What a Mendelian Randomization Funnel Plot Shows
In summary-data Mendelian randomization (MR), each genetic variant has an estimated association with the exposure and an estimated association with the outcome. A plot of these pairs can show whether their pattern is broadly compatible with a shared causal slope. A related, commonly used funnel plot displays each variant’s ratio estimate against its precision: weak or imprecisely estimated ratios tend to spread more widely, while precise ratios tend to cluster more closely.
These displays are useful as diagnostics, not as tests that certify the instrumental-variable assumptions. In particular, a visually symmetric plot does not establish that the variants affect the outcome only through the exposure. The introductory material on direct and correlated pleiotropy distinguishes those two ways that an instrument can violate this exclusion restriction; both matter when interpreting the plot.
A funnel is not guaranteed to have an exact geometric shape. Its appearance depends on the standard errors, the range of instrument strengths, and the plotting convention. The key idea is that uncertainty in a ratio estimate usually grows as the exposure association gets closer to zero. Thus, a cloud of ratio estimates can be wide at low precision and narrow at high precision. On the association scale, points compatible with one common causal effect tend to lie around a line through the origin, although sampling error and pleiotropic effects can move them away from it.
Assumptions and Construction
Before interpreting a plot, check that the plotted estimates are comparable. The effect allele must be aligned across the exposure and outcome estimates, the outcome and exposure scales must be stated, and each point must correspond to a clearly identified variant. If variants are correlated, treating them as independent points can make the display misleading; use an analysis and uncertainty calculation that account for their linkage disequilibrium, or use a suitably independent set of variants. In a two-sample setting with independent exposure and outcome samples, the estimation errors in the two associations can often be treated as independent. With sample overlap, that covariance may not be zero.
A common approximate standard error for the ratio uses the delta method. If the exposure association is estimated precisely relative to the outcome association, and the two estimates are independent, then \(SE(\hat{\theta}_j)\) is often approximated by \(SE(\hat{\beta}_{Yj})/|\hat{\beta}_{Xj}|\). If uncertainty in the exposure association is not negligible, the first-order variance approximation includes it:
The approximation follows by taking the first-order Taylor expansion of the ratio function \(g(x,y)=y/x\): its derivatives with respect to \(y\) and \(x\) are \(1/x\) and \(-y/x^2\), respectively. The variance of the resulting linear approximation gives the displayed expression. The simpler standard error formula drops the exposure-uncertainty term and the covariance term. It is therefore an approximation, not a universal formula; near-zero exposure associations can make ratio estimates unstable and the approximation poor.
Harmonize effect alleles, identify the variants, and retain \(\hat{\beta}_{Xj}\), \(\hat{\beta}_{Yj}\), and the standard errors needed for the chosen uncertainty calculation.
For each variant with a nonzero exposure estimate, calculate \(\hat{\theta}_j=\hat{\beta}_{Yj}/\hat{\beta}_{Xj}\) and \(1/SE(\hat{\theta}_j)\). State which standard-error approximation is being used.
Plot \(\hat{\beta}_{Yj}\) against \(\hat{\beta}_{Xj}\) to show association pairs and their implied slopes. Plot ratio estimates against precision to display the spread of the variant-specific estimates. Label axes, units, and any pooled slope or reference lines.
Describe the observed pattern, note influential or imprecise points, and treat asymmetry as a prompt for further investigation—not a diagnosis of one specific cause.
