Why Estimate a Mode of Variant-Specific Effects?
In Mendelian randomization (MR), each genetic variant can produce a ratio estimate of the causal effect of an exposure on an outcome. If some variants affect the outcome through pathways other than the exposure, their ratios may be displaced from the causal effect. The weighted mode estimator looks for the largest concentration of ratio estimates, rather than averaging all ratios or requiring valid variants to carry most of the total weight. It is useful when the largest cluster is credible as a group of valid instruments, even if that group is not a majority.
The key qualification is that a concentration in the observed ratios must correspond to valid instruments, not merely to a cluster created by shared pleiotropic effects. The estimator does not establish this from the ratios alone. Its causal interpretation depends on a substantive assumption about which variants form the modal group.
Assumptions and Construction
As with other MR estimators, the causal interpretation requires that the variants are associated with the exposure, are not associated with confounders of the exposure–outcome relation, and affect the outcome only through the exposure when they are valid. The mode approach relaxes the requirement that every instrument be valid. It still requires that the modal group is valid: a large cluster of invalid variants with similar direct effects can instead generate a misleading mode. The variants also need sufficiently strong exposure associations for their ratio estimates to be informative; weak instruments can make ratios unstable.
Let \(\hat\gamma_j\) and \(\hat\Gamma_j\) be estimated genetic associations, and let \(\hat\theta_j=\hat\Gamma_j/\hat\gamma_j\) be the estimated ratio for variant \(j\). Let \(w_j\) be a nonnegative weight, commonly based on the inverse estimated variance of \(\hat\theta_j\). Define the weighted kernel density estimate at a candidate effect \(b\) by
Here \(J\) is the number of variants, \(K\) is a kernel density function, and \(h>0\) is the bandwidth. A Gaussian kernel, for example, is \(K(u)=\exp(-u^2/2)/\sqrt{2\pi}\). The weights determine how much each ratio contributes to the density; the bandwidth determines how broadly nearby ratios are smoothed together. The maximizing value is an estimated causal effect, in the outcome units per exposure unit used for the associations.
Inverse-variance weights give more influence to ratios estimated more precisely. With independent, non-weak variants and negligible uncertainty in \(\hat\gamma_j\), a common approximation is \(w_j=1/SE(\hat\theta_j)^2\). In practice, the ratio standard error must reflect the estimation design; uncertainty in the exposure association can matter, particularly for weak instruments. When variants are correlated, treating their contributions as independent can misstate both the effective weighting and uncertainty. The dependence structure needs to be accounted for rather than ignored.
Why the Mode Targets the Causal Effect
The justification is a density argument. Ignoring sampling error for a moment, valid variants have \(\theta_j=\theta\), while invalid variants have \(\theta_j=\theta+\alpha_j/\gamma_j\). If zero is the most prominent weighted mode of the pleiotropic ratio effects \(\alpha_j/\gamma_j\), shifting every value by \(\theta\) shifts that mode to \(\theta\). The observed ratio estimates add sampling variation, and kernel smoothing approximates their weighted distribution. The estimate is therefore the location of its largest smoothed concentration.
This reasoning requires more than saying that “many” ratios look similar. In a finite set, a plurality means that one group has more weight than any competing group; it need not have more than half of the total weight. For the smoothed estimate to consistently target the causal effect, the corresponding population mode must be uniquely dominant and the ratio estimates and weights must be informative. If two clusters tie, or a different invalid cluster is more prominent, the target is ambiguous or incorrect.
