Information and Complexity in Statistical Modeling

The authors clearly have thought very deeply about the best way to present the material. All concepts are well defined and nicely argued and are followed by illustrative examples. Complete descriptions are given of all topics covered. Exercises are provided at the ends of most chapters, and the book is suitable as a supplemental text in a doctoral-level experimental design course or as the main text in a topics reading course. In Chapter 1 different types of stated choice experiments are explained, and the illustrations show the importance of the subject very well. Practical issues of running such experiments are beyond the scope of the book, but references are given. Chapter 2 gives a brief, but complete, review of the construction of fractional 2 and 3 factorial experiments. This chapter is the most technical one in the book, and readers may find it beneficial to consider it a review chapter, having learned the details at a more leisurely place elsewhere. Background material on design at this level likely is needed only for Chapter 8, but a basic knowledge of fractional factorial designs and factorial effects contrasts is needed throughout. Chapter 3 presents the multinomial logit model in detail, and various parameterizations for the choice experiment setting are explored. Under this model, the objective is to select a design for an experiment that is optimal for estimating the main effects and, if required, the two-factor interaction contrasts. Several optimality criteria are mentioned, and the D-optimality criterion is selected. (Doptimal designs minimize the volume of the confidence ellipsoid for the contrasts of interest.) Because the multinomial logit model is a nonlinear model, the D-optimal designs depend on the true values of the parameters. Rather than looking at plausible ranges of parameters and how the optimal designs vary, the authors obtain optimal designs under the null hypothesis that all effects are negligible. Although restrictive, this is a sensible choice because even in this situation, the optimal designs are not easy to obtain theoretically. Nonetheless, I would have liked to have seen a thorough investigation of robustness of these designs when prior information is available. Chapter 4 gives very careful derivations of the information matrices for estimating the factorial contrasts for choice sets in which all attributes have two levels (e.g., high/low, or present/absent). These derivations are combinatorially demanding, but the results are interesting and useful. For example, for choice sets of size 2 and main effects models, designs composed of all pairs of alternatives that are mirror images of each other, such as (low, low, high) compared with (high, high, low), are D-optimal. The competing class of designs is composed only of designs that have certain balance properties rather than all possible designs. The designs being compared have different numbers of distinct choice sets, which is different from most design situations. Methods of obtaining highly efficient smaller designs are given, together with ways of avoiding dominating alternatives. Chapters 3 and 4 form much of the groundwork of the book. The material in these two chapters is self-contained but fairly concentrated. If I were teaching from this book, I would allow ample time for covering this material. Having gained mastery of the multinomial logit model in Chapter 3 and knowledge of how to select a two-level D-optimal design in Chapter 4, the reader can then cover subsequent chapters more quickly. Chapters 5 and 6 extend the results of Chapter 4 to choice sets of size m, where m is fixed and >2, and to sets of attributes that may not have the same number of levels. In each case specific requirements for the form of the D-optimal designs are obtained and useful tables of optimal designs are presented. Chapters 7 and 8 round out the book by discussing additional topics. For example, the inclusion of a “none of the above” option in every choice set is considered, and its effect on the efficiency of the design is investigated. Methods of constructing optimal designs for different sized experiments are explored in Chapter 8; it is here that a more detailed background knowledge of experimental design, such as that of Chapter 2, is needed. The basic methodology involves selecting a starting design, d , and creating choice sets by adding generators to the treatment combinations in d . A number of other methods of construction found in the literature are compared and shown to perform less well in general under the D-optimality criterion. Designs obtained from the SAS macro do seem to be competitive in many settings, however. This book provides an excellent treatment of the structure of D-optimal designs for choice experiments. I enjoyed reading it and wished that more details on the topics covered briefly in Chapter 7 had been included. I found the print in the book too small for comfortable long-term reading, but this is the only drawback to an otherwise excellent text.