Design of Experiments (DOE)
Overview
DOE systematically varies process factors to identify their effects on responses. Full factorial tests all combinations; fractional factorial tests a strategic subset. Identifies main effects and interactions. More efficient than one-factor-at-a-time (OFAT) which misses interactions. Uses ANOVA for analysis.
When to Use
Trigger conditions:
- Identifying which process factors significantly affect quality/yield
- Optimizing process settings for target performance
- Screening many factors to find the vital few
When NOT to use:
- When the process is not stable (stabilize with SPC first)
- For observational data with no ability to manipulate factors
Algorithm
IRON LAW: One-Factor-At-A-Time (OFAT) MISSES Interactions
Changing one factor while holding others fixed cannot detect
interactions (where the effect of A depends on the level of B).
Full factorial or fractional factorial designs test ALL main effects
AND interactions in fewer runs than OFAT. A 2³ factorial (8 runs)
gives more information than 6 OFAT runs at lower cost.Phase 1: Input Validation
Define: response variable(s), factors (2-7 practical), levels per factor (usually 2 for screening, 3 for optimization), constraints, noise factors. Gate: Factors and levels defined, practical to run all experimental conditions.
Phase 2: Core Algorithm
Screening (many factors): 2^(k-p) fractional factorial. Choose resolution III+ (main effects not confounded with each other).
Optimization (few factors): 2^k full factorial or central composite design (CCD) for response surface.
- Generate design matrix (run order, factor level assignments)
- Randomize run order (critical for validity)
- Execute experiments, record responses
- Analyze: ANOVA for factor significance, effect plots, interaction plots
- If optimizing: fit response surface model, find optimal settings
Phase 3: Verification
Check: R² of model is adequate, residuals are normally distributed and random. Confirmation runs at predicted optimal settings match prediction. Gate: Model is significant, residuals OK, confirmation runs pass.
Phase 4: Output
Return significant factors, effects, and optimal settings.
Output Format
{
"significant_factors": [{"factor": "temperature", "effect": 12.5, "p_value": 0.001}, {"factor": "pressure", "effect": -8.2, "p_value": 0.008}],
"interactions": [{"factors": "temperature×time", "effect": 5.1, "p_value": 0.03}],
"optimal": {"temperature": 180, "pressure": 50, "time": 30, "predicted_response": 95.2},
"metadata": {"design": "2^3_full_factorial", "runs": 8, "replicates": 2, "r_squared": 0.94}
}Examples
Sample I/O
Input: 3 factors (temperature, pressure, time), each at 2 levels, response = yield Expected: 2³ = 8 runs + replicates. ANOVA reveals temperature and temp×pressure interaction are significant.
Edge Cases
| Input | Expected | Why |
|---|---|---|
| 7+ factors | Fractional factorial | Full factorial too expensive (2⁷=128 runs) |
| Factors with constraints | Constrained design | Some factor combinations may be physically impossible |
| Non-linear response | CCD or Box-Behnken | 2-level designs only fit linear models |
Gotchas
- Randomization is critical: Without randomization, time-varying factors (operator fatigue, ambient temperature) confound results. ALWAYS randomize run order.
- Replication vs repetition: Replication (re-setup and re-run) estimates error. Repetition (multiple measurements from one run) does not. Include true replicates.
- Alias structure: Fractional factorials confound some effects. Know which effects are aliased (confounded) before interpreting results.
- Center points: Adding center points to a 2-level design detects curvature (non-linearity) at minimal cost. Always include 3-5 center points.
- Practical significance vs statistical significance: A factor can be statistically significant (p<0.05) but practically unimportant (tiny effect). Focus on effect SIZE, not just p-values.
References
- For fractional factorial design tables, see
references/fractional-tables.md - For response surface methodology (RSM), see
references/rsm.md