Table 1.

Instrumental variable and other assumptions relevant for MR.

Assumption	Description
Instrumental variable assumptions
Relevance	The variant is associated with the outcome (${\gamma }_{Xj}\ne 0$); the variant does not need to be causal
Independence	The variant is not associated with any confounders (${\alpha }_{Cj}=0$)
Exclusion restriction	The variant is independent of the outcome given the exposure and all confounders (${\alpha }_{Yj}=0$)
Additional assumptions
Constant effect sizes
Same population parameters (multi-sample)	For multi-sample analyses, the (relevant) parameters are the same across all populations the different cohorts were drawn from
Same conditioning	The associations used are all conditioned on (relevantly) the same variables and in the same way, in terms of covariates included in analyses as well as selection effects (in multi-sample analysis)
No nonlinearities	Effect sizes for any causal effect or association are not dependent on the value of either of the two variables (as opposed to e.g., quadratic effect of causal variable, or with a binary outcome)
No interaction effects	Effect sizes for any causal effect or association are not dependent on the value of any other variable
Fully observed variables	The observed instance of each variable fully reflects the causally relevant instance of that variable; that is, it is observed without noise or rescaling relative to the causal instance

Which assumptions are required for a given MR analysis depends on the model used (see text).