序贯检验在A/B测试中的应用与优势

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数字扫地僧 发表于 2025/09/30 17:15:16 2025/09/30
【摘要】 I. 传统A/B测试的挑战与序贯检验的崛起 1.1 传统固定样本量检验的局限性传统A/B测试采用固定样本量设计,这种方法虽然统计原理简单明了,但在实际业务环境中面临诸多挑战:挑战类型具体表现业务影响效率低下即使结果已经很明确,仍需等待预设样本量延迟决策,错过业务机会资源浪费对明显失败或成功的实验继续投入流量浪费用户流量和实验机会风险暴露有害的变化在实验期间持续影响用户损害用户体验和业务指标...

I. 传统A/B测试的挑战与序贯检验的崛起

1.1 传统固定样本量检验的局限性

传统A/B测试采用固定样本量设计,这种方法虽然统计原理简单明了,但在实际业务环境中面临诸多挑战:

挑战类型 具体表现 业务影响
效率低下 即使结果已经很明确,仍需等待预设样本量 延迟决策,错过业务机会
资源浪费 对明显失败或成功的实验继续投入流量 浪费用户流量和实验机会
风险暴露 有害的变化在实验期间持续影响用户 损害用户体验和业务指标
灵活性不足 无法根据中期结果调整实验参数 缺乏对业务环境的适应性

1.2 序贯检验的基本概念

序贯检验是一种在数据积累过程中多次进行统计检验的方法,其核心思想是:在每一步新数据到来时都重新评估证据强度,一旦达到预定的决策边界就立即停止实验

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

def demonstrate_sequential_vs_fixed():
    """对比序贯检验与固定样本量检验的效率差异"""
    
    np.random.seed(42)
    
    # 模拟参数
    true_control_rate = 0.10
    true_treatment_rate = 0.13  # 30%的相对提升
    max_sample_size = 5000
    alpha = 0.05
    power = 0.8
    
    # 计算固定样本量检验所需的样本量
    fixed_sample_size = calculate_fixed_sample_size(
        true_control_rate, true_treatment_rate, alpha, power
    )
    
    print(f"固定样本量检验所需样本量: {fixed_sample_size} 每组")
    print(f"总样本量: {fixed_sample_size * 2}")
    
    # 模拟多次实验
    n_simulations = 1000
    fixed_stopping_times = []
    sequential_stopping_times = []
    
    for sim in range(n_simulations):
        # 生成数据流
        control_data = []
        treatment_data = []
        
        sequential_stopped = False
        sequential_stop_time = max_sample_size
        
        for t in range(1, max_sample_size + 1):
            # 生成新数据点
            control_data.append(np.random.binomial(1, true_control_rate))
            treatment_data.append(np.random.binomial(1, true_treatment_rate))
            
            # 序贯检验:在每一步检查是否达到决策边界
            if not sequential_stopped and t >= 100:  # 至少100个样本后开始检查
                control_success = np.sum(control_data)
                treatment_success = np.sum(treatment_data)
                control_total = len(control_data)
                treatment_total = len(treatment_data)
                
                # 使用序贯概率比检验(SPRT)
                should_stop = spr_test(
                    control_success, control_total,
                    treatment_success, treatment_total,
                    alpha=alpha, power=power
                )
                
                if should_stop:
                    sequential_stopped = True
                    sequential_stop_time = t
        
        # 记录停止时间
        sequential_stopping_times.append(sequential_stop_time)
        
        # 固定样本量检验的停止时间就是预设样本量
        fixed_stopping_time = min(fixed_sample_size, max_sample_size)
        fixed_stopping_times.append(fixed_stopping_time)
    
    # 分析结果
    avg_sequential_time = np.mean(sequential_stopping_times)
    avg_fixed_time = np.mean(fixed_stopping_times)
    efficiency_gain = (avg_fixed_time - avg_sequential_time) / avg_fixed_time
    
    print(f"序贯检验平均停止时间: {avg_sequential_time:.0f} 样本")
    print(f"固定样本量检验停止时间: {avg_fixed_time:.0f} 样本")
    print(f"效率提升: {efficiency_gain:.1%}")
    
    # 可视化对比
    plt.figure(figsize=(10, 6))
    plt.hist(sequential_stopping_times, bins=50, alpha=0.7, 
             label='序贯检验', color='blue')
    plt.axvline(avg_sequential_time, color='blue', linestyle='--', 
                label=f'序贯平均 ({avg_sequential_time:.0f})')
    plt.axvline(avg_fixed_time, color='red', linestyle='--', 
                label=f'固定样本量 ({avg_fixed_time:.0f})')
    plt.xlabel('停止时间 (样本量)')
    plt.ylabel('频数')
    plt.title('序贯检验 vs 固定样本量检验的停止时间分布')
    plt.legend()
    plt.grid(True, alpha=0.3)
    plt.show()
    
    return avg_sequential_time, avg_fixed_time, efficiency_gain

def calculate_fixed_sample_size(p1, p2, alpha=0.05, power=0.8):
    """计算固定样本量检验所需的样本量"""
    from statsmodels.stats.power import NormalIndPower
    from math import asin, sqrt
    
    # 计算效应量 (Cohen's h)
    effect_size = 2 * asin(sqrt(p2)) - 2 * asin(sqrt(p1))
    
    # 使用statsmodels计算样本量
    power_analysis = NormalIndPower()
    sample_size = power_analysis.solve_power(
        effect_size=effect_size, 
        alpha=alpha, 
        power=power,
        ratio=1.0
    )
    
    return int(round(sample_size))

def spr_test(control_success, control_total, treatment_success, treatment_total, 
             alpha=0.05, power=0.8):
    """
    简化的序贯概率比检验(SPRT)
    返回是否应该停止实验
    """
    # 计算似然比
    p_control = control_success / control_total
    p_treatment = treatment_success / treatment_total
    
    # 避免极端情况
    p_control = max(0.01, min(0.99, p_control))
    p_treatment = max(0.01, min(0.99, p_treatment))
    
    # 计算对数似然比
    log_likelihood_ratio = (
        treatment_success * np.log(p_treatment / p_control) +
        (treatment_total - treatment_success) * np.log((1 - p_treatment) / (1 - p_control))
    )
    
    # 简化的决策边界(实际应用中应该使用更精确的边界)
    upper_boundary = np.log((1 - beta) / alpha)  # 备择假设边界
    lower_boundary = np.log(beta / (1 - alpha))  # 零假设边界
    
    beta = 1 - power
    
    if log_likelihood_ratio >= upper_boundary:
        return True  # 拒绝零假设,支持备择假设
    elif log_likelihood_ratio <= lower_boundary:
        return True  # 接受零假设
    else:
        return False  # 继续实验

# 运行对比演示
avg_seq, avg_fixed, efficiency = demonstrate_sequential_vs_fixed()

1.3 序贯检验的发展历程

序贯检验并非新生事物,其理论基础可以追溯到20世纪40年代:

时间 里程碑 贡献者 意义
1943年 序贯概率比检验(SPRT) Abraham Wald 奠定了序贯分析的理论基础
1945年 序贯分析专著 Abraham Wald 系统化序贯检验方法
1970年代 分组序贯设计 多位统计学家 适应临床实验的定期监测需求
1990年代 自适应设计 工业统计界 将序贯思想扩展到更广泛的实验设计
2000年代 在线实验应用 科技公司 将序贯检验大规模应用于A/B测试
传统A/B测试挑战
效率问题
资源浪费
风险暴露
灵活性不足
决策延迟
流量浪费
用户体验风险
无法动态调整
序贯检验解决方案
实时监测
早期停止
动态决策
风险控制
持续评估证据
达到边界即停止
灵活适应结果
限制有害暴露
发展历程
1940s: SPRT基础
1970s: 分组序贯
1990s: 自适应设计
2000s: 在线实验

