Fitting relationships between exposure and standardized mortality ratios

Practical methods are described for formal fitting of simple (linear) relationships between exposures in several subgroups and standardized mortality (or morbidity) ratios (SMRs) for these groups, whether derived from a cohort study by the person-years method or otherwise. The observed events are assumed to have Poisson variation about the fitted relationships. Procedures are outlined for obtaining parameter estimates by iterative weighted least squares regression, which is not only equivalent to, but also the simplest form of maximum likelihood estimation. A test of fit is available in a chi 2 statistic for deviations of observed data from "fitted" values. A further test allows one to examine whether the SMR at zero exposure is different from unity. The concept of relative slopes is discussed, and methods are given for obtaining estimates and confidence intervals. Generalizations to more than one exposure, and to quadratic and other relationships, are explained. Steps for computing the basic fits are detailed, and instructions for adapting packaged computer programs given. Illustrations are provided from data of lung cancer SMRs in relation to exposure to asbestos and to smoking.