Additive and multiplicative models for relative survival rates

The relative survival rate is the ratio of the overall survival rate to that 'expected' for demographically similar individuals in a reference population. It is commonly used to estimate the effect of a particular disease on mortality, when the cause of death is not reliably known. The effect of the disease on the underlying hazards may be multiplicative or additive; the former case has been considered elsewhere but seems biologically less plausible than the latter. Both models are examined in this paper. The effect of the disease is assumed to be constant throughout the follow-up period, or to be piecewise constant within K follow-up intervals. Maximum likelihood estimates and related statistics are presented, as are simpler statistics based on moment estimators of the disease effect. The moment-based statistic for testing the homogeneity of r groups may be expressed as a sum of individual scores, which are shown to be closely related to the logrank scores when follow-up intervals are made arbitrarily small.