Optimal regression design for control in linear regression

Abstract

SIUMMARY A Bayesian decision theory approach to the choice of regression design is considered when it is intended to use the regression to help control the dependent variable at a chosen value, the control to be effected by choosing certain of the independent variables and fixing them at selected values. Some key word8: Bayesian decision theory; Control; D-optimum; Linear regression; Preposterior analysis; Regression design. 1. INTRODIUCTION This paper is concerned with the design of a regression experiment in which observations of a dependent variable are obtained at selected values of a set of independent variables. Using Bayesian decision theory, Lindley (1968) considered the analysis of data from a regression experiment in two situations, namely when it is desired to choose the best subset of independent variables (a) to predict a future value of the dependent variable, (b) to control the level of the dependent variable at a preassigned value. The first situation he termed a 'prediction problem' and the second a 'control problem'. The optimal choice of a designed regression experiment for Lindley's approach to the prediction problem was considered by Brooks (1972). In the present paper, the optimal choice of designed experiment for the control problem is considered, whe-n the regression is linear in the independent variables. To do this, a decision-theoretic analysis of the following sequence of decisions and events is considered. First, the size n of the experiment is decided, and then n sets of values X1, ..., X,