Multi-Factor Experimental Designs for Exploring Response Surfaces

Abstract

Suppose that a relationship $\eta = \varphi(\xi_1, \xi_2, \cdots, \xi_k)$ exists between a response $\eta$ and the levels $\xi_1, \xi_2, \cdots, \xi_k$ of $k$ quantitative variables or factors, and that nothing is assumed about the function $\varphi$ except that, within a limited region of immediate interest in the space of the variables, it can be adequately represented by a polynomial of degree $d$. A $k$-dimensional experimental design of order $d$ is a set of $N$ points in the $k$-dimensional space of the variables so chosen that, using the data generated by making one observation at each of the points, all the coefficients in the $d$th degree polynomial can be estimated. The problem of selecting practically useful designs is discussed, and in this connection the concept of the variance function for an experimental design is introduced. Reasons are advanced for preferring designs having a "spherical" or nearly "spherical" variance function. Such designs insure that the estimated response has a constant variance at all points which are the same distance from the center of the design. Designs having this property are called rotatable designs. When such arrangements are submitted to rotation about the fixed center, the variances and covariances of the estimated coefficients in the fitted series remain constant. Rotatable designs having satisfactory variance functions are given for $d = 1, 2$; and $k = 2, 3, \cdots, \infty$. Blocking arrangements are derived. The simplification in the form of the confidence region for a stationary point resulting from the use of a second order rotatable design is discussed.