In this chapter, we discuss methods for the estimation of causal effects using instrumental variables (IVs) with both continuous and binary outcomes. We focus attention on the case of a single continuous exposure variable, as this is the usual situation in Mendelian randomization studies; although the same methods could be used in the case of a single binary exposure. We explain for each method how to estimate a causal effect, and describe specific properties of the estimator. In turn, we consider the ratio of coefficients method, two-stage methods, likelihood-based methods, and semi-parametric methods. This order corresponds roughly to the complexity of the methods, with the simplest ones first. These methods are contrasted in terms of bias, coverage, efficiency, power, robustness to misspecification, and existence of finite moments. We have included a simple explanation of each method at first, and then further details for more technical readers. Also discussed are implementations of the methods using standard statistical software packages.


Methods for IV analysis range from the very simple (calculate the difference between two pairs of numbers and divide one by the other) to the more complicated. The development of complex methods has been driven by the desire to produce efficient estimates, for example by integrating data on multiple IVs, to allow for more flexible modelling assumptions, or to provide robustness against misspecification of modelling assumptions. Each method has its own advantages and disadvantages. The properties of many of these estimators will be discussed in the chapters to come in the specific contexts of weak instruments, binary outcomes, and evidence synthesis.

Relevant papers to chapter:

Whole chapter. S. Burgess, D.S. Small, S.G. Thompson. Comparison of statistical properties of estimates from instrumental variable methods used in Mendelian randomization. Submitted manuscript.

Section 4.2.3 (Non-collapsibility). S. Burgess. Identifying the odds ratio estimated by two-stage instrumental variable analysis with a logistic regression model. Statist. Med. 2013; 32(27):4726-4747.

Section 4.2.3 (Non-collapsibility). S. Burgess. Estimating and contextualizing the attenuation of odds ratios due to non-collapsibility. Comm. Stat. Theory 2015.

Section 4.3.3 (Bayesian methods). S. Burgess, S.G. Thompson, CRP CHD Genetics Collaboration. Bayesian methods for meta-analysis of causal relationships estimated using genetic instrumental variables. Statist. Med. 2010; 29(12):1298-1311.

Section 4.3.3 (Bayesian methods). S. Burgess, S.G. Thompson. Improving bias and coverage in instrumental variable analysis with weak instruments for continuous and binary outcomes. Statist. Med. 2012; 31(15):1582-1600.

Section 4.4.3 (Lack of identification for binary outcomes). S. Burgess, R. Granell, T.M. Palmer, J.A.C. Sterne, V. Didelez. Lack of identification in semiparametric instrumental variable models with binary outcomes. Am. J. Epidemiol. 2014; 180(1):111-119.