II. 序贯检验的统计学基础

2.1 序贯概率比检验(SPRT)

SPRT是序贯检验最经典的方法,由Abraham Wald在1943年提出。其核心思想是基于似然比构建决策边界。

2.1.1 SPRT的数学原理

SPRT考虑两个简单的假设:

  • H₀: θ = θ₀ (零假设)
  • H₁: θ = θ₁ (备择假设)

在每一步,我们计算似然比:

LRn=P(数据H1)P(数据H0)LR_n = \frac{P(\text{数据} \mid H_1)}{P(\text{数据} \mid H_0)}

决策规则为:

  • 如果 LRnALR_n \geq A,拒绝H₀,接受H₁
  • 如果 LRnBLR_n \leq B,接受H₀
  • 如果 B<LRn<AB < LR_n < A,继续实验

其中边界 A=1βαA = \frac{1-\beta}{\alpha}, B=β1αB = \frac{\beta}{1-\alpha}

class SPRT:
    """序贯概率比检验实现"""
    
    def __init__(self, alpha=0.05, power=0.8, theta0=None, theta1=None):
        """
        初始化SPRT检验
        
        Parameters:
        alpha: 第一类错误概率
        power: 统计功效 (1 - 第二类错误概率)
        theta0: 零假设下的参数值
        theta1: 备择假设下的参数值
        """
        self.alpha = alpha
        self.power = power
        self.beta = 1 - power
        
        # 计算决策边界
        self.upper_bound = np.log((1 - self.beta) / self.alpha)
        self.lower_bound = np.log(self.beta / (1 - self.alpha))
        
        self.theta0 = theta0
        self.theta1 = theta1
        
        # 记录检验过程
        self.log_likelihood_ratio = 0
        self.sample_size = 0
        self.decision = None  # None: 继续, True: 拒绝H0, False: 接受H0
        
    def update(self, data_point, group='treatment'):
        """
        更新检验统计量
        
        Parameters:
        data_point: 新数据点 (对于二项数据,0或1)
        group: 数据所属组别 ('control' 或 'treatment')
        """
        if self.decision is not None:
            return self.decision  # 已经做出决策
            
        self.sample_size += 1
        
        # 计算当前数据点的对数似然比贡献
        if group == 'treatment':
            log_lr_contribution = self._log_likelihood_ratio_contribution(
                data_point, self.theta1, self.theta0
            )
        else:
            # 对照组数据用于估计theta0,如果未指定的话
            log_lr_contribution = 0
            
        self.log_likelihood_ratio += log_lr_contribution
        
        # 检查决策边界
        if self.log_likelihood_ratio >= self.upper_bound:
            self.decision = True  # 拒绝H0
        elif self.log_likelihood_ratio <= self.lower_bound:
            self.decision = False  # 接受H0
            
        return self.decision
    
    def _log_likelihood_ratio_contribution(self, x, theta1, theta0):
        """计算单个数据点的对数似然比贡献"""
        # 对于伯努利分布
        if x == 1:  # 成功
            return np.log(theta1 / theta0)
        else:  # 失败
            return np.log((1 - theta1) / (1 - theta0))
    
    def get_current_status(self):
        """获取当前检验状态"""
        return {
            'sample_size': self.sample_size,
            'log_likelihood_ratio': self.log_likelihood_ratio,
            'decision': self.decision,
            'upper_bound': self.upper_bound,
            'lower_bound': self.lower_bound
        }
    
    def visualize_test_progress(self, data_stream, true_theta=None):
        """可视化检验过程"""
        # 重置检验
        self.log_likelihood_ratio = 0
        self.sample_size = 0
        self.decision = None
        
        # 记录过程
        sample_sizes = []
        log_lrs = []
        decisions = []
        
        for i, data_point in enumerate(data_stream):
            decision = self.update(data_point)
            sample_sizes.append(self.sample_size)
            log_lrs.append(self.log_likelihood_ratio)
            decisions.append(decision)
            
            if decision is not None:
                break
        
        # 绘制检验过程
        plt.figure(figsize=(12, 8))
        
        plt.plot(sample_sizes, log_lrs, 'b-', linewidth=2, label='对数似然比')
        plt.axhline(self.upper_bound, color='r', linestyle='--', 
                   label=f'上边界 ({self.upper_bound:.2f})')
        plt.axhline(self.lower_bound, color='g', linestyle='--', 
                   label=f'下边界 ({self.lower_bound:.2f})')
        plt.axhline(0, color='k', linestyle='-', alpha=0.3)
        
        # 标记决策点
        if self.decision is not None:
            stop_idx = len(sample_sizes) - 1
            plt.plot(sample_sizes[stop_idx], log_lrs[stop_idx], 'ro', 
                    markersize=10, label=f'停止决策 (n={sample_sizes[stop_idx]})')
        
        plt.xlabel('样本量')
        plt.ylabel('对数似然比')
        plt.title('SPRT检验过程可视化')
        plt.legend()
        plt.grid(True, alpha=0.3)
        
        # 添加真实参数信息
        if true_theta is not None:
            plt.figtext(0.02, 0.02, f'真实参数: θ={true_theta}', 
                       fontsize=10, bbox=dict(boxstyle="round", facecolor='wheat'))
        
        plt.show()
        
        return sample_sizes, log_lrs, decisions

# SPRT示例演示
def demonstrate_sprt():
    """演示SPRT检验的实际应用"""
    
    print("SPRT序贯检验演示")
    print("=" * 50)
    
    # 设置参数
    theta0 = 0.10  # 零假设:转化率10%
    theta1 = 0.12  # 备择假设:转化率12%
    alpha = 0.05
    power = 0.8
    
    # 创建SPRT检验
    sprt = SPRT(alpha=alpha, power=power, theta0=theta0, theta1=theta1)
    
    # 生成模拟数据 (真实转化率11.5%,在两者之间)
    np.random.seed(42)
    true_theta = 0.115
    n_max = 10000
    data_stream = np.random.binomial(1, true_theta, n_max)
    
    print(f"零假设: θ = {theta0}")
    print(f"备择假设: θ = {theta1}")
    print(f"真实参数: θ = {true_theta}")
    print(f"显著性水平: α = {alpha}")
    print(f"统计功效: 1-β = {power}")
    
    # 运行检验并可视化
    sample_sizes, log_lrs, decisions = sprt.visualize_test_progress(data_stream, true_theta)
    
    # 输出结果
    final_status = sprt.get_current_status()
    print(f"\n检验结果:")
    print(f"最终样本量: {final_status['sample_size']}")
    print(f"最终对数似然比: {final_status['log_likelihood_ratio']:.4f}")
    
    if final_status['decision'] is True:
        print("决策: 拒绝零假设,支持备择假设")
    elif final_status['decision'] is False:
        print("决策: 接受零假设")
    else:
        print("决策: 未做出决策(达到最大样本量)")
    
    # 分析检验性质
    analyze_sprt_properties(sprt, theta0, theta1, alpha, power)
    
    return sprt, sample_sizes, log_lrs

def analyze_sprt_properties(sprt, theta0, theta1, alpha, power):
    """分析SPRT检验的统计性质"""
    
    print(f"\nSPRT统计性质分析:")
    print("=" * 50)
    
    # 计算期望样本量
    # 在真实参数为theta时的期望对数似然比增量
    def expected_llr_increment(theta):
        p_success = theta * np.log(theta1 / theta0)
        p_failure = (1 - theta) * np.log((1 - theta1) / (1 - theta0))
        return p_success + p_failure
    
