FH05 · REFERENCE SHEET

DOE 实验设计公式总表 DOE Formula Sheet

把主效应、交互效应、2k 设计、别名结构、响应面与田口 S/N 比放在同一张速查表中。自上而下重要度递减：先背 T1，再熟练 T2，最后按需查 T3。 One working sheet for main and interaction effects, 2k designs, alias structure, response surfaces, and Taguchi S/N ratios. Importance drops as you move down: memorize T1, work fluently with T2, and pull T3 when the experiment calls for it.

覆盖范围 Coverage
T1 → T3
核心 DOE 工具链 core DOE stack
T1

必背 · 从因子变化读出效应 Memorize · Read Effects from Deliberate Factor Changes

4 张核心卡 4 core cards
T1-1

主效应 Main Effect

F38 深潜 → F38 deep dive →
EffectA = ȳ(A+) − ȳ(A−)

读作：因子在高水平的平均响应，减低水平的平均响应。Read it as: The factor's average response at the high level minus its average response at the low level.

  • ȳ(A+) A 在高水平时的平均响应 mean response with A at its high level
  • ȳ(A−) A 在低水平时的平均响应 mean response with A at its low level
何时用： Use it when: 量化因子 A 从低水平切到高水平时，平均响应改变多少。 You need the average response change when factor A moves from low to high.
⚠ ✗ 直接用单次 A+ 减单次 A− 当主效应 → ✓ 分别汇总所有 A+ 与 A− 组合的平均响应后再相减。 ✗ Subtract one A− run from one A+ run and call it the main effect → ✓ Average across every A+ and A− combination before taking the difference.
T1-2

交互效应 Interaction Effect

F38 深潜 → F38 deep dive →
EffectAB = ½[EffectA|B+ − EffectA|B−]

读作：B 变了以后，A 的效应变了多少——的一半。Read it as: Half of how much A's effect changes when B changes level.

EffectA|B+B 高时 A 的效应A's effect while B is high EffectA|B−B 低时 A 的效应A's effect while B is low ½[ · ]差值取半Take half the difference
  • EffectA|B+ B 为高水平时 A 的效应 effect of A while B is high
  • EffectA|B− B 为低水平时 A 的效应 effect of A while B is low
何时用： Use it when: 判断 A 的作用是否随 B 的水平改变；只有改变才叫交互。 You need to know whether the effect of A changes with the level of B; that change is the interaction.
⚠ ✗ 交互显著时仍单独解释主效应 → ✓ 先按另一因子的水平拆开解释条件效应。 ✗ Interpret main effects in isolation after a material interaction appears → ✓ Explain the conditional effect at each level of the other factor first.
T1-3

2k 全因子 2k Full Factorial

F38 深潜 → F38 deep dive →
Nruns = 2k × n

读作：每个因子两个水平，全组合就是 2 的 k 次方次试验。Read it as: With two levels per factor, the full set of combinations takes 2 to the kth power runs.

  • k 因子数，每个因子 2 水平 number of factors, each at 2 levels
  • n 每个处理组合的重复数 replicates per treatment combination
何时用： Use it when: 完整估计 k 个两水平因子的主效应与全部交互效应。 You need all main effects and interactions from k two-level factors without aliasing.
⚠ ✗ 把重复测量当成独立重复并低估误差 → ✓ 独立重复制备与随机化每轮试验。 ✗ Treat repeated readings of one setup as independent replicates → ✓ Re-create and randomize each experimental replicate independently.
T1-4

因子编码 Factor Coding

F38 深潜 → F38 deep dive →
xcoded = (actual − center) / half-range → −1/+1

读作：把实际值换算到 −1…+1，效应才能互相比大小。Read it as: Map actual settings onto −1 to +1 so effects can be compared on the same scale.

  • center 因子高、低水平的中点 midpoint between the factor's high and low settings
  • half-range 高、低水平间距的一半 half the distance between high and low settings
何时用： Use it when: 把不同单位、不同量级的因子统一映射到 −1/+1 编码空间。 You need factors with different units and scales mapped into the common −1/+1 design space.
⚠ ✗ 把实际单位系数直接当编码空间效应 → ✓ 先按中心值与半间距编码，再用 Effect = 2β 解读。 ✗ Read a natural-unit coefficient as a coded-space effect → ✓ Code by the center and half-range, then use Effect = 2β.
T2