Worked Association and Funnel Displays
Worked Example: Calculating and Plotting Variant-Specific Ratios
Suppose a hypothetical two-sample MR analysis has four independent, allele-harmonized variants. The exposure associations are treated as precise for this illustrative calculation, and the outcome standard error is \(0.04\) for each variant.
| Variant | \(\hat{\beta}_{Xj}\) | \(\hat{\beta}_{Yj}\) | \(SE(\hat{\beta}_{Yj})\) |
|---|---|---|---|
| 1 | 0.10 | 0.08 | 0.04 |
| 2 | 0.20 | 0.34 | 0.04 |
| 3 | 0.30 | 0.27 | 0.04 |
| 4 | 0.40 | 0.48 | 0.04 |
For variant 1, \(\hat{\theta}_1=0.08/0.10=0.80\), and the approximate standard error is \(0.04/0.10=0.40\), giving precision \(1/0.40=2.50\). Applying the same calculation gives ratios \(1.70,\ 0.90,\ 1.20\), standard errors \(0.20,\ 0.1333,\ 0.10\), and precisions \(5.00,\ 7.50,\ 10.0\), respectively. On the association-scale plot, the four points are \((0.10,0.08)\), \((0.20,0.34)\), \((0.30,0.27)\), and \((0.40,0.48)\). Their slopes from the origin are exactly the four ratios.
For reference, the inverse-variance weighted (IVW) through-origin slope, using outcome-association variances as weights, is
The ratios in the funnel display are not evenly spread around this reference: they are \(0.80,\ 1.70,\ 0.90,\ 1.20\), with the most precise estimate at \(1.20\). Four points are far too few to diagnose a pattern reliably. The association plot makes the same differences visible as departures from a line of slope \(1.163\); the funnel view adds information about how precisely each ratio is estimated. The numerical slope is in outcome-association units per exposure-association unit.
Why the IVW Reference Is a Through-Origin Slope
The through-origin fit is motivated by the idealized model \(\hat{\beta}_{Yj}\approx\theta\hat{\beta}_{Xj}\), when the variants satisfy the instrumental-variable assumptions and share a causal effect \(\theta\). Under a working model with independent outcome-association errors of known variances \(SE(\hat{\beta}_{Yj})^2\), the weighted sum of squared residuals for a candidate slope \(b\) is
Differentiating with respect to \(b\), setting the derivative to zero, and solving gives
This is the IVW slope in the stated working model. A funnel plot can show whether individual ratios scatter around a reference estimate, but the reference itself depends on the chosen estimator and its assumptions. A visually appealing cluster around an IVW line does not verify those assumptions.
Symmetry, Asymmetry, and What They Can Mean
Under a simple valid-instrument model, variant-specific outcome associations are proportional to exposure associations, apart from sampling error. Ratio estimates then tend to scatter around a common causal effect. In a ratio-versus-precision plot, less precise estimates may vary more widely, while precise estimates tend to cluster more tightly. This is the intuitive symmetric-funnel pattern. It is a pattern expected under a simplified model, not a required shape for every valid analysis: different standard errors, few variants, or a narrow range of precisions can make the funnel hard to recognize.
Worked Example: Asymmetry From a Directional Pleiotropic Shift
Consider a hypothetical set of variants with \(\hat{\beta}_{Xj}=0.05,\ 0.10,\ 0.20,\ 0.40\). Suppose their outcome associations follow \(\hat{\beta}_{Yj}=0.5\hat{\beta}_{Xj}+0.05\). The added \(0.05\) is a common direct shift in the outcome association, not an effect mediated by the exposure.
The outcome associations are \(0.075,\ 0.100,\ 0.150,\ 0.250\). The ratios are \(0.075/0.05=1.50\), \(0.100/0.10=1.00\), \(0.150/0.20=0.75\), and \(0.250/0.40=0.625\). The ratios are farther above \(0.5\) for weaker exposure associations and approach \(0.5\) as the exposure association grows. If the outcome standard errors are all \(0.03\), the approximate ratio standard errors are \(0.60,\ 0.30,\ 0.15,\ 0.075\), so precision increases from \(1.67\) to \(13.3\).
A ratio-versus-precision display would show a directional pattern rather than a cloud centered symmetrically around \(0.5\). On the association scale the points lie on a line with intercept \(0.05\), rather than on a line through the origin. This illustrates how directional pleiotropy can produce asymmetry. It does not mean that every asymmetric funnel has this cause: random variation, incorrect harmonization, outliers, or other model failures can also create an uneven pattern.
