Inverse Polynomials, a Useful Group of Multi-Factor Response Functions

Abstract

If x1, x2,..., xk represent the levels of k experimental factors and y is the mean response, then the inverse polynomial response function is defined by x1x 2 ⋯ xk/y = Polynomial in (x1, x2 ⋯, xk). Arguments are given for preferring these surfaces to ordinary polynomials in the description of certain kinds of biological data. The fitting of inverse polynomials under certain assumptions is described, and shown to involve no more labour than that of fitting ordinary polynomials. Complications caused by the necessity of fitting unknown origins to the xi are described and the estimation process illustrated by an example. The goodness of fit of ordinary and inverse polynomials to four sets of data is compared and the inverse kind shown to have some advantages. The general question of the value of fitted surfaces to experimental data is discussed.