Indirect corrections for confounding under multiplicative and additive risk models

We define a multiplicative model and an additive model for the hazards associated jointly with exposure and with the presence of a confounder like smoking. Under the multiplicative model, the crude relative risk may be adjusted indirectly, by means of a factor proposed by Axelson 1978, and implicitly by Cornfield et al. 1959 and Schlesselman 1978. We present corresponding indirect correction formulas under the additive risk model for the risk difference and for the excess relative risk. Conditions are established under which these corrections may be applied to age-adjusted rates from composite study populations. We demonstrate that indirect corrections may be no better than crude measures of risk if one assumes the wrong model for the joint action of the exposure and confounding factors. These results are illustrated on an example of occupational exposure to vermiculite. The limitations of the techniques are discussed.