The NOME assumption implies that the variance of the association between the instrumental variable (IV) and the exposure is small enough to be negligible and even treated as zero. Thus, there is no measurement error in this relationship. This becomes increasingly true as the sample size of the summary-level data used to source the association between each IV and the exposure (in a two-sample MR setting) tends towards infinity. Strong violation of the NOME assumption (usually by using weak IVs) can result in a regression dilution bias of the causal effect estimate towards the null.

This assumption becomes important when considering the most appropriate weights to use in MR analyses. Many two-sample MR methods weight the individual IV-level estimates by the inverse variance of the association between the IV and the outcome. These weights are called the "first-order" weights, as the variance used in the inverse variance weighting is approximated with the first term of the delta method used to calculate the causal effect estimate variance. The use of this first term in such weighting is equivalent to the NOME assumption, as it assumes that the variance of the association between the IV and the exposure is zero. Including the second term of the delta method approximating the variance of the causal effect estimate to these weights - so called "second-order" weights - or modified second-order weights can provide be more accurate given the inclusion of a parameter estimating the variance of the association between the IV and the exposure. See NOME adjustment.

## References

- Bowden J, Del Greco MF, Minelli C, Davey Smith G, Sheehan NA, Thompson JR. Assessing the suitability of summary data for two-sample Mendelian randomization analyses using MR-Egger regression: the role of the I2 statistic. Int J Epidemiol 2016: 45; 1961-1974.
- Bowden J, Del Greco M F, Minelli C, Davey Smith G, Sheehan N, Thompson J. A framework for the investigation of pleiotropy in two-sample summary data Mendelian randomization. Statistics in medicine 2017; 36: 1783-1802.

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

- Assortative mating
- Canalization
- Collider
- Collider bias
- Conditional F-statistic for multiple exposures
- Confounding
- Exclusion restriction assumption
- F-statistic
- Harmonization (in two-sample MR)
- Homogeneity Assumption
- Horizontal Pleiotropy
- Independence assumption
- INstrument Strength Independent of Direct Effect (InSIDE) assumption
- Intergenerational (or dynastic) effects
- Monotonicity assumption
- MR for testing critical or sensitive periods
- MR for testing developmental origins
- No effect modification assumption
- Non-linear MR
- Non-overlapping samples (in two-sample MR)
- Overfitting
- Pleiotropy
- Population stratification
- R-squared
- Regression dilution bias (attenuation by errors)
- Relevance assumption
- Reverse causality
- Same underlying population (in two-sample MR)
- Statistical power and efficiency
- Vertical pleiotropy
- Weak instrument bias
- Winner's curse