Table. Validation of the 3 Assumptions of Mendelian Randomization in Each Study.

Source	Assumption 1a	Assumptions 2 and 3b	Conclusion
Nordestgaard et al,21 2012	Strength of association between gene and BMI not estimated or reported from another study	No attempt was made to detect or adjust for pleiotropy.	None of the 3 assumptions validated
Fall et al,24 2013	Association between gene and BMI not tested for; assumed to be sufficient based on previous studies	Pleiotropy could not be tested for statistically.	Only a single genotype was used as the instrument. Considerable risk of bias due to pleiotropy
Holmes et al,18 2014	The F statistic was calculated to study the association between genes and BMI (F = 237).	Pleiotropy was not estimated.	Assumption 1 was validated. Pleiotropy was not tested for and is possibly present.
Hägg et al,20 2015	Random-effects meta-analysis was used to test for association between genetic score and BMI. A strong association was found (P = 2.77 × 10−107).	Association of individual adiposity SNPs with CHD using CARDIoGRAMplusC4D data were investigated; this suggested that large pleiotropic effects were unlikely.	Assumption 1 valid. Pleiotropy not specifically tested for and could be present.
Lyall et al,19 2017	F statistic calculated by the study was 2175.	MR-Egger analysis was conducted to detect and account for pleiotropy. The following covariates were used: Townsend deprivation index (P = .02), smoking status (P < .01), and alcohol intake (P < .001). These were adjusted for, and MR-Egger analysis did not suggest presence of unbalanced horizontal pleiotropy.	All 3 assumptions validated; pleiotropy was identified and adjusted for.
Dale et al,23 2017	Association between genes and BMI not estimated	MR-Egger regression was broadly consistent with conventional MR analysis, showing little evidence of pleiotropy.	Assumption 1 not validated. Pleiotropy was likely minimal.
Emdin et al,22 2017	Association between genes and WHRadjBMI not estimated in the study; F statistic reported from the UK Biobank was 1713.	Test for trend was performed across quartiles of the polygenic risk score for WHRadjBMI using logistic regression, with each potential confounder as the outcome. The association of the polygenic risk score with the following confounders was tested: smoking, alcohol use, physical activity, vegetable consumption, red meat consumption, and breastfeeding status as a child. No significant association was found. Five sensitivity analyses were also conducted, of which 4 were consistent with no pleiotropy.	Assumption 1 was considered valid based on data from the literature. Possible pleiotropy

Abbreviations: BMI, body mass index (calculated as weight in kilograms divided by height in meters squared); CARDIoGRAMplusC4D, Coronary Artery Disease Genome-Wide Replication and Meta-analysis plus the Coronary Artery Disease Genetics Consortium; CHD, coronary heart disease; DIAGRAM, Diabetes Genetics Replication and Meta-analysis; MR, mendelian randomization; SNP, single-nucleotide polymorphism; WHRadjBMI, waist to hip ratio adjusted for BMI.

^aGenotype must be associated with phenotype (obesity); validated in 4 studies.

^bAbsence of pleiotropy (ie, genotype should not be associated with confounders and should affect outcome only through the risk factor); verified in 3 studies.