Mendelian randomization uses genetic variants as instrumental variables to estimate the causal effect of a risk factor on an outcome using observational data. If a genetic variant is included in a Mendelian randomization study that does not satisfy the instrumental variable assumptions then the causal estimate from traditional instrumental variable methods will be biased. Since Mendelian randomization studies using publicly available summary level data (estimates and standard errors of the genetic associations with the risk factor and the outcome) from large consortia can be performed with relative ease and little expense, the popularity of Mendelian randomization in epidemiological studies has increased dramatically. As such, various methods have been developed in Mendelian randomization that use summary level data and account for possible violations of the instrumental variable assumptions. However, additional Mendelian randomization methods that account for violations in the instrumental variable assumptions are still required.

In this dissertation, we introduce robust methods for Mendelian randomization that downweight the contribution of genetic variants with heterogeneous causal ratio estimates. We extend the univariable MR-Egger method to the multivariable setting to account for both measured and unmeasured pleiotropic effects. We also explore the possibility of extending multivariable Mendelian randomization to the factorial setting to estimate statistical interaction effects. Finally, we apply some of the methods we have developed to perform a Mendelian randomization study to investigate the effect of adiposity and body composition measurements on asthma using data from UK Biobank and the GABRIEL consortium.