起源故事 · Neyman 与 Pearson 的"第二种错" Origin Story · Neyman and Pearson's "Second Kind of Mistake"
1933 年,波兰统计学家 Jerzy Neyman 和英国的 Egon Pearson(统计巨匠 Karl Pearson 之子)合作发表了划时代论文。 在他们之前,人们只盯着"误判有差别"这一种错(第一类错误 α)。Neyman 与 Pearson 指出:还有更隐蔽的第二种错 —— "真有差别却没发现"(第二类错误 β)。能避开这第二种错的能力,他们称之为功效 (power)。 从此,实验在开始前就要先问一句:我的样本量,够不够把真相捞上来? In 1933 Polish statistician Jerzy Neyman and the English Egon Pearson (son of the giant Karl Pearson) co-authored a landmark paper. Before them, statisticians watched only one mistake: calling a difference real when it isn't (Type I error, α). Neyman and Pearson pointed out a sneakier second mistake — missing a real difference (Type II error, β). The ability to avoid that second mistake they named power. From then on, every serious experiment opens with one question: is my sample big enough to fish the truth out?

1 两个世界的分布:它们分得开吗? Two Worlds, Two Distributions: Can You Tell Them Apart?

重叠中overlapping

左峰是"无差别"世界 (H0),右峰是"真有差别"世界 (H1)。绿色阴影 = 功效(成功检出),蓝色阴影 = β(漏检)。加大 n 看两峰分开、绿色变大。 The left peak is the "no-difference" world (H0); the right peak is the "real-difference" world (H1). Green shading = power (true detections); blue shading = β (misses). Crank up n and watch the peaks separate as the green area grows.

2 功效曲线:多少样本才爬过 80% 红线 Power Curve: How Many Samples Clear the 80% Line

横轴是样本量 n,纵轴是功效。曲线一开始陡升、后段趋平 —— 加样本的边际收益递减。虚线是 80% 红线,竖线是当前 n。 Sample size n runs along the horizontal axis, power up the vertical. The curve climbs steeply at first, then flattens — the marginal payoff of each extra sample dies off. The dashed line marks the 80% threshold; the vertical line marks your current n.

3 现实里的功效与样本量 Power and Sample Size in the Real World

实验设计先算 n:六西格玛 Improve 阶段,DOE 开跑前先做功效分析定样本量,避免做完才发现"看不出差别"。 Size the experiment first: in the Six Sigma Improve phase, run a power analysis to set sample size before launching a DOE — never finish the runs and only then realise "we can't tell the difference".
临床试验:监管要求功效 ≥ 80%,样本太少的阴性结果不被采信 —— 可能是真没效,也可能只是没测够。 Clinical trials: regulators demand at least 80% power. A negative result from an underpowered study buys no credit — it might mean the drug is useless, or simply that the trial was too small to know.
A/B 测试时长:要检出 1% 的转化提升,需要海量样本与足够天数,否则结论只是噪声。 A/B test duration: detecting a 1% lift in conversion takes a huge user base and enough calendar days. Cut it short and the "winner" is mostly noise.
避免假阴性浪费:功效不足的实验做了等于白做,既错过改善又浪费资源,是最贵的"省钱"。 Avoid the false-negative waste: an underpowered study is wasted effort — it misses the win and burns the budget. It is the most expensive form of "saving money".
一句话In One Line
功效不是越高越好就拼命加样本 —— 它是四个量的跷跷板:样本量 n、效应量 d、显著性 α、功效 1−β,定下三个,第四个就被锁死。 想检出更小的差别(d 小),就得付出更大的样本代价;想少漏检(功效高),要么加样本,要么放宽 α(但假阳性会涨)。 实验真正的智慧,是在动手之前就把这笔账算清 —— 而不是做完才发现网眼太大。 Power isn't "the higher the better, throw samples at it" — it is a four-way seesaw between sample size n, effect size d, significance α and power 1 − β. Pin down any three and the fourth is locked in. To catch smaller differences (small d) you pay in more samples. To miss fewer real effects (more power) you either spend more samples or loosen α — at the price of more false positives. The real craft is doing this arithmetic before you run the experiment, not discovering after the fact that the mesh was too wide.
常见误用Common Mistakes
把"不显著"当成"没差别"。可能只是功效不足漏检了,要看样本量够不够。 Reading "not significant" as "no difference". It may just be an underpowered miss — first ask whether the sample size was anywhere near enough.
实验做完才补算功效(事后功效)。样本量必须在开跑前算,事后功效几乎没意义。 Computing power after the fact (post-hoc power). Sample size must be sized before the experiment starts — retrospective power is almost meaningless.
为提功效一味放宽 α。α 放宽=假阳性增多,两类错误要平衡,不能只顾一头。 Cranking up α just to lift power. A looser α means more false positives. The two errors have to be balanced — never optimise one and ignore the other.

统计功效与样本量