Mendelian randomization analysis of case-control data using structural mean models

'Instrumental Variable' (IV) methods provide a basis for estimating an exposure's causal effect on the risk of disease. In Mendelian randomization studies, where genetic information plays the role of the IV, IV analyses are routinely performed on case-control data, rather than prospectively collected observational data. Although it is a well-appreciated fact that ascertainment bias may invalidate such analyses, ad hoc assumptions and approximations are made to justify their use. In this paper we attempt to explain and clarify why they may fail and show how they can be adjusted for improved performance. In particular, we propose consistent estimators of the causal relative risk and odds ratio if a priori knowledge is available regarding either the population disease prevalence or the population distribution of the IV (e.g. population allele frequencies). We further show that if no such information is available, approximate estimators can be obtained under a rare disease assumption. We illustrate this with matched case-control data from the recently completed EPIC study, from which we attempt to assess the evidence for a causal relationship between C-reactive protein levels and the risk of Coronary Artery Disease.