MR method proposed for two-sample MR settings that uses Bayesian statistics and a weakly informative prior to increase power in MR-Egger analyses for detecting horizontal pleiotropy.

In a two-sample MR framework, the inverse variance weighted (IVW) method is essentially equivalent to the MR-Egger method but with an optimistic informative Bayesian prior, which sets the intercept of the relationship of the instrumental variable (IV)-exposure and IV-outcome association to have a mean and variance of zero. Conversely, the MR-Egger method has a pessimistic non-informative Bayesian prior on the intercept of the IV-exposure and IV-outcome association with an infinite variance. Given the assumption that extreme departures from a zero intercept term (i.e., indicating balanced pleiotropy) are unlikely, the BMRE method alternatively includes a weakly informative Bayesian prior on the intercept to allow for some level of directional horizontal pleiotropy - a prior that is not as strict as assuming a zero mean and variance with the IVW method and not as lenient as assuming an infinite variance with the MR-Egger method. Under the Instrument Strength Independent of Direct Effect (InSIDE) assumption, the BMRE estimator with weakly informative priors on the intercept term increases power to detect the causal effect of the exposure on an outcome when detecting and accounting for horizontal pleiotropy.

## References

## Other terms in 'Pleiotropy-robust two-sample MR methods':

- Bayes MR
- Bayesian multi-instrument Mendelian randomization (MIMR)
- Bayesian network analysis
- Causal Analysis Using Summary Effect estimates (MR-CAUSE)
- Contamination mixture models
- Generalized Summary MR (GSMR)
- Genetic Instrumental Variable (GIV)
- Hierarchical joint Analysis of Marginal summary statistics (hJAM)
- Inverse variance weighted (IVW) random effects model
- Iterative Mendelian Randomization and Pleiotropy (IMRP)
- Leave-one-out analysis
- Median-based estimate
- Mode-based estimate
- MR accounting for Correlated and Idiosyncratic Pleiotropy (MRCIP)
- MR accounting for Linkage Disequilibrium and Pleiotropy (MR-LDP)
- MR Lasso
- MR Mixture (MRMix)
- MR using Robust regression (MR Robust)
- MR with penalized weights
- MR with regularization
- MR-Clust
- MR-Egger regression and extensions
- MR-Link
- MR-Path
- Multivariable MR (MVMR) and extensions
- Welch-weighted Egger regression (WWER)