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Optimal Experimental Designs for the Poisson Regression Model in Toxicity Studies

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Optimal Experimental Designs for the Poisson Regression Model in Toxicity Studies Optimal Experimental Designs for the Poisson Regression Model in Toxicity Studies Yanping Wang Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Statistics Eric P. Smith, Co-chair Keying Ye, Co-chair Christine Anderson-Cook Jeffrey B. Birch Raymond H. Myers July 23, 2002 Blacksburg, Virginia Keywords: Experimental Design, Design Optimality, Poisson Regression Model Copyright 2002, Yanping Wang Optimal Experimental Designs for the Poisson Regression Model in Toxicity Studies Yanping Wang (ABSTRACT) Optimal experimental designs for generalized linear models have received increasing attention in recent years. Yet, most of the current research focuses on binary data models especially the one-variable first-order logistic regression model. This research extends this topic to count data models. The primary goal of this research is to develop efficient and robust experimental designs for the Poisson regression model in toxicity studies. D-optimal designs for both the one-toxicant second-order model and the two-toxicant interaction model are developed and their dependence upon the model parameters is in- vestigated. Application of the D-optimal designs is very limited due to the fact that these optimal designs, in terms of ED levels, depend upon the unknown parameters. Thus, some practical designs like equally spaced designs and conditional D-optimal designs, which, in terms of ED levels, are independent of the parameters, are studied. It turns out that these practical designs are quite efficient when the design space is restricted. Designs found in terms of ED levels like D-optimal designs are not robust to parame- ters misspecification. To deal with this problem, sequential designs are proposed for Poisson regression models. Both fully sequential designs and two-stage designs are studied and they are found to be efficient and robust to parameter misspecification. For experiments that in- volve two or more toxicants, restrictions on the survival proportion lead to restricted design regions dependent on the unknown parameters. It is found that sequential designs perform very well under such restrictions. In most of this research, the log link is assumed to be the true link function for the model. However, in some applications, more than one link functions fit the data very well. To help identify the link function that generates the data, experimental designs for discrimination between two competing link functions are investigated. T-optimal designs for discrimination between the log link and other link functions such as the square root link and the identity link are developed. To relax the dependence of T-optimal designs on the model truth, sequential designs are studied, which are found to converge to T-optimal designs for large experiments. Dedication To my parents and my dear wife, Quan. iii Acknowledgments I would like to express my deepest appreciation to my advisors, Dr. Eric P. Smith and Dr. Keying Ye, for sharing their knowledge with me. Their mentoring, support and discipline throughout this endeavor have been invaluable. I would also like to thank Dr. Christine Anderson-Cook and Dr. Jeffrey B. Birch for their time and helpful suggestions. My special thanks go to Dr. Raymond H. Myers for his guidance and encouragement over all these years. iv Contents List of Figures viii List of Tables ix 1 Introduction and Literature Review 1 1.1 Background and Motivation . 1 1.2 Design Optimality . 2 1.3 Optimal Designs for Logistic Regression Model . 4 1.3.1 Locally Optimal Designs . 5 1.3.2 Bayesian Optimal Designs . 7 1.3.3 Minimax Approach . 8 1.3.4 Sequential Designs . 9 1.4 Optimal Designs for Poisson Regression Model . 10 1.4.1 Models, Assumptions and Notations . 10 1.4.2 Optimal Designs: A Review of Current Work . 12 1.4.3 Optimal Designs: Extensions . 14 1.5 Structure of the Dissertation . 15 2 Locally D-optimal Designs 17 2.1 Introduction . 17 2.2 One-toxicant Experiments: Second-order Model . 18 2.2.1 D-optimal Designs . 18 v 2.2.2 Equally Spaced Designs . 19 2.2.3 Parameter Misspecifications . 21 2.3 Two-toxicant Experiments: Interaction Model . 23 2.3.1 D-optimal Designs . 23 2.3.2 Conditional D-optimal Designs . 29 2.3.3 Parameter Misspecifications . 32 2.4 Ds-optimal Designs: Addressing ED Estimation . 36 2.4.1 Introduction . 36 2.4.2 One-toxicant Second-order Model . 39 2.4.3 Two-toxicant Interaction Model . 39 2.5 Summary . 42 3 Equivalence Theory 44 3.1 Introduction . 44 3.2 Equivalence Theory . 45 3.3 Optimality of Locally D-optimal Designs . 47 3.3.1 One-toxicant Second-order Model . 48 3.3.2 Two-toxicant Interaction Model . 