Why Use Radial Coordinates?
An IVW estimate combines variant-specific associations, but a single summary can conceal whether the variants tell a consistent story. Radial MR makes the contribution of each variant visible by expressing its exposure and outcome associations in units of the outcome-association standard error. In this coordinate system, the IVW estimate is a regression slope, and a variant’s distance from the fitted line corresponds to its contribution to heterogeneity.
The purpose is diagnostic: to locate variants whose associations are poorly aligned with the fitted IVW relation and to assess how much the estimate changes under a defensible sensitivity analysis. A large residual does not, by itself, establish horizontal pleiotropy, identify the source of the discrepancy, or justify removing the variant.
Radial Coordinates and the IVW Slope
For variant \(j\), let \(\hat{\beta}_{Xj}\) be its estimated association with the exposure, \(\hat{\beta}_{Yj}\) its estimated association with the outcome, and \(SE(\hat{\beta}_{Yj})\) the standard error of the outcome association. Assume the variants are independent, alleles are harmonized so that the associations refer to the same effect allele, and the outcome associations have approximately normal sampling errors. The first-order radial formulation treats the exposure associations as fixed, or sufficiently precise that their sampling uncertainty can be ignored. It also assumes that the outcome-association errors have no extra covariance across variants.
The phrase “unweighted regression” here refers to the transformed coordinates: division by the outcome standard error has already applied the inverse-variance scaling. Written in the original association units, the same estimate is the familiar first-order IVW estimator:
The radial residual is a standardized discrepancy between the observed outcome association and the association predicted by the fitted slope. Its square is nonnegative, so a large \(q_j\) means that variant contributes substantially to the total residual heterogeneity. The IVW weight in the ratio-estimate formulation is proportional to \(x_j^2=\hat{\beta}_{Xj}^2/SE(\hat{\beta}_{Yj})^2\): variants with greater radial leverage can have more influence on the fitted slope.
Why Radial Residuals Sum to Cochran’s Q
For the first-order IVW model, the fitted outcome association for variant \(j\) is \(\hat{\theta}_{IVW}\hat{\beta}_{Xj}\). Its standardized residual in the original association units is therefore \((\hat{\beta}_{Yj}-\hat{\theta}_{IVW}\hat{\beta}_{Xj})/SE(\hat{\beta}_{Yj})\), which is exactly \(r_j\). Squaring and summing these residuals gives the first-order Cochran heterogeneity statistic:
This identity explains the diagnostic role of radial coordinates: they do not create a different heterogeneity statistic; they display its component contributions. Under the model of independent, approximately normal outcome-association errors, with a common underlying slope and the first-order assumptions above, \(Q\) is approximately chi-squared with \(K-1\) degrees of freedom. The loss of one degree of freedom reflects estimation of the common slope. A small \(p\)-value is evidence against the common-slope model, not a test that identifies which assumption failed.
Worked Example: Transforming Associations into Radial Coordinates
Suppose three independent, harmonized variants have the following hypothetical summary associations. Exposure and outcome associations are in their respective measurement units; outcome standard errors are in outcome units.
| Variant | \(\hat{\beta}_{Xj}\) | \(\hat{\beta}_{Yj}\) | \(SE(\hat{\beta}_{Yj})\) |
|---|---|---|---|
| 1 | 0.10 | 0.04 | 0.02 |
| 2 | 0.20 | 0.10 | 0.04 |
| 3 | 0.30 | 0.12 | 0.03 |
The radial coordinates are \(x_1=0.10/0.02=5\), \(y_1=0.04/0.02=2\); \(x_2=0.20/0.04=5\), \(y_2=0.10/0.04=2.5\); and \(x_3=0.30/0.03=10\), \(y_3=0.12/0.03=4\). Thus,
The estimate is 0.4167 outcome units per exposure unit. The residuals are \(2-(0.4167)(5)=-0.0835\), \(2.5-(0.4167)(5)=0.4165\), and \(4-(0.4167)(10)=-0.167\), with small differences caused by rounding. Their squared contributions sum to approximately \(0.208\). This example illustrates both the scaling and the weighting: variant 3 has \(x_3^2=100\), compared with 25 for each of the other variants, because its exposure association is relatively large compared with its outcome standard error. The calculation uses the stated first-order assumptions; radial transformation does not remove the need to consider instrument validity or uncertainty in the exposure associations.
Reading Contributions and Identifying Potential Outliers
A radial plot places \(x_j\) horizontally and \(y_j\) vertically, with the fitted IVW line \(y=\hat{\theta}_{IVW}x\) through the origin. A point far from that line has a large absolute residual; its squared vertical distance is its contribution \(q_j\). Reporting the contributions alongside the plot is useful because a visual impression can be misleading when coordinates or scales differ.
A large contribution is a potential outlier signal, not an automatic exclusion rule. It can reflect a chance fluctuation, an unusual direct genetic effect, an allele-harmonization error, or an issue with the summary data or model. Conversely, a variant can have a modest residual yet still matter because its radial weight or leverage is high. Residual size and influence on the slope are related but are not identical.
Under a fixed common-slope model with known slope, a standardized residual would have approximately a standard normal distribution, and its square would have approximately a chi-squared distribution with one degree of freedom. In practice, the slope is estimated from the same variants, so residuals are not independent standard normal variables. Looking for the largest contribution among many variants also creates a multiple-comparisons problem. Thus, a per-variant threshold should not be treated as definitive without accounting for the diagnostic procedure and the number of variants examined.
