Application of the Instrumental Inequalities to a Mendelian Randomization Study With Multiple Proposed Instruments

Investigators often support the validity of Mendelian randomization (MR) studies, an instrumental variable approach proposing genetic variants as instruments, via. subject matter knowledge. However, the instrumental variable model implies certain inequalities, offering an empirical method of falsifying (but not verifying) the underlying assumptions. Although these inequalities are said to detect only extreme assumption violations in practice, to our knowledge they have not been used in settings with multiple proposed instruments.

We applied the instrumental inequalities to an MR analysis of the effect of maternal pregnancy vitamin D on offspring psychiatric outcomes, proposing four independent maternal genetic variants as instruments. We assessed whether the proposed instruments satisfied the instrumental inequalities separately and jointly and explored the instrumental inequalities' properties via simulations.

The instrumental inequalities were satisfied (i.e., we did not falsify the MR model) when considering each variant separately. However, the inequalities were violated when considering four variants jointly and for some combinations of two or three variants (two of 36 two-variant combinations and 18 of 24 three-variant combinations). In simulations, the inequalities detected structural biases more often when assessing proposed instruments jointly, although falsification in the absence of structural bias remained rare.

The instrumental inequalities detected violations of the MR assumptions for genetic variants jointly proposed as instruments in our study, although the instrumental inequalities were satisfied when considering each proposed instrument separately. We discuss how investigators can assess instrumental inequalities to eliminate clearly invalid analyses in settings with many proposed instruments and provide appropriate code.