us assume that the relationship between Y and (x1, x2) is modeled by a function f: i.e., y = f(x1,x2) + ε, where ε represent noise in response y. Let E(Y) = η = f(x1,x2). The surface represented by η = f(x1,x2) is called the response surface. In RSM the functional form of f is unknown. We generally try a linear function or a polynomial function to model this relation. When there is a variation of this relation from laboratory to laboratory, a mixed-effects polynomial model can be used and the method of maximum likelihood or marginal maximum likelihood can be used to estimate model parameters. Once the parameters have been estimated, we can use the estimated response surface to evaluate the values of x1 and x2 for a specific targeted value of y0—for example, a carcinogenetic threshold. A contour plot may help in this regard to estimate levels of x1 and x2 corresponding to a particular level of carcinogenic risk. The RMS method may not provide a reasonable solution for the true functional relationship over the entire space of the independent variables x1 and x2. In that case a small region for the independent variables is chosen and RMS is used sequentially. The interested reader is referred for further discussion on these issues to Box and Draper (2007).
Box, G. E. P., and N. Draper. 2007. Response surfaces, mixtures, and ridge analyses, 2nd edition. Wiley Series in Probability and Statistics. New York: John Wiley & Sons.
Box, G. E. P., J. S. Hunter, and W. G. Hunter. 2005. Statistics for experimenters: Design, innovation, and discovery, 2nd edition. New York: John Wiley & Sons.
Neter, J., M. H. Kutner, C. J. Nachtsheim, and W. Wasserman. 1996. Applied linear statistical models, 4th edition. Burr Ridge, IL: Irwin.