Response Surface Models with Mixed Effects

The purpose of this article is to demonstrate how to conduct a response surface investigation concerning a linear model that contains, in addition to fixed polynomial effects, a random effect representing an extraneous source of variation. The model also contains interactions involving the random effect. Estimates of the polynomial effects are obtained in a two-stage procedure. The variance components of the random effects are first estimated by applying the method of restricted maximum likelihood. The resulting values are then used to compute generalized least-squares estimates of the polynomial effects. This procedure leads to an expression for predicting the response within a certain region of interest, R. Optima of the predicted response are determined over portions of R in which the prediction variance falls within specified bounds. Finally, a procedure is described for establishing operating conditions that result in a predicted response close to some target value. A numerical example is given for illustrative purposes.