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

In this article, we assess the impact of case-control sampling on mendelian randomization analyses with a dichotomous 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 theoretical development and simulations, we compare the naive 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 structural mean models by double-logistic regression. Our results suggest that the naive 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.