Practice of Epidemiology Mendelian Randomization Studies for a Continuous Exposure Under Case-Control Sampling

In this article, we assess the impact of case-control sampling onmendelian randomization analyses with a dichot-omous disease outcome and a continuous exposure. The 2-stage instrumental variables (2SIV) method uses the prediction of the exposure given genotypes in the logistic regression for the outcome and provides a valid test and an approximation of the causal effect. Under case-control sampling, however, the first stage of the 2SIV procedure becomes a secondary trait association, which requires proper adjustment for the biased sampling. Through theo-retical development and simulations, we compare the naïve estimator, the inverse probability weighted estimator, and the maximum likelihood estimator for the first-stage association and, more importantly, the resulting 2SIV estimates of the causal effect. We also include in our comparison the causal odds ratio estimate derived from struc-tural meanmodels by double-logistic regression. Our results suggest that the naïve estimator is substantially biased under the alternative, yet it remains unbiased under the null hypothesis of no causal effect; the maximum likelihood estimator yields smaller variance and mean squared error than other estimators; and the structural mean models estimator delivers the smallest bias, though generally incurring a larger variance and sometimes having issues in algorithm stability and convergence. biased sampling; causal inference; instrumental variable; secondary trait association; 2-stage instrumental variables metho