Abstract: In science and engineering, there is often uncertainty in the linear model assumed for a response when an experiment is being designed. The errors in predictions made from a fitted model may then be more dependent on the systematic errors (bias) that arise from model misspecification than from errors related to the variance of the estimators of the unknown model coefficients. This may result in commonly used criteria, such as optimality, being inappropriate for the selection of a design. In this paper the true model is allowed to differ from the model assumed for design purposes by an additive contamination term which is a random variable. Design criteria are defined which involve properties of the resulting random bias. These criteria are applied to assumed linear models constructed from polynomials or from polynomial splines where the contamination is modeled by random variables that represent the unknown numbers, locations and coefficients of additional knots (or breakpoints). Designs are found using an exchange algorithm which incorporates Monte Carlo simulation to approximate properties of the bias distribution. When the expectation of the contamination is zero, theoretical results enable a reduction in the computationally intensive design search. The extension of the approach to the use of a mean squared error criterion is also considered.
Key words and phrases: Bias, design selection criteria, exchange algorithm, mean squared error, Monte Carlo simulation, optimal design, polynomial splines.