Scalable relative effectiveness models

Murdoch and Krewski (1988) introduced the concept of the relative effectiveness of dosing in models of carcinogenesis with time-dependent exposure. Under certain models, such as the Armitage--Doll multistage model with a single stage dose-dependent, the hazard H(t) from any pattern of dosing d(·) may be calculated under a constant dosing model using an equivalent constant dose d*(t) which is simply an average of the true dose rate weighted proportionally to the relative effectiveness at each time from 0 to t. The multistage model has the property of scalable relative effectiveness: the relative effectiveness function is the same (except for a rescaling of the time axis) for any time t. In this paper we consider general models with scalable relative effectiveness and arbitrary constant dose hazard functions. Not all such models are valid, and we derive a characterization of those that are: the constant dosing hazard may be proportional to any increasing function k(t) with k(0) = 0, and the relative effectiveness must increase slower than a fixed power of time.