Design of Experiments

Full Factorial Design with Mixed Level Counts

Generates the Cartesian product of all level combinations when factors have different level counts (e.g. 2 × 3 × 4).

Statusvalidated

Version1.0.0

Minitab equivalentStat > DOE > Factorial > Create Factorial Design (General full factorial option)

Description

Whereas the classic 2^k full factorial assumes every factor has exactly two levels, the general full factorial accommodates arbitrary level counts per factor. This is required as soon as a design mixes level counts (e.g. a 2-level categorical factor like Tool A/B together with a 3-level continuous factor like Low/Mid/High). The run count is the product of all level counts — grows quickly, so the user should consider it before selecting a full factorial.

Formulas

Run count

N = i = 1 \prod k n_{i}

Run count is the product of all factor level counts

Cartesian product

D = {0, \dots, n_{1} - 1} \times {0, \dots, n_{2} - 1} \times \dots \times {0, \dots, n_{k} - 1}

Design space as the Cartesian product of 0-indexed level sets

Coding

coded (j, l) = - 1 + \frac{2 l}{n _{j} - 1}

Level l (0-indexed) of a factor with n_j levels maps to evenly-spaced [-1, +1]

Assumptions

Factors are independent (no constraints between them)
Level count per factor ≥ 2

Limitations

Run count explodes: 4 factors at 5 levels → 625 runs. For large designs prefer fractional / D-optimal

References

Montgomery, D.C. (2017). Design and Analysis of Experiments, 9th Ed., Wiley — Chapter 5 (General Factorial Designs)