Concept

Centre-point — where it appears

A run at the middle of every factor's range, repeated, which supplies an estimate of pure error and a test for curvature. Without it a fitted plane cannot be distinguished from a curved surface sampled only at its corners.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

Four runs, and the term they cannot reach. Every run sits at a corner, so x₁² and x₂² are 1 at every run and both columns are copies of the intercept. The normal matrix is singular: the design has no information about curvature at all, and no analysis can recover it.

The design that cannot see a curve

A two-level factorial has every run at a corner, where every squared term equals one — so the column that would estimate curvature is a copy of the intercept, and the design has no information about it at all. A few runs at the centre buy one number back, and only one.

surface · Factorial

A central composite design, 13 runs. Adding 4 axial runs at ±√2 gives every factor three levels, which is the least that can estimate a squared term. The normal matrix now inverts, so each βᵢᵢ has an estimate of its own — and at exactly this axial distance the design is rotatable, which the next figure measures.

Three levels, and the ring where the design says the same thing

A central composite design puts its axial runs at ±α, and α is not a matter of taste. At F to the quarter the prediction variance depends only on how far a point is from the centre and not at all on which direction it lies in — a property with no simulation in it, exact or absent.

surface · Factorial

The Box–Behnken design in three factors: 15 runs, none at a corner. Twelve runs at the midpoints of the cube's edges and 3 at its centre. Every run holds one factor at zero, so no run puts all three factors at an extreme — which is what makes it runnable where a corner is not. The three panels are the design's coordinate projections, with repeated positions marked.

The design that refuses the corners

Box–Behnken runs three factors in fifteen runs and puts none of them at a corner, which is what makes it usable where a corner cannot be run. It predicts the corner 1.84 times worse than the seventeen-run design that goes there, and 1.31 times worse at the middle of a face, and all three numbers are matrix computations with no simulation in them.

design · Factorial

What a stated power leaves of the main effects. The main-effect efficiency of the best design reaching a power of 0.8 at level 0.05, as N|δ|² rises through 50, 60, 70, 80, 100, 130, 170, 220, 300, 400, 600, 1000. a saddle, two-degree test, the compound path: out of reach, out of reach, 0.806, 0.841, 0.881, 0.914, 0.937, 0.952, 0.966, 0.975, 0.984, 0.990. a bowl, two-degree test, the compound path: 0.782, 0.823, 0.851, 0.871, 0.898, 0.923, 0.941, 0.955, 0.967, 0.976, 0.984, 0.990. a bowl, its own one-degree contrast, the compound path: 0.827, 0.859, 0.880, 0.896, 0.918, 0.937, 0.952, 0.964, 0.973, 0.980, 0.987, 0.992. a bowl, a factorial with centre points: 0.914, 0.930, 0.940, 0.948, 0.959, 0.969, 0.976, 0.982, 0.987, 0.990, 0.993, 0.996.

A design chosen for a test

A screening experiment that wants its main effects and a check on curvature has been told to trade them with a weight nobody can state. It can state a power. Read against a power of 80%, the frontier collapses to one curve in runs times the squared size of the curvature, and a large enough trial pays almost nothing. But the curve belongs to a test. A factorial with a few centre points beats the entire frontier on a bowl, spending 4% of its main-effect information where the frontier spends 10%, and against a saddle of the same size it has exactly its 5% level.

criteria · Criterion

Named alongside it

The objects these essays reach for when they reach for this one.

Curvature Experimental design Factorial design Central composite design Prediction variance Rotatability Box behnken design Closed form D-optimality Degrees of freedom Design criterion Design measure

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