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.