Fixed- and Random-Effects IVW Under Pleiotropy

Why Compare Fixed- and Random-Effects IVW?

Inverse-variance weighted (IVW) Mendelian randomization combines the exposure and outcome association estimates for multiple genetic variants. Under a common-slope model, it estimates one association between the exposure and outcome across variants. Horizontal pleiotropy can make the variants’ outcome associations depart from that model: some departures may be consistent with sampling variation, while others may be larger or systematically aligned with the exposure associations.

Fixed-effect IVW treats the outcome-association standard errors as accounting for all residual variation around the common slope. Multiplicative random-effects IVW keeps the same slope estimate but allows greater residual dispersion to increase its standard error. That distinction matters: adjusting uncertainty for dispersion is not the same as correcting bias in the slope.

The standard first-order setup assumes independent variants, harmonized alleles, and no covariance between the exposure and outcome association estimates, as is often approximated in non-overlapping two-sample MR. It also ignores uncertainty in \(\hat{\beta}_{Xj}\) when constructing the weights. This approximation is more credible when instruments are sufficiently strong; the earlier material on instrument strength and the F statistic addresses that concern. The formulas below are not a correction for weak-instrument bias.

Fixed-Effect IVW as Weighted Regression

Fixed-effect IVW fits a weighted regression through the origin: the fitted outcome association for variant \(j\) is \(\theta\hat{\beta}_{Xj}\). The weights are \(1/SE_{Yj}^2\), so variants with more precise outcome associations contribute more to the estimate. The through-origin specification represents the assumption that there is no average intercept across variants; it does not establish that every variant satisfies the exclusion restriction.

Under the fixed-effect model, outcome-association errors are independent, have mean zero, and have variances \(SE_{Yj}^2\). Treating the exposure associations as fixed, the variance of the fitted slope is the reciprocal of the weighted sum of squared exposure associations. Thus the standard error is the square root of that reciprocal. A confidence interval commonly uses the normal approximation, for example \(\hat{\theta}_{\mathrm{IVW}}\pm1.96SE(\hat{\theta}_{\mathrm{IVW}})\).

Residual Dispersion and Multiplicative Random Effects

To assess how far the observed associations lie from the fitted common slope, define the weighted residual sum of squares

This quantity summarizes residual dispersion around the IVW line. Since one slope has been fitted, the residual degrees of freedom are \(K-1\). In this tutorial, the multiplicative random-effects convention estimates an overdispersion multiplier as \(\hat{\phi}=\max\{1,Q/(K-1)\}\). The lower bound of one means this convention increases uncertainty when dispersion is elevated but does not reduce the fixed-effect standard error when the observed dispersion is smaller. Other implementations may report a different scale convention, so it is important to state which is used.

The scaling follows directly from the model variance. If the residual variance for each outcome association is \(\phi SE_{Yj}^2\), with \(\phi\ge1\), then the variance of the weighted slope conditional on the exposure associations is \(\phi/\sum_j w_j\hat{\beta}_{Xj}^2\). Replacing \(\phi\) with its estimated value gives the random-effects standard error.

What Different Variant Ratios Mean

When variants have different ratios, fixed-effect IVW still returns a single weighted slope. It is not generally the ordinary average of those ratios. A variant with a larger exposure association, or a more precise outcome association, can have more influence. If variants have direct effects on the outcome, write their population associations as \(\beta_{Yj}=\theta\beta_{Xj}+\alpha_j\), where \(\alpha_j\) is the direct effect not mediated by the exposure. Substitution into the IVW formula shows how those effects can shift the estimate:

The second term is a weighted contribution from the direct effects. If it is nonzero, increasing the standard error with a random-effects multiplier does not remove it. Random-effects IVW can provide more cautious uncertainty around the same potentially shifted slope, but it does not make an invalid instrument valid.

Assumptions, Uses, and a Practical Comparison

Fixed-effect and multiplicative random-effects IVW share the same point estimate and the same central vulnerability to pleiotropic bias. Their difference concerns the variance model. Fixed-effect IVW assumes residual variation is adequately represented by the reported \(SE_{Yj}\) values. Multiplicative random-effects IVW allows the residual variance to be larger by a common multiplier. Neither approach, by itself, identifies which variants are pleiotropic or whether direct effects tend to align with exposure associations.

In the setting considered here, these methods are most useful for describing a common-slope summary and making their uncertainty assumptions explicit. When ratios disagree, a wider random-effects interval is a signal to avoid treating a fixed-effect standard error as automatically adequate. It is not, on its own, a diagnosis of the cause of the disagreement or a test of a particular pleiotropy mechanism. The dedicated material on Cochran’s Q develops formal heterogeneity testing.

Common Mistake

A common mistake is to reason, “The random-effects standard error is larger, so the random-effects estimate has corrected pleiotropy.” The correction is that multiplicative random-effects IVW retains the fixed-effect weighted slope and scales its uncertainty. If pleiotropic effects contribute a nonzero weighted term, the point estimate can remain biased; a larger interval merely reflects the chosen dispersion model. Conversely, similar fixed- and random-effects standard errors do not prove that every instrument satisfies the exclusion restriction.