The design that refuses the corners

Every arrangement in this field so far is built out of corners. The factorial is the corners, a fraction is half of them, and the central composite is the corners with axial runs bolted on.

A corner is the combination where every factor is at its extreme at once, and sometimes that is not a setting. Two reagents each at their maximum concentration in a mixture that has to sum to one. A temperature at the furnace’s limit with a hold time at the shift’s limit. A dose at the maximum tolerated combined with a duration at the maximum permitted. In each case the corner is not expensive, it is unavailable.

What it is made of #

The construction is one sentence. For each pair of factors, run the four combinations of that pair at ±1 with every other factor held at zero; then add centre runs.

At three factors that is three pairs times four combinations — twelve edge runs — plus the centres. The design has three levels in every factor, which is what a second-order model needs, and it reaches those levels without ever setting two factors to an extreme simultaneously.

The count grows as $2k(k-1)$ rather than as $2^{k}$, so at four factors it is twenty-four edge runs and at five it is forty. Against the central composite’s $2^{k} + 2k$ that is a win at five factors and above and a wash below:

So the run count is not the reason to choose it. At three factors it saves two runs out of seventeen, and at four it saves none. The reason is the corner, and the price of the corner is what the rest of this essay measures.

Why the run counts converge #

The table’s middle row is worth one paragraph, because two designs built on completely different principles arriving at twenty-seven runs at four factors looks like a coincidence and is arithmetic.

The Box–Behnken’s edge runs go as $2k(k-1)$ — four runs for each of the $k(k-1)/2$ pairs. The central composite’s non-centre runs go as $2^{k} + 2k$. At three factors those are 12 and 14; at four, 24 and 24; at five, 40 and 42; at six, 60 and 76.

So the composite design wins below four factors, they tie at four, and the Box–Behnken pulls away above — because a polynomial in $k$ eventually beats an exponential in it, and four is where the two cross. By six factors the gap is sixteen runs and by seven it is fifty-eight.

That is the honest scaling argument for the design, and it is an argument about large experiments. At three and four factors, which is where most response-surface work lives, the run count is not a reason to prefer either.

The price, at the place it does not go #

A design’s precision at a setting is a matrix computation on the design alone: no response, no noise, no seed. So the price of not visiting the corner is a number available before anything is measured.

The reading is direct. A prediction at the corner from the Box–Behnken design is as precise as a prediction from the central composite would be on 54% of its runs. An experimenter who runs the fifteen-run design and then wants a number at the corner is getting, at that setting, what nine runs of the other design would have given.

That is not a small penalty and it is not a surprising one. The design has no information about a place it never went, and the second-order model is extrapolating there from a surface fitted at the edges.

Which directions are dear, and which are cheap #

The corner is the extreme case and the whole profile is worth having, because it says the penalty is about direction rather than about distance.

The three ratios are 1.31, 1.74 and 1.84 going from face to edge to corner. The middle one is the informative one: the edge midpoints are the Box–Behnken’s own runs, and it is still 74% worse there than the design that also runs the corners.

That says the penalty is not localised at the corner. The central composite’s corner runs improve its prediction everywhere in the region, because a least-squares fit pools information across the whole design rather than answering each setting from the runs nearest it. Removing eight runs from the extremes of the space degrades the whole surface, and the degradation is largest where the removed runs were.

The corner is where every term is extreme at once #

Why the corner in particular is expensive is worth a paragraph, because it is the same reason the corner is often unavailable and the coincidence is not one.

Under the second-order model the prediction at a point is a linear combination of ten coefficients with weights $[1, x_1, x_2, x_3, x_1x_2, x_1x_3, x_2x_3, x_1^2, x_2^2, x_3^2]$. At the centre that vector is $[1,0,\dots,0]$ and the prediction loads on the intercept alone. At a corner every entry is ±1 — the main effects, the interactions and the squares are all at their extremes together — and the prediction loads on all ten coefficients at once.

So a prediction at a corner adds ten estimated quantities, and its variance is a sum of ten variances and ninety covariances. It is the single most demanding prediction the model can be asked for, at every design, and the optimal-design field found the same thing from the other direction: the setting where a catalogue design’s prediction variance attains its maximum is a corner, for every catalogue design it scored.

