Mendelian Randomization (MR)

A valid genetic instrument must satisfy three core conditions to produce unbiased causal estimates:

• Relevance (IV1): The variant is robustly associated with the exposure (e.g., F-statistic > 10)
• Independence (IV2): The variant is not associated with confounders of the exposure-outcome relationship
• Exclusion Restriction (IV3): The variant affects the outcome only through the exposure, not via alternative pathways Violation of any assumption, particularly horizontal pleiotropy, can invalidate the causal estimate.

Horizontal pleiotropy occurs when a genetic variant influences the outcome through pathways independent of the exposure, violating the exclusion restriction. Detection methods include:

• MR-Egger regression: Allows an intercept term to detect directional pleiotropy; the slope provides a pleiotropy-corrected causal estimate
• Weighted median estimator: Provides consistent estimates when up to 50% of the weight comes from invalid instruments
• MR-PRESSO: Detects and removes outlier variants driving pleiotropic bias
• Cochran's Q statistic: Tests for heterogeneity among variant-specific causal estimates

A study design that tests causal effects in both directions between two traits:

• Forward MR: Tests whether exposure A causally influences outcome B
• Reverse MR: Tests whether outcome B causally influences exposure A This approach helps disentangle complex, potentially bidirectional relationships—for example, determining whether BMI causally increases depression risk or whether depression causally increases BMI. Significant findings in both directions suggest a feedback loop rather than a unidirectional causal pathway.

An extension that estimates the direct causal effect of multiple correlated exposures simultaneously:

• Includes genetic variants associated with any of the exposures as instruments
• Estimates the independent effect of each exposure, conditioning on the others
• Addresses confounding between exposures that share genetic architecture Example: Estimating the direct effect of HDL cholesterol on coronary artery disease while accounting for LDL cholesterol and triglycerides, which are often genetically correlated.

A violation of the MR exclusion restriction where a genetic variant influences the outcome through pathways independent of the exposure under study. This is distinct from vertical pleiotropy, where the variant affects the outcome only through the exposure chain.

• Detected via MR-Egger regression intercept tests and Cochran's Q statistic heterogeneity assessment
• Balanced pleiotropy (random direction) can be tolerated by inverse-variance weighted methods
• Directional pleiotropy systematically biases causal estimates and requires robust sensitivity analyses
• MR-PRESSO identifies and removes outlier variants driving pleiotropic effects

A sensitivity analysis method that relaxes the strict exclusion restriction by allowing all genetic variants to exhibit directional pleiotropy, provided the pleiotropic effects are independent of variant-exposure associations (the InSIDE assumption). The slope provides a pleiotropy-corrected causal estimate.

• The intercept term formally tests for directional pleiotropy; a non-zero intercept signals bias
• Requires a minimum of 3 genetic instruments for identification
• Lower statistical power than inverse-variance weighted methods due to relaxed assumptions
• Best used as a sensitivity check alongside other robust methods rather than a primary analysis

The primary MR estimator that meta-analyzes Wald ratio estimates from individual genetic variants, weighting each by the inverse of its variance. Under the assumption that all variants are valid instruments with no horizontal pleiotropy, IVW provides the most statistically efficient causal estimate.

• Fixed-effects IVW assumes a single true causal effect across all variants
• Random-effects IVW accommodates balanced pleiotropy by allowing effect heterogeneity
• Cochran's Q statistic tests for heterogeneity; significant Q suggests pleiotropy or effect modification
• The IVW estimate is equivalent to regressing SNP-outcome effects on SNP-exposure effects with the intercept constrained to zero

Systematic bias toward the confounded observational association in MR estimates when genetic instruments explain only a small proportion of exposure variance. Weak instruments amplify violations of the exclusion restriction and produce imprecise, unreliable causal estimates.

• Quantified by the F-statistic; instruments with F < 10 are conventionally classified as weak
• In two-sample MR, weak instruments bias results toward the null rather than the confounded estimate
• Allele scores combining multiple variants increase instrument strength and mitigate this bias
• Weak instrument-robust methods include limited information maximum likelihood (LIML) and the Anderson-Rubin test