means, (100-ci)/2)
upper = np.percentile(bootstrap_means, ci + (100-ci)/2)
return lower, upper
boot_ci = bootstrap_ci(group_a)
print(f"\nBootstrap 95% CI for Group A: ({boot_ci[0]:.2f}, {boot_ci[1]:.2f})")
# 多重检验校正（Bonferroni）
num_tests = 4
bonferroni_alpha = 0.05 / num_tests
print(f"\nBonferroni Corrected Alpha: {bonferroni_alpha:.4f}")
print(f"Use this threshold for {num_tests} tests")
# 检验 8: Kruskal-Wallis 检验（非参数方差分析）
h_stat, p_kw = stats.kruskal(group1, group2, group3)
print(f"\nKruskal-Wallis Test (non-parametric ANOVA):")
print(f"H-statistic: {h_stat:.4f}, p-value: {p_kw:.4f}")
# 方差分析的效应量
f_stat, p_anova = stats.f_oneway(group1, group2, group3)
# 计算 eta-squared
grand_mean = np.mean([group1, group2, group3])
ss_between = sum(len(g) * (np.mean(g) - grand_mean)**2 for g in [group1, group2, group3])
ss_total = sum((x - grand_mean)**2 for g in [group1, group2, group3] for x in g)
eta_squared = ss_between / ss_total
print(f"\nEffect Size (Eta-squared): {eta_squared:.4f}")
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