Effects of informative censoring on the hazard function: Application to the proportional hazards regression model

The standard analyses of survival data involve the assumption that survival and censoring are independent. When censoring and survival are related, the phenomenon is known as informative censoring. This paper examines the effects of an informative censoring assumption on the hazard function and the estimated hazard ratio provided by the Cox model.^ The limiting factor in all analyses of informative censoring is the problem of non-identifiability. Non-identifiability implies that it is impossible to distinguish a situation in which censoring and death are independent from one in which there is dependence. However, it is possible that informative censoring occurs. Examination of the literature indicates how others have approached the problem and covers the relevant theoretical background.^ Three models are examined in detail. The first model uses conditionally independent marginal hazards to obtain the unconditional survival function and hazards. The second model is based on the Gumbel Type A method for combining independent marginal distributions into bivariate distributions using a dependency parameter. Finally, a formulation based on a compartmental model is presented and its results described. For the latter two approaches, the resulting hazard is used in the Cox model in a simulation study.^ The unconditional survival distribution formed from the first model involves dependency, but the crude hazard resulting from this unconditional distribution is identical to the marginal hazard, and inferences based on the hazard are valid. The hazard ratios formed from two distributions following the Gumbel Type A model are biased by a factor dependent on the amount of censoring in the two populations and the strength of the dependency of death and censoring in the two populations. The Cox model estimates this biased hazard ratio. In general, the hazard resulting from the compartmental model is not constant, even if the individual marginal hazards are constant, unless censoring is non-informative. The hazard ratio tends to a specific limit.^ Methods of evaluating situations in which informative censoring is present are described, and the relative utility of the three models examined is discussed. ^

Joint effects of pulmonary function and aerobic power on survival in a cohort of healthy adults

Objective measurements of physical fitness and pulmonary function are related individually to long-term survival, both in healthy people and in those who are ill. These factors are furthermore known to be related to one another physiologically in people with pulmonary disease, because advanced pulmonary disease causes ventilatory limitation to exercise. Healthy people do not have ventilatory limitation to exercise, but rather have ventilatory reserve. The relationship between pulmonary function and exercise performance in healthy people is minimal. Exercise performance has been shown to modify the effect of pulmonary function on mortality in people with chronic obstructive pulmonary disease, but the relationship between these factors in healthy people has not been studied and is not known. The purpose of this study is to quantify the joint effects of pulmonary function and exercise performance as these bear on mortality in a cohort of healthy adults. This investigation is an historical cohort study over 20 years of follow-up of 29,624 adults who had complete preventive medicine, spirometry and treadmill stress examinations at the Cooper Clinic in Dallas, Texas.^ In 20 years of follow-up, there were 738 evaluable deaths. Forced expiratory volume in one second (FEV$\sb1$) percent of predicted, treadmill time in minutes percent of predicted, age, gender, body mass index, baseline smoking status, serum glucose and serum total cholesterol were all significant, independent predictors of mortality risk. There were no frank interactions, although age had an important increasing effect on the risk associated with smoking when other covariates were controlled for in a proportional-hazards model. There was no confounding effect of exercise performance on pulmonary function. In agreement with the pertinent literature on independent effects, each unit increase in FEV$\sb1$ percent predicted was associated with about eight tenths of a percent reduction in adjusted mortality rate. The concept of physiologic reserve is useful in interpretation of the findings. Since pulmonary function does not limit exercise tolerance in healthy adults, it is reasonable to expect that exercise tolerance would not modify the effect of pulmonary function on mortality. Epidemiologic techniques are useful for elucidating physiological correlates of mortality risk. ^