Generalized Linear Models With Applications in Engineering and the Sciences

Non-normal data is commonplace in many engineering related experiments. A traditional approach has been to transform the data and then conduct the analysis using classical normal theory techniques. While data transformation is often very useful, the use of generalized linear models (GLMs) for non-normal data has gained widespread attention in recent years. GLMs make use of the natural distribution of data and accommodate a wide range of non-linear functions when modeling. Many statistical software packages (SAS, SPSS, SPLUS, etc.) have implemented GLMs within their framework. Although the use of GLMs has received a great deal of attention in the biological and biomedical fields for close to 20 years, only recently have engineering disciplines begun to embrace these techniques (Hamada and Nelder, 1997; Myers and Montgomery, 1997). This is the first book on GLMs written by authors who are primarily associated with engineering related applications. Combined, the authors have over 85 years of experience in statistical consulting and training. All three authors have produced well-received texts in such areas as experimental design, modeling, and engineering statistics. Throughout the book, the authors continually keep in mind that their readers are students of statistics, whether this may be practitioners, graduate students, or even Ph.D. level statisticians who desire familiarity with an area outside of their specific expertise. The text begins with a nearly 100-page review of linear and non-linear regression. Although it is stated in the preface that the material is appropriate for anyone who has completed a course in regression analysis, the essentials of regression analysis are thoroughly covered in the first three chapters. Chapter 2, “Linear Regression Models”, is 50 pages and covers material on estimation (least squares, weighted least squares, and maximum likelihood), inference, and diagnostic checking. The topical format of Chapter 2 “Estimation, Inference, and Diagnostic Checking” could be viewed as a template for Chapters 3–6. The benefit of this is that the reader is afforded the opportunity of understanding the intuitive genesis of GLMs. As with all of the topics covered in the text, the authors illustrate concepts with well-chosen examples. When encountered with a non-linear model, a natural approach may be to simply transform the model into a linear model and analyze the transformed model. Recognizing this, the authors begin Chapter 3, “Nonlinear Regression Models,” by taking the time to show the reader the potential perils of doing this. It would be easy enough for the authors to explain this as multiplicative versus additive errors and then “leave it to the reader” to figure out the subtleties. However, the authors take a data set and analyze it both ways, showing the subtleties. Another highlight of Chapter 3 arrives in the explanation of parameter estimation in non-linear regression. It is well known that for non-linear regression, when the parameters are estimated via least squares, the score equations have no simple closedform solutions and some iterative technique is used to find a solution. This is clear when one works out the messy derivatives involved with least squares of a non-linear model, but often, intuition stops here. The authors once again appeal for a higher intuition and give a nice explanation of the geometry of linear versus non-linear least squares. For illustrative purposes, they consider a two variable linear and non-linear model and then show the sum of squared residuals contour plot versus the two model parameters. It is clear why a unique solution exists for least squares of linear model but not why non-unique solutions exist in the nonlinear model setting. The illustration is clear and sets the stage for complicated design topics discussed in the final chapter. Logistic and Poisson regression are perhaps the most common applications of GLMs. Chapter 4, “Logistic and Poisson Regression Models”, is devoted to the discussion of these models. In the section on estimation, the authors not only relate the discussion back to estimation in the classic linear model but they also demonstrate that maximum likelihood and weighted least squares result in the same score equation, thereby providing the same parameter estimates. Regarding inference, a careful explanation is provided on the differences between Wald and likelihood inference. Example data sets are analyzed throughout the chapter using SAS PROC LOGISTIC and PROC GENMOD. One particularly interesting data set deals with an electronic circuit card assembly by a wave-soldering process. The response is Poisson and is characterized by overdispersion. Ignoring overdispersion can result in overly optimistic standard errors of the parameter estimates. This concept is well-illustrated as the authors analyze the data