Why estimator choice depends on the failure mechanism
Mendelian randomization (MR) uses genetic variants as instrumental variables to estimate the effect of an exposure on an outcome. The usual causal assumptions are relevance (the variants predict the exposure), independence (they are independent of confounders of exposure and outcome), and exclusion restriction (they affect the outcome only through the exposure). The exclusion restriction is the assumption most directly threatened by horizontal pleiotropy: a variant affects the outcome through a pathway other than the exposure.
Different estimators tolerate different patterns of invalid instruments. No estimator is protected against every violation. In particular, an estimator that tolerates a few outlying variants may still fail when most instruments have direct effects in the same direction. Choosing a method therefore requires identifying a plausible failure mechanism, checking the method's assumptions, and interpreting estimates as sensitivity analyses rather than selecting whichever result is most appealing.
These summary-data methods generally rely on suitably harmonized associations and on variants that are independent, or on accounting for their linkage disequilibrium. Their estimands are on the scale of the supplied associations. For example, if the outcome association is a log odds ratio, an MR slope is interpreted on the corresponding log-odds scale per unit of exposure; the outcome scale does not itself establish that the instruments are valid.
What each estimator protects against
Inverse-variance weighted (IVW) estimation combines ratio information by fitting a weighted regression of outcome associations on exposure associations with the intercept constrained to zero. With independent variants and outcome-association variances treated as known, its common first-order form is:
If all instruments are valid and the model is otherwise appropriate, IVW can be efficient. It can also be consistent under a form of balanced pleiotropy in which the weighted direct effects average to zero in the relevant regression. It is not generally protected from directional pleiotropy: a nonzero weighted average direct effect can shift its slope.
MR-Egger fits the same weighted regression but estimates an intercept rather than fixing it at zero. A nonzero intercept can represent average directional pleiotropy. Its slope has a different protection: the Instrument Strength Independent of Direct Effect (InSIDE) assumption requires instrument strength to be independent of the direct effects. MR-Egger also relies on the No Measurement Error (NOME) condition, or an adequate correction for measurement error in the exposure associations. It often has low precision, and weak instruments can attenuate its slope toward zero.
The weighted median takes the weighted median of the variant-specific ratio estimates, usually using approximate inverse-variance weights proportional to \(\hat\beta_{Xj}^{\,2}/s_{Yj}^{\,2}\). It is consistent if more than half of the total weight comes from valid instruments. The weighted mode estimates the most prominent cluster (mode) of ratio estimates. Its key condition, often called the zero modal pleiotropy assumption (ZEMPA), is that the largest relevant cluster of instruments by weight consists of valid instruments. Neither method needs every variant to be valid, but neither is guaranteed to work if its validity condition fails.
Robust approaches include methods that limit the influence of large residuals or outlying ratio estimates. MR-RAPS (robust adjusted profile score) is designed for many weak instruments and can accommodate balanced, mean-zero pleiotropic effects, with a robust loss helping to limit outlier influence. Its model still requires appropriate assumptions about the distribution and average of pleiotropic effects; it does not turn widespread directional pleiotropy into a valid design. Robust regression variants likewise depend on the pattern of contamination and on the valid instruments carrying enough information to anchor the fit.
| Estimator | Key protection or assumption | Important vulnerability |
|---|---|---|
| IVW | All instruments valid, or appropriately balanced direct effects | Directional pleiotropy can bias the weighted slope |
| MR-Egger | InSIDE; NOME or correction for exposure-association measurement error | Low power, weak-instrument dilution, and InSIDE violations |
| Weighted median | More than 50% of total weight is valid | Can fail when invalid instruments hold at least half the weight |
| Weighted mode | The largest ratio cluster by weight is valid (ZEMPA) | Can fail if the dominant cluster is invalid or clusters are poorly separated |
| MR-RAPS / robust regression | Model-specific assumptions; robust fitting limits selected forms of influence | Robustness to outliers does not correct systematic, widespread invalidity |
Why the median tolerates a minority of invalid weight
The weighted-median condition is about total weight, not simply the number of variants. Suppose the valid instruments' ratio estimates are centered on \(\theta\), and valid instruments carry more than half the total weight. The weighted median is the point at which at least half the weight lies on either side. Invalid estimates with less than half the total weight cannot, by themselves, move that point beyond the range needed to pass through the central valid weight. With sampling variation, the median need not equal \(\theta\) in a particular sample; the condition supports consistency as information grows, not exactness for every data set.
