Fixed-Effect And Multiplicative Random-Effects IVW

Why distinguish the two IVW models?

Inverse-variance weighted (IVW) Mendelian randomization combines variant-specific information about an exposure and an outcome. Its weighted estimate can be the same under two commonly used models, while its reported uncertainty differs. The distinction matters when the variant-specific outcome associations scatter around the fitted causal relationship more than their reported standard errors predict: a multiplicative random-effects model allows that extra dispersion to widen the standard error.

The comparison concerns uncertainty, not a new way to remove bias. A larger standard error can reflect unexplained variation among variants, but it does not establish that the variants satisfy the instrumental-variable assumptions or that the fitted causal estimate is unbiased.

Summary-data model and assumptions

Suppose \(K\) genetic variants have been harmonized so their exposure and outcome associations refer to the same effect allele. Write \(\hat\beta_{Xj}\) and \(\hat\beta_{Yj}\) for the estimated associations of variant \(j\) with the exposure and outcome, and \(s_{Yj}\) for the standard error of \(\hat\beta_{Yj}\). The usual first-order IVW calculation treats the exposure associations as fixed and uses the outcome-association standard errors as weights.

These models assume the variants are independent, as is approximately intended when instruments have been clumped to limit linkage disequilibrium. They also use a linear relationship through the origin across the included variants, and assume the standard errors \(s_{Yj}\) adequately describe the sampling variability up to the common multiplier \(\phi\). For a causal interpretation, the instrumental-variable assumptions are still required: variants must be relevant to the exposure, independent of relevant confounders, and affect the outcome only through the exposure (or meet the conditions needed for the chosen IVW interpretation). The random-effects variance model does not replace those assumptions.

Estimate, standard error, and dispersion

Let \(w_j=1/s_{Yj}^{\,2}\). IVW minimizes the weighted residual sum of squares over \(\theta\). Differentiating that sum gives the same point estimate under either model, because multiplying every weight by the same factor does not change the minimizer.

The fixed-effect standard error follows by treating the exposure associations as fixed: the estimate is a weighted linear combination of outcome associations, so its variance is the reciprocal of \(\sum_j w_j\hat\beta_{Xj}^{\,2}\). Under the multiplicative model, that variance is multiplied by \(\phi\), giving \(SE_{\mathrm{RE}}=\sqrt{\phi}\,SE_{\mathrm{FE}}\).

The residual dispersion is summarized by the weighted residual sum of squares

There is one fitted parameter, \(\theta\), so the residual degrees of freedom are \(df=K-1\). Under the stated independent normal-error model, \(Q/\phi\) has a chi-squared distribution with \(K-1\) degrees of freedom. Consequently, \(Q/(K-1)\) estimates \(\phi\). A common conservative convention for an overdispersion adjustment is to use \(\hat\phi=\max\{1,Q/(K-1)\}\), so estimated underdispersion does not shrink the fixed-effect standard error. Then

Some implementations report the untruncated estimate \(Q/(K-1)\) as the multiplier. That convention can yield a standard error below the fixed-effect value when \(Q<K-1\). State which convention is used when reporting results; the examples here use the conservative cap at 1.

Why the dispersion estimate has \(K-1\) degrees of freedom

A short linear-algebra argument explains both the residual degrees of freedom and the scale estimate. Define \(z_j=\hat\beta_{Yj}/s_{Yj}\) and \(a_j=\hat\beta_{Xj}/s_{Yj}\). Then the weighted fit is an ordinary one-parameter regression of \(z_j\) on \(a_j\) through the origin, and \(Q\) is the squared length of the residual vector. Fitting one coefficient projects the \(K\)-dimensional data vector onto a one-dimensional subspace, leaving a residual subspace of dimension \(K-1\).

Under the multiplicative model, the standardized errors have variance \(\phi\) in each independent direction. The expected squared length in the \(K-1\) residual directions is therefore \((K-1)\phi\). Thus \(E(Q)=(K-1)\phi\), which motivates estimating the variance multiplier by \(Q/(K-1)\). This is a scale estimate, not a procedure that changes the fitted coefficient.

Interpretation, diagnostics, and alternatives

For a fixed-effect analysis, the uncertainty calculation assumes the observed scatter is adequately explained by the reported outcome-association standard errors. Multiplicative random effects retain the same relative weights and the same estimate, but inflate the variance by a common factor when \(Q/(K-1)>1\). A confidence interval is a measure of uncertainty under the specified model; it is not a probability statement that the fixed causal effect lies in a particular computed interval.

The value of \(Q\) is useful as a diagnostic of fit, but it is not a verdict on the cause of the scatter. A large value may arise from chance, underestimated standard errors, violations of the simple linear relationship, or variant-specific effects not represented by the model. It does not by itself identify which variants are problematic, prove horizontal pleiotropy, or show that the causal estimate is biased. Conversely, a small \(Q\), particularly with few variants, does not prove that the assumptions hold.

The multiplicative approach is convenient when the aim is to retain the IVW point estimate but avoid treating excess scatter as if it were absent. Its limitation follows from that same feature: it widens uncertainty without correcting a systematic shift in the estimate. It also relies on a common dispersion factor and does not account for dependence among variants. The calculations here assume independent variants; when that assumption is not reasonable, the covariance among associations must be incorporated rather than treated as a common scale increase.

Common mistake

A frequent error is to see \(\hat\phi>1\) and multiply the IVW estimate itself by \(\sqrt{\hat\phi}\). That changes the estimated causal effect and is not the multiplicative random-effects procedure. The correction is to leave \(\hat\theta_{\mathrm{IVW}}\) unchanged and multiply its standard error by \(\sqrt{\hat\phi}\); the resulting interval becomes wider while remaining centered at the same estimate. Another error is to interpret the wider interval as evidence that the estimate is unbiased. It only reflects an uncertainty adjustment under the stated model.