起源故事 · Fisher 与农业试验田
Origin Story · Fisher and the Agricultural Plots
1920 年代,统计学家 R.A. Fisher 在英国 Rothamsted 农业试验站研究施肥。庄稼一季只长一次,没法慢慢"一次试一个"。
他想出革命性的安排:把肥料、品种、灌溉同时安排在不同地块,再用方差分析把各因子的贡献拆开。这就是实验设计(DOE)的诞生。
2ᵏ 全因子是它最朴素的形态 —— 每个因子取"高/低"两档,跑遍所有组合,于是主效应和交互作用一次性全部现形。
In the 1920s, the statistician R.A. Fisher was studying fertilizer trials at the Rothamsted Experimental Station in England. Crops grow once a season — there is no time to "vary one factor at a time."
His radical move: vary fertilizer, variety, and irrigation simultaneously across different plots, then use analysis of variance to peel each factor's contribution apart. That was the birth of Design of Experiments (DOE).
The 2ᵏ full factorial is its plainest form — each factor takes a "high" and a "low" level, you run every combination, and every main effect and every interaction comes out in one pass.
1 实验立方体:八个角,跑八次 The Design Cube: Eight Corners, Eight Runs
2³ = 82 主效应斜率 + A×B 交互图 Main-Effect Slopes + A×B Interaction Plot
左:每个因子从低到高的主效应斜率,越陡影响越大。右:A×B 交互图 —— 两条线平行说明无交互;叉开说明 A 的效果取决于 B 在高还是低,这是 OFAT 永远看不到的。 Left: the main-effect slope of each factor from low to high — the steeper, the stronger. Right: the A×B interaction plot — parallel lines mean no interaction; lines that fan apart mean A's effect depends on whether B is high or low. OFAT can never see this.
3 现实里的 2ᵏ 设计 2ᵏ Designs in the Real World
工艺参数优化:注塑温度×保压×冷却时间,几次试验锁定最优组合,告别拍脑袋试错。
Process tuning: injection temperature × hold pressure × cool time — a handful of runs locks in the best combination, no more gut-feel guesswork.
配方筛选:哪种添加剂真有用?2ᵏ 一次比出主效应,把无关因子尽早剔除。
Formulation screening: which additive actually matters? A 2ᵏ run ranks main effects in one pass and weeds the inert ones out early.
焊接质量:电流×速度×气体流量对强度的交互作用,单变量试验完全察觉不到。
Weld quality: current × travel speed × shielding-gas flow interact on joint strength — single-variable trials miss the effect entirely.
替代试错:用结构化设计替代"调一下看一下",试验次数砍半、信息量翻倍。
Replace tweak-and-look: a structured design beats "twist a knob, peek at the gauge" — half the runs, twice the information.
一句话In One Line
OFAT 一次只动一个因子,看似严谨,实则既低效又有盲区 —— 它永远撞不见交互作用。
2ᵏ 全因子把试验摆在立方体的角上,让每个因子的高低各占一半,于是每个主效应都用上了全部数据(隐藏的重复),
交互作用也自动浮现。代价是因子一多,2ᵏ 会爆炸(7 个因子要 128 次),那就上部分因子设计取舍分辨率 —— 但入门,先把这个立方体玩透。
OFAT looks disciplined but is actually slow and blind — it will never stumble onto an interaction.
A 2ᵏ design places runs on the corners of a cube so every factor's high and low each see half the runs. Every main effect therefore uses all of the data (hidden replication), and interactions fall out for free.
The catch: 2ᵏ explodes with k (seven factors = 128 runs), so once factor counts climb, trade resolution for size with a fractional factorial. But first, master this cube.
常见误用Common Mistakes
继续用 OFAT 找最优。OFAT 看不见交互,DOE 同时估主效应与交互,更少试验、更全信息。
Hunting for the optimum with OFAT. OFAT is blind to interactions. A factorial estimates main effects and interactions together — fewer runs, more information.
因子太多硬上全因子。2ᵏ 随 k 指数爆炸,多因子先用部分因子设计做筛选。
Brute-forcing a full factorial with too many factors. 2ᵏ blows up exponentially with k — screen first with a fractional factorial.
不设中心点就断定线性。加中心点检验曲率,有弯曲再上响应面(下一页)。
Assuming linearity with no center points. Add center points to test for curvature — if the surface bends, move on to RSM (next page).