Weak instruments in multivariable Mendelian randomization: methods and practice

The method of multivariable Mendelian randomization uses genetic variants to instrument multiple exposures, to estimate the effect that a given exposure has on an outcome conditional on all other exposures included in a linear model. Unfortunately, the inclusion of every additional exposure makes a weak instruments problem more likely, because we require conditionally strong genetic predictors of each exposure. This issue is well appreciated in practice, with different versions of F-statistics routinely reported as measures of instument strength. Less transparently, however, these F-statistics are sometimes used to guide instrument selection, and even to decide whether to report empirical results. Rather than discarding findings with low F-statistics, weak instrument-robust methods can provide valid inference under weak instruments. For multivariable Mendelian randomization with two-sample summary data, we encourage use of the inference strategy of Andrews (2018) that reports both robust and non-robust confidence sets, along with a statistic that measures how reliable the non-robust confidence set is in terms of coverage. We also propose a novel adjusted-Kleibergen statistic that corrects for overdispersion heterogeneity in genetic associations with the outcome.