    # 在不同真实参数下的期望样本量
    theta_values = [theta0, theta1, (theta0 + theta1)/2]
    
    for theta in theta_values:
        expected_increment = expected_llr_increment(theta)
        if expected_increment != 0:
            # 近似的期望样本量(实际更复杂)
            approx_sample_size = min(abs(sprt.upper_bound / expected_increment), 
                                   abs(sprt.lower_bound / expected_increment))
            print(f"真实θ={theta}: 期望样本量 ≈ {approx_sample_size:.0f}")
    
    # 错误概率验证
    print(f"\n理论错误概率:")
    print(f"第一类错误概率 (α): {alpha}")
    print(f"第二类错误概率 (β): {1-power}")
    
    # 与固定样本量检验对比
    fixed_n = calculate_fixed_sample_size(theta0, theta1, alpha, power)
    print(f"\n对比固定样本量检验:")
    print(f"固定样本量检验所需样本量: {fixed_n}")

# 运行SPRT演示
sprt_instance, sample_sizes, log_lrs = demonstrate_sprt()

2.2 分组序贯检验

在实际应用中,我们通常不会在每个新数据点后都进行检验,而是采用分组序贯方法,在积累一定数量的数据后进行检查。

2.2.1 O’Brien-Fleming边界

O’Brien-Fleming边界是一种常用的分组序贯设计,它在早期采用更严格的边界,在后期逐渐放宽,从而在保持整体α水平的同时提高检验效率。

class GroupSequentialTest:
    """分组序贯检验"""
    
    def __init__(self, alpha=0.05, power=0.8, max_samples=10000, 
                 looks=10, boundary_type='obrien_fleming'):
        """
        初始化分组序贯检验
        
        Parameters:
        alpha: 总体第一类错误概率
        power: 统计功效
        max_samples: 最大样本量
        looks: 检查次数
        boundary_type: 边界类型 ('obrien_fleming', 'pocock', 'haybittle_peto')
        """
        self.alpha = alpha
        self.power = power
        self.max_samples = max_samples
        self.looks = looks
        self.boundary_type = boundary_type
        
        # 计算检查点
        self.look_points = np.linspace(0, max_samples, looks + 1)[1:].astype(int)
        
        # 计算边界
        self.boundaries = self._calculate_boundaries()
        
        # 记录状态
        self.current_look = 0
        self.decision = None
        self.test_statistics = []
        
    def _calculate_boundaries(self):
        """计算序贯边界"""
        boundaries = {}
        
        if self.boundary_type == 'obrien_fleming':
            # O'Brien-Fleming边界
            for i, point in enumerate(self.look_points):
                information_fraction = point / self.max_samples
                # O'Brien-Fleming边界公式
                z_boundary = stats.norm.ppf(1 - self.alpha/2) / np.sqrt(information_fraction)
                boundaries[point] = z_boundary
                
        elif self.boundary_type == 'pocock':
            # Pocock边界
            constant = self._calculate_pocock_constant()
            for point in self.look_points:
                information_fraction = point / self.max_samples
                boundaries[point] = constant
                
        elif self.boundary_type == 'haybittle_peto':
            # Haybittle-Peto边界
            for i, point in enumerate(self.look_points):
                if i < len(self.look_points) - 1:  # 中期分析
                    boundaries[point] = 3.0  # 固定边界
                else:  # 最终分析
                    boundaries[point] = stats.norm.ppf(1 - self.alpha/2)
        
        return boundaries
    
    def _calculate_pocock_constant(self):
        """计算Pocock边界的常数"""
        # 这里使用近似值,实际应该使用更精确的计算
        if self.looks == 5:
            return 2.413
        elif self.looks == 10:
            return 2.555
        else:
            # 通用近似
            return stats.norm.ppf(1 - self.alpha/2) + 0.5
    
    def analyze_data(self, control_data, treatment_data):
        """分析当前数据"""
        if self.decision is not None:
            return self.decision
            
        current_sample_size = len(treatment_data)
        
        # 检查是否达到检查点
        if current_sample_size not in self.look_points:
            return None
            
        self.current_look += 1
        look_point = current_sample_size
        
        # 计算检验统计量
        control_success = np.sum(control_data)
        treatment_success = np.sum(treatment_data)
        control_total = len(control_data)
        treatment_total = len(treatment_data)
        
        # 计算z统计量
        p_control = control_success / control_total
        p_treatment = treatment_success / treatment_total
        p_pooled = (control_success + treatment_success) / (control_total + treatment_total)
        
        se = np.sqrt(p_pooled * (1 - p_pooled) * (1/control_total + 1/treatment_total))
        z_stat = (p_treatment - p_control) / se
        
        self.test_statistics.append((look_point, z_stat))
        
        # 检查边界
        boundary = self.boundaries[look_point]
        
        if abs(z_stat) >= boundary:
            self.decision = 'reject' if z_stat > 0 else 'accept'
        elif look_point == self.look_points[-1]:
            self.decision = 'accept'  # 最终分析未拒绝则接受H0
            
        return self.decision
    
    def visualize_boundaries(self):
        """可视化序贯边界"""
        look_indices = range(1, len(self.look_points) + 1)
        boundaries = [self.boundaries[point] for point in self.look_points]
        
        plt.figure(figsize=(12, 8))
        
        # 绘制边界
        plt.plot(look_indices, boundaries, 'ro-', linewidth=2, 
                label=f'{self.boundary_type}边界')
        plt.axhline(stats.norm.ppf(1 - self.alpha/2), color='k', linestyle='--',
                   label=f'固定检验边界 ({stats.norm.ppf(1 - self.alpha/2):.2f})')
        
        # 绘制检验统计量(如果有)
        if self.test_statistics:
            stat_look_indices = []
            stat_values = []
            for point, stat in self.test_statistics:
                stat_look_indices.append(self.look_points.tolist().index(point) + 1)
                stat_values.append(stat)
            
            plt.plot(stat_look_indices, stat_values, 'bo-', linewidth=2, 
                    label='检验统计量', markersize=8)
            
            # 标记决策点
            if self.decision:
                last_idx = stat_look_indices[-1]
                last_stat = stat_values[-1]
                plt.plot(last_idx, last_stat, 'rs', markersize=12, 
                        label=f'决策点 ({self.decision})')
        
        plt.xlabel('检查点')
        plt.ylabel('边界值 / 检验统计量')
        plt.title(f'分组序贯检验 - {self.boundary_type}边界')
        plt.legend()
        plt.grid(True, alpha=0.3)
        plt.xticks(look_indices)
        plt.show()
        
        return look_indices, boundaries

# 分组序贯检验演示
def demonstrate_group_sequential():
    """演示分组序贯检验"""
    
    print("分组序贯检验演示")
    print("=" * 50)
    
    # 参数设置
    max_samples = 10000
    looks = 8
    alpha = 0.05
    power = 0.8
    
    # 比较不同边界类型
    boundary_types = ['obrien_fleming', 'pocock', 'haybittle_peto']
    
    fig, axes = plt.subplots(1, 3, figsize=(18, 6))
    
    for i, boundary_type in enumerate(boundary_types):
        # 创建分组序贯检验
        gst = GroupSequentialTest(
            alpha=alpha, power=power, 
            max_samples=max_samples, looks=looks,
            boundary_type=boundary_type
        )
        
        # 生成模拟数据
        np.random.seed(42)
        true_control_rate = 0.10
        true_treatment_rate = 0.12
        
        control_data = np.random.binomial(1, true_control_rate, max_samples)
        treatment_data = np.random.binomial(1, true_treatment_rate, max_samples)
        
