 Center-point curvature contrast

In a two-factor design with four corner observations and $n_0$ independent center replicates, the difference between the corner mean and center mean has variance ${\sigma }^2(1/4+1/n_0)$. Under a full quadratic regression surface it estimates ${\beta }_{11}+{\beta }_{22}$; the linear and interaction terms average to zero at the corners. With $n_0\ge 2$, the center pure error estimate makes its squared standardized contrast an $F_{1,n_0-1}$ test under a first-order mean model. Zero contrast does not exclude curvature, since the two squared-term coefficients may cancel.