[数据分析]Beyond A-B Test：多变量测试与交互作用分析

import numpy as np
import pandas as pd
import statsmodels.api as sm
from statsmodels.formula.api import ols
import matplotlib.pyplot as plt
import seaborn as sns
from scipy import stats

# 设置随机种子确保结果可重现
np.random.seed(42)

# 生成模拟数据
n_per_group = 2000  # 每组样本量
total_n = n_per_group * 8  # 总样本量

# 创建因子水平
image_style = np.repeat(['realistic', 'white_bg'], total_n // 2)
discount_display = np.tile(np.repeat(['percentage', 'amount'], total_n // 4), 2)
button_color = np.tile(np.repeat(['blue', 'green'], total_n // 8), 4)

# 创建模拟转化数据（基于设定的真实效应）
base_rate = 0.12  # 基础转化率

# 设定主效应
image_effect = 0.02  # 白色背景比真实场景高2%
discount_effect = 0.015  # 金额显示比百分比高1.5%
color_effect = 0.01  # 绿色按钮比蓝色高1%

# 设定交互效应
image_discount_interaction = 0.03  # 白色背景和金额显示的交互效应
image_color_interaction = 0.005  # 白色背景和绿色按钮的交互效应
discount_color_interaction = 0.002  # 金额显示和绿色按钮的交互效应
three_way_interaction = 0.004  # 三阶交互效应

# 生成转化概率
prob = base_rate + \
       (image_style == 'white_bg') * image_effect + \
       (discount_display == 'amount') * discount_effect + \
       (button_color == 'green') * color_effect + \
       ((image_style == 'white_bg') & (discount_display == 'amount')) * image_discount_interaction + \
       ((image_style == 'white_bg') & (button_color == 'green')) * image_color_interaction + \
       ((discount_display == 'amount') & (button_color == 'green')) * discount_color_interaction + \
       ((image_style == 'white_bg') & (discount_display == 'amount') & (button_color == 'green')) * three_way_interaction

# 添加随机噪声
prob = prob + np.random.normal(0, 0.01, total_n)
prob = np.clip(prob, 0, 1)  # 确保概率在0-1之间

# 生成转化结果
conversion = np.random.binomial(1, prob)

# 创建DataFrame
df = pd.DataFrame({
    'image_style': image_style,
    'discount_display': discount_display,
    'button_color': button_color,
    'conversion': conversion
})

# 查看前几行数据
print(df.head())

# 计算各组的转化率
grouped_stats = df.groupby(['image_style', 'discount_display', 'button_color']).agg({
    'conversion': ['mean', 'count', 'std']
}).round(4)

print("各组转化率统计:")
print(grouped_stats)

# 绘制各因子水平的转化率
plt.figure(figsize=(15, 5))

plt.subplot(1, 3, 1)
sns.barplot(x='image_style', y='conversion', data=df, ci=95)
plt.title('转化率 by 图片风格')

plt.subplot(1, 3, 2)
sns.barplot(x='discount_display', y='conversion', data=df, ci=95)
plt.title('转化率 by 折扣显示方式')

plt.subplot(1, 3, 3)
sns.barplot(x='button_color', y='conversion', data=df, ci=95)
plt.title('转化率 by 按钮颜色')

plt.tight_layout()
plt.show()

# 绘制交互作用图
plt.figure(figsize=(15, 5))

plt.subplot(1, 3, 1)
sns.pointplot(x='image_style', y='conversion', hue='discount_display', data=df, ci=95)
plt.title('图片风格 vs 折扣显示方式')

plt.subplot(1, 3, 2)
sns.pointplot(x='image_style', y='conversion', hue='button_color', data=df, ci=95)
plt.title('图片风格 vs 按钮颜色')

plt.subplot(1, 3, 3)
sns.pointplot(x='discount_display', y='conversion', hue='button_color', data=df, ci=95)
plt.title('折扣显示方式 vs 按钮颜色')

plt.tight_layout()
plt.show()

# 进行三因素方差分析
model = ols('conversion ~ C(image_style) * C(discount_display) * C(button_color)', data=df).fit()
anova_table = sm.stats.anova_lm(model, typ=2)

print("方差分析结果:")
print(anova_table.round(4))

# 计算效应大小（η²）
anova_table['eta_sq'] = anova_table['sum_sq'] / sum(anova_table['sum_sq'])
print("\n效应大小(η²):")
print(anova_table['eta_sq'].round(4))

# 简单效应分析 - 以图片风格和折扣显示方式的交互为例
print("简单效应分析: 图片风格在不同折扣显示方式下的效应")

# 在百分比折扣条件下比较图片风格
percentage_data = df[df['discount_display'] == 'percentage']
percentage_effect = stats.ttest_ind(
    percentage_data[percentage_data['image_style'] == 'white_bg']['conversion'],
    percentage_data[percentage_data['image_style'] == 'realistic']['conversion']
)
print(f"百分比折扣条件下，图片风格的简单效应: t = {percentage_effect.statistic:.4f}, p = {percentage_effect.pvalue:.4f}")

# 在金额折扣条件下比较图片风格
amount_data = df[df['discount_display'] == 'amount']
amount_effect = stats.ttest_ind(
    amount_data[amount_data['image_style'] == 'white_bg']['conversion'],
    amount_data[amount_data['image_style'] == 'realistic']['conversion']
)
print(f"金额折扣条件下，图片风格的简单效应: t = {amount_effect.statistic:.4f}, p = {amount_effect.pvalue:.4f}")

# 最佳组合识别
best_combination = grouped_stats[('conversion', 'mean')].idxmax()
best_rate = grouped_stats[('conversion', 'mean')].max()
print(f"\n最佳组合: {best_combination}, 转化率: {best_rate:.4f}")

# 创建综合结果可视化
plt.figure(figsize=(12, 8))

# 主效应和交互作用热图
heatmap_data = df.groupby(['image_style', 'discount_display', 'button_color'])['conversion'].mean().unstack().unstack()
sns.heatmap(heatmap_data, annot=True, fmt='.3f', cmap='YlOrRd', center=base_rate)
plt.title('不同实验条件下的平均转化率')
plt.tight_layout()
plt.show()

# 三因素交互作用图
fig, ax = plt.subplots(1, 2, figsize=(15, 6))

for i, color in enumerate(['blue', 'green']):
    color_data = df[df['button_color'] == color]
    color_pivot = color_data.groupby(['image_style', 'discount_display'])['conversion'].mean().unstack()
    color_pivot.plot(kind='bar', ax=ax[i])
    ax[i].set_title(f'按钮颜色: {color}')
    ax[i].set_ylabel('转化率')
    ax[i].legend(title='折扣显示方式')

plt.tight_layout()
plt.show()

# 生成报告摘要
print("="*50)
print("多变量测试分析报告摘要")
print("="*50)
print(f"总样本量: {len(df)}")
print(f"总体转化率: {df['conversion'].mean():.4f}")
print("\n显著效应:")
for effect, p in anova_table['PR(>F)'].items():
    if p < 0.05:
        significance = "***" if p < 0.001 else "**" if p < 0.01 else "*"
        print(f"- {effect}: p = {p:.4f} {significance}")

print(f"\n最佳组合: {best_combination}")
print(f"预计提升: {(best_rate - df['conversion'].mean()) / df['conversion'].mean() * 100:.2f}%")