Response surface methodology

Response surface methodology fits local regression approximations to a quantitative response as experimental settings change. A first-order model estimates a local gradient; augmenting the design permits quadratic regression, curvature assessment and local optimization. Extrapolating a fitted surface outside the experimentally supported region requires caution.

For a two-factor factorial design with coded corners $(\pm 1,\pm 1)$, observe each of the four corners once and add $n_0\ge 2$ independent center runs at $(0,0)$. Assume independent Gaussian errors with common variance ${\sigma }^2>0$. Write ${\overline{Y}}_F$ for the corner average, ${\overline{Y}}_0$ for the center average, and $S_0^2$ for the unbiased center-run sample variance. Then the center-point curvature contrast isFor the full quadratic regression mean ${\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_{12}{x}_1{x}_2+{\beta }_{11}{x}_1^2+{\beta }_{22}{x}_2^2$, the contrast has expected value ${\beta }_{11}+{\beta }_{22}$. The corner average cancels the linear and cross-product terms, while both squared coordinates equal one at every corner and zero at the center.

Under the null ${\beta }_{11}+{\beta }_{22}=0$,The independent normal contrast and center residuals justify this exact F-distribution; a first-order mean model alone would not imply it without the error assumptions. The center observations supply pure error, while the extra corner-to-center comparison detects one aggregate curvature direction.^[ref]

Zero contrast does not exclude curvature. The mean $x_1^2-x_2^2$ has a saddle shape but gives the same mean at all four corners and the center. A central composite design adds axial runs that can distinguish the separate squared terms.

For a fitted local first-order response $\hat{m}(x)={\hat{\beta }}_0+{\hat{b}}^{T}x$, the unit direction maximizing its directional derivative in coded Euclidean distance is $d=\hat{b}/\Vert \hat{b}\Vert$, by the Cauchy-Schwarz inequality. Run sequential feasible settings $x_0+hd$, increase $h$ while measured response improves, and refit near promising settings. Physical increments are $h_{j}h{d}_j$, so coded and unscaled physical gradients need not give the same experimental path. A near-zero gradient calls for further modeling rather than division by zero.

A coded experimental variable rescales a physical setting ${\xi }_j$ around its center $c_j$, dividing by a chosen positive half-range $h_j$. The endpoints $c_j\pm h_j$ become $\pm 1$, giving dimensionless factorial design coordinates. The choice of scale affects the metric used for steepest ascent in response surface methodology.

A rotatable design for a specified regression model has variance of a fitted regression mean depending on $x$ only through its Euclidean distance from the design center. It concerns prediction precision, not rotational symmetry of the mean response itself.

For a full quadratic regression model in $d\ge 2$ variables, consider all $2^d$ factorial corners, the $2d$ axial points $\pm \alpha e_i$, and center runs giving a nonsingular information matrix of an experimental design. Sign symmetry cancels odd moments and permutation symmetry makes the second coordinate moments equal. Fourth-order rotational balance requires, for distinct $i,j$,The factorial corners contribute $2^d$ to both sums, axial runs contribute $2{\alpha }^4$ only to the first, and center runs contribute zero. Thus $2^d+2{\alpha }^4=3\cdot 2^d$, givingFor $d=2$, $\alpha =\sqrt{2}$; for $d=3$, $\alpha =8^{1/4}$. Center replication can estimate pure error but does not change this fourth-moment equation. If a fractional factorial portion is used, its own moments and estimability must be checked; simply substituting the original factor count into the full-factorial formula is not justified. In one dimension rotational invariance only means reflection symmetry, so the distinct-coordinate fourth-moment equation does not impose the displayed choice.

A central composite design combines a factorial portion with axial points $\pm \alpha e_j$ and center runs. For a full $2^d$ factorial design, $d$ quantitative factors and $n_0\ge 1$ center runs, its run count is $2^d+2d+n_0$. The factorial portion supplies main effects and interaction terms, while the axial runs allow the individual squared terms of a full quadratic regression model to be estimated. Additional center replicates estimate pure error.

For two factors and $n_0=5$, this gives $4+4+5=13$ runs. The four corners are $(\pm 1,\pm 1)$, the four axial points are $(\pm \alpha ,0)$ and $(0,\pm \alpha )$, and the remaining five runs repeat $(0,0)$. Choosing $\alpha =\sqrt{2}$ makes this full-factorial design rotatable. Choosing $\alpha =1$ keeps the axial points on the faces and uses only three levels per factor, but does not give that rotatability in dimension at least two.

Circumscribed designs place the axial points outside the original factorial cube; inscribed designs rescale the cube inward to respect physical factor limits. A fractional factorial portion can reduce runs, but must retain sufficient resolution of a fractional factorial design to estimate the desired interactions: axial points have zero cross-products and cannot separate interactions that remain aliased in the factorial portion. The institutional handbook describes augmentation from a full or resolution V factorial portion.^[ref]