On Some Methodological Issues in the Analysis of Survival Data from Prospective-type Experiments

Let X denote age, T be the survival time since occurrence of a certain initial event at age x, and z' = (z z z) be a vector ,.., l' 2' ... , S of characteristics measured on individual aged x. The purpose of this paper is to construct survival models for prospective-type experiments, emphasizing that the original lifetime distribution, SX(x; !), plays an essential role and cannot be ignored. The survival distribution since ~ initial event at age x (e.g. entry to the study, onset of a disease) is the future lifetime distribution, ST(tlx; !), and is closely related to SX(x;~. In the original population, this would be Sx(x+tlx>x; !). Exponential hazard rate models are suggested and justified, to some extent, in regard to their mathematical properties and simple biological interpretation of their parameters. When the covariables are linear functions of age, there is a close relation to Cox's model, and an approximate relationship with the multiple logistic linear model. The discussion is extended to competing risk analysis with several causes of death. Non-Gompertzian models are constructed by introducing non linear functional dependence of z's on x, or by allowing for interaction between the z's and x. The relationship between the z's and x can be estimated from repeated measurements of z's on each individual. The techniques of model building are illustrated by several examples. *) This work was supported by U.S. National Heart, Lung and Blood Institute contract NIH-NHLI-712243 from the National Institutes of Health. -21• INTRODUCTION 1.1. Probability Models in Follow-Up Studies In the last two decades there has been considerably increased interest in prognostic factors of mortality from chronic diseases, especially those classified as being of malignant or cardiovascular type. There is a vast number and variety of relevant studies, but here we restrict ourselves to two major types: epidemiologic prospective follow-up studies and clinical trials. Let ~'= (zl' .•. , zS) denote a vector of S characteristics, named briefly concomitant variables or covariab1es, which are considered as potential prognostic factors; let x be the age, and t the survival time since occurrence of a cer~n initial event, such as onset of a disease or entry to the study, for an individual at age x. The probabilistic and statistical problems lie in constructing a model of the survival distribution function (SDF), ST(tlx;!) = Pr{T>tlx; !}, or equivalently, a model for the hazard rate, AT(tlx;!) fT(tlx; !)!ST(tlx; !), where fT(tlx; !) is the probability density function, and estimating the parameters, including those which represent the contributions of concomitant variables. Two models have been commonly used: ~. (i) The logistic linear model q(Tlx; !) * * log~) • y + ~ox + ~ 'z , p\Tlx; !J .... (1.1) where q(Tlx; z) and p(Tlx;!) 1-q(Tlx;!) are the probabilities of death and survival, respectively, over a specified period T, and (ii) Cox's (1972) exponential hazard rate model