Inverse-Variance Weighting (IVW)

IVW operates under the fixed-effect assumption that all genetic instruments estimate the same true causal effect. It computes a weighted average of individual Wald ratio estimates, where the weight assigned to each variant is the inverse of the variance of its causal estimate.

• Weighting formula: w_j = 1 / var(β̂_j)
• Pooled estimate: β̂_IVW = Σ(w_j · β̂_j) / Σ(w_j)
• Variants with smaller standard errors (typically those with stronger SNP-exposure associations) dominate the pooled result
• This approach maximizes statistical power when the fixed-effect assumption holds

IVW relies on the three standard instrumental variable assumptions being satisfied for every genetic variant in the analysis:

• Relevance (IV1): Each genetic variant is robustly associated with the exposure of interest
• Independence (IV2): No genetic variant is associated with confounders of the exposure-outcome relationship
• Exclusion Restriction (IV3): Each variant affects the outcome only through the exposure, with no horizontal pleiotropy

Violation of IV3 is the most common threat to IVW validity, motivating the development of pleiotropy-robust extensions.

When heterogeneity exists among causal estimates, IVW can be extended with random-effects models:

• Multiplicative random effects: Scales standard errors by √Q/(L-1), where Q is Cochran's Q statistic and L is the number of instruments. Point estimates remain unchanged, but confidence intervals widen
• Additive random effects: Adds an estimated between-instrument variance component (τ²) to each variant's variance before weighting. This can alter both point estimates and precision
• The choice between models depends on whether heterogeneity is believed to reflect balanced pleiotropy (multiplicative) or genuine variation in causal effects (additive)

IVW serves as the reference method against which pleiotropy-robust approaches are compared:

• MR-Egger regression: Relaxes the IVW assumption by allowing an unconstrained intercept. When the intercept is zero, MR-Egger collapses to the IVW estimate
• Weighted median estimator: Provides consistent estimates when at least 50% of the weight comes from valid instruments, offering robustness where IVW would be biased
• MR-PRESSO: Explicitly detects and removes outlier variants before re-running IVW
• Concordance across these methods strengthens causal inference; discordance signals potential pleiotropy

IVW is most commonly applied in the two-sample Mendelian randomization framework, where SNP-exposure and SNP-outcome associations come from independent GWAS datasets:

• Input data: GWAS summary statistics for both exposure and outcome
• Harmonization step: Align effect alleles across datasets before analysis
• First-order weighting: Uses only first-order (delta method) variance approximations, assuming negligible measurement error in SNP-exposure associations
• No Measurement Error (NOME) assumption: Violated when instruments are weak, leading to regression dilution bias toward the null
• Software implementations include the TwoSampleMR and MendelianRandomization R packages

IVW is susceptible to weak instrument bias, which biases the pooled estimate toward the confounded observational association in two-sample designs:

• F-statistic threshold: An F-statistic > 10 for each variant or the mean F-statistic across instruments is the standard safeguard
• F = (β̂_exposure / SE_exposure)²: Measures the strength of the SNP-exposure association
• Weak instruments inflate the variance of individual Wald estimates, but IVW weights them more heavily (inverse of variance), paradoxically amplifying their influence
• When mean F < 10, consider robust IVW methods, allele score approaches, or penalized weighting schemes