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

A design with two levels per factor cannot estimate a squared term, so the fix is a third level, and the question is where to put it. That sounds like a matter of taste. It is not: there is exactly one axial distance at which the finished design has a property nothing else about it can supply, the property is checkable to machine precision, and a design either has it or does not.

What the design is made of #

Three groups of runs, each doing a different job.

The factorial points, at every combination of ±1, estimate the main effects and the interaction and carry all of the information about them. They are the design from the previous essay, unchanged.

The axial points, at (±α, 0) and (0, ±α), give each factor a third level. They are the only runs that let the two squared coefficients be told apart: at a corner both xᵢ² are 1, and at an axial run one of them is α² and the other is 0.

The centre points, replicated, estimate σ without reference to any model and supply the curvature contrast. They are also the runs that decide how well the design predicts its own middle, which turns out to be the second half of this essay.

Thirteen runs in two factors, twenty in three, thirty-one in four — the design grows as $2^k + 2k + n_c$, and the factorial part can be a fraction, which is what keeps it affordable past four factors.

The one property α decides #

Here is the quantity to look at, and it is one this site has not used before.

For any design and any point x in the region, least squares gives a fitted value ŷ(x) whose variance is σ²·x′(X′X)⁻¹x, where x is the model vector [1, x₁, x₂, x₁x₂, x₁², x₂²]. Multiply by N and divide by σ² and the result is the scaled prediction variance — the precision of a prediction, per run, in units that let designs of different sizes be compared.

It is a function of the design alone. No response, no noise, no seed: the arrangement of the runs determines how well the fitted surface is known at every point in the region, before anything is measured. That makes it exactly the kind of quantity this site likes — a property that can be checked rather than estimated — and the check is a walk around a circle.

At α = √2 the prediction variance around that circle is constant to 1.1 × 10⁻¹⁵ of its own value, which is machine precision. At α = 1 it varies by 59.9% and at α = 2 by 37.6%. A design that is not rotatable predicts a point towards a corner differently from a point the same distance away along an axis, and it does so for no reason anyone chose — the difference is a side effect of where the axial runs were put.

The value that makes it flat is F^(1/4), the fourth root of the number of factorial points: √2 at two factors, 4^(1/4)·… — for three factors with a full factorial, 8^(1/4) = 1.682, and for four, 2. The derivation is a statement about the design’s moments through fourth order matching those of a sphere, and the check here does not use it: the ring is walked and the numbers are compared.

Two runs a parameter, at every number of factors #

The growth rule $2^k + 2k + n_c$ is usually quoted against nothing, and the comparison that makes it meaningful is the number of parameters it has to estimate.

A second-order model in k factors has $(k+1)(k+2)/2$ coefficients: 6 at two factors, 10 at three, 15 at four. The designs are 13, 20 and 31 runs — so the centre-point counts implied are five, six and seven, and the ratio of runs to parameters is

2.17, 2.00 and 2.07.

Two runs a parameter, held to within a tenth across a design that has more than doubled in size. That is the property that makes the arrangement usable rather than the geometry, and it is what the fractional factorial exists to preserve: a full factorial at five factors would be 32 runs against 21 parameters before any axial or centre runs, and the ratio only stays near two because the corners can be halved.

The rotatable design is very nearly a sphere, and exactly one at two factors and four #

Rotatability fixes the axial distance at $\alpha = 2^{k/4}$, which is 1.414, 1.682 and 2.000 at two, three and four factors. Set that beside where the factorial points sit: a corner at $(\pm1,\ldots)$ is at distance $\sqrt{k}$ from the centre, which is 1.414, 1.732 and 2.000.

At two factors the two coincide exactly — all eight non-centre runs lie on one circle of radius $\sqrt2$. At four factors they coincide exactly again, at radius 2. At three the axial runs sit three per cent inside the corners, and at five factors they would sit outside them.

The coincidence is not a numerical accident. $2^{k/4} = \sqrt{k}$ requires $k\log 2 = 2\log k$, which holds at exactly $k = 2$ and $k = 4$ and nowhere else.

