Mendelian randomization: How biased is a simple odds ratio estimator?

Mendelian randomization is an application of the technique of instrumental variables, using genetic variants as the instruments. Instrumental variable methods are straightforward for linear models, but epidemiologists often study binary outcomes and use odds ratios to quantify effects. Many early applications of Mendelian randomization dichotomized genotype and estimated the causal log odds ratio for a unit increase in exposure by dividing the genotype-disease log odds ratio by the difference in mean exposure between genotypes. This simple estimator is known to be inconsistent, but it is not clear whether its bias is large enough to be of practical importance. We study the large-sample bias of this estimator in a simple model with a continuous exposure, a single confounder and interpretable parameters. We focus on a range of parameter values designed to reflect scenarios in which Mendelian randomization is applied, including realistic values for the degree of confounding and the strength of the causal effect. We evaluate this estimator and the causal odds ratio using numerical integration and also obtain approximate analytic expressions to check results and build insight. We find that the bias of this simple estimator of the odds ratio is no more than around 10% in scenarios typical of those in which Mendelian randomization is likely to be useful. Nevertheless, better methods such as structural mean models have recently been developed and we recommend their use when the individual-level data they require is available. The simple estimator still has a role in meta-analysis based on summary data.