起源故事 · Fisher vs Neyman-Pearson 的世纪论战 Origin Story · Fisher vs Neyman-Pearson, the Century-Long Feud
统计学有两大门派。R.A. Fisher 主张显著性检验:只设一个 H0,算出 p 值,p 小就说证据反对 H0 —— 灵活、连续、不预设拒绝点。 Jerzy Neyman 与 Egon Pearson 则提出假设检验:明确摆出 H0 与 H1,事先定好 α 和拒绝域,把检验当成一套决策规则,并引入了第二类错误 β 与功效。 两派为此论战数十年、互不相让。而今天教科书里那套"设 H0/H1、定 α、算 p、比临界值"的流程,其实是两家思想被后人缝合在一起的混合体。 Statistics has two rival schools. R.A. Fisher championed significance testing: state only H0, compute a p-value, and let a small p speak as evidence against H0 — flexible, continuous, no preset cutoff. Jerzy Neyman and Egon Pearson answered with hypothesis testing: spell out both H0 and H1 up front, fix α and a rejection region in advance, and treat the test as a decision rule — and they introduced Type II error β and power along the way. The two camps argued for decades, neither willing to yield. The textbook recipe you know today — "state H0 and H1, fix α, compute p, compare to a critical value" — is a stitched-together hybrid later generations cobbled out of both traditions.

1 拖动判定线:α 与 β 一升一降,永不和解 Drag the Decision Line: α and β Never Settle Down Together

拖那条黑线 → Drag the black line →

灰色是 H0(无效应)分布,蓝色是 H1(真有效应)分布。黑色判定线右侧是"拒绝 H0"区。红色面积是 α(H0 真却被拒),蓝色面积是 β(H1 真却没拒)。把线往右拖,红区缩小、蓝区却长大 —— 这就是两类错误的死结。 Grey is the H0 (no-effect) distribution; blue is the H1 (real-effect) distribution. Right of the black line is the "reject H0" region. The red area is α (H0 is true but rejected); the blue area is β (H1 is true but not rejected). Drag the line right and red shrinks while blue grows — that is the deadlock between the two errors.

2 四种结局:真相 × 你的判定 Four Outcomes: Truth × Your Verdict

对角线是正确判定(绿);两个角是错误:弃真 α 与 取伪 β。法庭上 H0 = "无罪推定":α 是冤枉好人,β 是放走坏人 —— "宁纵勿枉"就是宁愿要大 β、也要压低 α。 The diagonal is the right call (green); the off-diagonal corners are the mistakes — Type I (α) and Type II (β). In a courtroom H0 = "presumed innocent": α convicts the innocent, β lets the guilty walk free — "better acquit a hundred guilty than convict one innocent" is just keeping α small at the cost of a larger β.

3 现实里的两类错误 The Two Errors in the Real World

质量验收:α = 生产者风险(好批被拒退),β = 消费者风险(坏批放行)。抽样方案就是在两者间权衡。 Acceptance sampling: α = producer's risk (a good batch gets rejected); β = consumer's risk (a bad batch slips through). The whole sampling plan is just trading these two off.
医学诊断:α = 假阳性(误报有病、白做手术),β = 假阴性(漏诊、错过治疗)。两者代价天差地别。 Medical diagnostics: α = false positive (you're told you're sick and undergo needless surgery); β = false negative (the disease is missed and treatment never starts). The costs are not remotely the same.
司法审判:H0 = 无罪。α = 冤枉无辜,β = 放走真凶。"宁纵勿枉"= 刻意把 α 压到极低。 Criminal trials: H0 = "innocent". α = convicting the innocent; β = acquitting the guilty. "Better to let ten guilty go free than convict one innocent" is a deliberate choice to drive α as low as possible.
改善验证:β 太大 = 改善真有效却被判"没差",白干。这就是为什么要先算样本量、保证功效。 Improvement validation: a large β means a real improvement is dismissed as "no difference" and the work goes uncredited. That is exactly why you compute sample size up front to guarantee enough power.
一句话In One Line
假设检验的本质是在两类错误之间做权衡的决策框架。判定线一移,α 与 β 必然反向变化 —— 同一笔证据下,你只能选择更怕哪种错。 想同时压低 α 和 β,唯一的办法不是挪线,而是把两条分布拉得更开:要么真实效应更大,要么加大样本量让分布变窄。 这正是下一页功效与样本量的主题:先定好 α、想抓住的效应量和目标功效,反推出该抽多少样本 —— 检验从"画线的艺术"变成"算样本的工程"。 At heart, hypothesis testing is a decision framework that trades one error against the other. Move the line and α and β move in opposite directions — given the same evidence, all you really pick is which mistake you fear more. Want to push α and β down at the same time? Moving the line won't do it — you have to pull the two distributions further apart, either by a bigger real effect or by raising the sample size so each bell gets narrower. That is exactly the subject of the next page on power and sample size: fix α, the effect size you want to catch, and the target power, then back out how many samples you need — testing turns from "the art of drawing a line" into "the engineering of sizing a sample".
常见误用Common Mistakes
把 p 值当成"H0 为真的概率"。p 是"若 H0 为真,看到这么极端数据的概率",不是 H0 的概率。 Reading the p-value as "the probability that H0 is true". p is "assuming H0 is true, the probability of seeing data this extreme" — it is not the probability of H0 itself.
不显著就断言"两者相等"。没拒绝 H0 ≠ 接受 H0;可能只是样本太少、功效不足(β 大)。 Treating "not significant" as "the two are equal". Failing to reject H0 is not the same as accepting it — the sample may simply be too small and power too low (a large β).
只盯 α、完全不管 β。把 α 设得极小却忽视功效,会把真实改善判成无效,白白浪费。 Watching α and ignoring β entirely. Driving α down to almost nothing without minding power dooms real improvements to "no effect" verdicts — a costly way to seem rigorous.

假设检验框架