Concept

Box behnken design — where it appears

A three-level response-surface design whose runs sit at the midpoints of a cube's edges and at its centre, never at a corner. It is built for the case where no factor combination may put every factor at an extreme at once, and it pays for that by predicting the corner it never visits much less precisely.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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 one lost run costs a 16-run factorial fitting 11 coefficients. Every run is worth the same: dropping any one multiplies every coefficient's variance by 1.2000, which is 1 + 1/(N − p) with N = 16 and p = 11, and gives every pair of coefficients a correlation of 0.1667 where the complete design had exactly zero.

The run that did not happen

Lose one run from any orthogonal design and every coefficient's variance is multiplied by exactly 1 + 1/(N − p), and every pair of coefficients acquires a correlation of exactly 1/(N − p + 1) where there was none. The price is set by the design's spare capacity and by nothing else, and a saturated design cannot survive it at all.

design · Factorial

Named alongside it

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

Central composite design Experimental design Factorial design Study design Centre point Correlation Curvature Degrees of freedom Fractional factorial Least squares Orthogonality Prediction variance

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