An extension of the two-stage least squares (TSLS) estimator for one-sample MR analyses but, instead of including the predicted values from the first stage regression (i.e., between the instrumental variable (IV) and exposure) in the second stage (i.e., the model including the outcome), the first-stage residuals are included.
The first stage of the model involves the residuals of the regression between the IV and exposure to be generated. Then, in the second stage, regression is conducted for the outcome of interest, replacing the exposure with the residuals from the first stage regression. Like the two-stage predictor substitution method, the two-stage residual inclusion estimator method will obtain identical results as the TSLS method for linear models. However, unlike the two-stage predictor substitution, for non-linear models, two-stage residual inclusion estimators are consistent and unlikely to be biased in the presence of endogeneity (i.e., unmeasured confounders or measurement error in regression covariates). Therefore, this method is favoured over the two-stage predictor substitution method for non-linear instrumental variable (IV) models where endogeneity is likely. See Two-stage predictor substitution.
References
- Terza JV, Basu A, Rathouz PJ. Two-stage residual inclusion estimation: Addressing endogeneity in health econometric modeling. Journal of Health Economics 2008; 27:531-543
- Sanderson E, Glymour MM, Holmes MV, Kang H, Morrison J, Munafò MR, Palmer T, Schooling MC, Wallace C, Zhao Q, Davey Smith G. Mendelian randomization. Nat Rev Methods Primers 2022; 2: 6.
Other terms in 'One-sample MR methods':
- Generalized Method of Moments (GMM) estimator
- MR with a time-to-event outcome
- Non-parametric methods with bounds of causal effect
- Polygenic risk score approach
- Structural Mean Models (SMMs)
- Two-stage least squares (TSLS)
- Two-stage least squares (TSLS) with binary outcomes
- Two-stage predictor substitution estimators
- Within-family MR