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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 heteroscedasticity-robust standard errors for these estimates. We compared several different forms of the standard error for linear and logistic TSRI estimates in simulations and in real-data examples. Among others, we consider standard errors modified from the approach of Newey (1987), Terza (2016), and bootstrapping. In our simulations Newey, Terza, bootstrap, and corrected 2-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 standard errors 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 standard errors have correct type I error under the null. Researchers should report TSRI estimates with modified standard errors instead of reporting unadjusted or heteroscedasticity-robust standard errors.

Original publication




Journal article


Am J Epidemiol

Publication Date





1104 - 1114


2-stage predictor substitution estimators, 2-stage residual inclusion estimators, Mendelian randomization, causal inference, instrumental variables, Bias, Body Mass Index, Causality, Computer Simulation, Confounding Factors (Epidemiology), Data Interpretation, Statistical, Diabetes Mellitus, Genetic Predisposition to Disease, Genotype, Humans, Hypertension, Least-Squares Analysis, Linear Models, Logistic Models, Mendelian Randomization Analysis