Worked Examples
Worked Example: A Plurality of Valid Instruments
Suppose five independent variants have hypothetical ratio estimates, with inverse-variance weights shown below. Assume the first four are valid and the fifth has a direct effect.
| Variant | Ratio estimate | Weight |
|---|---|---|
| 1 | 0.20 | 4 |
| 2 | 0.22 | 3 |
| 3 | 0.19 | 5 |
| 4 | 0.21 | 4 |
| 5 | 0.55 | 1 |
The four ratios near 0.20 have total weight \(4+3+5+4=16\), compared with weight 1 for the ratio at 0.55. For a Gaussian kernel with \(h=0.04\), evaluate the weighted density at \(b=0.203\). The standardized distances for the first four ratios are \(0.075,\ -0.425,\ 0.325,\ -0.175\); the fifth is \(-8.675\). Substitution gives
The distant fifth ratio contributes essentially zero at this location. Numerical maximization of the density gives a mode close to 0.203. If the causal effect is expressed, for example, in outcome standard-deviation units per exposure standard deviation, the estimate is about 0.203 outcome standard deviations per exposure standard deviation. This interpretation depends on the assumption that the cluster near 0.20 is the valid modal group; the arithmetic alone cannot verify that assumption.
Worked Example: Bandwidth Changes the Shape and Location
Consider five hypothetical ratio estimates with weights: \(0.20,0.21,0.22\), each with weight 2, and \(0.48,0.50\), each with weight 2. The total weight is 10. There are two clusters, with total weights 6 and 4. For a Gaussian kernel and \(h=0.02\), the weighted kernel contributions at \(b=0.21\), before the common density multiplier \(1/(h\sqrt{2\pi}\sum w_j)\), are
The second cluster is so far away, relative to this bandwidth, that it contributes negligibly. At \(b=0.49\), the contribution is approximately \(2(2e^{-0.5^2/2})=3.530\). The density therefore has a slightly higher peak near the first cluster for this bandwidth. The common multiplier is \(1/(0.02\sqrt{2\pi}\cdot10)\approx1.995\), so the density at \(0.21\) is approximately \(5.530(1.995)=11.03\).
Now use \(h=0.15\). At \(b=0.21\), the weighted kernel sum is approximately \(2(1+e^{-0.01^2/(2(0.15)^2)}+e^{-0.01^2/(2(0.15)^2)})+ 2(e^{-0.27^2/(2(0.15)^2)}+e^{-0.29^2/(2(0.15)^2)})\approx6.70\). At \(b=0.25\), the same sum is approximately \(2(e^{-0.05^2/(2(0.15)^2)}+e^{-0.04^2/(2(0.15)^2)}+e^{-0.03^2/(2(0.15)^2)})+ 2(e^{-0.23^2/(2(0.15)^2)}+e^{-0.25^2/(2(0.15)^2)})\approx6.90\). Thus the broad bandwidth smooths the two clusters together and moves the higher density away from the narrow-band peak near 0.21. A full numerical search locates the mode; these evaluations illustrate why its location and even the number of visible peaks depend on \(h\). A bandwidth should not be chosen just to produce a preferred causal estimate.
Worked Example: Interpreting the Scale of the Estimate
Suppose a hypothetical analysis estimates an exposure–outcome mode of 0.30, where the genetic associations with the outcome are measured in log-odds units and those with the exposure in standard-deviation units. The estimated effect is 0.30 log-odds units per one standard-deviation increase in exposure. On the odds-ratio scale, the corresponding ratio is \(\exp(0.30)\approx1.350\): under the MR assumptions, a one-standard-deviation increase in exposure corresponds to an estimated odds ratio of about 1.35.
This is a scale conversion, not a claim that the mode establishes the assumptions. If the modal cluster is invalid, or if ratios are poorly estimated, the numerical conversion remains correct but the causal interpretation does not follow.
Bandwidth, Uncertainty, and Diagnostics
A small bandwidth preserves local detail. It can reveal distinct clusters, but may create a peak around one or a few ratios, making the estimate highly sensitive to individual variants and sampling noise. A large bandwidth suppresses small irregularities but may merge distinct clusters, shift the mode, or obscure a meaningful plurality. Consequently, “more precise” does not mean “use the smallest bandwidth”: a sharply defined peak can be an artefact of undersmoothing.
