Correcting the standard errors of two-stage residual inclusion estimators for Mendelian randomization studies

Mendelian randomization studies use genotypes as instrumental variables to test for and estimate the causal effects of modifiable risk factors on outcomes. Two-stage residual inclusion (TSRI) estimators have been used when researchers are willing to make parametric assumptions. However, researchers are currently reporting uncorrected or heteroskedasticity robust standard errors (SEs) for these estimates. We compare several different forms of the SE for linear and logistic TSRI estimates in simulations and in real data examples. Amongst others we consider SEs modified from the approach of Newey (1987), Terza (2016), and bootstrapping. In our simulations Newey, Terza, bootstrap, and corrected two-stage least squares (in the linear case) standard errors gave the best results in terms of coverage and type I error. In the real data examples the Newey SEs were 0.5% and 2% larger than the unadjusted standard errors for the linear and logistic TSRI estimators respectively. We show that TSRI estimators with modified SEs have correct type I error under the null. Researchers should report TSRI estimates with modified SEs instead of reporting unadjusted or heteroskedasticity robust SEs.