That is also why the corner is often the setting that cannot be run. A corner is where the process is under every stress at once, which is what makes it both physically impossible and statistically expensive, and the two are the same fact about the geometry seen from two sides.

What it buys back #

The comparison so far is one-sided, and the design would not be in every response-surface catalogue if it were only worse.

Three things are genuinely in its favour, and the first is the one that decides.

It can be run. Where the corner is unavailable, the central composite is not a design with a worse number attached — it is not a design at all. The relevant comparison is then against a face-centred composite, which pulls the axial runs in to ±1 and is what the essay on three levels and a ring measured as not rotatable at any number of centre runs: its prediction variance around a unit circle varies by 59.9% between its best and worst directions.

It uses three levels, not five. The central composite at α = 1.682 asks each factor to take the values −1.682, −1, 0, 1 and 1.682. The Box–Behnken asks for −1, 0 and 1. For a factor that is a categorical setting with three achievable states — three suppliers, three machine speeds, three protocol variants — the composite design cannot be built and this one can.

Its runs are all at the same distance from the centre. Every edge midpoint is at radius $\sqrt2 = 1.4142$, and that is the only non-zero radius in the design: the twelve non-centre runs lie exactly on one sphere. The central composite has two — its corners at $\sqrt3 = 1.7321$ and its axial runs at $2^{3/4} = 1.6818$, three per cent apart and not equal.

So on the one geometric property rotatability is a statement about, the Box–Behnken is the tidier arrangement at three factors and the central composite is not. That is the reverse of the ordering everything else in this essay produces, and it is the reason the two designs are both in the catalogues rather than one having replaced the other.

The centre runs decide more here #

The number of centre runs is a free parameter in both designs, and in the Box–Behnken it is doing more work, because the design has fewer distinct distances from the centre to interpolate between.

With one centre run the Box–Behnken predicts its own middle at 13.000 and its worst point at 20.313 — a design that is poor at the place an optimum is most likely to be. Three centre runs bring the middle to 5.000, five bring it to 3.400, and the worst point drifts up to 23.163 because the scaling is by the total number of runs and the extra centres do nothing for the outside.

The central composite shows the same trade from a worse starting point: at one centre run its worst prediction is 14.825 and it is at the centre, not the corner. That is the same defect the two-factor version of this design has at one centre run, where it predicts its own middle at 9.00 against 4.22 at radius one.

So the rule that comes out of both is the same and it is worth stating once for the field: a second-order design with too few centre runs is worst at its own middle, which is the one place every such design is built to look at.

Where this leaves a reader choosing between the two #

The arithmetic supports a short rule, and the short rule is not “the fewer runs the better”.

If the corner cannot be run, there is no comparison to make. The Box–Behnken is one of two designs available and the other is the face-centred composite, which is not rotatable at any number of centre runs and varies by 59.9% around a unit circle. Choose between those two, on rotatability, and the Box–Behnken wins.

If the corner can be run and predictions near it matter, run it. Two extra runs out of seventeen buy a factor of 1.84 at the corner and 1.74 at the edges, which is the best-value pair of runs anywhere in this field.

If the corner can be run and the optimum is expected near the middle, the choice is close and turns on the centre runs rather than on the geometry. At three centre runs the two designs predict the centre at 5.000 and 5.645, and the fifteen-run design is ahead — by 11%, on two fewer runs, at the one setting a response-surface experiment is most likely to be read at.

That last case is the common one and it is the reason the two designs coexist in the catalogues. The honest summary is that they differ where they differ — at the extremes of the region — and agree in the middle, and an experimenter who knows which of the two they will be reading has already made the choice.

A design is a set of places, and a model is what joins them #

One general thing comes out of all of this and it is worth extracting, because it is the reason the corner penalty is larger than an intuition about interpolation would suggest.

A design does not predict a setting from the runs near it. Least squares fits one surface to every run at once, so the variance at any point is a quadratic form in the whole inverse information matrix, and removing runs from one part of the region changes the prediction everywhere. The Box–Behnken’s penalty at the face midpoints — 1.31, at settings it has runs within one factor’s extreme of — is the visible form of that.

