Improving bias and coverage in instrumental variable analysis with weak instruments for continuous and binary outcomes Stephen Burgess Simon G. Thompson Department of Public Health and Primary Care, University of Cambridge * November 14, 2011 Abstract Causal estimates can be obtained by instrumental variable analysis using a two-stage method. However, these can be biased when the instruments are weak. We introduce a Bayesian method, which adjusts for the first-stage residuals in the second-stage regression and has much improved bias and coverage properties. In the continuous outcome case, this adjustment reduces median bias from weak instruments to close to zero. In the binary outcome case, bias from weak instruments is reduced and the estimand is changed from a marginal population-based effect to a conditional effect. The lack of distributional assumptions on the posterior distribution of the causal effect gives a better summary of uncertainty and more accurate coverage levels than methods which rely on the asymptotic distribution of the causal estimate. These properties are discussed in the context of Mendelian randomization. Key words: instrumental variable, causal inference, Mendelian randomization, weak instrument, non-collapsibility, Bayesian methods 1 Introduction A common research question of interest is to estimate the change in an outcome caused by a unit change in a risk factor. When observational rather than experimental data is used, bias from endogeneity of the risk factor, due to unmeasured confounding and reverse causation, prevents a causal interpretation of the association between risk factor and outcome [1]. The method of instrumental variables can be used for * Address: Department of Public Health & Primary Care, Strangeways Research Laboratory, Worts Causeway, Cambridge, CB1 8RN, UK. Telephone: +44 1223 740002. Correspondence to: sb452@medschl.cam.ac.uk. This work was supported by the UK Medical Research Council (grant U.1052.00.001). 1