This book aims to be a “new introductory course in statistics, aimed at students in a wide range of programmes.” What is new about this textbook is a unified focus on both Bayesian and frequentist approaches and a focus on modeling. The first component allows side-by-side contrasts with the different types of analyses, with the goal of allowing more modern analyses and up-to-date curriculum, more relevant for big data and data science. This approach leverages recent advancements in technology such as MC Bayesian methods and parallel programming. Using a shared example for both approaches enhances readers’ comprehension of the distinction between the two methodologies, but may be challenging for the statistical beginner. A wide range of examples are used throughout the book, suitable for the different backgrounds of the readers.
The initial chapter provides a very brief overview of statistical modeling, statistical analysis, and statistical inference. We do wish more books started with these big picture ideas. We find many of our tertiary students are not as versed in the idea of “model” as we often assumed, and how this idea manifests itself in many disciplines. Then the book presents several research studies that will be the focused on in the text, as well as the “StatLab database.” If anything, we wish the book went into more detail initially on these datasets, but we appreciate the consistency of revisiting the same dataset to enhance student understanding of the evolution of statistical methods. For instance, Chapter 3.14 discusses a linear regression of the number of patients on the number of beds using a Maximum Likelihood (ML) fitted model. In Chapter 18.3.1, the same dataset is revisited, applying the ML fitted model to the log-scaled data. The author then proposes models for both mean and variance for the same dataset. Similarly, the author illustrates the evolution of general linear regression models from single parameters to double GLM using the same examples. Consistent use of the same dataset aids readers in grasping diverse models and their applications. Chapter 5 provides an overview of issues of bias in sample surveys, again providing important context from real-world applications.
The meat of the text begins in Chapter 6, a full treatment of probability. While this may feel like a bit of a detour from the real-world data examples just discussed, the text tries to balance this formal treatment of probability with interesting examples such as Geiger counters, ESP testing, screening tests, and the case of Sally Clark. Some material, such as the sums of independent random variables, is still presented a bit in isolation.
Chapter 7 covers statistical inference for discrete distributions. The primary motivation is survey sampling and both model-based inference theories and likelihood functions are introduced. This leads to discussion of the binomial distribution, including sufficient and ancillary statistics, and maximum likelihood estimation. The presentation is at a higher mathematical level than many of our introductory statistics students would appreciate. The middle of Chapter 7 lives up to the promise of contrasting frequentist and Bayesian approaches, though the discussion seems to lose some focus on the data-centric questions. This sequence of discussion is repeated for the Poisson distribution and multinomial distributions. The instructor may wish to supplement this discussion with more applications and discussion of research articles employing these methods. It is not clear to us how quickly our introductory students would catch up to conjugate prior distributions, bootstrapping, posterior draw, etc. Chapter 7 ends with a brief section on sampling without replacement, but again without a link back to how this would modify the analysis in a genuine research study. The Agresti and Caffo (Citation2000) alternative prior distribution is discussed, but we feel much more discussion could be included on how this approach straddles both frequentist and Bayesian approaches, and when the analysis does/does not change appreciably.
Chapter 8 compares binomials, motivated by randomized clinical trials. Again, both frequentist and Bayesian approaches are highlighted and we especially appreciated the discussion of other measures of treatment difference. This chapter should be accessible to students after Chapter 7 and has a stronger ties to genuine applications.
Chapter 9 focuses on data visualization. This approach, postponing detailed discussion of quantitative data, makes sense, but the discussion covering histograms, empirical distribution functions, and probability models is rather limited.
Chapter 10 addresses statistical inference for single-parameter continuous distributions like the exponential, Gaussian (with known scale parameter), and uniform distributions. The chapter begins by introducing the frequentist approach to constructing confidence intervals based on transformed data, followed by a presentation of the Bayesian approach. Hypothesis testing using both approaches is then explored. Chapter 11 extends this discussion to two parameter distributions such as the Gaussian, Lognormal, Weibull, and Gamma. However, the presentation heavily leans towards mathematical intricacies, making it challenging for introductory statistics students. The examples, though present, lack detailed step-by-step explanations, hindering the reader’s ability to grasp the concepts effectively. This absence of comprehensive worked-out steps could potentially limit the suitability of the text as an introductory textbook. The content appears to be more advanced, surpassing the expected level for introductory statistics students.
