Instrumental Variable (IV)

The genetic variant must be robustly associated with the exposure. This is the only empirically verifiable assumption.

• Assessed via the F-statistic from the regression of exposure on variant; a rule of thumb is F > 10 to avoid weak instrument bias.
• Weak instruments inflate standard errors and bias causal estimates toward the confounded observational association in two-sample MR.
• Genome-wide significant variants (p < 5 × 10⁻⁸) from large GWAS are typically selected to satisfy this condition.

The genetic variant must be independent of confounders of the exposure-outcome relationship.

• This is fundamentally untestable and must be justified by domain knowledge.
• Violations occur when the variant influences traits that affect both exposure and outcome (e.g., a variant affecting both BMI and smoking behavior).
• Negative control outcomes (traits known not to be causally affected by the exposure) can provide supporting evidence for independence.

The genetic variant must affect the outcome only through the exposure, not via any alternative causal pathway.

• Horizontal pleiotropy is the primary violation, where a variant influences the outcome through a pathway independent of the exposure.
• Methods like MR-Egger regression and the weighted median estimator provide sensitivity analyses robust to some forms of pleiotropy.
• Biological plausibility and colocalization analyses help assess whether the variant acts through the hypothesized exposure.

For binary exposures, the instrument must affect the exposure in the same direction for all individuals in the population.

• This assumption is required for interpreting IV estimates as a local average treatment effect (LATE) rather than an average treatment effect.
• Violations occur if there are 'defiers'—individuals whose exposure status changes opposite to the variant's effect.
• In genetic MR, the monotonicity assumption is often considered plausible due to the fixed germline nature of genetic variants.

Standard IV methods assume a linear, homogeneous causal effect of the exposure on the outcome across the population.

• Effect modification by the instrument (gene-environment interaction) violates this assumption.
• The NO Measurement Error (NOME) assumption requires negligible imprecision in SNP-exposure associations, approximated by high F-statistics.
• Violations can be addressed using summary data-based MR methods that model heterogeneity explicitly.

A robust MR study reports a suite of sensitivity analyses to assess robustness to assumption violations.

• Cochran's Q test: Detects heterogeneity among individual variant causal estimates, indicating potential pleiotropy.
• MR-PRESSO: Identifies and removes outlier variants driving pleiotropic bias.
• Leave-one-out analysis: Sequentially removes each variant to ensure no single SNP drives the overall estimate.
• Steiger directionality test: Confirms the hypothesized causal direction (exposure → outcome) is correct.

A systematic bias that occurs when a third variable influences both the exposure and the outcome, creating a spurious association. In observational epidemiology, unmeasured confounders—such as socioeconomic status or lifestyle factors—routinely distort effect estimates.

• Confounding by indication: Patients with more severe disease receive treatment, biasing results
• Residual confounding: Remaining bias after imperfect measurement or adjustment
• IV analysis breaks this structure by isolating variation in the exposure that is independent of confounders

The canonical estimation method for instrumental variable analysis. The first stage regresses the exposure on the instrument to obtain predicted values; the second stage regresses the outcome on these predicted values to recover the causal effect.

• First stage F-statistic > 10: Rule of thumb for sufficient instrument strength
• Weak instruments produce biased estimates and inflated standard errors
• 2SLS assumes linear, homogeneous treatment effects unless extended with interaction terms

The assumption that the instrumental variable affects the outcome exclusively through its effect on the exposure. Violations occur when the instrument has pleiotropic effects—influencing the outcome through alternative biological pathways.

• Horizontal pleiotropy: Genetic variant affects outcome via a pathway independent of the exposure
• Vertical pleiotropy: Variant affects a mediator downstream of the exposure, which is permissible
• MR-Egger regression and weighted median estimators provide sensitivity analyses when this assumption is suspect

A statistical artifact where instruments with low explanatory power for the exposure produce causal estimates that are biased toward the confounded observational association in finite samples. Even with large sample sizes, weak instruments amplify any violation of the exclusion restriction.

• F-statistic: Primary diagnostic; values below 10 signal potential weak instrument problems
• LIML (Limited Information Maximum Likelihood): More robust to weak instruments than 2SLS
• Allele scores: Aggregating multiple weak variants into a single instrument can improve strength

Formal graphical models that encode assumptions about causal relationships between variables. DAGs make the identification conditions for instrumental variables visually explicit, showing whether a proposed instrument satisfies the back-door criterion for confounder control.

• Nodes represent variables; directed edges represent causal effects
• Collider bias: Conditioning on a common effect of exposure and outcome induces spurious association
• DAGitty and other software tools automate identification of valid instruments from graph structures