The mode estimator’s sampling uncertainty is not obtained merely by reading off the width of the kernel. A practical approach is to resample independent variants, recompute the ratios, weights, and mode, and summarize the resulting bootstrap distribution. If variants are correlated, resampling them independently is inappropriate; resampling or modeling must respect their dependence. A bootstrap distribution with several separated peaks, substantial spread, or many resamples whose modes jump between clusters signals instability. Such behavior should be reported rather than hidden by a single standard error.
Useful checks include plotting the weighted ratio distribution and its density over a reasonable range of bandwidths, inspecting how much total weight lies near the estimated mode, and checking whether one variant dominates the local peak. These are sensitivity checks, not tests that prove zero-modal pleiotropy. Also consider whether instrument strength, ratio uncertainty, or LD makes the apparent clusters unreliable. The earlier material on the weighted median gives a useful contrast: it relies on a majority of the weight coming from valid instruments, whereas the mode relies on the most prominent weighted cluster being valid.
Comparison and Limitations
| Estimator | Core robustness condition | Main sensitivity |
|---|---|---|
| Inverse-variance weighted | Typically requires all instruments to be valid or pleiotropic effects not to bias the combined estimate. | Invalid variants can shift the weighted average. |
| Weighted median | More than half the total weight must come from valid instruments. | It can fail when invalid instruments hold a majority of the weight. |
| Weighted mode | The largest weighted ratio cluster must correspond to valid instruments. | Bandwidth, competing clusters, and sparse or unstable modes. |
The mode can remain useful when the valid instruments are a plurality but less than half the total weight. Its costs are that it can be less precise than an estimator using all valid information, and that the result can depend materially on smoothing choices. There may be no clear dominant cluster; several clusters can have similar weight; or the dominant cluster can consist of invalid variants with similar direct effects. In these situations, the estimate may be unstable or systematically misleading. Agreement among estimators is informative only alongside their assumptions: similar answers do not prove that those assumptions hold.
Common Mistakes
- Requiring a majority of valid weight. That is the weighted median’s key robustness condition. The mode requires the valid group to be the largest weighted cluster, which may be less than half the total weight.
- Assuming the largest observed cluster is valid by definition. The zero-modal-pleiotropy assumption is a substantive condition, not a conclusion from a density plot. An invalid cluster can be the largest.
- Treating bandwidth as a cosmetic setting. Bandwidth changes smoothing, and can alter the number and location of modes. Report sensitivity across defensible choices.
- Calling the mode the average ratio. It is the location maximizing a weighted, smoothed density. It need not equal an arithmetic or inverse-variance weighted average.
- Ignoring dependence or weak instruments. Correlated variants and unstable ratios can make both the apparent density and uncertainty misleading if not handled appropriately.
Check Your Understanding
Answer the questions, then compare your reasoning with the solutions.
- Can a valid modal group have less than half the total weight? State the condition it must meet instead.
- If \(\theta=0.4\) and the modal pleiotropic ratio effect is \(0.1\), where is the population modal ratio located, and does zero-modal pleiotropy hold?
- What is the likely effect of making the bandwidth extremely small? Name one diagnostic sign of instability.
- Why does a mode estimate of 0.25 not by itself establish a causal effect of 0.25?
- Why should variants in LD not automatically be resampled independently to estimate uncertainty?
Solutions. (1) Yes. The valid group must have greater weighted prominence than any competing group; it need not exceed half of all weight. (2) The modal ratio is \(0.4+0.1=0.5\). Zero-modal pleiotropy does not hold because the modal pleiotropic ratio effect is not zero. (3) An extremely small bandwidth can produce narrow peaks around individual ratios and make the estimate sensitive to single variants; bootstrap modes that jump between peaks are one warning sign. (4) The estimate has a causal interpretation only if the MR assumptions hold, including the assumption that the valid instruments form the weighted modal group. (5) LD means variants are dependent, so independent resampling fails to preserve their joint variation and can misstate uncertainty.