Design and analyze factorial experiments to identify significant process factors and optimize settings. Use this skill when the user needs to systematically test factor effects, optimize a manufacturing process, or determine which variables matter most — even if they say 'which factors affect quality', 'optimize process settings', or 'design an experiment'.
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.
Trigger conditions:
When NOT to use:
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.
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.
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.
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.
Return significant factors, effects, and optimal settings.
{
"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}
}
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.
| 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 |
references/fractional-tables.mdreferences/rsm.md