= Center-point curvature contrast
{title2=$\widehat\gamma=\bar Y_F-\bar Y_0$}
For a two-factor <factorial design> with coded corners $(\pm1,\pm1)$, observe each of the four corners once and add $n_0\geq2$ <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> is
$$
\widehat\gamma=\overline Y_F-\overline Y_0,\qquad
\operatorname{Var}(\widehat\gamma)=\sigma^2(1/4+1/n_0).
$$
For the full <quadratic regression> <mean> $\beta_0+\beta_1x_1+\beta_2x_2+\beta_{12}x_1x_2+\beta_{11}x_1^2+\beta_{22}x_2^2$, 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 $\beta_{11}+\beta_{22}=0$,
$$
F=\frac{\widehat\gamma^2}{S_0^2(1/4+1/n_0)}\sim F_{1,n_0-1}.
$$
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.<Optimization Designs (NIST)>[Slide 10.18, Curvature Check with Center Points]{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.
Back to article page