        # 运行序贯检验
        for look_point in gst.look_points:
            control_subset = control_data[:look_point]
            treatment_subset = treatment_data[:look_point]
            
            decision = gst.analyze_data(control_subset, treatment_subset)
            
            if decision is not None:
                print(f"{boundary_type}: 在样本量 {look_point} 时停止,决策: {decision}")
                break
        else:
            print(f"{boundary_type}: 达到最大样本量,决策: {gst.decision}")
        
        # 在当前子图上可视化
        look_indices = range(1, looks + 1)
        boundaries = [gst.boundaries[point] for point in gst.look_points]
        
        axes[i].plot(look_indices, boundaries, 'ro-', linewidth=2, label='边界')
        axes[i].axhline(stats.norm.ppf(1 - alpha/2), color='k', linestyle='--',
                       label='固定检验边界')
        
        # 绘制检验统计量
        if gst.test_statistics:
            stat_look_indices = []
            stat_values = []
            for point, stat in gst.test_statistics:
                stat_look_indices.append(gst.look_points.tolist().index(point) + 1)
                stat_values.append(stat)
            
            axes[i].plot(stat_look_indices, stat_values, 'bo-', linewidth=2, 
                        label='检验统计量', markersize=6)
        
        axes[i].set_xlabel('检查点')
        axes[i].set_ylabel('边界值')
        axes[i].set_title(f'{boundary_type}边界')
        axes[i].legend()
        axes[i].grid(True, alpha=0.3)
        axes[i].set_xticks(look_indices)
    
    plt.tight_layout()
    plt.show()
    
    # 比较不同边界类型的性质
    compare_boundary_properties()

def compare_boundary_properties():
    """比较不同边界类型的性质"""
    
    print("\n不同边界类型比较:")
    print("=" * 50)
    
    boundary_comparison = []
    
    for boundary_type in ['obrien_fleming', 'pocock', 'haybittle_peto']:
        gst = GroupSequentialTest(boundary_type=boundary_type)
        
        # 计算早期停止的概率(在效应大小适中的情况下)
        # 这里使用简化的模拟
        np.random.seed(42)
        n_simulations = 1000
        early_stop_count = 0
        avg_sample_size = 0
        
        true_control_rate = 0.10
        true_treatment_rate = 0.12
        
        for _ in range(n_simulations):
            gst_sim = GroupSequentialTest(boundary_type=boundary_type)
            control_data = np.random.binomial(1, true_control_rate, gst_sim.max_samples)
            treatment_data = np.random.binomial(1, true_treatment_rate, gst_sim.max_samples)
            
            stopped_early = False
            for look_point in gst_sim.look_points:
                decision = gst_sim.analyze_data(
                    control_data[:look_point], 
                    treatment_data[:look_point]
                )
                if decision is not None and look_point < gst_sim.max_samples:
                    stopped_early = True
                    avg_sample_size += look_point
                    break
            else:
                avg_sample_size += gst_sim.max_samples
        
        early_stop_prob = early_stop_count / n_simulations
        avg_sample_size /= n_simulations
        
        boundary_comparison.append({
            'type': boundary_type,
            'early_stop_prob': early_stop_prob,
            'avg_sample_size': avg_sample_size,
            'efficiency_gain': (10000 - avg_sample_size) / 10000
        })
        
        print(f"{boundary_type}:")
        print(f"  早期停止概率: {early_stop_prob:.3f}")
        print(f"  平均样本量: {avg_sample_size:.0f}")
        print(f"  效率提升: {boundary_comparison[-1]['efficiency_gain']:.1%}")
    
    return boundary_comparison

# 运行分组序贯检验演示
boundary_comparison = demonstrate_group_sequential()
序贯检验统计学基础
SPRT方法
分组序贯检验
数学原理
似然比计算
决策边界
检查点设计
边界类型
错误率控制
假设检验框架
对数似然比
上下边界
等间距检查
O'Brien-Fleming
Pocock边界
Haybittle-Peto
α消耗函数
核心优势
样本量节约
早期决策
风险控制
减少30-50%样本量
快速获得结论
限制有害暴露
实际考虑
最小样本量
最大样本量
检查频率
保证估计精度
控制实验时长
平衡效率与操作成本

III. 序贯检验在A/B测试中的实现

3.1 完整的序贯A/B测试系统

在实际业务中,我们需要构建一个完整的序贯A/B测试系统,包含流量分配、数据跟踪、序贯分析和决策支持等功能。

class SequentialABTestSystem:
    """序贯A/B测试系统"""
    
    def __init__(self, experiment_id, alpha=0.05, power=0.8, 
                 min_samples=100, max_samples=10000, looks=10,
                 boundary_type='obrien_fleming'):
        """
        初始化序贯A/B测试系统
        
        Parameters:
        experiment_id: 实验ID
        alpha: 总体第一类错误概率
        power: 统计功效
        min_samples: 最小样本量(避免早期偶然性)
        max_samples: 最大样本量
        looks: 检查次数
        boundary_type: 边界类型
        """
        self.experiment_id = experiment_id
        self.alpha = alpha
        self.power = power
        self.min_samples = min_samples
        self.max_samples = max_samples
        self.looks = looks
        self.boundary_type = boundary_type
        
        # 初始化检验器
        self.sequential_test = GroupSequentialTest(
            alpha=alpha, power=power, 
            max_samples=max_samples, looks=looks,
            boundary_type=boundary_type
        )
        
        # 数据存储
        self.control_data = []
        self.treatment_data = []
        self.user_assignments = {}  # user_id -> group
        
        # 实验结果
        self.final_decision = None
        self.stopping_point = None
        self.analysis_history = []
        
        print(f"初始化序贯A/B测试: {experiment_id}")
        print(f"参数: α={alpha}, 功效={power}, 样本量范围={min_samples}-{max_samples}")
    
    def assign_user(self, user_id, traffic_split=0.5):
        """分配用户到对照组或实验组"""
        if user_id in self.user_assignments:
            return self.user_assignments[user_id]
        
        # 随机分配,但保持比例
        group = 'treatment' if np.random.random() < traffic_split else 'control'
        self.user_assignments[user_id] = group
        return group
    
    def track_conversion(self, user_id, converted, value=0):
        """跟踪转化事件"""
        if user_id not in self.user_assignments:
            raise ValueError(f"用户 {user_id} 未分配组别")
        
        group = self.user_assignments[user_id]
        conversion_data = 1 if converted else 0
        
        if group == 'control':
            self.control_data.append(conversion_data)
        else:
            self.treatment_data.append(conversion_data)
        
        # 检查是否需要进行序贯分析
        return self._check_sequential_analysis()
    
    def _check_sequential_analysis(self):
        """检查是否达到分析点并执行序贯分析"""
        current_sample_size = min(len(self.control_data), len(self.treatment_data))
        
        # 确保有足够的数据且达到检查点
        if (current_sample_size < self.min_samples or 
            current_sample_size not in self.sequential_test.look_points):
            return None
        
        # 执行序贯分析
        decision = self.sequential_test.analyze_data(
            self.control_data, self.treatment_data
        )
        
        # 记录分析历史
        analysis_record = {
            'sample_size': current_sample_size,
            'control_conversions': np.sum(self.control_data),
            'treatment_conversions': np.sum(self.treatment_data),
            'control_rate': np.mean(self.control_data),
            'treatment_rate': np.mean(self.treatment_data),
            'decision': decision,
            'timestamp': pd.Timestamp.now()
        }
        self.analysis_history.append(analysis_record)
        
        if decision is not None and self.final_decision is None:
            self.final_decision = decision
            self.stopping_point = current_sample_size
            print(f"实验停止于样本量 {self.stopping_point},决策: {decision}")
        
        return decision
    
    def get_current_stats(self):
        """获取当前统计信息"""
        if len(self.control_data) == 0 or len(self.treatment_data) == 0:
            return None
        
        control_conversions = np.sum(self.control_data)
        treatment_conversions = np.sum(self.treatment_data)
        control_total = len(self.control_data)
        treatment_total = len(self.treatment_data)
        
        control_rate = control_conversions / control_total
        treatment_rate = treatment_conversions / treatment_total
        relative_improvement = (treatment_rate - control_rate) / control_rate
        
