Hazard rate models with covariates

Many problems, particularly in medical research, concern the relationship between certain covariates and the time to occurrence of an event. The hazard or failure rate function provides a conceptually simple representation of time to occurrence data that readily adapts to include such generalizations as competing risks and covariates that vary with time. Two partially parametric models for the hazard function are considered. These are the proportional hazards model of Cox (1972) and the class of log-linear or accelerated failure time models. A synthesis of the literature on estimation from these models under prospective sampling indicates that, although important advances have occurred during the past decade, further effort is warranted on such topics as distribution theory, tests of fit, robustness, and the full utilization of a methodology that permits non-standard features. It is further argued that a good deal of fruitful research could be done on applying the same models under a variety of other sampling schemes. A discussion of estimation from case-control studies illustrates this point.