Factorial design matrix

1. Factorial Design in Experimental
Studies
• Matrix Construction and Analysis
• Presented by: Your Name
• Date: 26 September 2025

2. Introduction to Factorial Design
• - Factorial design studies the effect of two or
more factors simultaneously.
• - Useful to analyze main effects and
interaction effects.
• - Applications: Materials Science, Engineering,
Chemistry, Biology.

3. Key Concepts
• - Factor: Independent variable being studied
• - Level: Value/settings of a factor (e.g., Low,
High)
• - Main Effect: Effect of a single factor
• - Interaction Effect: Combined effect of two or
more factors

4. Factorial Design Notation
• - 2^k design: k factors, 2 levels each (-1, +1)
• - 3^k design: k factors, 3 levels (-1, 0, +1)
• - Number of runs: levels^factors

5. Constructing a Factorial Design
Matrix
• 1. Determine number of factors (k) and levels.
• 2. Calculate total runs = levels^k.
• 3. Assign levels for each factor systematically:
• - Alternate first factor every half runs
• - Second factor every quarter runs, etc.
• 4. Include interaction columns by multiplying
factor columns.

6. Example: 2^2 Factorial Design
• Factors: A = Temperature, B = Pressure
• | Run | A | B |
• |-----|---|---|
• | 1 | -1| -1|
• | 2 | +1| -1|
• | 3 | -1| +1|
• | 4 | +1| +1|
• Interaction AB = A*B
• | Run | A | B | AB |

7. Example: 2^3 Factorial Design
• Factors: A, B, C
• | Run | A | B | C | AB | AC | BC | ABC |
• |-----|---|---|---|----|----|----|-----|
• | 1 | -1| -1| -1| +1 | +1 | +1 | -1 |
• | 2 | +1| -1| -1| -1 | -1 | +1 | +1 |
• | 3 | -1| +1| -1| -1 | +1 | -1 | +1 |
• | 4 | +1| +1| -1| +1 | -1 | -1 | -1 |
• | 5 | -1| -1| +1| +1 | -1 | -1 | +1 |
• | 6 | +1| -1| +1| -1 | +1 | -1 | -1 |

8. Steps to Compute Matrix
Systematically
• 1. Determine number of runs
• 2. Assign coded levels (-1, +1, 0)
• 3. Fill the matrix using alternating patterns
• 4. Compute interaction columns by
multiplying factors

9. Tips
• - For >3 factors, use software (Excel, Python,
Minitab)
• - Fractional factorial designs reduce the
number of experiments
• - Always check main and interaction effects

10. Summary
• - Factorial design helps study multiple factors
simultaneously
• - Design matrix lists all combinations of factor
levels
• - Interaction terms are obtained by multiplying
factors
• - Systematic construction ensures no runs are
missed