@bullet a Comparison of the Logistic Risk Function and the Proportlonal Hazards Model in Prospective Epidemiologic Studies

A COMPARISON OF THE LOGISTIC RISK FUNCTION AND THE PROPORTIONAL HAZARDS MODEL IN PROSPECTIVE EPIDEMIOLOGIC STUDIES. M.S. Green and M.J. Symons, University of North Carolina at Chapel Hill, North Carolina 27514. 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 followup period is short and the disease is generally rare at all levels of the risk factors, 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 contrasted. These results are illustrated utilizing a previously published data set on metastatic cancer of the breast. With increasing follow. ,. up time, the results from logistic regression became biased and less reliable. KEY WORDS; Survival Methods, Relative Risk, Rate Ratio, Odds Ratio, Breast Cancer *Postscript; The relationship is slightly more general. By inspection of equation (8), (9), and (10), could be a Weibull or Gompertz hazard, for example. 1.0 Introduction and Sketch of Results The prospective epidemiologic studies on chronic disease that have been carried out in recent years have employed a number of different approaches in the analysis of the data generated. In these studies, a number of variables thought to be possible risk factors for the disease are measured on each individual in a selected cohort at a specified point in time. The cause specific morbidity and/or mortality is then recorded for each individual during the follow-up period. The data generated from such studies thus consist of values on a dichotomous dependent variable, e.g., presence or absence of disease, alive or dead, and baseline readings of the postulated risk factors. Some years ago, Cornfield [lJ suggested the use of the multiple logistic risk function for analysis of such data, and in 1967 Truett, Cornfield and Kannel [2J published a more extensive analysis of the Framingham Heart Study data using this model. Since that time logistic analysis has been widely applied to prospective epidemiologic studies of chronic disease [3,4J and the results of these analyses have contributed considerably to the current knowledge and understanding of the risk factors associated with various chronic diseases. However, there is an aspect of logistic analysis which challenges its integrity as a model for survival data. This feature is that logistic analysis does not utilize information on the point in time during the follow-up period at which a positive response, i.e., manifestation of disease or death, occurs. Then, for example, a response occurring near the beginning of the follow-up period is given the same weight in the ~ analysis as one occurring near the end of that period. As a result, the significance of the contribution of the postulated risk factor to disease