The following are some the books on survival analysis that I have found useful. There are of course many other good ones not listed.

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- Modelling Survival Data in Medical Research, by Collett (2nd edition 2003)
- Survival and Event History Analysis: A Process Point of View, by Aalen, Borgan and Gjessing (2008)
- Handbook of Survival Analysis, edited by Klein, van Houwelingen, Ibrahim and Scheike (2014)
- Statistical Models Based on Counting Processes, by Andersen, Borgan, Gill and Keiding (1993)

**’Modelling Survival Data in Medical Research, by Collett (2nd edition 2003)’**

This is the survival text book I bought while doing my MSc in Medical Statistics. It provides a thorough coverage of all the main methods and principles needed for survival analysis. As suggested by the title, methods are demonstrated throughout by application to medical examples. As well as core topics such as the Kaplan-Meier survival function estimator, log rank test, Cox model, etc, the second edition I have (there is now a third) includes coverage of additional topics such as accelerated failure time models, models for interval censored data, and sample size calculations for survival studies. The book is very good for the applied statistician in that a lot of emphasis is given to model diagnostics and recommendations about the relative advantages and disadvantages of different methods.

**’Survival and Event History Analysis: A Process Point of View, by Aalen, Borgan and Gjessing (2008)’**

This book serves as an excellent introduction to survival and event history analysis methods. Its mathematical level is moderate. Aalen did pioneering work in his PhD thesis on using the theory of counting processes to derive results for the statistical properties of many survival analysis methods, and this book emphasizes this approach. Indeed, the authors write that part of their motivation for this book is that the counting process theory had been somewhat absent from most survival analysis text books (an exception being this book), due to the apparent technical nature of the theory. They argue that conceptually the counting process theory, at least at a high level, is not terribly difficult to understand, and that because it provides such an elegant theory for the statistical properties of lots of the methods in use, an understanding of the theory is highly desirable.

The first chapter introduces through examples the basic concepts involved in survival and event history analysis, and gives an intuitive high level introduction to the theory of counting processes. A more detailed exposition of the latter is then given in the second chapter. The third chapter then covers the non-parametric Nelson-Aalen estimator of the cumulative hazard function, the Kaplan-Meier estimator, and non-parametric tests of equality of survival functions. The fourth chapter then considers semiparametric regression models, including Cox’s model and Aalen’s additive hazards model, with proofs of their statistical properties which exploit the counting process theory. Chapter 6 is a fascinating exposition of the implications of unobserved between subject variation, otherwise known as frailty in survival analysis. It is this chapter (and attending a course by the book’s authors) which was the basis of my previous blog post on interpreting changes in hazard and hazard ratios. The remaining chapters, which I have read to a lesser extent, cover multivariate survival data, models for recurrent event data, causality, first passage time models and models for dynamic frailty.

**’Handbook of Survival Analysis, edited by Klein, van Houwelingen, Ibrahim and Scheike (2014)’**

This book is another in the recent CRC Press series of handbooks of modern statistical methods. Like the others in the series, it contains contributed chapters from a wide range of leading authors in the field. I have only recently obtained this book, and so have not read it extensively. The range of topics covered is though extensive, and in particular many topics are included which may not be included in more standard survival analysis texts.

The first part covers various regression modelling approaches for classical right censored survival data, while the second considers methods for competing risks. The third is on model selection and validation, including a chapter by Quigley and Xu on their work on proportional hazards models when the proportional hazards assumption does not hold. Part four covers other types of censoring, including that induced by nested case-control and case-cohort study designs, and interval censoring. The fifth part covers multivariate survival data, while the last part covers topics relevant for clinical trials, including a chapter on group sequential methods.

For those conducting research on methods in survival analysis, the book is likely to be very relevant as an up to date tour of the current state of play.

**’Statistical Models Based on Counting Processes, by Andersen, Borgan, Gill and Keiding (1993)’**

This 700+ page tome is a technical and comprehensive exposition of the theory of counting processes applied to statistical models of among other things, survival and event histories. It gives a rigourous description of this theory, illustrated with ample examples throughout. The Kaplan-Meier estimator of the survival curve, Nelson-Aalen cumulative hazard estimator, and non-parametric tests (e.g. log rank) are introduced, and their statistical properties derived using the elegant theory of counting processes. Regression models are then covered, both parametric and semi-parametric (including Cox’s proportional hazards model). There are also chapters on frailty models and asymptotic efficiency, the latter building on recent (at the time) work on semiparametric theory.

I bought this book quite cheaply a few years ago and had not really read it to any extent, largely because I was put off by the heavy going maths. However, after reading Aalen, Borgan and Gjessing’s book quite extensively recently, I have at last started getting into the book, in relation to the statistical properties of weighted log rank tests. I think it is probably fair to say that this book is not suited to applied researchers looking to learn about survival analysis methods in order to apply them. But for those wanting to get to the heart of the theoretical basis for the majority of the statistical methods used for survival analysis today, it is the go to reference.