Integer-Valued, Minimax Robust Designs for Estimation and Extrapolation in Heteroscedastic, Approximately Linear Models

Abstract

We present our findings on a new approach to robust regression design. This approach differs from previous investigations into this area in three respects: The use of a finite design space, the use of simulated annealing to carry out the numerical minimization problems, and in our search for integer-valued, rather than continuous, designs. We present designs for the situation in which the response is thought to be approximately polynomial. We also discuss the cases of approximate first- and second-order multiple regression. In each case we allow for possible heteroscedasticity and also obtain minimax regression weights. The results are extended to cover extrapolation of the regression response to regions outside of the design space. A case study involving dose-response experimentation is undertaken. The optimal robust designs, which protect against bias as well as variance, can be roughly described as being obtained from the classical variance-minimizing designs by replacing replicates with clusters of observations at nearby but distinct sites.

Key Words:

• Bioassay
• Carcinogen
• Dose response
• Efficient rounding
• Finite design space
• Fisher consistency
• Logistic model
• Polynomial regression
• Probit model
• Quota rounding
• Second-order design
• Simulated annealing
• Weighted least squares