Worked Example: A Symmetric-Looking Plot With Invalid Instruments
Now suppose the true causal effect is \(0.5\), but each variant also has a direct effect on the outcome equal to \(0.3\hat{\beta}_{Xj}\). Then
Every point lies exactly on a line through the origin with slope \(0.8\), and every ratio is \(0.8\). A funnel plot could look perfectly symmetric and show no scatter at all, even though the direct effects violate the exclusion restriction and the plotted slope does not equal the stated causal effect \(0.5\). The plot alone cannot distinguish this pattern from a valid model with causal effect \(0.8\). This is why visual symmetry is not evidence that instruments are valid.
Diagnostics, Alternatives, and Limitations
Read the display alongside the data and the model, rather than as a stand-alone verdict. First check for allele-orientation mistakes: a sign reversal can move a point to the opposite side of a plot. Check whether a few variants dominate the range, whether exposure associations are close to zero, and whether the apparent symmetry is driven by only a small number of imprecise points. Report which precision definition and ratio standard errors were used. If the exposure estimates are uncertain or the two association estimates are correlated, the simple ratio standard error may misrepresent the funnel’s horizontal ordering.
A funnel plot is also sensitive to selection and dependence. Selecting variants because they have strong observed exposure associations can affect their apparent distribution; correlated variants can create clusters that look like many independent observations. A small number of instruments provides little visual resolution, while a large number can make minor departures look striking. Inspecting asymmetry does not replace checks of instrument strength, allele harmonization, or the assumptions underlying the causal estimate.
The plot complements, rather than substitutes for, other approaches. Cochran’s \(Q\) provides a formal summary of dispersion relative to a specified model, but does not identify the cause of heterogeneity. MR-Egger regression estimates a slope and intercept under its own assumptions, including InSIDE for its usual causal interpretation; the funnel’s visual asymmetry is not equivalent to an MR-Egger test. Robust estimators and outlier-focused methods can assess sensitivity in different ways, but each has its own assumptions and failure modes. The appropriate response to an unusual funnel is to investigate with relevant analyses and substantive knowledge, not to select whichever estimate looks most reassuring.
Common Mistake: Treating Symmetry as Proof
Wrong reasoning: “The ratios are symmetric around the pooled estimate, so there cannot be pleiotropy and the instruments are valid.”
Correction: Symmetry describes the observed pattern, not whether every variant satisfies the exclusion restriction. Pleiotropic effects can be balanced, proportional to exposure associations, too small to see, or masked by sampling variation. Treat symmetry as limited diagnostic information and assess validity using the assumptions and evidence relevant to the study.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A variant has \(\hat{\beta}_{X}=0.25\), \(\hat{\beta}_{Y}=0.10\), and \(SE(\hat{\beta}_{Y})=0.05\). Under the approximation that exposure-association uncertainty is negligible, calculate its ratio estimate, standard error, and precision.
- Why can a small exposure association make a ratio estimate imprecise?
- What does an asymmetric ratio-versus-precision pattern suggest, and what does it not prove?
- How can a perfectly symmetric plot coexist with invalid instruments?
Solutions. (1) The ratio is \(0.10/0.25=0.40\). Its approximate standard error is \(0.05/0.25=0.20\), and its precision is \(1/0.20=5.00\). (2) Dividing by a small \(\hat{\beta}_{X}\) magnifies outcome-association uncertainty; in the approximation, \(SE(\hat{\theta})=SE(\hat{\beta}_{Y})/|\hat{\beta}_{X}|\). (3) Asymmetry can flag directional pleiotropy or other problems, but does not identify a unique cause or prove that pleiotropy is present. (4) Direct effects can be proportional to exposure associations, shifting all ratios to a common value while violating the exclusion restriction; visual symmetry cannot distinguish that case from a valid common-slope model.