Response Surface Design

A response curve is a mathematical function representing the relationship between the mean of a response variable and the level of an explanatory variable. The curve might be linear, a higher degree polynomial, or some nonlinear function. A response surface is a mathematical function representing the relationship between the mean of a response variable and the levels of several explanatory variables. The surface might be linear, a higher order polynomial, or a nonlinear function. In an experiment, the explanatory variables, or treatment factors, are under the control of the experimenter, and the level of each must be chosen for each experimental run. A response surface design is the set of combinations of the levels of the explanatory variables used in a particular experiment.

There are many reasons experimenters might want to fit response surface models, such as (a) to gain a simple mathematical and graphical description of the effects of the treatment factors on the response, (b) to identify the factors that have important effects on the response, (c) to estimate the effects of changing the levels of the factors from standard operating conditions, or (d) to enable estimation of the levels of the factors that optimize (maximize, minimize, or achieve a target level of) the response. The use of economical response surface designs can help researchers answer research questions precisely with considerably less time and cost than more ad hoc methods of experimentation. Response surface designs are particularly useful when combined with the experimenters’ expert knowledge and previous results.

In many areas of research, especially applied research, questions arise that can be answered by experimenting to determine the effects of several factors on one or more responses of interest. In such cases, the use of factorial-type experiments, in which all factors are studied in the same experiment, have long been known to be the most economical way of experimenting. In standard [Page 1274]factorial designs, it is usually assumed that the factors are qualitative, or at least that their effects will be modeled in terms of main effects and interactions, ignoring the quantitative nature of the levels.

In contrast, if the factors represent continuous variables, it is natural to fit a response surface model. Occasionally, mechanistic information will be available to determine the specific response surface function that should be used, but more often a purely empirical polynomial response surface model will be used to approximate the unknown function. Statistical methods for empirical modeling, collectively known as response surface methodology, were developed by George Box and his colleagues at Imperial Chemical Industries, in England, in the 1950s and have been further refined since then. This entry describes useful designs for fitting and checking first-order and second-order models and then considers complications that can arise.