49 3.4 Optimality of Locally Ds-optimal Designs . 50 3.4.1 One-toxicant Second-order Model . 52 3.4.2 Two-toxicant Interaction Model . 54 3.5 D-optimal Designs for Multi-toxicant Interaction Model . 54 3.5.1 Three-toxicant Interaction Model . 56 3.5.2 Multi-toxicant Interaction Model . 58 3.6 Summary . 58 4 Sequential Designs 60 4.1 Introduction . 60 4.2 Sequential Designs . 61 vi 4.2.1 Algorithm . 61 4.2.2 An example . 63 4.2.3 Evaluation of Sequential Designs . 65 4.2.4 Restricted Design Space . 69 4.2.5 Inference Based on Sequential Designs . 79 4.3 Two-stage Designs . 82 4.3.1 Two-stage Procedure . 82 4.3.2 An Example of Two-stage Designs . 85 4.3.3 Evaluation of Two-stage Designs . 86 4.4 Summary . 92 5 Optimal Designs for Link Discrimination 95 5.1 Introduction . 95 5.2 T-optimality . 97 5.3 Locally T-optimal Designs . 99 5.4 Sequential Designs . 101 5.4.1 Algorithm . 101 5.4.2 Evaluation . 106 5.5 Summary . 108 6 Future Research 109 Bibliography 111 Appendices 115 A Proofs and Derivations 115 Vita 125 vii List of Figures 1.1 Mean Function of Poisson Regression Model . 11 2.1 Structure of D-optimal Design for Two-toxicant Interaction Model . 27 3.1 D-optimality of Designs for One-toxicant Second-order Model . 49 3.2 D-optimality of Designs for Two-toxicant Interaction Model . 51 3.3 Ds-optimality of Designs for One-toxicant Second-order Model . 53 3.4 Ds-optimality of Designs for Two-toxicant Interaction Model . 55 3.5 Structure of D-optimal Design for Three-toxicant Interaction Model . 57 4.1 An Example of Sequential Design . 64 4.2 Determination of Simulation Size for Sequential Design . 67 4.3 Restricted Design Regions for Different r .................... 71 4.4 An Example of Sequential Design on Restricted Region . 74 4.5 Determination of Simulation Size for Two-stage Design . 88 4.6 Sample Allocation for Two-stage Design . 90 5.1 Examples of Competing Link Functions . 96 5.2 T-optimality of Designs for Link Discrimination . 102 5.3 An Example of Sequential Design for Link Discrimination . 105 5.4 T-efficiency of Sequential Designs . 107 viii List of Tables 1.1 Some Optimal Designs for One-toxicant First-order Model . 13 2.1 D-optimal Designs for One-toxicant Second-order Model . 20 2.2 D-efficiency of Equally Spaced Designs for One-toxicant Second-order Model 21 2.3 D-efficiency of Equally Spaced Designs under Parameter Misspecification . 22 2.4 D-optimal Designs for Two-toxicant Interaction Model . 26 2.5 D-efficiency of Conditional D-optimal and 5-point Designs . 30 2.6 D-efficiency of Bayesian D-optimal Design for Synergism . 32 2.7 D-efficiency of Conditional D-optimal Design under Parameter Misspecification 33 2.8 D-efficiency of Conditional D-optimal Design under Parameter Misspecifica- tion on Restricted Regions . 35 2.9 Ds-optimal Designs for One-toxicant Second-order Model . 40 2.10 Ds-optimal Designs for Two-toxicant Interaction Model . 41 3.1 D-optimal Designs for Three-toxicant Interaction Model . 58 4.1 An Example of Sequential Design . 63 4.2 D-efficiency of Sequential Designs . 69 4.3 D-optimal Designs on Restricted Region . 72 4.4 Penalized D-efficiency of Sequential and Conditional D-optimal Designs on Restricted Region . 78 4.5 Coverage of Confidence Regions Based on Expected and Observed Information Matrices . 81 4.6 An Example of Two-stage Design . 85 ix 4.7 D-efficiency of Two-stage Designs . 91 4.8 D-efficiency of Sequential, Two-stage and Conditional D-optimal Designs . 93 5.1 T-optimal Designs for One-toxicant First-order Model . 103 x Chapter 1 Introduction and Literature Review 1.1 Background and Motivation Over the past few decades, a large portion of scientific research has been driven by environ- mental concerns and health issues. Environmental studies focus on assessing the effect of pollutants on the delicate balance of nature’s ecosystems while much of the research in the health arena centers on eradication of serious illness plaguing the world’s populations. The toxicity study is a common thread that runs through both environmental and health-related research. A typical toxicity study involves application of different concentrations of the toxi- cants under investigation to organisms and then measuring the effect. Indicators of effect include mortality, growth effect, reproduction impairment and tumor incidence. Very often, these responses cannot be appropriately modelled as normal random variables in the anal- ysis of effect. Two representative examples are binary responses and count responses. In such situations, classical linear models are not applicable and generalized linear models are usually used for analysis. Generalized linear models for both binary and count data have been fully studied in the literature (McCullagh and Nelder, 1987; Myers, Montgomery and Vining, 2002).
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