A comparison of the logistic risk function and the proportional hazards model in prospective epidemiologic studies

The logistic regression and proportional hazards models are each currently being used in the analysis of prospective epidemiologic studies examining risk factors in chronic disease applications. The advantages and disadvantages of each are yet to be fully described. However, a theoretical relationship between the two models has been documented. In this paper the conditions under which results from the two models approximate one another are described. It is shown that where the follow-up period is short and the disease is generally rare, the regression coefficients of the logistic model approximate those of the proportional hazards model with a constant underlying hazard rate. Since under the same conditions the likelihood functions approximate one another, the regression coefficients have similar estimated standard errors. Further, estimation of relative risk with these models is constrasted. These results are illustrated utilizing a previously published data set on metastatic cancer of the breast. With increasing follow-up time, the logistic regression coefficients become uncertain and less reliable.