Design of Experiments

CCD with Categorical Factors (Cartesian × Continuous CCD)

Extends the classical central composite design to plans with mixed continuous/categorical factors — every categorical level combination gets a full CCD over the continuous factors.

Statusvalidated

Version1.0.0

Minitab equivalentStat > DOE > Response Surface > Create Response Surface Design (Categorical-factor option)

Description

The classical CCD assumes every factor is continuous and can take meaningful axial points (±α). For a categorical factor like Tool = {Steel, HSS, Carbide}, axial points are meaningless (there's no ‘Steel × 1.4’ value). The extension documented here cuts the problem structurally: the CCD is generated over the continuous factors only, and the result is Cartesian-multiplied by all categorical level combinations. Every categorical cell thus gets its own full CCD — corner points, axial points, and center points. This is the clean math for quadratic modelling in mixed-factor RSM.

Formulas

Run count

N = N_{CCD} (k_{cont}) \cdot j \in cat \prod ∣ L_{j} ∣

Run count = (CCD over continuous) × (product of all categorical level counts)

Cartesian construction

D = {[r_{cont}; c] : r_{cont} \in CCD (k_{cont}), c \in \bigtimes_{j} L_{j}}

Design matrix as union of all (r_cont, c) pairs

Assumptions

At least 2 continuous factors (CCD needs ≥ 2 axes)
Categorical factors have finite level sets

Limitations

Run count grows multiplicatively in categorical levels — 2 ternary categories × CCD(3) = 9 × 17 = 153 runs
Estimates only categorical main effects and categorical × continuous interactions — not categorical × categorical

References

Box, G.E.P. & Wilson, K.B. (1951). On the experimental attainment of optimum conditions, JRSS B 13:1-45
Myers, R.H., Montgomery, D.C., Anderson-Cook, C.M. (2016). Response Surface Methodology, 4th Ed., Wiley — Chapter 7.6 (Designs with Categorical Factors)