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For the general normal linear model, minimizing $\Vert Y-X\beta {\Vert }^2$ gives the least-squares normal equations $X^{T}X\hat{\beta }=X^{T}Y$. Full column rank makes $X^{T}X$ invertible, soThe mean and covariance follow by substituting $Y=X\beta +ϵ$ and using $\mathbb{E}ϵ=0$, $\mathrm{Cov}(ϵ)={\sigma }^{2}I$. Normal errors also give the exact multivariate normal distribution of this estimator.

Use coded experimental variablesTheir origin is the middle of the allowed rectangle, and the stated endpoints become $\pm 1$. The four corners and $n_0$ center runs give mutually orthogonal columns $1,x_1,x_2$, withWriting ${\bar{Y}}_0=n_0^{-1}{\sum }_{i=5}^{4+n_0}{Y}_i$ when $n_0>0$, the ordinary least squares estimates areTheir variances are ${\sigma }^2/(4+n_0)$, ${\sigma }^2/4$, ${\sigma }^2/4$, with zero covariances.

The repeated center runs estimate pure error without requiring the fitted mean surface to be correct:They also permit a center-point curvature contrast. Let ${\bar{Y}}_F=(Y_1+Y_2+Y_3+Y_4)/4$ and $\hat{\gamma }={\bar{Y}}_F-{\bar{Y}}_0$. Under the first-order model, $\mathbb{E}\hat{\gamma }=0$ and $\mathrm{Var}(\hat{\gamma })={\sigma }^2(1/4+1/n_0)$. Under a quadratic surface, its expectation is ${\beta }_{11}+{\beta }_{22}$, since the linear and $x_1{x}_2$ terms average to zero. Thusunder the first-order normal model. With only one center run there is no pure-error degree of freedom. A zero center contrast does not exclude all curvature, because the two quadratic coefficients can cancel. The corners also estimate an interaction contrast; the full Lack-of-fit F-test compares the first-order fit with the five setting means, with two lack-of-fit and $n_0-1$ pure-error degrees of freedom.

If the first-order model is adequate and $\hat{b}=({\hat{\beta }}_1,{\hat{\beta }}_2{)}^T\ne 0$, steepest ascent in response surface methodology uses the unit coded direction $d=\hat{b}/\Vert \hat{b}\Vert$. The Cauchy-Schwarz inequality proves that it maximizes the predicted directional derivative. Run sequential settings $x=hd$ for increasing feasible $h$, corresponding to physical increments $(5h{d}_1,2.5h{d}_2)$ from the center. Stop or reduce the steps when measured yield ceases to improve, and refit locally near promising settings. Respect the allowed rectangle; if a boundary is reached, investigate feasible boundary directions rather than extrapolating outside it. A near-zero gradient offers no first-order ascent direction and calls for further modeling.

If curvature is indicated, add the four axial settings $(1,0),(-1,0),(0,1),(0,-1)$, together with further center replication. This face-centered central composite design stays inside the allowed physical ranges and separates the two squared coordinates, which the corners alone cannot do. Fit the full quadratic regression surfaceWrite its fitted quadratic part as $x^{T}Bx$, where $B_{11}={\hat{\beta }}_{11}$, $B_{22}={\hat{\beta }}_{22}$ and $B_{12}=B_{21}={\hat{\beta }}_{12}/2$. Its gradient and Hessian are $\hat{b}+2Bx$ and $2B$. If $B$ is nonsingular, the stationary setting isIf $B$ is negative definite and $x_{\ast }$ is feasible, it is the fitted maximum. Otherwise compare feasible stationary settings, optima on each edge obtained from the restricted one-variable quadratic, and the corners. A singular or poorly determined $B$ calls for attention to ridge directions and more runs. Convert any chosen coded setting back using ${\xi }_1=95+5x_1$, ${\xi }_2=37.5+2.5x_2$, and confirm its yield experimentally. The fitted local surface guides the search; it does not establish an untested physical optimum.