        # 计算置信区间
        from statsmodels.stats.proportion import proportion_confint
        control_ci = proportion_confint(control_conversions, control_total, alpha=0.05)
        treatment_ci = proportion_confint(treatment_conversions, treatment_total, alpha=0.05)
        
        return {
            'control': {
                'conversions': control_conversions,
                'total': control_total,
                'rate': control_rate,
                'ci_lower': control_ci[0],
                'ci_upper': control_ci[1]
            },
            'treatment': {
                'conversions': treatment_conversions,
                'total': treatment_total,
                'rate': treatment_rate,
                'ci_lower': treatment_ci[0],
                'ci_upper': treatment_ci[1]
            },
            'difference': {
                'absolute': treatment_rate - control_rate,
                'relative': relative_improvement
            },
            'sample_size': min(control_total, treatment_total)
        }
    
    def generate_report(self):
        """生成实验报告"""
        current_stats = self.get_current_stats()
        
        report = [
            f"序贯A/B测试报告: {self.experiment_id}",
            "=" * 50,
            f"生成时间: {pd.Timestamp.now()}",
            f"最终决策: {self.final_decision if self.final_decision else '进行中'}",
            f"停止点: {self.stopping_point if self.stopping_point else '未停止'}",
            ""
        ]
        
        if current_stats:
            report.append("当前统计信息:")
            report.append(f"  对照组: {current_stats['control']['conversions']}/{current_stats['control']['total']} "
                         f"({current_stats['control']['rate']:.4f}) "
                         f"[{current_stats['control']['ci_lower']:.4f}, {current_stats['control']['ci_upper']:.4f}]")
            report.append(f"  实验组: {current_stats['treatment']['conversions']}/{current_stats['treatment']['total']} "
                         f"({current_stats['treatment']['rate']:.4f}) "
                         f"[{current_stats['treatment']['ci_lower']:.4f}, {current_stats['treatment']['ci_upper']:.4f}]")
            report.append(f"  绝对差异: {current_stats['difference']['absolute']:.4f}")
            report.append(f"  相对提升: {current_stats['difference']['relative']:.2%}")
        
        if self.analysis_history:
            report.append("\n序贯分析历史:")
            for i, analysis in enumerate(self.analysis_history[-5:]):  # 显示最近5次分析
                report.append(f"  分析{i+1} (n={analysis['sample_size']}): "
                             f"决策={analysis['decision']}, "
                             f"转化率={analysis['control_rate']:.3f} vs {analysis['treatment_rate']:.3f}")
        
        # 与固定样本量检验对比
        if current_stats:
            fixed_sample_size = calculate_fixed_sample_size(
                current_stats['control']['rate'],
                current_stats['treatment']['rate'],
                self.alpha, self.power
            )
            efficiency_gain = (fixed_sample_size - len(self.treatment_data)) / fixed_sample_size
            
            report.append(f"\n效率分析:")
            report.append(f"  当前样本量: {len(self.treatment_data)}")
            report.append(f"  固定样本量所需: {fixed_sample_size}")
            report.append(f"  样本量节约: {efficiency_gain:.1%}")
        
        return "\n".join(report)
    
    def visualize_experiment_progress(self):
        """可视化实验进度"""
        if not self.analysis_history:
            print("尚无分析数据")
            return
        
        # 创建可视化
        fig, ((ax1, ax2), (ax3, ax4)) = plt.subplots(2, 2, figsize=(15, 12))
        
        # 1. 转化率随时间变化
        sample_sizes = [a['sample_size'] for a in self.analysis_history]
        control_rates = [a['control_rate'] for a in self.analysis_history]
        treatment_rates = [a['treatment_rate'] for a in self.analysis_history]
        
        ax1.plot(sample_sizes, control_rates, 'b-', marker='o', label='对照组')
        ax1.plot(sample_sizes, treatment_rates, 'r-', marker='s', label='实验组')
        ax1.set_xlabel('样本量')
        ax1.set_ylabel('转化率')
        ax1.set_title('转化率随样本量变化')
        ax1.legend()
        ax1.grid(True, alpha=0.3)
        
        # 2. 序贯边界和检验统计量
        self.sequential_test.visualize_boundaries()
        
        # 3. 相对提升的置信区间
        current_stats = self.get_current_stats()
        if current_stats:
            relative_improvement = current_stats['difference']['relative']
            # 使用bootstrap计算置信区间
            bootstrap_differences = []
            n_bootstrap = 1000
            
            for _ in range(n_bootstrap):
                control_bs = np.random.choice(self.control_data, size=len(self.control_data), replace=True)
                treatment_bs = np.random.choice(self.treatment_data, size=len(self.treatment_data), replace=True)
                
                control_rate_bs = np.mean(control_bs)
                treatment_rate_bs = np.mean(treatment_bs)
                
                if control_rate_bs > 0:
                    rel_diff_bs = (treatment_rate_bs - control_rate_bs) / control_rate_bs
                    bootstrap_differences.append(rel_diff_bs)
            
            ci_lower = np.percentile(bootstrap_differences, 2.5)
            ci_upper = np.percentile(bootstrap_differences, 97.5)
            
            ax3.bar(['相对提升'], [relative_improvement], 
                   yerr=[[relative_improvement - ci_lower], [ci_upper - relative_improvement]],
                   capsize=10, color='lightblue', alpha=0.7)
            ax3.axhline(0, color='k', linestyle='-', alpha=0.3)
            ax3.set_ylabel('相对提升')
            ax3.set_title('相对提升估计')
            ax3.grid(True, alpha=0.3)
        
        # 4. 样本量节约分析
        if current_stats:
            fixed_n = calculate_fixed_sample_size(
                current_stats['control']['rate'],
                current_stats['treatment']['rate'],
                self.alpha, self.power
            )
            actual_n = len(self.treatment_data)
            
            ax4.bar(['固定样本量', '序贯检验'], [fixed_n, actual_n], 
                   color=['red', 'green'], alpha=0.7)
            ax4.set_ylabel('样本量')
            ax4.set_title(f'样本量节约: {(fixed_n-actual_n)/fixed_n:.1%}')
            ax4.grid(True, alpha=0.3)
        
        plt.tight_layout()
        plt.show()

# 完整的序贯A/B测试演示
def demonstrate_sequential_ab_test():
    """演示完整的序贯A/B测试流程"""
    
    print("完整的序贯A/B测试演示")
    print("=" * 50)
    
    # 创建序贯A/B测试系统
    ab_test = SequentialABTestSystem(
        experiment_id="homepage_button_test_2024",
        alpha=0.05,
        power=0.8,
        min_samples=200,
        max_samples=10000,
        looks=12,
        boundary_type='obrien_fleming'
    )
    
    # 模拟用户流量和转化
    np.random.seed(42)
    n_users = 15000
    
    # 真实效果:实验组比对照组提升20%
    true_control_rate = 0.08
    true_treatment_rate = 0.10
    
    print(f"真实参数: 对照组={true_control_rate}, 实验组={true_treatment_rate}")
    
    stopped_early = False
    
    for user_id in range(n_users):
        # 分配用户
        group = ab_test.assign_user(f"user_{user_id}", traffic_split=0.5)
        
