Radial Mendelian Randomization

Why Use Radial Coordinates?

With many genetic variants, a conventional scatterplot can make the most important discrepancies hard to see: variants with less precise outcome estimates contribute less to the inverse-variance weighted fit, but an ordinary plot gives them the same visual weight. Radial Mendelian randomization rescales each variant’s exposure and outcome association by the standard error of its outcome association. In the resulting coordinates, the inverse-variance weighted (IVW) fit and its residual contributions have a direct geometric interpretation.

A large radial residual signals that a variant’s outcome association is poorly aligned with the fitted model relative to its reported uncertainty. This is evidence of statistical disagreement, not proof of horizontal pleiotropy or of any other particular cause. As with other Mendelian randomization diagnostics, interpretation depends on the model and assumptions.

The units of the original association estimates depend on how the exposure and outcome are measured. The radial coordinates are scaled quantities, not causal effects with the original exposure-to-outcome units. The slope \(\hat\theta\), however, retains the usual interpretation and units of the corresponding MR estimate.

Radial IVW: The Same Estimate in a Different Geometry

For independent variants, the usual first-order IVW estimate, treating the exposure-association estimates as fixed and using outcome-association uncertainty as the weights, is

The right-hand side is the slope of a regression of \(v_j\) on \(u_j\) constrained to pass through the origin. This equality follows by substituting \(u_j=\hat\beta_{Xj}/s_{Yj}\) and \(v_j=\hat\beta_{Yj}/s_{Yj}\) into the weighted IVW formula. Thus radial coordinates change how the calculation is displayed, not the estimate itself.

Under the null model that all variants conform to the fitted IVW relationship, the radial residuals are expected to be small relative to their uncertainty. The sum of their squares is the familiar heterogeneity statistic:

With independent variants and the fixed-effect model, \(Q_{\mathrm{IVW}}\) is approximately distributed as chi-square with \(m-1\) degrees of freedom under the model. This is the same Cochran’s Q statistic used in the earlier material on heterogeneity testing. A large \(Q_{\mathrm{IVW}}\) indicates that the residual variation exceeds what the reported sampling errors alone would predict. It does not identify why the variants disagree.

Radial MR-Egger

MR-Egger allows a nonzero intercept to represent average directional pleiotropy under its assumptions. In the original association scales, it fits a weighted regression of \(\hat\beta_{Yj}\) on \(\hat\beta_{Xj}\), with weights \(1/s_{Yj}^{2}\):

Dividing each fitted relationship and its residual by \(s_{Yj}\) gives the radial form

Therefore, the radial MR-Egger fit is a regression with no ordinary intercept and with two predictors, \(u_j\) and \(z_j\). The coefficient of \(u_j\) is the MR-Egger slope; the coefficient of \(z_j\) is the original-scale intercept \(\alpha\). The fitted radial value is \(\hat\theta u_j+\hat\alpha z_j\). It would be incorrect to fit an ordinary radial line with a single constant intercept when the \(s_{Yj}\) differ: the original-scale intercept becomes \(\alpha/s_{Yj}\) after scaling.

This regression representation requires the usual MR-Egger conditions for interpreting its slope, including the InSIDE assumption, and the no-measurement-error approximation for exposure associations used in standard MR-Egger. It also relies on appropriate variant harmonisation and, in this simple formulation, independent variants. Radial coordinates do not repair violations of these assumptions.

Residuals, Leverage, and Outlier Screening

A raw radial residual is useful for seeing which variants contribute most to \(Q\), since variant \(j\)’s contribution is \(e_j^2\). But residuals from a fitted regression are affected by leverage: observations that help determine the fitted line can have smaller residuals even when their underlying discrepancy is substantial. For IVW, the leverage of variant \(j\) is \(h_{jj}=u_j^2/\sum_k u_k^2\). A commonly used leverage-adjusted residual is

For MR-Egger, form the design matrix with columns \(u_j\) and \(z_j\). Its projection matrix is \(H=Z(Z^\mathsf{T}Z)^{-1}Z^\mathsf{T}\), where \(Z\) denotes that design matrix, and \(h_{jj}\) is its \(j\)th diagonal element. The same adjustment uses the residual from the Egger fit and this leverage. Under a correctly specified Gaussian model with known standard errors, the standardized residual has a standard normal marginal distribution; in practice, estimated standard errors and departures from the model make threshold-based flags approximate diagnostics.

When examining many variants, comparing each residual with an unadjusted 1.96 threshold inflates the chance of at least one false flag. One simple option is a Bonferroni screen: for \(m\) variants and a two-sided familywise level \(\alpha\), flag a variant only if its two-sided marginal p-value is below \(\alpha/m\), equivalently if \(|r_j|\) exceeds the corresponding standard normal quantile. This is a conservative screening rule, not a guarantee that a flagged variant is biologically or causally aberrant.

Assumptions, Comparisons, and Failure Modes

The basic radial IVW calculations above assume independent variants, validly harmonised association estimates, and outcome standard errors that provide appropriate inverse-variance weights. The usual first-order formulation also treats \(\hat\beta_{Xj}\) as fixed; weak instruments or material uncertainty in those estimates can undermine that approximation. With correlated variants, the diagonal weighting and independent-residual arguments do not apply as written; the covariance-aware IVW formulation is needed instead. MR-Egger adds its own requirements, particularly InSIDE and the no-measurement-error approximation for the exposure associations.

Radial analysis is principally a way to express fitted models and inspect their residual structure. It does not replace a global heterogeneity test, and a large residual does not distinguish pleiotropy from data error, association-estimation error, or other model departures. Nor does a small residual prove that the instrumental-variable assumptions hold. A practical analysis should examine the overall heterogeneity, the pattern of residuals, leverage, and sensitivity of the estimate, while checking data harmonisation and instrument strength.

A common mistake is to interpret the largest radial residual as proof that its variant is pleiotropic and then remove it before reporting an estimate. The correction is to call it a candidate outlier, check leverage and data quality, report the prespecified diagnostic and multiplicity handling, and present any exclusion-based estimate as a sensitivity analysis. Data-driven removal can make uncertainty appear smaller and can change the target analysis.

Radial coordinates are useful because they put residual size and inverse-variance weighting into one view. Their strength is diagnostic clarity; their limitation is that diagnostics cannot establish why variants disagree. Causal conclusions still require the substantive assumptions of the chosen MR estimator.