Choice of second-order response surface designs for logistic and Poisson regression models

Abstract

Response surface methodology is widely used for process development and optimisation, product design, and as part of the modern framework for robust parameter design. For normally distributed responses, the standard second-order designs such as the central composite design and the Box-Behnken design have relatively high D and G efficiencies. In situations where these designs are inappropriate, standard computer software can be used to construct D-optimal and I-optimal designs for fitting second-order models. When the response distribution is either binomial or Poisson, the choice of an appropriate design is not as straightforward. We illustrate the construction of D-optimal second-order designs for these situations and show that they are considerably better choices than the standard designs. We present an example applying this approach to optimisation of an etching process.

Separation in D‐optimal experimental designs for the logistic regression model

“…For standard linear models, D ‐optimal designs are constructed for a specific sample size n by choosing n design points, which are not necessarily unique, to maximize the determinant of X T X , where is the design matrix expanded to model form. For linear models, D ‐optimal designs are functions only of the design points …”

“…However, the logistic regression model is a type of generalized linear model, which is nonlinear in the model parameters. To construct a D ‐optimal design for a generalized linear model, the n design points must be chosen to maximize the determinant of X T V X , where is a diagonal weight matrix dependent on the specific generalized linear model . For the logistic regression model, the diagonal elements of V are Since X T V X is dependent on the unknown model parameters β , constructing optimal designs for the logistic model is encumbered by the dependence on the unknown parameters.…”

“…An augmentation strategy may also be used, where a small design is initially used to obtain estimates of β . Once the estimates are in place, additional runs are added to the design to maximize | X T V X | …”

“…One may also specify a prior distribution for β , denoted as f ( β ), to model the uncertainty in the parameter values. This prior distribution is then incorporated into the design criterion, which now involves choosing design points to maximize the expectation of the logarithm of | X T V X | . Mathematically, this design criterion can be written as and was initially proposed by Chaloner and Larntz to construct optimal designs for a single factor logistic regression model.…”

An Expository Paper on Optimal Design

“…The method described in Gotwalt et al (2009) can be used to construct D-optimal designs for this experiment. For examples of designed experiments for generalized linear models, see Johnson and Montgomery (2009) and Myers et al (2010).…”