So the design a rotatability argument produces is, at the two most common sizes, the same design a sphere argument would produce — every run at one radius, plus replicates at the middle. Two quite different requirements landing on one arrangement is the sort of agreement worth noticing, and it is also a warning: at three factors and at five they part company, and a design chosen for one of the two reasons is not the design the other reason wants.

Why rotatability is worth wanting #

The property is easy to state and easy to dismiss as aesthetic, so it is worth being concrete about what it buys.

A second-order design is run because the optimum is expected to be somewhere in the region and its location is not known. If the location were known there would be nothing to estimate. So the design is being asked to predict well in a neighbourhood, and it has no information about which direction from the centre matters more.

A non-rotatable design has made that decision anyway. At α = 1 the corners are further from the centre than the axial runs, so the design predicts corner directions better and axis directions worse — by a factor of nearly 1.6 at radius 1 — and an experimenter who then finds the optimum along an axis has an interval wider than they would have had from the same thirteen runs arranged differently.

Rotatability is the statement that the design has made no such choice. It is not that it predicts everything equally well; it is that its precision depends only on distance, which is the only thing the experimenter has an opinion about.

The other route to the same number #

A prediction variance computed from a matrix is an algebraic claim about a simulation nobody ran, and this site does not accept those alone.

So it is checked the other way: twenty thousand experiments on the thirteen-run design, each fitting the six-parameter model to pure noise, and the variance of the fitted value at four points recorded.

The second and third rows are the same distance from the centre in different directions, and the simulation reproduces their equality as well as their value. Rotatability is visible in the counted numbers, not only in the algebra.

The tolerance on that comparison is worth a sentence, because it is the sort of thing that gets set by eye. The variance of a sample variance from twenty thousand normal draws is 2/n of its own square, so its standard error is about 1% and four of them is 4%. That is the tolerance the check uses: tight enough that a wrong inverse or a mistranscribed model vector fails it, loose enough that the seed does not decide the outcome. A comparison of two routes is only as good as the width it is allowed, and a width chosen from the simulation’s own arithmetic is the only kind that is not negotiable after the fact.

What the centre runs buy here is not power #

The previous essay had centre runs buying degrees of freedom for the curvature test. In a second-order design they buy something else, and it is easier to see than to name.

With one centre run the design predicts its own centre at 9.00 and a point at radius 1 at 4.22: it is worst at the middle, by more than a factor of two. That is a design pointing its precision away from the place an optimum is most likely to be.

Adding centre runs pulls the middle down fast — 3.67 at three, 2.60 at five — and pushes the outside up slowly, because the scaling is by N and the extra runs are not helping the outside at all. Somewhere in between, the two are equal, and that design is called uniform precision.

Five centre runs, in a coding #

Every response-surface table gives the uniform-precision design for two factors as five centre runs, and the numbers above put the crossing between three and four. Both are right, and the disagreement is worth following because it is entirely notation.

Design tables are written in a coding where the sum of x² over the runs equals N — the design is standardised so that “a unit distance from the centre” means the same thing whatever arrangement is being described. This site codes the factorial at ±1, which is the coding an experimenter writes down on a run sheet, and the two differ by a factor of √(Σx²/N) that depends on how many centre runs there are.

For the thirteen-run design that factor is 0.784. Read at radius 0.784, the design’s prediction variance is 2.708 against 2.600 at the centre — equal to within 4%, which is what “uniform precision” means and where the five comes from. Read at radius 1, where the corners actually are, the same design gives 2.600 against 3.494 and is not uniform at all.

Neither number is wrong and neither is a rounding. What is wrong is treating a design catalogue’s constant as a property of an arrangement rather than of an arrangement plus a way of writing it down. That is a small instance of something this site has met at larger scale — the Engle–Granger statistic read against a t table, where the number and the table were each correct and belonged to different distributions — and the general repair is the same: recompute the constant in the units the work is actually done in.

When the axial runs cannot go there #

The α = 1 design in every figure above is not a straw man. It is the face-centred central composite design, it is what gets run whenever the factors cannot be pushed beyond their stated limits — a concentration that cannot exceed saturation, a temperature at the limit of the equipment, a dose already at the maximum tolerated — and it is not rotatable, at any number of centre runs, by construction.