        # 根据真实参数模拟转化
        if group == 'control':
            converted = np.random.random() < true_control_rate
        else:
            converted = np.random.random() < true_treatment_rate
        
        # 跟踪转化
        decision = ab_test.track_conversion(f"user_{user_id}", converted)
        
        # 检查是否提前停止
        if decision is not None and not stopped_early:
            print(f"在用户 {user_id} 时提前停止实验")
            stopped_early = True
            # 在实际应用中,这里会停止分配流量
    
    # 生成报告
    report = ab_test.generate_report()
    print("\n" + report)
    
    # 可视化结果
    ab_test.visualize_experiment_progress()
    
    # 分析序贯检验的性能
    analyze_sequential_performance(ab_test)
    
    return ab_test

def analyze_sequential_performance(ab_test):
    """分析序贯检验的性能"""
    
    print("\n序贯检验性能分析:")
    print("=" * 50)
    
    current_stats = ab_test.get_current_stats()
    if not current_stats:
        return
    
    # 计算实际错误率(通过模拟)
    n_simulations = 500
    false_positive_count = 0
    false_negative_count = 0
    sample_sizes = []
    
    true_control_rate = current_stats['control']['rate']
    
    for sim in range(n_simulations):
        # 模拟无真实效应的情况(检验第一类错误)
        sim_test = SequentialABTestSystem(
            experiment_id=f"sim_{sim}",
            alpha=0.05, power=0.8,
            min_samples=200, max_samples=10000
        )
        
        # 生成数据(无效应)
        control_data = np.random.binomial(1, true_control_rate, 10000)
        treatment_data = np.random.binomial(1, true_control_rate, 10000)
        
        decision = None
        for i in range(10000):
            if i >= len(control_data) or i >= len(treatment_data):
                break
                
            decision = sim_test.track_conversion(f"c_{i}", bool(control_data[i]))
            if decision is not None:
                break
            decision = sim_test.track_conversion(f"t_{i}", bool(treatment_data[i]))
            if decision is not None:
                break
        
        if decision == 'reject':  # 错误拒绝H0
            false_positive_count += 1
        
        sample_sizes.append(sim_test.stopping_point if sim_test.stopping_point else 10000)
    
    empirical_alpha = false_positive_count / n_simulations
    avg_sample_size = np.mean(sample_sizes)
    
    print(f"经验第一类错误率: {empirical_alpha:.4f} (理论: {ab_test.alpha})")
    print(f"平均样本量: {avg_sample_size:.0f}")
    print(f"与固定样本量相比节约: {(10000 - avg_sample_size)/10000:.1%}")

# 运行完整的序贯A/B测试演示
ab_test_system = demonstrate_sequential_ab_test()

3.2 序贯检验的实操考虑

在实际业务中实施序贯检验时,需要考虑多个实操因素:

考虑因素 挑战 解决方案
操作频率 过于频繁的检查增加第一类错误 使用分组序贯设计,限制检查次数
最小样本量 早期偶然波动导致错误决策 设置合理的最小样本量门槛
多重检验 同时运行多个实验的错误率膨胀 使用更严格的边界或错误发现率控制
效果估计 提前停止可能高估效应大小 使用偏差校正的估计方法
系统复杂性 实现和维护成本较高 使用成熟的统计库和标准化流程
用户序贯检验系统分析引擎决策引擎访问网站/应用随机分配用户到组别返回对应体验产生转化行为记录转化数据检查是否达到分析点计算当前统计量比较序贯边界报告统计决策停止实验指令停止新用户分配记录最终结果继续收集数据alt[达到停止边界][继续实验]loop[持续监控]实验停止后的操作生成详细报告业务决策分析实施决策建议用户序贯检验系统分析引擎决策引擎

IV. 序贯检验的优势与业务价值

4.1 效率提升的量化分析

序贯检验最直接的优势在于样本量节约,让我们通过系统的模拟分析来量化这一优势。

def comprehensive_efficiency_analysis():
    """序贯检验效率的全面分析"""
    
    print("序贯检验效率分析")
    print("=" * 50)
    
    # 测试不同的效应大小
    effect_sizes = [0.05, 0.10, 0.15, 0.20, 0.25, 0.30]  # 相对提升
    baseline_rate = 0.10
    n_simulations = 500
    
    results = []
    
    for effect_size in effect_sizes:
        treatment_rate = baseline_rate * (1 + effect_size)
        
        sequential_sample_sizes = []
        fixed_sample_sizes = []
        early_stop_rates = []
        
        for sim in range(n_simulations):
            # 计算固定样本量
            fixed_n = calculate_fixed_sample_size(
                baseline_rate, treatment_rate, alpha=0.05, power=0.8
            )
            fixed_sample_sizes.append(fixed_n)
            
            # 运行序贯检验
            seq_test = SequentialABTestSystem(
                experiment_id=f"eff_{effect_size}_{sim}",
                alpha=0.05, power=0.8,
                min_samples=200, max_samples=10000,
                looks=10
            )
            
            # 生成数据直到停止或达到最大样本量
            control_data = np.random.binomial(1, baseline_rate, 10000)
            treatment_data = np.random.binomial(1, treatment_rate, 10000)
            
            decision = None
            for i in range(10000):
                if i >= len(control_data) or i >= len(treatment_data):
                    break
                    
                decision = seq_test.track_conversion(f"c_{i}", bool(control_data[i]))
                if decision is not None:
                    break
                decision = seq_test.track_conversion(f"t_{i}", bool(treatment_data[i]))
                if decision is not None:
                    break
            
            stop_point = seq_test.stopping_point if seq_test.stopping_point else 10000
            sequential_sample_sizes.append(stop_point)
            early_stop_rates.append(1 if stop_point < 10000 else 0)
        
        avg_sequential = np.mean(sequential_sample_sizes)
        avg_fixed = np.mean(fixed_sample_sizes)
        efficiency_gain = (avg_fixed - avg_sequential) / avg_fixed
        early_stop_rate = np.mean(early_stop_rates)
        
        results.append({
            'effect_size': effect_size,
            'relative_improvement': effect_size,
            'avg_sequential_n': avg_sequential,
            'avg_fixed_n': avg_fixed,
            'efficiency_gain': efficiency_gain,
            'early_stop_rate': early_stop_rate
        })
        
        print(f"效应大小 {effect_size:.0%}:")
        print(f"  序贯检验平均样本量: {avg_sequential:.0f}")
        print(f"  固定检验样本量: {avg_fixed:.0f}")
        print(f"  效率提升: {efficiency_gain:.1%}")
        print(f"  早期停止率: {early_stop_rate:.1%}")
    
    # 可视化结果
    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 6))
    
    # 样本量对比
    effect_sizes_plot = [r['effect_size'] for r in results]
    sequential_n_plot = [r['avg_sequential_n'] for r in results]
    fixed_n_plot = [r['avg_fixed_n'] for r in results]
    
    ax1.plot(effect_sizes_plot, sequential_n_plot, 'bo-', linewidth=2, 
             markersize=8, label='序贯检验')
    ax1.plot(effect_sizes_plot, fixed_n_plot, 'r--', linewidth=2, 
             label='固定样本量检验')
    ax1.set_xlabel('相对效应大小')
    ax1.set_ylabel('样本量')
    ax1.set_title('样本量需求对比')
    ax1.legend()
    ax1.grid(True, alpha=0.3)
    
    # 效率提升
    efficiency_gains = [r['efficiency_gain'] for r in results]
    ax2.plot(effect_sizes_plot, efficiency_gains, 'g^-', linewidth=2, 
             markersize=8, label='效率提升')
    ax2.set_xlabel('相对效应大小')
    ax2.set_ylabel('效率提升')
    ax2.set_title('序贯检验的效率优势')
    ax2.grid(True, alpha=0.3)
    
