所属模块：
M1 · 数学与统计基础 (Mathematics & Statistics Fundamentals)| 专题分类：假设检验 (Hypothesis Testing)| 难度等级：Easy
一、核心一句话结论 (One-Sentence Summary)
α = 弃真概率；β = 取伪概率；功效 = 1-β = 正确检出真实效应的概率。
Type I error $alpha$ is false positive (rejecting true $H_0$); Type II error $beta$ is false negative (failing to reject false $H_0$); Power $1-beta$ is the probability of correctly detecting a real effect.
二、核心考点要义 (Key Insights)
- 📌 功效随效应量、样本量、α 上升
- 📌 常规要求 power ≥ 0.8
English Insights:
– Type I Error ($alpha$): $P(text{Reject } H_0 mid H_0 text{ is True})$; standard industrial threshold is $alpha = 0.05$.
– Type II Error ($beta$): $P(text{Fail to Reject } H_0 mid H_1 text{ is True})$; standard industrial threshold is $beta = 0.20$.
– Statistical Power ($1 – beta$): Probability of detecting a true effect; standard target is $80%$ or $90%$.
– Fundamental tradeoff: Lowering $alpha$ shifts critical threshold outward, automatically increasing $beta$ unless sample size $N$ is increased.
三、核心数学原理与机理推导 (Mathematical Principles & Derivation)
$$alpha=Pr(text{reject}mid H_0),quad text{power}=1-beta=Pr(text{reject}mid H_1)$$
两类错误的完整对照：第一类错误（弃真）——H₀ 为真却拒绝了它，概率为 α（显著性水平），这是’误报’；第二类错误（取伪）——H₁ 为真却未拒绝 H₀，概率为 β，这是’漏报’。功效 power=1−β 是正确检出真实效应的概率。四者的关系可写成 2×2 表：判决/真相的组合中，正确拒绝的概率是 power，正确不拒绝的概率是 1−α。关键性质：固定 n 时 α 与 β 反向变动（降低 α 会抬高 β）；要同时降低两者只能增大样本量或效应量。
📖 查看英文严格数学推导 (English Mathematical Derivation)
For testing $H_0: mu = 0$ vs $H_1: mu = delta > 0$ with test statistic $Z = frac{bar{X}}{sigma/sqrt{N}}$: Rejection region is $Z > z_{1-alpha}$. Under $H_0$, $Z sim mathcal{N}(0, 1)$, giving Type I error $P(Z > z_{1-alpha} mid H_0) = alpha$. Under $H_1$, $Z sim mathcal{N}left(frac{delta}{sigma/sqrt{N}}, 1right)$. The Type II error is $beta = P(Z le z_{1-alpha} mid H_1) = Phileft(z_{1-alpha} – frac{delta}{sigma/sqrt{N}}right)$. Setting this to $z_beta$ yields the fundamental relationship: $z_{1-alpha} + z_{1-beta} = frac{delta}{sigma/sqrt{N}}$, linking significance level, power, effect size, and sample size.
四、工业级落地权衡与工程考量 (Industrial Trade-offs)
实践要点：① 功效分析的四个输入——显著性水平 α、期望功效（通常 0.8）、效应量（MDE，最小可检测效应）、方差 σ²；给定其中三个可解出第四个（通常是样本量 n）。A/B 测试上线前必须做功效分析，否则可能跑了一个注定检不出效果的实验。② 功效不足的严重后果——低功效实验不仅容易漏掉真实效应，还会使已检出的显著结果被高估（’winner’s curse’）：在低功效下，只有效应量被随机高估的样本才能越过显著性阈值，故观测到的效应系统性偏大，这解释了为什么很多小样本研究的效应无法被大样本复现。③ 多重比较的放大——同时检验 m 个假设时，至少一次假阳性的概率为 1−(1−α)^m，m=20 时达 64%，故需 Bonferroni（控制 FWER）或 BH（控制 FDR）校正。
⚙️ 查看英文落地权衡分析 (English Systems & Trade-offs)
In online experimentation: (1) If launching a risky feature (e.g. Redesigning core checkout UI), Type I error is dangerous, so one might tighten $alpha = 0.01$. (2) If screening hundreds of candidate algorithms in early funnel stages, Type II error is costlier (missing a breakthrough), so one might tolerate $alpha = 0.10$ to boost power. The only way to decrease both $alpha$ and $beta$ simultaneously is to increase sample size $N$ or reduce metric variance via CUPED.
五、常见面试避坑陷阱 (Common Pitfalls & Traps)
- ⚠️ 认为 α 与 β 可以同时任意降低（需增大样本量）
- ⚠️ 忽略低功效导致的效应量高估（winner’s curse）
English Pitfalls:
– Concluding an experiment was successful simply because it had a high statistical power target (power is a pre-experiment planning metric, not a post-hoc evaluation).
– Calculating ‘post-hoc observed power’ using the observed sample effect size (which is mathematically redundant with the p-value and provides zero new information).
六、高频深度面试追问与预测 (Follow-Up Questions)
- 功效分析的四个输入是什么？
- Why is post-hoc power considered mathematically invalid and misleading in experimental analysis?
- 为什么功效不足会导致’显著的结果不可靠’？
- How does the Neyman-Pearson Lemma prove that likelihood ratio tests maximize power for simple hypotheses?
七、知识图谱对齐 (Knowledge Graph Anchor)
- 🔗 关联底层卡片：
数理统计假说检验、P 值、I/II 类错误与统计功效(Hypothesis Testing, P-Values, Power & Type I/II Error) - 🗺️ 知识图谱模块：
数理基础思维导图
🔬 算法科学家与机器学习深度考察全量题库 (Science Depth)
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