This is one of a set of additional (to the core) IV assumptions that are required for a well-defined causal parameter.

If the genetic IV does not modify the effect of the exposure on the outcome within levels of the exposure and for all levels of the exposure, then this assumption holds and IV estimate is consistent for the ACE even if the effect of the exposure on the outcome is heterogeneous (see Homogeneity assumption). When there is biological understanding, a Gene-Environment (G×E) interaction study can provide evidence of causality. In more general, MR studies biological evidence may provide evidence against effect modification being plausible. Monotonicity (a weaker assumption) might be used but for some MR causal estimates. It is difficult in practice to know how important violation of these additional IV assumptions are in MR studies. Large GWAS collaborations increasingly combine results from many studies (though mostly from European original populations) and show consistency of association across these studies for variants defined as being associated with the exposure at genome-wide significance (variants used in most MR studies), suggesting homogeneity may exist for several MR studies in European populations. Non-parametric methods that provide bounds of causal effect estimates requiring only the core IV assumptions may be applicable for some MR studies Triangulation of results with other (non-MR) methods is likely to improve causal inference from MR studies.

## References

- Sheehan NA, Didelez V. Epidemiology, genetic epidemiology and Mendelian randomisation: more need than ever to attend to detail. Human Genetics 2019.
- Swanson SA, Hernán MA. The challenging interpretation of instrumental variable estimates under monotonicity. International Journal of Epidemiology 2017;47:1289-1297.
- Small DS, Tan Z, Ramashai RR, Lorch SA, Brookhart MA. Instrumental Variable Estimation with a Stochastic Monotonicity Assumption. Statistical Science 2017;32:561-579.

## Other terms in 'Sources of bias and limitations in MR':

- Assortative mating
- Canalization
- Collider
- Collider bias
- Confounding
- Dynastic effects
- Exclusion restriction assumption
- Harmonization failure (in two-sample MR)
- Homogeneity Assumption
- Horizontal Pleiotropy
- Independence assumption
- InSIDE assumption (in two-sample MR using aggregate data)
- Monotonicity assumption
- MR for testing critical or sensitive periods
- MR for testing developmental origins
- Non-linear effects
- Non-overlapping samples (in two-sample MR)
- Overfitting
- Pleiotropy
- Population stratification
- Regression dilution bias (attenuation by errors)
- Relevance assumption
- Reverse causality
- Same underlying population (in two-sample MR)
- Statistical power/efficiency
- Vertical Pleiotropy
- Weak instrument bias
- Winner's curse