An extension of the F-statistic traditionally used to test for the presence of weak instrument bias in MR analyses with one exposure, the conditional F-statistic allows testing for the presence of weak instrument bias in an MR analysis where there are multiple exposures.

In MR analyses, the first-stage F-statistic (or just "F-statistic") can be used as an indicator of the strength of the association between the genetic instrumental variable (IV) and exposure of interest and the extent of the relative bias that is likely to occur in estimating the causal effect of the exposure on the outcome using the genetic IV. Similarly, the conditional F-statistic can be used as an indicator of the strength of association between the genetic IV and each exposure conditional on the other exposures. This is particularly helpful and is most predominantly used within multivariable MR settings. As the conditional F-statistic has the same distribution as the conventional F-statistic, the same general rule, where a condictional F-statistic >10 indicates that the level of weak instrument bias is small, can be used. However, as with the conventional F-statistic, the conditional F-statistic should not be used to select IVs for multiple exposures but, instead, should be used as a test for weak instrument bias. For example, a conditional F-statistic of <10 does not indicate that any IV should not be used in an MR analysis but, instead, it should be noted in the analysis that weak instrument bias should be a considered limitation. See F-statistic.

## References

- Sanderson E, Spiller W, Bowden J. Testing and correcting for weak and pleiotropic instruments in two-sample multivariable Mendelian randomization. Statistics in Medicine 2021; 40: 5434-5452.
- Sanderson E, Windmeijer F. A weak instrument F-test in linear IV models with multiple endogenous variables. Journal of Econometrics 2016; 190: 212-221

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

- Assortative mating
- Canalization
- Collider
- Collider bias
- 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
- NO Measurement Error (NOME) 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