Multi-state models are stochastic processes that represent transitions of an individual between discrete states.  In statistical modelling, they are used for two general kinds of data.  Firstly, data where the state is known at any time during an individual's follow-up.   Secondly, data where the state is only known at a finite series of times.  In literature on multi-state models, it is not always instantly clear which of the two situations we are in, but the first case has perhaps been treated more extensively in textbooks (e.g. Beyersmann et al. 2011, Andersen et al. 1993).

This is the first book devoted to multi-state models for intermittently-observed data that I know of.  As the author of the "msm" R package for this class of models and data, I have seen how common and widespread they are, being used in areas including medicine, epidemiology, biology, ecology, economics, finance and engineering.  These methods have only previously been covered in scattered papers, software manuals and book chapters, so it's great to finally see a full textbook.

This book is motivated by applications to demography, disease and survival. This is shown by the book's title, and the structure, which starts with standard survival analysis for times to death.   This focus on one area is probably a sensible choice to keep the book clear and concise, though inexperienced practitioners in other areas may face an extra hurdle.  For example, all the examples have an absorbing state representing death, when this is not necessary in a general multi-state model.

"Interval-censored" times of disease onset or state change are introduced in the third chapter. These are also called "intermittently observed" or "panel" data in other literature.  Multi-state models with general state-transition patterns are then described.  An important later chapter covers estimation of expected time spent in states such as healthy life.   A range of advanced topics such as frailty models, Bayesian inference, hidden Markov models and aggregate data are introduced fairly briefly, though with enough detail to get the substance across, and references to other resources.  In the Bayesian chapter I would have liked to see more detail of how to specify informative priors on an intuitive scale.  Also the JAGS Bayesian modelling software has useful built-in functions for this class of models (an "msm" module, including a matrix exponential function) which wasn't acknowledged.

The applicability of the models to real data is stressed throughout.  Each new model is illustrated through one of several running examples related to long-term illness or ageing.   The applied interpretations of the examples are discussed clearly.  Though in the results tables, I would have preferred estimates of parameters such as hazard ratios to be presented on a natural scale, as they would be in a medical journal, rather than a log scale.  Likewise, transition intensities are more easily interpreted in terms of sojourn times and next-state probabilities.  While less mathematically convenient, transforming results to their most interpretable scale would be a simple step to bring theory and practice closer together.

The models advocated are flexible enough to cover all typical applications.  Dependence of transition rates on age or time is emphasised, and modelling this dependence is made straightforward through a novel piecewise-constant approximation method.  This is an advance over methods available in current software.   Sensible modelling choices, such as parsimony constraints on model parameters, are illustrated through the examples.

The writing style strikes a good balance between readability and mathematical rigour.  Each new topic is generally begun with an approachable explanation, with formal definitions following later.    A helpful appendix gives some useful algebraic results and example R implementations of the non-standard methods.  The level is approximately suitable for a postgraduate statistics student or applied statistician.   Applied practitioners without formal statistical training should therefore be warned that the statistical modelling is treated from first mathematical principles.  The intended audience seems to be people accustomed to writing custom code based on algebraic definitions.  The book even presents details of an optimisation algorithm used for maximum likelihood estimation, when efficient general-purpose optimisers are available in most statistical software.

I'll be recommending this book to users of the "msm" package and my students, especially anyone modelling chronic diseases or age-dependent conditions.    I hope that the software will eventually catch up with some of the advanced methods presented here, so that they become routinely used in practice!


Bibliography:

Beyersmann, J., Allignol, A. and Schumacher, M., 2011. Competing risks and multistate models with R. Springer

Andersen, P.K., Borgan, O., Gill, R.D. and Keiding, N., 1993. Statistical models based on counting processes. Springer