This book studies so-called frailty models intended for time-to-event data with a cluster structure. Random effects models have long been used with success to take into account correlation in for example longitudinal data analysis, for a recent book, see Diggle et al. (2002). Such models do also exist for time-to-event data and they are called frailty models. In its simplest form, a frailty is an unobserved random proportionality factor that modifies the hazard function of an individual or of related individuals. The term frailty itself was introduced by Vaupel et al. (1979) in univariate survival models. An individual with a large value of this unobserved random variable is more prone to experience the event of interest thus being more ''frail''.
The author's ambition is to provide a thorough presentation of the most current techniques used in this area of time-to-event analysis with emphasis on analysis of real data sets. The book is intended for students and applied statisticians. Code of the programs used in the examples is freely available from the publisher's web site, which the applied statistician certainly may find useful.
The three main chapters are Chapters 2, 4 and 5. Chapter 2 deals with parametric proportional hazards models with gamma frailty. This is good place to start as it is possible to get an explicit expression for the observed hazard and, since the baseline hazard function is specified using a parametric family, it is straightforward to do ordinary maximum likelihood analysis. Chapter 5 deals with the more challenging model where the baseline hazard function is left unspecified. Estimation is carried out using either the EM-algorithm or a penalized partial likelihood approach. Large sample properties are difficult to get at, but a breakthrough was made by Murphy (1994Murphy ( , 1995) ) and Parner (1998) for the (shared) gamma frailty model. Recently Zeng and Lin (2007) considered maximum likelihood estimation in very general hazard models and derived large sample properties. Ibrahim, Zeng, and Chen (2009) studied a general class of gamma frailty hazards models and gave results on how to test the null of the frailty variance being zero, which is a non-trivial problem. These more technical in-depth analyses of semi-parametric frailty models are not dealt with in detail, which is in line with the book being aimed at students and applied statisticians. The more theoretically interested reader needs to consult the journal literature. Chapter 4 describes different frailty distributions comparing their properties such as their ability to describe early and/or late dependence. Effective goodness-of-fit tools in this area do not yet seem to be developed. Unfortunately, the current book does not add much on this either. The authors report results from simulations to judge the effect of using a misspecified frailty model, however.
The parts on Bayesian analyses are very readable and I also enjoyed the part in Chapter 1 where the frailty model is used to describe selection rather than correlation, although the univariate situation is not followed up in later parts of the book. The book also contains a chapter on alternative methods such as marginal models if estimation of the correlation parameter is not of primary interest.
Another important book on multivariate time-to-event data and an alternative to the current book is Hougaard (2000). It covers many of the same topics and also some additional ones. Hougaard ( 2000) is perhaps less suited as a textbook for students, however.
To summarize, this book gives a good description of frailty models. It is well written and its many real applications and the availability of computer code make it a valuable resource for the applied statistician who needs to analyze clustered time-to-event data. The book uses only classical statistical techniques and should therefore be fairly easy to read for students who already have some basic r 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim