DOE / optimal design

Optimal and mixture designs

A factorial design needs a neat box and a clean fraction. Real experiments often have an irregular region, a fixed budget, or components that must sum to one. This searches a candidate set for the runs that maximise |X′X| — the same algorithm that produces a Scheffé mixture design once you constrain the region to the simplex.

RegionBoxModel6 termsSweeps3

D-efficiency

0.462

9 runs for 6 parameters

log |X′X|

8.55

Larger is better; the search maximises it

Search stability

Stable

6 random starts · spread of log|X′X|

Peak variance

0.81

Scaled by σ²: prediction variance at the worst candidate

Design matrix

Coded levels from −1 to +1 for each factor.

9 runs

Design matrix
Run	x1	x2	Variance
1	-1.00	-1.00	0.81
2	0.00	-1.00	0.56
3	1.00	1.00	0.81
4	-1.00	1.00	0.81
5	1.00	-1.00	0.81
6	0.00	0.00	0.56
7	1.00	0.00	0.56
8	0.00	1.00	0.56
9	-1.00	0.00	0.56

Variance is the scaled prediction variance x′(X′X)⁻¹x at that run.

Prediction variance across the region

Rows run from x₂ = +1 down to −1; columns from x₁ = −1 to +1. Pale cells are where the design predicts badly.

Worst-case prediction variance against the run budget

The first run that makes the model estimable sets the floor; each extra run cuts the worst prediction variance.

Canonical equations

The exact relationships implemented by this engine.

maximise |X′X| · D-optimality: the smallest joint confidence region.

|X′X|^{1/p} / n · One for an orthogonal design with unit columns.

x′(X′X)⁻¹x · Scaled prediction variance at a point.

Σx = 1, no intercept · Scheffé form; squares are collinear.

Show model notation and computationOpen the symbolic form, the variable definitions, and the step-by-step derivation when you need the detail.

A clear path from inputs to results.

Step-by-step methodWorked exampleAssumptions explained

The formula

\max |X'X| \;;\; D\text{-eff} = \frac{|X'X|^{1/p}}{n} \;;\; \mathrm{Var}(\hat y) = \sigma^2 x'(X'X)^{-1}x

Reference: Coordinate exchange (Meyer & Nachtsheim, 1995); Scheffé mixture models (Scheffé, 1958).

Worked example

Two factors, full quadratic, nine runs from 49 candidates

Inputs

Two factors, box region, seven-level grid: 49 candidate points.
Full quadratic model: intercept, two linear, one interaction, two squares.
Nine runs, search seed 20261.

Calculation path

1
The search starts from nine random candidates and exchanges each row for the best alternative.
2
It converges in three sweeps to the eight points at the corners and axis midpoints plus the centre.
3
log |X′X| is 8.55 and the D-efficiency is 0.462 on the raw model scale.
4
The prediction variance peaks at the corners, which is exactly where the design has no runs between them.

Result

Nine runs · log|X′X| 8.55 · the factorial-plus-star pattern a CCD would give.

Interpreting the result

A D-efficiency below one does not mean a bad design: it means the model columns are not orthogonal on their raw scale, which the squared terms never are. What matters is that the value is stable when you change the seed. If it moves, the search is finding different local optima and you should not trust a single run of it.

Assumptions & limitations

Coordinate exchange finds a local optimum, not a proven global one.

The candidate set defines the region; the design can only choose from it.

Mixtures drop the intercept and the squared terms because Σx = 1 makes them collinear.

Prediction variance is scaled by σ² and reported at the candidate points.

Frequently asked questions

What is D-optimality?

It maximises the determinant of X′X, which minimises the volume of the joint confidence region for the coefficients. In practice it means the design pins down every term with as little confounding as possible.

When should I use a mixture design?

Whenever the factors are proportions that must sum to one: blends, formulations, portfolio weights. The Scheffé form drops the intercept because it is redundant under the constraint.

Why is the efficiency 0.46 rather than 1?

Because the model includes squared terms whose columns are not centred. Efficiency is only exactly one when X′X is a multiple of the identity, which happens for an orthogonal design with unit-scaled columns.

Why change the seed?

Coordinate exchange is a local search. A different start can land in a different optimum; if the efficiency barely moves, the design is stable and you can use it.