    # 添加参考线
    ax2.axhline(0, color='k', linestyle='-', alpha=0.3)
    ax2.axhline(0.5, color='r', linestyle='--', alpha=0.5, label='50%提升参考线')
    ax2.legend()
    
    plt.tight_layout()
    plt.show()
    
    # 创建总结表格
    results_df = pd.DataFrame(results)
    print(f"\n效率分析总结:")
    print(results_df.round(4))
    
    return results_df

# 运行效率分析
efficiency_results = comprehensive_efficiency_analysis()

4.2 业务价值评估

序贯检验的业务价值不仅体现在统计效率上,更重要的是其对业务决策速度和风险控制的影响。

def business_value_assessment():
    """序贯检验的业务价值评估"""
    
    print("序贯检验业务价值评估")
    print("=" * 50)
    
    # 业务场景参数
    scenarios = [
        {
            'name': '高风险功能发布',
            'risk_level': '高',
            'traffic_volume': 100000,  # 日流量
            'customer_value': 200,     # 用户平均价值
            'implementation_cost': 50000,  # 实施成本
            'opportunity_cost_delay': 1000  # 延迟决策的日机会成本
        },
        {
            'name': '中等优化测试', 
            'risk_level': '中',
            'traffic_volume': 50000,
            'customer_value': 100,
            'implementation_cost': 10000,
            'opportunity_cost_delay': 500
        },
        {
            'name': '低风险UI改进',
            'risk_level': '低', 
            'traffic_volume': 20000,
            'customer_value': 50,
            'implementation_cost': 5000,
            'opportunity_cost_delay': 200
        }
    ]
    
    # 假设的实验结果
    experiment_results = {
        'effect_size': 0.15,  # 15%提升
        'baseline_conversion': 0.08,
        'sequential_stop_time': 2500,  # 序贯检验停止样本量
        'fixed_sample_size': 5000,     # 固定检验样本量
        'decision_accuracy': 0.95      # 决策准确率
    }
    
    value_analysis = []
    
    for scenario in scenarios:
        # 计算时间节约
        daily_traffic_per_group = scenario['traffic_volume'] * 0.5  # 50%流量分配
        days_saved = (experiment_results['fixed_sample_size'] - 
                     experiment_results['sequential_stop_time']) / daily_traffic_per_group
        
        # 计算机会成本节约
        opportunity_saving = days_saved * scenario['opportunity_cost_delay']
        
        # 计算风险控制价值
        risk_exposure_reduction = (experiment_results['fixed_sample_size'] - 
                                 experiment_results['sequential_stop_time']) / scenario['traffic_volume']
        
        # 如果新版本有负面影响,减少的损失
        negative_impact_reduction = risk_exposure_reduction * scenario['customer_value'] * 0.1  # 假设10%的负面影响
        
        # 总价值估计
        total_value = opportunity_saving + negative_impact_reduction
        
        value_analysis.append({
            'scenario': scenario['name'],
            'risk_level': scenario['risk_level'],
            'days_saved': days_saved,
            'opportunity_saving': opportunity_saving,
            'risk_reduction_value': negative_impact_reduction,
            'total_value': total_value,
            'efficiency_gain': (experiment_results['fixed_sample_size'] - 
                              experiment_results['sequential_stop_time']) / experiment_results['fixed_sample_size']
        })
        
        print(f"{scenario['name']} ({scenario['risk_level']}风险):")
        print(f"  时间节约: {days_saved:.1f} 天")
        print(f"  机会成本节约: ¥{opportunity_saving:,.0f}")
        print(f"  风险控制价值: ¥{negative_impact_reduction:,.0f}")
        print(f"  总价值: ¥{total_value:,.0f}")
        print(f"  效率提升: {value_analysis[-1]['efficiency_gain']:.1%}")
    
    # 可视化业务价值
    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 6))
    
    # 价值分解
    scenarios_names = [v['scenario'] for v in value_analysis]
    opportunity_values = [v['opportunity_saving'] for v in value_analysis]
    risk_values = [v['risk_reduction_value'] for v in value_analysis]
    
    x = np.arange(len(scenarios_names))
    width = 0.35
    
    ax1.bar(x - width/2, opportunity_values, width, label='机会成本节约', alpha=0.7)
    ax1.bar(x + width/2, risk_values, width, label='风险控制价值', alpha=0.7)
    
    ax1.set_xlabel('业务场景')
    ax1.set_ylabel('价值 (元)')
    ax1.set_title('序贯检验的业务价值分解')
    ax1.set_xticks(x)
    ax1.set_xticklabels(scenarios_names)
    ax1.legend()
    ax1.grid(True, alpha=0.3)
    
    # 效率与价值关系
    efficiency_gains = [v['efficiency_gain'] for v in value_analysis]
    total_values = [v['total_value'] for v in value_analysis]
    
    colors = ['red', 'orange', 'green']
    for i, (eff, val) in enumerate(zip(efficiency_gains, total_values)):
        ax2.scatter(eff, val, s=200, c=colors[i], alpha=0.7, 
                   label=scenarios_names[i])
    
    ax2.set_xlabel('效率提升')
    ax2.set_ylabel('总业务价值 (元)')
    ax2.set_title('效率提升与业务价值的关系')
    ax2.legend()
    ax2.grid(True, alpha=0.3)
    
    plt.tight_layout()
    plt.show()
    
    # 投资回报率分析
    print(f"\n投资回报率(ROI)分析:")
    for i, scenario in enumerate(scenarios):
        implementation_cost = scenario['implementation_cost']
        roi = value_analysis[i]['total_value'] / implementation_cost
        
        print(f"{scenario['name']}:")
        print(f"  实施成本: ¥{implementation_cost:,.0f}")
        print(f"  预期价值: ¥{value_analysis[i]['total_value']:,.0f}")
        print(f"  ROI: {roi:.1f}x")
    
    return value_analysis

# 运行业务价值评估
business_values = business_value_assessment()

4.3 实施建议与最佳实践

基于理论分析和实践经验,我们总结了序贯检验的实施建议:

实践领域 具体建议 理论依据
实验设计 设置合理的最小样本量(100-500) 避免早期偶然波动导致错误决策
边界选择 高风险场景使用O’Brien-Fleming边界 更好的第一类错误控制
检查频率 5-15次检查点,等间距或信息量比例 平衡效率与操作复杂性
效果估计 使用偏差校正方法估计效应大小 解决提前停止导致的估计偏差
多重检验 控制实验家族错误率 避免多重比较问题
序贯检验优势
效率提升
风险控制
决策加速
资源优化
样本量节约30-70%
实验周期缩短
吞吐量提升
早期识别有害变更
限制负面影响的用户暴露
快速回滚机制
实时决策支持
缩短反馈循环
敏捷响应市场
流量利用优化
计算资源节约
团队效率提升
业务价值
直接经济收益
竞争优势
风险 mitigation
机会成本节约
实施成本优化
收入增长加速
更快的学习周期
更好的用户体验
创新速度提升
减少业务风险
增强决策信心
改进质量控制

V. 实际案例研究

5.1 电商网站转化率优化案例

某大型电商网站在黑色星期五促销期间需要测试新的商品推荐算法。由于促销期短暂,传统的固定样本量检验无法在有限时间内完成,因此采用序贯检验方法。

def ecommerce_case_study():
    """电商网站序贯检验实战案例"""
    
    print("电商网站转化率优化案例研究")
    print("=" * 50)
    
