An Enhanced Recursive Stopping Rule for Steepest Ascent Searches in Response Surface Methodology

In traditional Response Surface Methods (RSM), running a first order experimental design is usually followed by a steepest ascent search using the vector of parameter estimates as an approximation to the gradient. In the presence of variability in the responses, there is a need for a stopping rule to determine the optimal point in the search direction. Two formal stopping rules have been proposed in the literature, Myers and Khuri's (MK) stopping rule and Del Castillo's Recursive Parabolic Rule (RPR). The first rule requires the specification of an initial guess on the number of steps required to reach the optimum, while the performance of the second rule has only been studied for quadratic responses. This paper proposes some modifications to the recursive parabolic rule in order to increase its robustness for non-quadratic responses. The modifications consist of using only a fraction of the experiments made along the search, the estimation of all the parameters in the parabolic model, and the utilization of a coding convention for the regressors. The performance of the two rules, together with classical rules of stopping after 1, 2 or 3 consecutive drops in the response, are studied through simulation under non-quadratic and non-normally distributed responses. We show, by example, that the new rule is easy to use in practice, keeping one of the main attractions of this RSM technique. It was observed that the original recursive parabolic rule stops before the optimum under non-quadratic behavior, while the modified parabolic rule and the MK rule perform satisfactorily under most of the simulated conditions.