D-Optimal Designs for Weighted Linear Regression and Binary Regression Models

This thesis is concerned with D-optimal designs primarly for binary response or weighted linear regression models. Its principle aim is to prove (using geometric and other arguments) that D-optimal designs have two support points for two parameter models depending on one design variable for all possible design intervals. We also extend established results, for Gamma, Beta and Normal density weight functions. The first aim of this work is to prove Ford, Torsney and Wu (1992) conjectures for a variety of such models. We also extend these results to higher dimensions. This is based on a parameter-dependent transformation to a weighted regression model and results will be extended to other such models. Chapter 1 mainly gives an introduction to the study for linear and nonlinear Optimal designs for regression models. Chapter 2 leads on with D-optimal designs for binary regression models which depend on two parameters and one covariate x in a design region, say X. It mainly deals with the following three cases: (a) X is a unbounded, (b) X is a bounded interval and (c) X is bounded at one end only. We first establish that only two support points are needed and then establish their values. The above conjecture is confirmed for most models using a transformation to weighted regression design. Chapter 3 presents Weighted Linear Regression and D-optimal Designs for the particular case of a Three Parameter Model with two design variables under a transformation to a weighted linear regression when the design space is rectangular. We first show that we have a four-point design for many of the weight functions considered. We also have an explicit solution for the optimal weights. An appropriate extension of the above conjecture is confirmed. Consideration of more realistic constraints on two design variables in Chapter 4 leads, under a transformation, to bounded design spaces in the shape of polygons. We establish results about D-optimal designs for such spaces. Chapter 5 widens the scope of the thesis, by considering more general models and, in particular, multiparameter binary regression models. Here again, we establish the existence of an explicit solution for the optimal weights for the rectangular case of the design space and further extensions of the conjecture. Chapter 6 extends the ideas of Chapter 2 by applying them to Contingent Valuation Studies. We illustrate one type of Contingent Valuation (CV) study, namely a dichotomous choice CV study with the design variable being a 'Bid' value. Respondents are asked if they are willing to pay this value for some service or amenity. We focus on both dichotomous choice (or single bounded) CV's and on double bounded CV's (in which a second bid is offered). Finally, Chapter 7 presents our conclusions and ideas for future work