The corollary runs the other way and is more useful. Runs placed at the extremes of the region improve predictions in the middle, which is why a design with no extreme runs at all — a cluster near the centre — is a poor design even for predicting the centre. That is the arithmetic behind the optimal-design field’s finding that the certified design puts its weight at the boundary and almost none in the interior.

So the choice between these two designs is not a choice about where predictions will be wanted. It is a choice about which extremes are reachable, and the reachable extremes decide the precision everywhere.

What the catalogues say, and what they leave out #

Both designs are in every response-surface catalogue, usually with a sentence saying the Box–Behnken “avoids the extreme corners” and the central composite “requires runs outside the factor ranges”. Both sentences are true and neither is a quantity.

The quantity the catalogues leave out is the one this essay is about: how much worse the prediction is at the settings the design does not visit, and how far that penalty reaches back into the region. At three factors it is 1.84 at the corner, 1.74 at the edges the design does run, and 1.31 at the face midpoints — so the answer to “how far does it reach” is everywhere, at a magnitude that falls off slowly.

That is a general hazard with design catalogues and it is not specific to these two entries. A catalogue lists arrangements and their construction rules; the property an experimenter needs is the prediction variance over the region they care about, which is a function of the arrangement and the region and the model. The same complaint arrives from the optimal-design side, where the rotatable composite design — admired in this field for a property it genuinely has on a circular region — turns out to be the worst design in the table when it is scored on a square one, by a factor of nearly five.

The repair is the same in both places and it is cheap: compute the prediction variance for the design in hand, over the region in hand, before running it. It is a matrix inverse and a walk over a grid, and it takes less time than reading the catalogue entry.

What is claimed here, and what is not #

The claim is what refusing the corners costs: that the Box–Behnken design in three factors uses fifteen runs against the central composite’s seventeen and predicts the corner 1.84 times worse, the edge midpoint 1.74 times worse and the face midpoint 1.31 times worse; that the run counts converge at four factors and cross at five; and that too few centre runs make either design worst at its own centre.

Every prediction variance here is $N\mathbf{x}'(\mathbf{X}'\mathbf{X})^{-1}\mathbf{x}$ computed from the design’s own coordinates. There is no response, no noise and no simulation anywhere in the essay, which is what makes the numbers properties of the arrangement rather than estimates of them.

What stays out: the four- and five-factor comparisons beyond their run counts, where the same profile would have to be walked in four and five dimensions and the grid gets expensive; designs for a mixture, where the factors sum to a constant and the whole geometry is a simplex rather than a cube — that is the setting where the corner is most often unavailable and it deserves its own arithmetic; and D-efficiency, which the optimal-design field computes for arrangements like these and which ranks them on a different criterion from the one used here.

The grid used for the worst-prediction figures is nine points a side, which is stated because it is an instrument rather than an exact maximisation: the maximum of a quadratic form over a cube sits at a vertex only when the form is convex in every direction, and these designs are not.

Still open: what one missing run does #

This essay compares two complete designs, and a design is a plan. What arrives is a dataset, and datasets lose runs — a sample is contaminated, an instrument fails, a subject withdraws, a batch is scrapped.

The question nobody in this field has asked is what that costs, and the answer turns out to be exact rather than a matter of degree. For any orthogonal design, losing one run multiplies every coefficient’s variance by $1 + 1/(N-p)$ and gives every pair of coefficients a correlation of $1/(N-p+1)$ where the complete design had exactly zero — so the price is set by the design’s spare capacity and by nothing else, and a saturated design cannot survive it at all. That is what one lost run costs.

The check, and the refusal #

Two claims are gated and they are of different kinds. That the Box–Behnken design puts zero runs at any corner, which is a count and is the design’s definition rather than a property of it. And that both designs can predict everywhere on the line being walked, which is what makes the comparison about precision rather than about estimability — a design that could not fit the model at all would produce no curve, and a figure with one curve missing would have been read as a scale problem.

The refusal is the central composite’s own count: it is required to run eight corners, exactly. If that check ever passed for the Box–Behnken or failed for the composite, the two designs would not be the two designs the essay compares, and every ratio in it would be a ratio between something else. A check on a design’s definition sounds like a check that cannot fail — which is why it is written as an equality on a count rather than as an inspection, and why the two halves are required in opposite directions.