Design of Experiments

D-Optimal Design with Mixed Level Counts

D-optimal search that constrains categorical factors to their discrete levels — continuous factors keep the 5-level grid for quadratic models.

Statusvalidated

Version1.0.0

Minitab equivalentStat > DOE > Response Surface > Create Response Surface Design (D-Optimal with multi-level factors)

Description

Before Phase 4, the coordinate-exchange search (for quadratic models) used a uniform 5-level grid [-1, -0.5, 0, 0.5, 1] for ALL factors. For categorical factors that produced fractional values (-0.5, 0.5) which correspond to no real level — mathematically meaningless. The fix: per-factor candidate sets chosen by factor kind — continuous factors keep the 5-level grid, categorical factors are restricted to their evenly-spaced discrete codings (-1, …, +1 according to their level count). The linear path via point exchange was already correct (`fullFactorialGeneral` as the candidate source).

Formulas

D-criterion

D = det (X^{'} X)

Determinant of the information matrix — maximised to maximise coefficient precision

Per-factor candidate set

L_{j} = {{- 1, - \frac{1}{2}, 0, \frac{1}{2}, 1} {- 1 + \frac{2 k}{n _{j} - 1} : k = 0, \dots, n_{j} - 1} j continuous, quadratic j categorical

Per factor j either the 5-level grid or the discrete level codings, depending on the kind

Assumptions

Model is linear (point exchange) or quadratic (coordinate exchange)
Run count ≥ number of model parameters

Limitations

D-optimal solution is local — multi-start (default 10) mitigates the risk of getting stuck in a local optimum
Coordinate exchange is O(maxIter · nRuns · k · |L|) — compute-intensive for very large designs

References

Atkinson, A.C., Donev, A.N. & Tobias, R.D. (2007). Optimum Experimental Designs, with SAS, Oxford University Press
Myers, R.H., Montgomery, D.C., Anderson-Cook, C.M. (2016). Response Surface Methodology, 4th Ed., Wiley — Chapter 8 (Optimum Designs)