The book by Visser and Speekenbrink (Reference Visser and Speekenbrink2022) provides an integrated framework for finite mixture (FM) and hidden Markov (HM) models. FM models (e.g., Frühwirth-Schnatter et al., Reference Frühwirth-Schnatter, Celeux and Robert2019; McLachlan et al., Reference McLachlan, Lee and Rathnayake2019) have been proposed since many years and nowadays have become very popular due to the rise of many classification and clustering challenges as a consequence of the increasing availability of large and complex data, especially in behavioral science; see, among others, Everitt and Hand (Reference Everitt and Hand1981). These models are typically specified by formulating the conditional distribution of the response variables given mixture components having certain probabilities, also known as weights, so that individuals assigned to the same component share a common distribution of these variables. The Gaussian FM model (see McLachlan and Peel Reference McLachlan and Peel2000, for a thorough illustration) is the most common model-based clustering method for analyzing continuous responses that also allows for classification, which is “...the process of assigning group membership labels to unlabeled observations. In this context, a group may be a class or a cluster...” (McNicholas, Reference McNicholas2016).
HM models are dependent mixture models proposed for analyzing time-series data; see, among others, MacDonald and Zucchini (Reference MacDonald and Zucchini1997), and the review provided in Ephraim and Merhav (Reference Ephraim and Merhav2002). Then, they have been extensively developed and used for the analysis of longitudinal data starting from the work of Wiggins (Reference Wiggins1973); see, among others, Bartolucci et al. (Reference Bartolucci, Farcomeni and Pennoni2013) for detailed illustrations and extensions of these models and Visser and Speekenbrink (Reference Visser and Speekenbrink2014) for a related discussion. HM models assume that the response variables depend on a latent process that follows a Markov chain, typically of first order and homogeneous over time. Over the last two decades, these models have had a profound impact on applied research for analyzing multivariate longitudinal data in particular. Their popularity is due to mathematical tractability that allows them to handle the temporal structure of many observed phenomena. In this framework, it is also possible to account for missing data, which frequently arise under different structures. HM models provide a practical model-based dynamic clustering method (Bouveyron et al., Reference Bouveyron, Celeux, Murphy and Raftery2019) for identifying latent group structures, also represented in terms of trajectories.
The book of Visser and Speekenbrink (Reference Visser and Speekenbrink2022) is a compendium of these models with examples based on simulated and real data arising from social sciences research fields, especially psychology. The authors are expert researchers who have contributed to the theoretical and computational development of the models at issue for many years. The book links theory and applications through the open-source statistical environment R (R Core Team, 2024).
1. Book Contents
Each chapter of the book provides a general introduction to specific models and methods, along with applications based on basic R functions and certain R packages. The output from the used software is discussed in some detail. A GitHub repository at https://depmix.github.io/hmmr/ conveniently collects all R codes and data proposed in the book. These are also available as an R package called hmmr. The authors make extensive use of the functions of the package depmixS4 (Visser and Speekenbrink, Reference Visser and Speekenbrink2010) developed and currently maintained by them.
Chapter 1 introduces basic R commands and illustrates the data used in the book. This chapter also discusses the main research questions related to proposed datasets that can be answered through these models.
Chapter 2 covers FMs, and latent class models, along with computational methods for estimating them and, specifically, maximum likelihood estimation using the expectation–maximization algorithm. This chapter also covers parameter identification and model selection (i.e., choosing the number of mixture components or latent classes).
Chapter 3 provides an overview through empirical applications of the univariate and multivariate FMs and latent class models. Bootstrap methods are introduced to compute standard errors.
Chapter 4 introduces HM models, along with maximum likelihood methods for estimating them. This chapter also covers how the models can accommodate different missing data mechanisms. Details on numerical optimization are illustrated with the package depmixS4.
Chapter 5 focuses on univariate HM models with empirical applications related to psychology, which have been investigated in previous authors’ articles.
Chapter 6 illustrates applications of the HM models to multivariate time-series based on the R packages developed by the authors.
Chapter 7 concludes the book with some advanced topics in HM models, such as higher-order serial dependencies and Bayesian estimation. It illustrates these topics with reproducible R code examples. The problem of obtaining standard errors for assessing the uncertainty of the estimated parameters is treated in particular detail.
2. Intended Audience
Basic elements of statistical inference and computational procedures are a preliminary requirement, and therefore, the book is ideal for instructors, graduate and postgraduate students, and applied researchers. Throughout the book, the authors illustrate the implementation of the methods with several data examples, codes, visualization of data, and results using an expository style. The book is intended to guide the reader step by step on how to apply basic versions of FM and HM models and keep the text generally accessible to a broad audience, mainly from sociology, psychology, and economics, who have not used these models before and would apply them.
3. Suggested Improvements for Future Editions
While we believe that the book by Visser and Speekenbrink (Reference Visser and Speekenbrink2022) is a valuable contribution to the development of knowledge and applications of FM, latent class, and HM models, there are some issues that we hope the authors could consider for future editions of the book. First, the initial introduction to the software could be avoided and the authors may refer to specific references. At the same time, the authors could clearly define the different data structures (i.e., cross-section, time-series, and longitudinal data), which can be analyzed by the models illustrated in the book, and they could also avoid to illustrate Gaussian FM models using time-series data characterized by serial dependence.
Second, while the book introduces FM and HM models, along with their interpretation, a similar introduction is absent for the latent class model, although this is one of the topics of the book, and this could be added. Certain concepts of Bayesian inference could be dealt with more detail, as well as how to include covariates in HM models, which could be part of one of the advanced sections.
Third, the authors could add some important references about popular models proposed in the literature, such as the multilevel latent class model (Vermunt Reference Vermunt2003) and latent class item response theory models, which may be employed to perform hierarchical clustering (Bartolucci et al., Reference Bartolucci, Bacci and Gnaldi2015). Certain R packages mentioned in the book, such as poLCA (Linzer and Lewis, Reference Linzer and Lewis2011), should be properly cited, and the authors could also mention some other relevant packages for estimating FM models, such as flexmix (Grün and Leisch, Reference Grün and Leisch2008).
Finally, the first three chapters present an interesting section on further readings useful for scholars exploring more advanced versions of the proposed models. However, this section is not present in the last three chapters. Due to the gained popularity of the HM models, it could be really useful to provide more details for the readers in the later chapters as well. Moreover, the last chapter could be stronger if it will also refer to many of the other popular statistical software to estimate FM models, such as Latent GOLD (Vermunt and Magidson, Reference Vermunt and Magidson2021), Stata (GLAMM, (Rabe-Hesketh et al., Reference Rabe-Hesketh, Skrondal and Pickles2004), and other R packages to estimate FM or HM models such as Mclust (Scrucca et al., Reference Scrucca, Fraley, Murphy and Raftery2023) and LMest (Bartolucci et al., Reference Bartolucci, Pandolfi and Pennoni2017).