Worked Example: A Large Contribution in a Four-Variant Analysis
Consider four hypothetical, independent, harmonized variants, each with outcome-association standard error 0.05. Let the exposure associations be 0.10, 0.20, 0.15, and 0.25, and let the outcome associations be 0.05, 0.10, 0.075, and 0.40. Assume the first-order radial model is appropriate for this illustration.
The radial coordinates are \((2,1)\), \((4,2)\), \((3,1.5)\), and \((5,8)\). The fitted slope is
The residual for variant 4 is \(8-(1.0093)(5)=2.9535\), so its contribution is approximately \(q_4=(2.9535)^2=8.724\). The other residuals are approximately \(-1.0185\), \(-2.0370\), and \(-1.5278\), giving contributions of approximately 1.037, 4.150, and 2.334. Therefore, \(Q\approx 16.25\). With \(K-1=3\) degrees of freedom, the chi-squared reference gives \(p\approx 0.0010\), which is evidence of heterogeneity under the stated model. Variant 4 is the largest individual contributor, but the other variants also contribute; the diagnostic does not prove that variant 4 is invalid or that it alone explains the heterogeneity.
Comparing Estimates Before and After Exclusion
A useful sensitivity analysis compares the estimate from all prespecified variants with an estimate after excluding a variant for a reason that can be defended independently of merely wanting a preferred result. Examples of such reasons include a documented harmonization problem or a prespecified diagnostic rule followed by investigation of the flagged variant. The comparison should report the change in estimate and uncertainty, not only whether a significance threshold was crossed.
For first-order fixed-effect IVW under independent-variant assumptions, the estimated standard error is \(1/\sqrt{\sum_j x_j^2}\). This describes sampling uncertainty under the fitted model; it does not adjust for searching through variants and choosing an exclusion based on their residuals. A post-exclusion interval can therefore understate uncertainty if the selection process is ignored. In addition, removing a variant changes the set of instruments and can change the estimand when genetic effects are heterogeneous.
Worked Example: Quantifying the Effect of a Justified Exclusion
Use the four-variant data from the preceding example. Suppose a separate audit finds that the fourth variant’s outcome association was incorrectly harmonized, providing a data-based reason to correct or exclude it. Before exclusion, the IVW estimate is 1.0093 and \(\sum_jx_j^2=54\), so its fixed-effect standard error is \(1/\sqrt{54}=0.1361\). A conventional 95% interval is
After excluding variant 4, the remaining outcome-to-exposure ratios are all 0.5. The estimate is \[ \hat{\theta}_{IVW} =\frac{2(1)+4(2)+3(1.5)}{2^2+4^2+3^2} =\frac{14.5}{29} =0.5000. \] The standard error is \(1/\sqrt{29}=0.1857\), giving the conventional interval \(0.5000\pm1.96(0.1857)=(0.1361,\ 0.8639)\). The estimate shifts from 1.009 to 0.500 outcome units per exposure unit, and the interval becomes wider because the removed variant contributed substantial radial weight. These intervals are conditional on the respective fixed-effect models; the second does not account for the fact that a diagnostic was examined or for the exclusion decision. The comparison demonstrates sensitivity to the variant, not proof that the post-exclusion estimate is the causal effect.
Assumptions and Practical Interpretation
Radial diagnostics inherit the assumptions of the IVW model used to create them. In particular, variants should be correctly aligned, sufficiently independent for the diagonal-variance calculation, and analyzed on compatible scales. The first-order version treats the exposure associations as fixed and uses outcome standard errors for scaling. When uncertainty in exposure associations is material, this approximation can be inadequate; a radial plot does not repair weak-instrument or measurement-error problems. Correlated variants also require a covariance-aware analysis rather than treating their contributions as independent.
The overall \(Q\) statistic and individual \(q_j\) values answer related but distinct questions. \(Q\) summarizes the total discrepancy from a common slope. The \(q_j\) values show how that discrepancy is distributed across variants. Neither establishes whether the discrepancy is caused by horizontal pleiotropy, nor whether a particular variant violates an instrumental-variable assumption. A large contribution is best used to prompt checks of data quality, biological context, and sensitivity of the estimate.
Confirm allele harmonization, units, standard errors, and whether the independence and first-order assumptions are plausible.
Calculate the radial IVW slope, residuals, individual contributions, and total \(Q\); use a plot to see the pattern as well as the largest contribution.
Check flagged variants for data issues and interpret their residuals in light of leverage, multiplicity, and the assumptions of the model.
Give estimates and uncertainty before and after any justified exclusion, and state that the comparison is diagnostic rather than proof of causality.
Common Mistake
A common mistake is to remove whichever variant has the largest radial residual and then describe the remaining estimate as the corrected causal answer. That reasoning treats a model-dependent diagnostic as if it identified an invalid instrument, ignores the chance of extreme residuals when many variants are inspected, and overlooks the uncertainty introduced by choosing an exclusion. The correction is to investigate the flagged variant, apply a defensible and transparently reported rule, and present the all-variant and sensitivity estimates without claiming that either is automatically causal.
Check Your Understanding
Answer each question using the definitions and assumptions in this tutorial.
- For one variant, \(\hat{\beta}_{Xj}=0.12\), \(\hat{\beta}_{Yj}=0.06\), and \(SE(\hat{\beta}_{Yj})=0.03\). What are its radial coordinates?
- Why does the sum of squared radial residuals equal the first-order Cochran \(Q\) statistic?
- What does a large \(q_j\) indicate, and what does it not establish?
- Why can an outlier search across many variants make an unadjusted per-variant threshold misleading?
- After excluding a flagged variant, why should the new estimate not automatically be called the corrected causal estimate?