Deriving Inverse-Variance Weighted Estimation

From Variant-Specific Ratios to a Shared Estimate

With several genetic variants as instruments, each variant can yield a Wald ratio: its estimated association with the outcome divided by its estimated association with the exposure. If the variants are valid instruments and represent a common causal effect, these ratios should estimate the same underlying quantity, apart from sampling error. In practice, their precision can differ substantially. Inverse-variance weighted (IVW) estimation combines them by giving more influence to more precise ratios.

Let \(\hat\beta_{Xj}\) and \(\hat\beta_{Yj}\) denote the estimated associations of variant \(j\) with the exposure and outcome, respectively, for \(j=1,\ldots,J\). The variant-specific Wald ratio is \(\hat\theta_j=\hat\beta_{Yj}/\hat\beta_{Xj}\), provided \(\hat\beta_{Xj}\ne 0\). The target \(\theta\) is the causal effect on the outcome for a one-unit increase in the exposure, under the causal assumptions for Mendelian randomization. Its units are outcome units per exposure unit.

The definition alone does not determine which variance estimates to use. The usual IVW estimator in two-sample Mendelian randomization uses weights based on the outcome-association standard errors, treating the estimated exposure associations as fixed for this calculation. The derivation below makes that approximation explicit.

Assumptions and Derivation

For the standard derivation, suppose the variants are independent, their alleles have been harmonized so that the exposure and outcome associations refer to the same effect allele, and each variant is valid for the causal effect being estimated. In particular, the model has no direct variant effect on the outcome outside the exposure pathway. Write \(\beta_{Xj}\) and \(\beta_{Yj}\) for the corresponding population associations. Under a common causal effect, \(\beta_{Yj}=\theta\beta_{Xj}\). Assume the estimated outcome associations have approximately normal sampling errors with known or estimated variances \(\sigma_{Yj}^{2}\), independent across variants. For this derivation, treat the exposure-association estimates \(\hat\beta_{Xj}\) as fixed and nonzero, and use \(\hat\beta_{Yj}=\theta\hat\beta_{Xj}+\varepsilon_j\), where \(\varepsilon_j\) has mean zero and variance \(\sigma_{Yj}^{2}\).

Dividing this model by \(\hat\beta_{Xj}\) shows how the ratio estimates arise:

Thus the approximate inverse-variance weight is \(w_j=\hat\beta_{Xj}^{2}/\sigma_{Yj}^{2}\). To derive the combined estimate, choose the value of \(\theta\) that minimizes the weighted sum of squared differences between the observed ratios and a shared effect:

Differentiating with respect to \(\theta\), setting the derivative to zero, and solving gives

Under the stated independent, normal-error model with variances treated as known, this is also the maximum-likelihood estimator. Its estimated standard error is \(1/\sqrt{\sum_j w_j}\). In applications, the variances are estimated, so this standard error and any associated normal-approximation interval are themselves approximations.

The same derivation gives an equivalent regression expression. Substituting \(\hat\theta_j=\hat\beta_{Yj}/\hat\beta_{Xj}\) and \(w_j=\hat\beta_{Xj}^{2}/\sigma_{Yj}^{2}\) yields

This is the weighted least-squares slope from regressing the outcome associations \(\hat\beta_{Yj}\) on the exposure associations \(\hat\beta_{Xj}\), with the intercept constrained to zero and regression weights \(1/\sigma_{Yj}^{2}\). Indeed, the regression objective is

So, under these weights, minimizing the regression objective is exactly minimizing the weighted ratio objective. The regression is not an unrelated alternative: it is an algebraic form of the same estimator.

What the Weights Do—and Do Not—Mean

The squared exposure association appears in the weight because a given amount of outcome-association uncertainty translates into less ratio uncertainty when the exposure association is larger. The weight also increases when the outcome-association standard error decreases. These weights therefore describe estimated precision under the stated approximation; they are not direct measures of instrument validity or proof that a variant has no pleiotropic effect.

The standard formula treats the exposure-association estimates as fixed. If their uncertainty is material, a first-order delta-method variance for a single ratio, assuming independent exposure and outcome association estimates, includes an additional term:

If the two association estimates are not independent, the delta-method variance also includes a covariance term. Using the fuller ratio variances still defines a precision-weighted mean of ratios, but its weights are no longer generally \(\hat\beta_{Xj}^{2}/SE(\hat\beta_{Yj})^{2}\). Consequently, it is not generally the same as the usual weighted regression through the origin. The equivalence proved above depends on the outcome-based weights, not merely on the name “inverse-variance weighted.”

Checking the Model and Avoiding Misinterpretation

IVW is most straightforward to interpret when the variants estimate a common causal effect and the assumptions supporting their use as instruments are plausible. The calculation itself cannot establish those assumptions. Before interpreting the combined estimate, check allele harmonization, the signs and scales of the associations, and whether the variants are sufficiently independent for the variance formula used. A reversed effect allele can change the signs of associations; misaligned exposure and outcome effects can produce misleading ratios. A very small exposure association also makes a ratio unstable and its approximation less reassuring.

A useful descriptive check is to inspect the individual ratio estimates and their uncertainty, alongside the fitted through-origin regression. Markedly different ratios or large residuals from the common-effect line suggest that the shared-effect model may not describe the data well. Such patterns can arise from sampling variability, invalid instruments, or other departures from the model; they do not, by themselves, identify the cause. Formal heterogeneity procedures and alternative estimators require additional methods and assumptions.

Other summaries answer different questions. An unweighted mean gives every ratio equal influence, regardless of its estimated precision. A weighted regression with a free intercept allows a nonzero intercept, so its slope is not generally the standard through-origin IVW estimate. If variants are correlated, the independent-variant derivation and simple sum of weights do not apply unchanged; the dependence must be reflected in the covariance structure. These alternatives should not be substituted silently for the estimator derived here.

• Mistake: “IVW is the ordinary average of the ratios.” Correction: it is a weighted average, with weights determined by the chosen ratio-variance approximation.
• Mistake: “Any inverse-variance weighting is equivalent to regression through the origin.” Correction: the usual equivalence requires weights \(\hat\beta_{Xj}^{2}/SE(\hat\beta_{Yj})^{2}\).
• Mistake: “A precise IVW estimate verifies the instrumental-variable assumptions.” Correction: precision quantifies sampling uncertainty under the model; it does not demonstrate instrument validity or a causal interpretation.