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 togeher with the author’s reply.
Reviewer: D. R. Cox