A Bayes decision theoretic approach to the optimal design of screens

Abstract

An item may be said to reach a standard suitable for use if it has some prescribed attributes. Supposet hat a variable 2: measurest he standard and TE, qT. if an item has the desired attributes. The variable -T may be very expensive to measure and so, some cheaper to measure screening variables, X say, correlated to I may be used to classify items. The purpose of screen design is to determine CX, the region of X space, for which an item should be said to reach the standard. If the error probabilities of classifying an item based on X are very high it may be economical to measure IT. Chapter 2 deals with this idea in the context of a very simple two-stage set-up in which, at the first stage of the screen a univariate screening variable X is measured. Some items are sentenced as acceptable or unacceptable, and the remainder are passed on to the second stage at which T is determined. The optimal screen is found that minimises cost, where costs are given for misclassifying items and for measuring the variables. The variable T is assumed binary and the model for TIX is a probit regression model. In designing a two-stage screen, Chapter 3 considers: (a) a general stochastic structure for (1, X), (b) a general loss function set up for misclassification costs and (c) assumes no fixed form for the screen. Also in Chapter 3, we consider a scenario in which a statistical goal or constraint is imposed in addition to the decision-theoretic target of minimising expected cost. In Chapter 4 we consider a sequential screen that operates as follows. At each stage of a sequence a covariate is measured and items may be accepted as suitable, discarded or passed on to the next stage. At the final stage the performance variable T is measured. Returning to the simple one-stage screen based solely on measuring covariates, Chapter 5 poses the question of how many and which covariates to include as part of the screen.