Optimal Design of Experiments offers a rare blend of linear algebra, convex analysis, and statistics. The optimal design for statistical experiments is first formulated as a concave matrix optimization problem. Using tools from convex analysis, the problem is solved generally for a wide class of optimality criteria such as D-, A-, or E-optimality. The book then offers a complementary approach that calls for the study of the symmetry properties of the design problem, exploiting such notions as matrix majorization and the Kiefer matrix ordering. The results are illustrated with optimal designs for polynomial fit models, Bayes designs, balanced incomplete block designs, exchangeable designs on the cube, rotatable designs on the sphere, and many other examples.
The increasing cost of research means that scientists are in more urgent need of optimal design theory to increase the efficiency of parameter estimators and the statistical power of their tests. The objectives of a good design are to provide interpretable and accurate inference at minimal costs. Optimal design theory can help to identify a design with maximum power and maximum information for a statistical model and, at the same time, enable researchers to check on the model assumptions. This Book: Introduces optimal experimental design in an accessible format. Provides guidelines for practitioners to increase the efficiency of their designs, and demonstrates how optimal designs can reduce a study’s costs. Discusses the merits of optimal designs and compares them with commonly used designs. Takes the reader from simple linear regression models to advanced designs for multiple linear regression and nonlinear models in a systematic manner. Illustrates design techniques with practical examples from social and biomedical research to enhance the reader’s understanding. Researchers and students studying social, behavioural and biomedical sciences will find this book useful for understanding design issues and in putting optimal design ideas to practice.
The essential introduction to the theory and application of linear models—now in a valuable new edition Since most advanced statistical tools are generalizations of the linear model, it is neces-sary to first master the linear model in order to move forward to more advanced concepts. The linear model remains the main tool of the applied statistician and is central to the training of any statistician regardless of whether the focus is applied or theoretical. This completely revised and updated new edition successfully develops the basic theory of linear models for regression, analysis of variance, analysis of covariance, and linear mixed models. Recent advances in the methodology related to linear mixed models, generalized linear models, and the Bayesian linear model are also addressed. Linear Models in Statistics, Second Edition includes full coverage of advanced topics, such as mixed and generalized linear models, Bayesian linear models, two-way models with empty cells, geometry of least squares, vector-matrix calculus, simultaneous inference, and logistic and nonlinear regression. Algebraic, geometrical, frequentist, and Bayesian approaches to both the inference of linear models and the analysis of variance are also illustrated. Through the expansion of relevant material and the inclusion of the latest technological developments in the field, this book provides readers with the theoretical foundation to correctly interpret computer software output as well as effectively use, customize, and understand linear models. This modern Second Edition features: New chapters on Bayesian linear models as well as random and mixed linear models Expanded discussion of two-way models with empty cells Additional sections on the geometry of least squares Updated coverage of simultaneous inference The book is complemented with easy-to-read proofs, real data sets, and an extensive bibliography. A thorough review of the requisite matrix algebra has been addedfor transitional purposes, and numerous theoretical and applied problems have been incorporated with selected answers provided at the end of the book. A related Web site includes additional data sets and SAS® code for all numerical examples. Linear Model in Statistics, Second Edition is a must-have book for courses in statistics, biostatistics, and mathematics at the upper-undergraduate and graduate levels. It is also an invaluable reference for researchers who need to gain a better understanding of regression and analysis of variance.
This book collects the proceedings of the 10th Workshop on Model-Oriented Design and Analysis (mODa). A model-oriented view on the design of experiments, which is the unifying theme of all mODa meetings, assumes some knowledge of the form of the data-generating process and naturally leads to the so-called optimum experimental design. Its theory and practice have since become important in many scientific and technological fields, ranging from optimal designs for dynamic models in pharmacological research, to designs for industrial experimentation, to designs for simulation experiments in environmental risk management, to name but a few. The methodology has become even more important in recent years because of the increased speed of scientific developments, the complexity of the systems currently under investigation and the mounting pressure on businesses, industries and scientific researchers to reduce product and process development times. This increased competition requires ever increasing efficiency in experimentation, thus necessitating new statistical designs. This book presents a rich collection of carefully selected contributions ranging from statistical methodology to emerging applications. It primarily aims to provide an overview of recent advances and challenges in the field, especially in the context of new formulations, methods and state-of-the-art algorithms. The topics included in this volume will be of interest to all scientists and engineers and statisticians who conduct experiments.
This textbook provides a concise introduction to optimal experimental design and efficiently prepares the reader for research in the area. It presents the common concepts and techniques for linear and nonlinear models as well as Bayesian optimal designs. The last two chapters are devoted to particular themes of interest, including recent developments and hot topics in optimal experimental design, and real-world applications. Numerous examples and exercises are included, some of them with solutions or hints, as well as references to the existing software for computing designs. The book is primarily intended for graduate students and young researchers in statistics and applied mathematics who are new to the field of optimal experimental design. Given the applications and the way concepts and results are introduced, parts of the text will also appeal to engineers and other applied researchers.
This book provides a comprehensive treatment of the design of blocked and split-plot experiments. The optimal design approach advocated in the book will help applied statisticians from industry, medicine, agriculture, chemistry and many other fields of study in setting up tailor-made experiments. The book also contains a theoretical background, a thorough review of the recent work in the area of blocked and split-plot experiments, and a number of interesting theoretical results.
This carefully edited collection synthesizes the state of the art in the theory and applications of designed experiments and their analyses. It provides a detailed overview of the tools required for the optimal design of experiments and their analyses. The handbook covers many recent advances in the field, including designs for nonlinear models and algorithms applicable to a wide variety of design problems. It also explores the extensive use of experimental designs in marketing, the pharmaceutical industry, engineering and other areas.
Optimal Design for Nonlinear Response Models discusses the theory and applications of model-based experimental design with a strong emphasis on biopharmaceutical studies. The book draws on the authors’ many years of experience in academia and the pharmaceutical industry. While the focus is on nonlinear models, the book begins with an explanation of the key ideas, using linear models as examples. Applying the linearization in the parameter space, it then covers nonlinear models and locally optimal designs as well as minimax, optimal on average, and Bayesian designs. The authors also discuss adaptive designs, focusing on procedures with non-informative stopping. The common goals of experimental design—such as reducing costs, supporting efficient decision making, and gaining maximum information under various constraints—are often the same across diverse applied areas. Ethical and regulatory aspects play a much more prominent role in biological, medical, and pharmaceutical research. The authors address all of these issues through many examples in the book.
There has been an enormous growth in recent years in the literature on discrete optimal designs. The optimality problems have been formulated in various models arising in the experimental designs and substantial progress has been made towards solving some of these. The subject has now reached a stage of completeness which calls for a self-contained monograph on this topic. The aim of this monograph is to present the state of the art and to focus on more recent advances in this rapidly developing area. We start with a discussion of statistical optimality criteria in Chapter One. Chapters Two and Three deal with optimal block designs. Row-column designs are dealt with in Chapter Four. In Chapter Five we deal with optimal designs with mixed effects models. Repeated measurement designs are considered in Chapter Six. Chapter Seven deals with some special situations and Weighing designs are dis cussed in Chapter Eight. We have endeavoured to include all the major developments that have taken place in the last three decades. The book should be of use to research workers in several areas including combinatorics as well as to the experimenters in diverse fields of applications. Since the details of the construction of the designs are available in excellent books, we have only pointed out the designs which have optimality proper ties. We believe, this will be adequate for the experimenters.
