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Mendelian randomization (MR) is an increasingly popular approach to estimating causal effects. Although the assumptions underlying MR cannot be verified, they imply certain constraints, the instrumental inequalities, which can be used to falsify the MR conditions. However, the instrumental inequalities are rarely applied in MR. We aimed to explore whether the instrumental inequalities could detect violations of the MR conditions in case studies analyzing the effect of commonly studied exposures on coronary artery disease risk. Using 1077 single nucleotide polymorphisms (SNPs), we applied the instrumental inequalities to MR models for the effects of vitamin D concentration, alcohol consumption, C-reactive protein (CRP), triglycerides, high-density lipoprotein (HDL) cholesterol, and low-density lipoprotein (LDL) cholesterol on coronary artery disease in the UK Biobank. For their relevant exposure, we applied the instrumental inequalities to MR models proposing each SNP as an instrument individually, and to MR models proposing unweighted allele scores as an instrument. We did not identify any violations of the MR assumptions when proposing each SNP as an instrument individually. When proposing allele scores as instruments, we detected violations of the MR assumptions for 5 of 6 exposures. Within our setting, this suggests the instrumental inequalities can be useful for identifying violations of the MR conditions when proposing multiple SNPs as instruments, but may be less useful in determining which SNPs are not instruments. This work demonstrates how incorporating the instrumental inequalities into MR analyses can help researchers to identify and mitigate potential bias. [ABSTRACT FROM AUTHOR]