zbMATH — the first resource for mathematics

Regression models and life-tables. With discussion by F. Downton, Richard Peto, D. J. Bartholomew, D. V. Lindley, P. W. Glassborow, D. E. Barton, Susannah Howard, B. Benjamin, John J. Gart, L. D. Meshalkin, A. R. Kagan, M. Zelen, R. E. Barlow, Jack Kalbfleisch, R. L. Prentice and Norman Breslow, and a reply by D. R. Cox. (English) Zbl 0243.62041
The statistical analysis is considered of observations on non-negative variables subject to censoring on the right. That is, for each individual we may observe either the value of the random variable \(T\) or that \(T\) exceeds some given value, not necessarily the same for all individuals. Such data arise commonly in medical, actuarial and industrial contexts. For simplicity, call \(T\) a failure time. Further it is assumed that there is available for each individual a vector of explanatory variables which may influence \(T\). Possible approaches to the analysis are reviewed. Primarily the paper deals with a model in which the age-specific failure rate (hazard function) has the form \[ \exp(\beta_1z_1+\dots+\beta_pz_p)\lambda_0(t), \] where \(\lambda_0(\cdot)\) is an arbitrary unknown function, \(\beta_1,\dots, \beta_p\) are unknown parameters and \((z_1,\dots, z_p)\) is the vector of explanatory variables. A modified likelihood function is obtained for inference about \(\beta_1,\dots, \beta_p\) by arguing conditionally on the observed failure times. From this likelihood tests and confidence regions are obtained. In the special case of a two-sample problem with proportional hazards, the test of the null hypothesis of zero difference reduces to a generalization to censored data of the most efficient two-sample rank test for exponential distributions. A number of generalizations are considered and the relation with stochastic models discussed.
Discussion of the paper by 15 contributors is included together with the author’s reply.
Reviewer: D. R. Cox

62J05 Linear regression; mixed models
62N05 Reliability and life testing
62F10 Point estimation
Full Text: Link