A SMM is a semi-parametric estimation method designed for analysing data from randomized control trials (RCTs) where there is incomplete compliance. Additive and multiplicative SMMs were originally developed for instrumental variable (IV) estimation of the effects of time-varying exposures on outcomes of interest using counterfactuals to characterize the consequences of between-subject heterogeneity in the treatment effect.

SMMs can be applied to MR, such that the first stage regression model does not need to be specified. In the context of MR (and, indeed other IV approaches), IV analyses estimate the average causal effect of an exposure on an outcome and these estimators can be represented by parameters of particular structural mean models. In particular, various SMMs (e.g., additive or multiplicative SMMs) can be used within one-sample MR settings to estimate the causal effect of an exposure on an outcome. For this method to provide a valid causal estimate of the exposure on the outcome, genetic variants must satisfy the MR assumptions. However, there is no need to specify the first stage association (e.g., between the genetic variant and exposure) model. This method can also be extended to handle binary outcomes.

## References

- Burgess S, Small DS, Thompson SG. A review of instrumental variable estimators for Mendelian randomization. Stat Methods Med Res 2017; 26: 2333-2355.
- 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.
- Hernán MA, Robins JM. Instruments for Causal Inference An Epidemiologist's Dream? Epidemiology 2006; 17: 360-372
- Bowden J, Vansteelandt S. Mendelian randomization analysis of case-control data using structural mean models. Statistics in Medicine 2011; 30: 678-694.

## 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
- Two-stage least squares (TSLS)
- Two-stage least squares (TSLS) with binary outcomes
- Two-stage predictor substitution estimators
- Two-stage residual inclusion estimators
- Within-family MR