Public health interest often centres around the causal effect of an exposure on a particular outcome. For ethical, practical, or financial reasons, randomized controlled trials (RCTs) to investigate such an effect may not be possible and inferences must be drawn from observational data. These may be distorted by reverse causation, measurement errors, or the presence of socioeconomic and behavioural confounding factors that are difficult to measure even if known. Mendelian randomization is an instrumental variable (IV) method that permits estimation of the exposure-outcome causal effect in the presence of confounding. The assumptions required, i.e. all variables are continuous and all relationships between variables are linear, are often violated in epidemiological applications where the outcome (e.g. disease status) is typically binary and hence cannot have a linear relationship with the exposure. This is especially relevant to case-control data where the exposure-outcome relationship is usually logistic. The aim of this project is to develop theoretically sound methods for dealing with non-linear models and to investigate the practical implications of weakening, or even violating, some of the more restrictive conditions, given the type of data that we expect to arise in these settings. Theory must inform practice but applications should drive the theoretical investigation. This is not always the case and we view this as one of the main strengths of our proposal.

IV methods for binary outcomes are often used in econometrics to estimate causal effects when the exposure is subject to measurement error. We will investigate the adaptability of these approaches to epidemiological settings. Relevant methods also exist for RCT data when the effect for a particular subgroup is of interest. We will clarify when these are useful in Mendelian randomization applications and what conditions are required. The causal literature has shown that bounds on the average causal effect can be calculated without making restrictive distributional or relational assumptions. We will extend these to measures of casual effect that are more likely in epidemiological settings (e.g. causal odds ratios) and identify the data conditions under which they will be useful in practice. In order to get more precise estimates, data are often combined from different epidemiological studies. This pooling of data raises all the classic issues associated with meta-analysis and existing meta-analytic methods will be extended to take account of these. Finally, models that allow for more complex biological pathways will be developed to cater for multiple exposure and disease endpoints.