This argument also shows the limitation: one very strong invalid variant can contribute more weight than several weaker valid variants. Counting variants alone is not a substitute for checking their weights and the estimator's assumptions.
Worked Example: An Outlying Ratio Pulls IVW Away From the Median
Consider three hypothetical independent variants with exposure associations, outcome associations and outcome standard errors as shown. Assume the associations are harmonized and sufficiently precise for this illustration. The first two ratios agree; the third is larger and might reflect an invalid instrument.
| Variant | \(\hat\beta_{Xj}\) | \(\hat\beta_{Yj}\) | \(s_{Yj}\) |
|---|---|---|---|
| 1 | 0.10 | 0.050 | 0.020 |
| 2 | 0.20 | 0.100 | 0.025 |
| 3 | 0.15 | 0.120 | 0.030 |
The ratio estimates are \(0.050/0.10=0.50\), \(0.100/0.20=0.50\), and \(0.120/0.15=0.80\). For the first-order IVW weights, \(w_j=1/s_{Yj}^2\), so the weights are 2500, 1600 and 1111.1. Equivalently, the slope calculation can use the common rescaling \(w_j\hat\beta_{Xj}^2\), giving ratio weights 25, 64 and 25.
Thus the IVW estimate is:
The weighted median is \(0.50\), because the two ratios at 0.50 together carry weight \(25+64=89\), which exceeds half the total weight \(114/2=57\). This example illustrates how one discordant ratio can pull IVW upward while the weighted-median condition still holds. It does not prove that the third variant is invalid: sampling variation or other model problems could also cause disagreement.
Directional pleiotropy and weak strength require different checks
Directional pleiotropy means direct effects do not average to zero in the relevant sense. MR-Egger's intercept can reveal evidence of a nonzero average direct effect, but the intercept is not a universal pleiotropy test: low power can hide directional effects, and the slope's protection depends on InSIDE. A symmetric funnel plot or a small Egger intercept does not establish that every instrument satisfies exclusion restriction.
Weak instruments create another problem. In two-sample MR without sample overlap, weak-instrument bias often tends toward the null; with overlap, bias can move toward the confounded observational association. The direction and size depend on the design. MR-Egger is particularly susceptible to measurement error in \(\hat\beta_{Xj}\), which can attenuate its slope. MR-RAPS is a candidate when many instruments are weak and its pleiotropy model is plausible. A robust loss can reduce the effect of outliers, but cannot compensate for instruments that collectively encode a systematic direct effect.
Worked Example: Egger Intercept With a Directional Component
Suppose four hypothetical variants have exposure associations \(0.10, 0.20, 0.30, 0.40\), and outcome associations \(0.07, 0.12, 0.17, 0.22\), each with outcome standard error \(0.01\). Assume independent variants and that exposure-association estimates are precise enough for NOME in this illustration. The weighted regression has equal weights because all four outcome standard errors are equal.
The points lie exactly on \(\hat\beta_{Yj}=0.02+0.50\hat\beta_{Xj}\). For example, at \(\hat\beta_X=0.30\), the fitted outcome association is \(0.02+0.50(0.30)=0.17\). Therefore the MR-Egger intercept is 0.02 outcome-association units per allele, and the slope is 0.50 outcome-association units per exposure-association unit.
In this constructed example, the nonzero intercept represents a common directional component. If InSIDE holds and NOME is adequate, the Egger slope targets the causal effect despite that component. The data alone do not verify InSIDE, and four perfectly aligned points make this an arithmetic illustration rather than persuasive evidence about a real causal effect.
Worked Example: Weak Associations Suggest a Different Sensitivity Analysis
Consider four hypothetical exposure associations with standard errors: \(0.03\) (SE \(0.04\)), \(0.04\) (SE \(0.04\)), \(0.05\) (SE \(0.04\)), and \(0.06\) (SE \(0.04\)). A rough single-variant strength diagnostic is \(F_j=(\hat\beta_{Xj}/s_{Xj})^2\), where \(s_{Xj}\) is the exposure-association standard error.