Chapter 12 focuses on model assessment, which is explained using the same datasets from previous chapters. The assessment of Gaussian, Lognormal, Exponential, Weibull and Gamma models are discussed. It seems statistical software is being used to generate 95% critical regions and the CDF plots. It would be valuable to provide additional details on the software’s application in the assessment process.
Chapter 13 explores statistical inference for the multinomial distribution, extending from the binomial distribution. Beginning with the multinomial likelihood function, the chapter again discusses both frequentist and Bayesian analyses. However, the in-depth discussion on the criticisms of the Haldane prior might be challenging for introductory-level readers. The chapter provides valuable insights into why the frequentist bootstrap is more commonly employed than the Bayesian bootstrap, illustrated through the multinomial example. Despite its richness, the content may pose difficulty for beginners to fully grasp.
Chapter 14 provides further discussion about model assessment and model averaging. For general model comparison, the author compares the Bayesian and frequentist approaches to model comparison with known parameters and with unknown parameters. After discussion on log-likelihood, how the Bayesian and frequentist use the deviance as a tool of inference is discussed. The author gives an example on model selection via deviance for the phone lifetime data. The author also points out the technical difficulty with the comparison of the multinomial deviance.
Chapter 15 is about regression models with a Gaussian response variable, and their maximum likelihood and Bayesian analysis. The Vitamin K dataset, which appears first in Chapter 3, is consistently employed throughout the chapter. The chapter also discusses common machine learning extensions to ridge regression, the lasso, and principal component regression (PCA) within the context of statistical modeling. The model robust analysis provides valuable insights, however, it may be challenging for beginners. The rest of the chapter discusses further applications of multiple regression models including ANOVA and ANOCOVA. Additional examples are presented to show the broader scope of application. It would be beneficial for the students to provide more details with these examples.
Chapter 16 uses the maximum likelihood EM algorithm and the Bayesian Data Augmentation algorithm to analyze many kinds of incomplete data. It starts with the introduction of different types of missingness, then explores different approaches to address these missing data. While the theoretical part of EM is not covered in the book, several references are provided for advanced readers seeking further insights.
Chapter 17 extends the Gaussian multiple regression framework to generalized linear models (GLMs) and further extension of GLMs. The chapter starts with using the regression models in the exponential family of distributions. Then the author talks about the GLM algorithm and package, followed by the Bayesian package development. The chapter covers binary response models and transformations, providing multiple examples to show the practical applications. Poisson regression is discussed via the application of fish species data.
Chapter 18 is about the extensions of GLMs to two parameter distributions (e.g., extending an example of gamma regression from Chapter 17), revisiting several data sets introduced in earlier chapters. Segmented regression is introduced in this chapter with two examples. GLM is also extended to a nonlinear situation through Heterogeneous Regressions. The last two topics covered in this chapter are neural networks and social networks. The brief introduction of various GLM extensions in this chapter is beneficial to readers as it provides a comprehensive overview, offering a broader perspective of the model.
In general, we appreciate the focus on applications and comparing and contrasting analyses for the same dataset, especially the Bayesian/Frequentist contrasts, but feel the illustration of many of these examples is too concise. We feel students (or at least their instructors) would benefit from more detailed steps such as showing values substituted into the formulas and/or the use of specific software packages, rather than relying on formulas and final answers. This, along with the advanced level of mathematical discussion throughout the text, makes it more suitable in our opinion for graduate study or as a valuable reference text rather than for students just beginning their statistical journeys. Numerous typos in the text can also make the discussion difficult for readers to follow (e.g., location of some of the tables and figures).
Statistics Department, California Polytechnic State University
San Luis Obispo, CA
Beth Chance
Statistics Department, California Polytechnic State University
San Luis Obispo, CA
[email protected]
Reference
- Agresti, A., and Caffo, B. (2000), “Simple and Effective Confidence Intervals for Proportions and Difference of Proportions Result from Adding Two Successes and Two Failures,” The American Statistician, 54, 280–288. DOI: 10.1080/00031305.2000.10474560.