Lectures on Survival Analysis

Preface. These notes, though mainly written after the course was given, follow quite closely my lectures at the 22nd Saint FlourÉcole d' ´ Eté de Calcul des Probabilités, 8–25 July 1992. I am really grateful to the organisers and the participants who have shaped the lecture notes in many ways. The vague title is a cover-up for the more honest 'topics in and around survival analysis which interest me at the moment, with an audience of French probabilists in mind'. Accordingly, the main theme of the lectures—to my mind the fundamental notion in survival analysis—is product-integration, and to begin with I have tried to cover its basic theory in fair detail. Probabilistic connections are emphasized. The next group of lectures study the Kaplan-Meier or product-limit estimator: the natural generalisation, for randomly censored survival times, of the empirical distribution function considered as nonparametric estimator of an unknown distribution. Using product-integration, the asymptotics of the Kaplan-Meier estimator are treated in two different ways: firstly, using modern empirical process theory, and secondly, using mar-tingale methods. In both approaches a simple identity from product-integration, the Duhamel equation, does all the real work. Counting processes lurk in the background of the martingale approach though they are not treated here at length; the interested reader is urged to follow them up in the book Statistical models based on counting processes (1993); the book is referred to as 'ABGK' in the sequel. I also neglect statistical issues such as asymptotic optimality theory, partly with my audience in mind, and partly because this subject is still very fluid with, in my opinion, interesting developments ahead; the reader is referred in the meantime to Section IV.1.5 and Chapter VIII of ABGK. However beneath the surface statistical ideas, especially involving nonparametric maximum likelihood, are ever-present and give the real reason for many otherwise surprising results. Neglected in the written notes are applications, though, in the real lectures, illustrations taken from ABGK were prominent. Most of this part of the course covers classical material, though there are also new results. One of the most striking is the proof, using discrete time martingales, of Stute and Wang's (1993) very recent Glivenko-Cantelli theorem for the Kaplan-Meier 2 estimator. I suspect the techniques used here could find applicability in many other problems in survival analysis of a sequential or combinatorial nature. Another striking result is the use of van der Laan's identity (on estimation …