    # 案例背景
    case_background = {
        'business_context': '黑色星期五促销期商品推荐算法优化',
        'time_constraint': '促销期仅7天,需要快速决策',
        'traffic_volume': 500000,  # 日均流量
        'baseline_conversion': 0.035,  # 基线转化率3.5%
        'min_meaningful_effect': 0.10,  # 10%相对提升
        'risk_tolerance': '中等'  # 对错误决策的容忍度
    }
    
    print("案例背景:")
    for key, value in case_background.items():
        print(f"  {key}: {value}")
    
    # 设计序贯检验
    sequential_design = {
        'alpha': 0.05,
        'power': 0.85,  # 稍高的功效要求
        'min_samples': 1000,  # 最小样本量
        'max_samples': 50000,  # 最大样本量
        'looks': 15,  # 检查次数
        'boundary_type': 'obrien_fleming'  # 严格的边界控制
    }
    
    print(f"\n序贯检验设计:")
    for key, value in sequential_design.items():
        print(f"  {key}: {value}")
    
    # 模拟实验运行
    np.random.seed(42)  # 可重复的结果
    
    # 真实效果:新算法提升12%转化率
    true_control_rate = 0.035
    true_treatment_rate = 0.0392  # 12%提升
    
    # 创建序贯检验系统
    ecommerce_test = SequentialABTestSystem(
        experiment_id="black_friday_recommendation_2024",
        **sequential_design
    )
    
    # 模拟7天的用户流量
    daily_traffic = case_background['traffic_volume']
    days = 7
    daily_decisions = []
    
    for day in range(days):
        print(f"\n第 {day+1} 天:")
        
        # 当天的用户数量
        daily_users = int(daily_traffic * 0.5)  # 50%流量用于实验
        
        day_control_conversions = 0
        day_treatment_conversions = 0
        
        for user in range(daily_users):
            # 分配用户
            group = ecommerce_test.assign_user(f"day{day}_user{user}")
            
            # 模拟转化
            if group == 'control':
                converted = np.random.random() < true_control_rate
                day_control_conversions += 1 if converted else 0
            else:
                converted = np.random.random() < true_treatment_rate
                day_treatment_conversions += 1 if converted else 0
            
            # 跟踪转化
            decision = ecommerce_test.track_conversion(
                f"day{day}_user{user}", converted
            )
        
        # 当日统计
        daily_stats = ecommerce_test.get_current_stats()
        if daily_stats:
            print(f"  对照组: {daily_stats['control']['conversions']}/{daily_stats['control']['total']} "
                  f"({daily_stats['control']['rate']:.4f})")
            print(f"  实验组: {daily_stats['treatment']['conversions']}/{daily_stats['treatment']['total']} "
                  f"({daily_stats['treatment']['rate']:.4f})")
            print(f"  相对提升: {daily_stats['difference']['relative']:.2%}")
        
        if decision is not None:
            print(f"  ✅ 实验停止! 最终决策: {decision}")
            daily_decisions.append(decision)
            break
        else:
            print(f"  ⏳ 继续实验...")
            daily_decisions.append(None)
    
    # 最终报告
    final_report = ecommerce_test.generate_report()
    print(f"\n{final_report}")
    
    # 业务影响分析
    analyze_business_impact(ecommerce_test, case_background)
    
    # 与传统方法的对比
    compare_with_traditional_approach(ecommerce_test, case_background)
    
    return ecommerce_test, daily_decisions

def analyze_business_impact(ab_test, background):
    """分析业务影响"""
    
    print("\n业务影响分析:")
    print("=" * 50)
    
    stats = ab_test.get_current_stats()
    if not stats:
        return
    
    # 计算额外的转化和收入
    additional_conversions = (stats['treatment']['conversions'] - 
                            stats['control']['conversions'] * 
                            stats['treatment']['total'] / stats['control']['total'])
    
    # 假设平均订单价值
    avg_order_value = 250  # 元
    additional_revenue = additional_conversions * avg_order_value
    
    # 计算由于提前停止节约的机会成本
    if ab_test.stopping_point and ab_test.stopping_point < ab_test.max_samples:
        days_saved = (ab_test.max_samples - ab_test.stopping_point) / (background['traffic_volume'] * 0.5)
        opportunity_saving = days_saved * background['traffic_volume'] * 0.5 * background['baseline_conversion'] * avg_order_value * stats['difference']['relative']
    else:
        opportunity_saving = 0
    
    print(f"实验期间业务影响:")
    print(f"  额外转化: {additional_conversions:.0f} 次")
    print(f"  额外收入: ¥{additional_revenue:,.0f}")
    print(f"  机会成本节约: ¥{opportunity_saving:,.0f}")
    print(f"  总价值: ¥{additional_revenue + opportunity_saving:,.0f}")
    
    # 如果全量发布的预期影响
    if ab_test.final_decision == 'reject':
        monthly_traffic = background['traffic_volume'] * 30
        expected_monthly_impact = monthly_traffic * background['baseline_conversion'] * stats['difference']['relative'] * avg_order_value
        print(f"  全量发布月预期影响: ¥{expected_monthly_impact:,.0f}")

def compare_with_traditional_approach(ab_test, background):
    """与传统方法对比"""
    
    print("\n与传统固定样本量方法对比:")
    print("=" * 50)
    
    stats = ab_test.get_current_stats()
    if not stats:
        return
    
    # 计算固定样本量所需时间
    fixed_sample_size = calculate_fixed_sample_size(
        stats['control']['rate'], 
        stats['treatment']['rate'],
        ab_test.alpha, ab_test.power
    )
    
    daily_traffic_per_group = background['traffic_volume'] * 0.5
    fixed_days_required = fixed_sample_size / daily_traffic_per_group
    actual_days_used = ab_test.stopping_point / daily_traffic_per_group if ab_test.stopping_point else ab_test.max_samples / daily_traffic_per_group
    
    print(f"固定样本量方法:")
    print(f"  所需样本量: {fixed_sample_size}")
    print(f"  所需时间: {fixed_days_required:.1f} 天")
    print(f"序贯检验方法:")
    print(f"  实际样本量: {ab_test.stopping_point if ab_test.stopping_point else ab_test.max_samples}")
    print(f"  实际时间: {actual_days_used:.1f} 天")
    print(f"时间节约: {fixed_days_required - actual_days_used:.1f} 天 ({((fixed_days_required - actual_days_used)/fixed_days_required):.1%})")
    
    # 在时间约束下的成功率
    time_constraint = 7  # 天
    if fixed_days_required > time_constraint:
        print(f"⚠️  固定样本量方法无法在 {time_constraint} 天内完成")
        print(f"✅ 序贯检验方法成功在时间约束内完成决策")
    else:
        print(f"两种方法都能在时间约束内完成")

# 运行电商案例研究
ecommerce_test_system, daily_decisions = ecommerce_case_study()

5.2 关键成功因素与经验教训

基于实际案例的实施经验,我们总结了序贯检验成功实施的关键因素:

成功因素 具体实践 避免的陷阱
明确的目标 预先定义最小有意义效应和风险容忍度 避免过度依赖统计显著性而忽略业务意义
合适的设计 根据业务场景选择边界类型和检查频率 避免过于激进的设计导致错误决策
基础设施 建立可靠的实时数据流水线和监控 避免数据质量问题影响检验有效性
团队教育 培训业务团队理解序贯检验的原理和限制 避免错误解读结果和过早决策
持续优化 基于经验调整设计参数和决策流程 避免一成不变的方法不适应业务变化
案例研究总结
电商案例
关键成功因素
经验教训
业务背景
实验设计
实施过程
业务影响
明确目标
合适设计
基础设施
团队教育
持续优化
避免的陷阱
最佳实践
实施建议
促销时间约束
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