Principles of Survival Analysis

Principles of Survival Analysis 2 Preface When I first tried to learn about survival analysis I found there to be an unwelcome gap in literature. There are many textbooks and papers that give the formulas of survival analysis, explain how one should use standard statistical software packages, and give examples of applications of standard methods to real data. Then there are hardcore statistics papers that often focus on very mathematical/technical questions and are often written in the language of measure theory. I could not find good textbooks that sit in the middle, to explain in detail the conceptual and mathematical basis of the formulas of survival analysis. Where do all these formulas come from? What assumptions were made? How exactly are key quantities defined? The traditional survival analysis methods such as proportional hazards regression and Kaplan-Meier risk estimators were perfect for their time (the 1970s), when each university had just one computer (that filled several rooms, and was probably slower than today's average laptop) and mathematical methods had to be simple in order to be applied to real data. However, nowadays the use of this traditional methodology is increasingly inappropriate. In modern biomedicine we have new problems and new ambitions: we want to use the wealth of new data for personalised medicine, but face complex heterogeneous diseases and cohorts, and a vast dimensional mismatch between the number of (e.g. genetic) covariates and the number of patients on which we have data. I would say that at this moment the pressing problems in survival analysis are not measure-theoretic, they are more basic. We need to develop new methods, that do not deviate unnecessarily from the traditional ones (and preferably include these in special simplifying limits), but can handle the big questions of today – individualised prediction, cohort heterogeneity and dimensional mismatch. To do this we need to understand and review in full detail the principles and mathematical derivations of traditional survival analysis, and rebuild the edifice where this is needed to accommodate the new questions that we want survival analysis to answer. These lecture notes are written as an attempt to fill the above gap. I try to map out and explain the definitions, assumptions and derivations of the main methods in survival analysis. The style is that of the physicist who appreciates that there is a time and place for investigating mathematical subtleties like Lebesgue measures, noncommuting limits, and distribution …