Four corners and 5 runs at the centre

The centre runs add two things at once. They estimate σ from replicates at one setting, which assumes nothing about the surface, and they supply the one contrast that sees curvature — the corner mean minus the centre mean, which estimates Σβᵢᵢ. It is one number: the design still cannot say which factor the curvature is in. 5 centre runs give 4 degrees of freedom for the pure-error estimate, and that is what sets the test's power.

Every design here has the same four corners and differs only in how many runs sit at the centre. The line is the non-central t on one fewer degrees of freedom than there are centre runs, at non-centrality Σβᵢᵢ divided by σ√(1/factorial runs + 1/centre runs); the points are 6,000 simulated experiments each. Power goes from 32% at 3 centre runs to 92% at 16, and none of that came from the factorial.

Each point is the stationary point of one fitted quadratic, from one central composite design run on the same true surface — which has a maximum at (0.91, 0.35), marked. 8% of the fits are saddles rather than maxima, so their stationary point is not an optimum of anything, and 39% land outside the region the design explored. The median distance from the centre is 1.18 against a true 0.98.

Walked around a circle of radius 1 at 72 angles, with no simulation anywhere in it: the scaled prediction variance is a matrix computation on the design. At α = √2 it is flat to 7.6e-16 of its own value — the design says the same thing in every direction — and at α = 1 it varies by 47%, so a prediction towards a corner is worth measurably less than one along an axis.

Each walk fits a plane to the same four-corner factorial, takes its gradient as a direction, and steps along it until a run comes in below the one before. The true optimum is the cross. 80% of the walks stop before the best point on their own path — not because the direction was wrong, but because one noisy run is enough to stop them, and the direction error costs only 3.9% of the available gain.

One dataset, one fitted quadratic, and two answers to "where is the best setting". The delta method reports 0.80 ± 0.46, a finite interval it will report whatever the data does. Fieller's set is 0.49 to 1.76, because the curvature here has t = -4.04. The true optimum is at 0.75.

The defining relation is I = ABCD, so the resolution is 4. A is estimated as A + BCD; B is estimated as B + ACD; C is estimated as C + ABD; D is estimated as D + ABC. Each of those is an identity about the design rather than an approximation about the data.

Each main effect's least-squares estimate is the sum of its whole alias chain, so with every two- and three-factor interaction equal to 0.8 the reported values are A = 5.80 against a true 5, B = 2.80 against a true 2, C = -0.20 against a true -1, D = 3.80 against a true 3. Nothing here is simulated: the bias is an expectation computed from the design.

the full factorial: 32 runs, nothing confounded; half — E = ABCD: 16 runs, resolution 5; half — E = AB: 16 runs, resolution 3; quarter — D = AB, E = ACD: 8 runs, resolution 3. Two fractions of the same size can differ by a whole level of resolution, so the number of runs does not say what a design can estimate.

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.

Box–Behnken (15 runs) against the central composite (17 runs), both under the full second-order model in three factors, along the line from the centre towards a corner. At the far end they read 20.937 and 11.388. No simulation: the scaled prediction variance is a matrix computation on the design alone.

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.

Six orthogonal designs, from a 8-run factorial fitting 7 coefficients to a 32-run one fitting 6. The counted inflation and the closed form 1 + 1/(N − p) agree to 8.9e-16 at every one of them, and the induced correlation is 1/(N − p + 1) by the same arithmetic. A saturated design, with no spare runs at all, cannot survive losing one.

One eigenvalue held at −3 and the other swept from −2 to 2, so the truth is a maximum on the left and a saddle on the right and the change happens at exactly zero. At an eigenvalue of −0.25 — a genuine maximum — the fit reports a saddle on 26.4% of studies; at +0.25 — a genuine saddle — it reports a maximum on 25.1%. The standard error of a squared coefficient under this design is 0.3791, and the region of confusion is about that wide either side of zero.

One fitted surface. Its stationary point is at a radius of 2.289 and the fit calls the shape a maximum. The ridge is the best setting at each radius, found by the Lagrange condition (B̂ − μI)x = −ĝ/2; the fitted response rises along it from 59.93 at the centre to 62.10 at the edge. The true optimum is at (0.4, 0.3).

A truth whose own optimum is inside the region, at (0.4, 0.3). At σ = 2 the fitted stationary point is outside on 24.9% of studies and more than three units out on 11.8%, and the best point inside the region is on its boundary on 47.3%.

The true optimum is worth 62.348. At σ = 2 the fit predicts 62.679 at the setting it recommends and the truth there is 61.821 — a gap of 0.858, which is 0.72 of the prediction's own standard error. The setting itself gives up 0.527 against the best available.