Full Factorial Design
A full factorial design is a DOE (Design of Experiments) approach that tests every possible combination of factor levels, providing complete information about all main effects and all interaction effects — with no confounding.
Structure
- For $k$ factors, each at $n$ levels, a full factorial requires $n^k$ experimental runs.
- Example: 3 factors at 2 levels each ($2^3$) = 8 runs. Each factor is tested at its low and high level in all possible combinations with the other factors.
- Example: 4 factors at 2 levels ($2^4$) = 16 runs.
- Example: 3 factors at 3 levels ($3^3$) = 27 runs.
The $2^k$ Full Factorial
The most common type in semiconductor manufacturing — each factor has only 2 levels (low/−1 and high/+1):
| Run | Factor A | Factor B | Factor C |
|---|---|---|---|
| 1 | − | − | − |
| 2 | + | − | − |
| 3 | − | + | − |
| 4 | + | + | − |
| 5 | − | − | + |
| 6 | + | − | + |
| 7 | − | + | + |
| 8 | + | + | + |
What Full Factorial Reveals
- All Main Effects: The individual impact of each factor.
- All 2-Factor Interactions: How pairs of factors interact (A×B, A×C, B×C).
- All Higher-Order Interactions: 3-factor (A×B×C), 4-factor, etc. Usually negligible in practice.
- No Confounding: Every effect is estimated independently — no ambiguity about which factor or interaction caused an observed change.
Advantages
- Complete Information: No confounding, no aliasing — all effects fully resolved.
- Model Fitting: Enables fitting a complete regression model relating inputs to outputs.
- Inference Quality: The highest-quality DOE for understanding factor effects.
Disadvantages
- Exponential Growth: The number of runs grows rapidly: $2^5$ = 32, $2^7$ = 128, $2^{10}$ = 1,024. Beyond 5–6 factors, full factorials become impractical.
- Wafer Cost: Each run in semiconductor DOE typically consumes one or more wafers — expensive for large designs.
- Time: Processing and measuring many wafers takes significant fab time.
When to Use Full Factorial
- Few Factors (2–5): The number of runs is manageable.
- Interactions Expected: When you suspect significant interactions between factors.
- Final Optimization: For the final, detailed study after a screening DOE has identified the important factors.
Full factorial is the gold standard of DOE designs — it provides complete, unaliased information, and should be used whenever the number of factors allows a practical run count.