常用 · 建模、缩减试验与识别曲率 Working Set · Model, Reduce Runs, and Detect Curvature

4 张常用卡 4 working cards
T2-5

效应 ↔ 回归系数 Effects and Regression Coefficients

F38 深潜 → F38 deep dive →
y = β0 + Σβixi + Σβijxixj + ε
coded space: βi = Effecti/2
  • βi 编码空间中的主效应系数 main-effect coefficient in coded space
  • βij 编码空间中的二因子交互系数 two-factor interaction coefficient in coded space
何时用： Use it when: 把 DOE 效应表写成可预测、可检验的回归模型。 You need to turn a DOE effect table into a predictive, testable regression model.
T2-6

对比与效应 Contrast and Effect

F38 深潜 → F38 deep dive →
Effect = Contrast / (n·2k−1)
Contrast = Σ(±yi)
  • ± 由目标主效应或交互列决定的符号 sign pattern from the target main-effect or interaction column
  • n·2k−1 每侧响应的总重复权重 total replicate weight on each side of the contrast
何时用： Use it when: 从标准符号列和各次响应手算主效应或交互效应。 You need to calculate a main or interaction effect directly from the signed design column.
T2-7

部分析因与分辨率 Fractional Factorials and Resolution

F41 深潜 → F41 deep dive →
2k−p design
R-III: main ↔ 2FI  ·  R-IV: main clear  ·  R-V: 2FI clear
  • p 相对全因子缩减的 2 的幂次 power-of-two reduction from the full factorial
  • R-III / IV / V R-III 主效应混交互、R-IV 主效应干净、R-V 二阶交互也干净 R-III aliases main effects with interactions; R-IV clears main effects; R-V also clears two-factor interactions
何时用： Use it when: 因子多、试验预算有限，并能接受由分辨率明确限定的混杂结构。 There are many factors, the run budget is tight, and the resolution's alias structure is acceptable.
T2-8

中心点弯曲检验 Center-Point Curvature Check

F39 深潜 → F39 deep dive →
compare ȳfactorial with ȳcenter
significant difference → curvature → RSM
  • ȳfactorial 因子角点响应的平均值 mean response across the factorial corner points
  • ȳcenter 中心点重复响应的平均值 mean response across replicated center points
何时用： Use it when: 检查两水平线性模型是否遗漏曲率；差异显著就升级 RSM。 You need to detect curvature missed by a two-level linear model and escalate to RSM when it is material.
T3

进阶 · 曲面优化、稳健设计与别名 Advanced · Surface Optimization, Robust Design, and Aliases

3 张进阶卡 3 advanced cards
T3-9

响应面二阶模型 Second-Order Response Surface Model

F39 深潜 → F39 deep dive →
y = β0 + Σβixi + Σβiixi2 + ΣΣβijxixj + ε
  • βii 因子 i 的二次曲率系数 quadratic curvature coefficient for factor i
  • βij 因子 i 与 j 的交互系数 interaction coefficient for factors i and j
何时用： Use it when: 已发现曲率，需要寻找响应最优区或稳健操作窗口。 Curvature is present and you need an optimum or a robust operating window.
T3-10

田口 S/N 比 Taguchi S/N Ratios

F40 深潜 → F40 deep dive →
larger: −10·log10[(1/n)Σ(1/yi2)]
smaller: −10·log10[(1/n)Σyi2]
nominal: 10·log10(ȳ2/s2)
  • larger / smaller 望大／望小质量特性 larger-is-better and smaller-is-better characteristics
  • nominal 望目质量特性 nominal-is-best characteristic
何时用： Use it when: 把均值表现与噪声波动合并为稳健设计的优化指标。 You need one robust-design objective that balances mean performance against noise variation.
T3-11

别名结构 Alias Structure

F41 深潜 → F41 deep dive →
I = ABC  ⇒  A·I = A·ABC = BC
A ↔ BC
  • I = ABC 定义该半分数设计的生成元 generator defining the half fraction
  • A ↔ BC A 与 BC 混杂，数据无法把两者分开 A is aliased with BC, so the data cannot separate them
何时用： Use it when: 在部分析因试验前写清哪些效应彼此混杂，并据此选择分辨率。 You need the confounding map before running a fraction and must choose a defensible resolution.
相关教学页 Related teaching pages
21 · DOE 投石机 21 · DOE catapult 51 · 2k 全因子 51 · 2k factorial 52 · 响应面 52 · Response surface 53 · 田口稳健设计 53 · Taguchi robust design
其他公式总表 Other formula sheets
FH01 流程能力 FH01 Capability FH02 假设检验 FH02 Hypothesis Testing FH03 方差分析 FH03 ANOVA FH04 回归 FH04 Regression FH05 DOE FH06 SPC FH07 MSA

DOE 实验设计公式总表