What it costs is now a number rather than a caveat: the prediction variance around a circle of radius 1 varies by 59.9% between its best direction and its worst. The design predicts towards the corners well and along the axes badly, because along the axes the furthest run is at 1 and towards the corners the nearest is at √2.

Two things follow that are worth having.

The cost is not a loss of information; it is a loss of neutrality. A face-centred design estimates its six coefficients perfectly respectably. What it cannot do is treat directions alike, so an optimum found along an axis comes with a wider interval than the same optimum found on a diagonal, and nothing in the output says which one happened.

And it changes the arithmetic of the centre runs. With α = 1 the design has fewer distinct levels and less spread, so the squared terms are estimated less precisely from the same number of runs — which is the real reason face-centred designs are usually given more centre points than the tables recommend, rather than the reason usually given, which is uniform precision.

Anyone running one should say so. “Central composite design” without the axial distance is the same omission as a p-value without its sample size: the number is correct, the label is correct, and the reader cannot tell which of two quite different objects is being described.

Three factors, and where the arithmetic stops being free #

At two factors every version of this design is cheap enough that the choice barely matters. Past two it matters quickly, and the reason is that 2ᵏ grows and 2k does not.

At three factors the full design is $8 + 6 + n_c$ = twenty runs at α = 8^(1/4) = 1.682. At four it is $16 + 8 + n_c$ = thirty-one at α = 2. At five, a full factorial takes the total to fifty-two, and the standard answer is to use a half fraction for the factorial part — which changes F, and therefore changes the rotatable α, to 16^(1/4) = 2 rather than 32^(1/4) = 2.38.

That last point is the one worth carrying, because it is where a design gets built wrong in practice. The rotatable axial distance is a function of the number of factorial points actually run, not of the number of factors. Fractionating the factorial to save runs and then taking α from a table indexed by k gives a design that is not rotatable and was believed to be, and the belief is invisible: every coefficient still has an estimate, every standard error is still computed, and the only symptom is a prediction variance that depends on direction in a way nobody asked for.

Which is why the check here walks the ring rather than comparing α against a table. A property that can be measured directly on the design should be, and this one costs a matrix inverse and seventy-two evaluations.

What the design cannot fix #

Three things this arrangement does not buy, and they are worth listing because a rotatable uniform-precision design has an air of finality about it.

It does not make the second-order model true. Every quantity in this essay is computed under the assumption that the response is quadratic in the region. If it is not — if there is a ridge with a twist in it, or a discontinuity, or a third-order term — then the prediction variance is still exactly what the matrix says and the predictions are still wrong, because the variance of a fitted value says nothing about the bias of the model that fitted it. The centre runs are the only defence, and what they support is a lack-of-fit test rather than a repair.

It does not choose the region. All of the above is in coded units, and the coding is a decision about how wide to make the design. Too narrow and the response barely changes across it, so every coefficient is small relative to σ; too wide and the quadratic stops being a good description. Nothing in the arithmetic of the design speaks to that choice, which is the largest one an experimenter makes here.

And it does not locate the optimum. The design estimates six coefficients well. The thing the experiment is for is a function of those coefficients — a ratio, with a curvature in the denominator — and how badly that behaves even when every coefficient is well estimated is the next essay.

Two claims and one refusal #

The check behind this essay walks five rings at seventy-two angles each, for three axial distances, and requires the rotatable design to be flat to within 10⁻¹² of its own value.

The refusal beside it is the half that matters, and it is the shape this site keeps returning to: the same measurement is run on designs at α = 1 and α = 2 and is required to fail — to report a relative spread above 5% — because a rotatability check that reports every design as rotatable is measuring its own arithmetic. The two non-rotatable designs come out at 59.9% and 37.6%, which is a long way from either 0 or each other.

And there is a second refusal in the uniform-precision claim, which is why it is stated in both codings. The check asserts that five centre runs give uniform precision in one of them and that they do not in the other, so a future reader who finds one of those numbers surprising will find the other one beside it rather than discovering the discrepancy in a design catalogue years later.