A Basis for the Selection of a Response Surface Design

Abstract The general problem is considered of choosing a design such that (a) the polynomial f(ξ) = f(ξ1, ξ2, · · ·, ξ k ) in the k continuous variables ξ' = (ξ1, ξ2, · · ·, ξ k ) fitted by the method of least squares most closely represents the true function g(ξ1, ξ2, · · ·, ξ k ) over some “region of interest” R in the ξ space, no restrictions being introduced that the experimental points should necessarily lie inside R; and (b) subject to satisfaction of (a), there is a high chance that inadequacy of f(ξ) to represent g(ξ) will be detected. When the observations are subject to error, discrepancies between the fitted polynomial and the true function occur: i. due to sampling error (called here “variance error”), andii. due to the inadequacy of the polynomial f(ξ) exactly to represent g(ξ) (called here “bias error”). To meet requirement (a) the design is selected so as to minimize J, the expected mean square error averaged over the region R. J contains two components, one associated entirely with variance error and the other associated entirely with bias error. There is a class of designs which satisfy requirement (a). To meet requirement (b) we select from this class a subclass for which the “non-centrality term” in the expectation of the residual sum of squares in the analysis of variance is large. This leads to a sensitive test of goodness of fit. In this paper the theory is applied to the particular case where f(ξ) is a polynomial of first degree and g(ξ) a polynomial of second degree; that is, the experimenter is hopefully fitting a first degree equation over the region R in the circumstances where the true function is really quadratic. The somewhat unexpected conclusion is reached that, at least in the cases considered, the optimal design in typical situations in which both variance and bias occur is very nearly the same as would be obtained if variance were ignored completely and the experiment designed so as to minimize bias alone. Particular examples of the class of optimal designs derived are fractional by replicated two-level factorial designs (in which no two-factor interaction is confounded with the main effect) with added center points. Particular examples of the class of optimal designs derived are fractional by replicated two-level factorial designs (in which no two-factor interaction is confounded with the main effect) with added center points.