Retrospective studies and failure time models

SUMMARY The proportional hazards failure time model of Cox (1972) is adapted to the retrospective epidemiological study, in which cases and controls are sampled from a population in order to relate disease incidence rates to exposure to suspected risk factors. The method permits simultaneous study of several exposure variables, that may be discrete or continuous or a mixture of both, in the presence of additional explanatory variables and competing risks. The proportional hazards framework provides a fresh look at the concepts of matching, choice of controls, confounding and effect modification. The relative risk is a common and useful measure of association between a chronic disease and suspected risk factors. It is given by the ratio of disease incidence rates for variously exposed individuals. Use of this measure involves the often implicit assumption that the ratios of incident rates are relatively constant with respect to age and other personal char- acteristics. The proportional hazards model introduced by Cox (1972) gives a probabilistic formulation for the constant relative risk concept. It has been discussed both in relation to survival studies (Kalbfleisch & Prentice, 1973; Breslow, 1975) and in relation to prospective epidemiological studies (Breslow, 1977). The prospective epidemiological study involves follow- ing disease-free individuals having various exposure levels forward in time to observe disease incidence. Such studies, unfortunately, are frequently too long term and expensive to be feasible, especially if diseases of low incidence are under study. The retrospective case- control study attempts to circumvent these difficulties by selecting, from a well-defined population, cases with the study disease for comparison with a control sample of persons who are disease-free or have some other diagnosis. Exposure histories and other personal data are determined retrospectively by interview or other means. This paper proposes to adapt the proportional hazards model for use in case-control investigations. This approach enables one to associate multiple qualitative and quantitative exposure variables with one or more diseases in a general regression framework. Interaction terms in the regression equation lead to relaxation and testing of the constant relative risk assumption.