The four values are \(0.03^2/0.04^2=0.5625\), \(0.04^2/0.04^2=1.000\), \(0.05^2/0.04^2=1.5625\), and \(0.06^2/0.04^2=2.250\). These associations are weak by this diagnostic. An analysis relying on precise exposure associations, especially uncorrected MR-Egger, is therefore concerning. If there are many such instruments and the mean-zero pleiotropy model is plausible, MR-RAPS is a relevant sensitivity estimator. This calculation does not establish that MR-RAPS is unbiased: its modeling assumptions and the study's overlap and variant-dependence structure still require scrutiny.
Diagnostics, alternatives, and limitations
The earlier material on MR-PRESSO, radial MR, MR-Lasso, contamination-mixture estimation, influence diagnostics, funnel plots and CAUSE provides complementary ways to inspect outliers, heterogeneity, or correlated pleiotropy. These analyses address different structures; none replaces stating the assumptions of the estimator used for the causal estimate. Useful warning signs include strong heterogeneity, a small number of variants dominating the weights, marked asymmetry or outliers in ratio patterns, and large changes in the estimate when influential variants are removed. Such signs motivate investigation, not automatic deletion.
Compare estimates on the same harmonized variant set and compatible scale where possible. An IVW estimate that differs from a weighted median may indicate influential variants or directional pleiotropy, but it may also reflect uncertainty. MR-Egger can be much less precise than IVW, so a wide interval is not evidence that its estimate is equivalent to another one. Likewise, agreement among methods is reassuring only to the extent that their assumptions fail differently; shared violations can make several estimators agree and still be wrong.
A practical decision is scenario-based:
Assess relevance, independence, exclusion restriction, variant harmonization, linkage disequilibrium handling, and possible sample overlap before interpreting estimator-specific robustness.
For a few outliers, consider robust or influence-resistant methods; for directional pleiotropy with plausible InSIDE, consider MR-Egger; for a minority of invalid weight, consider the weighted median; for a valid dominant cluster, consider the weighted mode.
Inspect exposure-association precision and overlap. If many instruments are weak, consider a method designed for that regime, such as MR-RAPS, only under its pleiotropy assumptions.
Report whether estimates are compatible within uncertainty, investigate influential variants and model fit, and explain which assumptions make each result credible.
There is no data-driven rule that identifies the true estimator in every setting. Robust methods trade efficiency for protection under specified contamination models; median and mode methods require their own valid-instrument conditions; Egger can be imprecise; and IVW is most efficient when its assumptions hold. If the invalid variants dominate or share a directional pathway correlated with instrument strength, these alternatives may all be misleading.
Common mistake
A common mistake is to select the estimator with the smallest p-value, or to describe the weighted median as “robust” without stating its condition. This reasoning confuses a numerical result with evidence that its assumptions hold. The correction is to state the relevant assumption for each estimate, examine strength and pleiotropy diagnostics, and describe disagreement and uncertainty. A sensitivity estimator is not a repair for an unidentified causal effect.
Check Your Understanding
Answer each question, then compare with the solutions below.
- What condition allows the weighted median to remain consistent when some instruments are invalid?
- Under which key assumption does the MR-Egger slope address directional pleiotropy, and what additional issue can weaken it?
- Five instruments have equal weights, and three are valid. Does the weighted-median validity condition hold?
- Why does use of a robust loss in MR-RAPS not guarantee protection from directional pleiotropy?
- IVW and MR-Egger estimates differ, and the Egger interval is wide. What can and cannot be concluded from this pattern?
Solutions. (1) More than half of the total weight must come from valid instruments. (2) InSIDE must hold; measurement error in exposure associations, or violation of NOME without correction, can attenuate the slope, while weak instruments can also reduce reliability. (3) Yes: with equal weights, three of five instruments contribute 60% of the weight, exceeding 50%. (4) Robust loss limits the influence of selected outliers, but systematic direct effects with a nonzero mean can violate the underlying pleiotropy model. (5) The difference is a reason to examine assumptions, strength, influence and heterogeneity. The wide Egger interval means its estimate is imprecise; it does not prove that either estimate is correct, nor that their underlying effects differ.