» Counting Processes and Survival Analysis (Wiley Series in Probability and Statistics)

Counting Processes and Survival Analysis (Wiley Series in Probability and Statistics) Reviews

Customer Rating:
Summary: Well-written and Informative
Comment: This is one of the best treatments I have read on the martingale theory of the analysis of survival data. This material really gets you under-the-hood of proportional hazards modeling and details when the theory is applicable and where things can break down in the models. This is positively a "must-own" for advanced researchers working with survival data and a "good-to-have" desk reference for anyone dealing with survival data.

Chapter 0 provides a meaningful introduction, and the authors use real-world survival data sets to set the stage for the basic concepts. They like the PBC liver study data set a lot and use it frequently through the text. There is some great material in this short chapter, including a formal definition of the hazard function, a nice overview of the Kaplan-Meier estimator, and an introduction of the Cox model with a very nice, intuitive treatment of the derivation of Cox's partial likelihood function. The authors also set the stage for the martingale theoretic treatment and give three motivating (and compelling) reasons for the theory.

Chapter 1 covers the basics from stochastic analysis that are required for the remainder of the book. Basic definitions and concepts like filtration, conditional expectation, the definition of a martingale and the Doob-Meyer decomposition are covered. No prior knowledge of stochastic analysis is assumed. However, a good understanding of measure theory is very helpful (something along the lines of the first four chapters for Rudin's Real and Complex Analysis). The chapter wraps up with the martingale transformation theorem.

The main aim of Chapter 2 is to establish quadratic variation properties for continuous compensators of counting processes. This material is heavily used in the asymptotic Brownian motion material in Chapter 5 (where a large part of the story rests on the limiting behavior of quadratic variation). To get there, a number of localization results are established. The Optional Sampling Theorem is stated and used (the proof is referenced out to the literature). The main workhorse, the Optional Stopping Theorem is established as a nice application of optional sampling.

Chapter 3 is a wonderful, rigorous treatment of the survival estimators and test statistics that we know and love and always wondered why these are vaguely true. The main result is the consistency of the Kaplan-Meier estimator, which foreshadows the consistency results for the Cox regression estimator established in Chapter 8.

The proportional hazards model and multiplicative intensity models are the main focus of Chapter 4. The modeling framework is introduced, basic concepts such as uninformative censoring are introduced and the method of partial likelihoods is explored in depth. The chapter just has great little pearls sprinkled throughout, including martingale properties for Breslow's estimator for baseline hazard and a number of modeling building diagnostic techniques. There is also a very nice set of graphs on the martingale residual technique of assessing functional form of continuous covariates.

Chapter 5 is the core of the book and develops the asymptotic limit results, including the martingale central limit theorem for counting processes. The chapter is nearly self-contained, with the occasional reference to one of the classical probability texts like Chung or Billingsley. Proofs that could prove a distraction to the main thread are placed in the appendix.

Chapters 6, 7 and 8 provide very nice applications of the martingale central limit theorem. These include: building confidence bands, establishing large sample properties of test statistics and putting Cox's technique of partial likelihoods on solid footing by establishing by establishing consistency and asymptotic normality. As an important final topic for consideration, asymptotic efficiency of the Cox estimator is explored. It is somewhat disappointing that no formal theorem establishing conditions for asymptotic efficiency is presented.

As a wish list item for the next edition, it would be nice to see a chapter or two introducing multivariate survival analysis (competing risks), and the role of Markov processes. For the definitive work on the topic of multivariate survival analysis, I recommend Andersen, Borgan, Gill and Kieding's Statistical Models Based on Counting Processes (Springer Series in Statistics).

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