Relative risk and acceleration in lung cancer

For a substance that increases the relative risk of disease, it does not necessarily follow that the proportion of cases due to exposure to the substance is the same as the attributable fraction in the exposed. An alternative explanation is that the substance has accelerated the occurrence of disease and, therefore, played a role in all cases. When the incidence of disease with time follows the Weibull distribution, it is well known that the proportional hazards model and the accelerated failure time model are equivalent. The purpose of this paper is to provide a numerical illustration of the relationship between the relative risk and the acceleration time of occurrence of cases. A Weibull distribution is a good approximation for lung cancer death rates up to the age of 80 years. The numerical relationship between the relative risk and the time by which cases are accelerated is given for lung cancer deaths occurring at ages of 40-75 years with relative risks of 1.01-3. As an example, for a death due to lung cancer at age 60 years in a smoker, relative risks of 2 and 1.1 due to occupational exposure to a substance correspond to accelerations of 5.2 years and 8 months, respectively.