Mode-Based MR and the ZEMPA Assumption

Why Use a Mode-Based Estimate?

Inverse-variance weighted Mendelian randomization (IVW) combines variant-specific information, but a group of invalid instruments can pull the estimate away from the causal effect. Mode-based MR takes a different approach: it looks for the most prominent cluster of variant-specific ratio estimates. If the cluster corresponding to valid instruments is the modal cluster, the method can identify the causal effect even when valid instruments are not a majority.

This robustness depends on an assumption about which cluster is largest, not on the absence of all horizontal pleiotropy. The assumption is the Zero Modal Pleiotropy Assumption (ZEMPA). Before using it, we need to define the ratio estimates and how a mode is estimated from them.

In practice, the observed ratio estimate is \(\hat r_j=\hat\Gamma_j/\hat\gamma_j\). The ratio estimates are noisy, so a mode is usually estimated by smoothing them with a kernel density. This estimates where the ratio estimates are most concentrated, rather than simply selecting the most repeated exact value.

Simple and Weighted Mode Estimates

For a bandwidth \(h>0\), use the Gaussian kernel \(K(u)=(2\pi)^{-1/2}\exp(-u^2/2)\). The simple-mode density gives every variant equal weight. The weighted-mode density gives more weight to variants whose ratio estimates are more precise. A standard choice for the weighted estimate uses weights \(w_j=1/\widehat{\operatorname{Var}}(\hat r_j)\). Multiplying all weights by the same positive constant does not change the location of the mode.

The simple-mode estimate is the value of \(t\) that maximizes the first density; the weighted-mode estimate maximizes the second. Since the Gaussian kernel and the normalizing constants are common across variants, one can locate either estimate by maximizing the corresponding sum of kernel contributions. A ratio close to \(t\) contributes more than a distant ratio, and the bandwidth determines how quickly that contribution declines.

The weights require an estimate of each ratio’s variance. A first-order delta-method approximation, with \(\hat\gamma_j\ne 0\), is

When exposure associations are treated as sufficiently precise under the NOME approximation, the term involving their variance may be ignored; in a two-sample setting with independent association estimates, the covariance term is zero. These are approximations, not identities. If exposure-association uncertainty is substantial, weights based only on outcome-association uncertainty can misrepresent ratio precision.

What ZEMPA Assumes

Write the ratio-scale pleiotropic deviation as \(u_j=\alpha_j/\gamma_j\). The population ratio is then \(\beta+u_j\). ZEMPA says that zero is the modal value of these deviations: the most prominent cluster of instruments has no pleiotropic deviation on the ratio scale. Equivalently, the modal cluster of population ratio estimates is at \(\beta\). Since instruments with \(\alpha_j=0\) satisfy exclusion, this is often described as the assumption that the largest cluster of instruments is valid.

The basic justification is a shift of the distribution. If the distribution of \(U=\alpha_j/\gamma_j\) has density \(f_U\), then the ratio distribution \(R=\beta+U\) has density \(f_R(r)=f_U(r-\beta)\). A shift moves the location of the mode by the same amount. Thus, if the unique mode of \(U\) is zero, the mode of \(R\) is \(\beta\).

The kernel estimate approximates this population mode from observed ratios. In the standard large-sample setting, consistency requires a well-defined, unique, separated population mode, sufficient regularity of the ratio distribution, and a bandwidth that shrinks while still smoothing enough observations. For an ordinary unweighted density estimate, under standard conditions such as a bounded uniformly continuous density and a bounded kernel with suitable tail behavior, \(h\to 0\) and \(nh/\log n\to\infty\) ensure uniform consistency of the kernel density estimate; together with a unique, separated population mode, this supports consistency of the estimated global mode. This justification explains the target; it does not prove that ZEMPA holds in a particular application.

Assessing the Estimate in Practice

Mode-based MR is useful because a modal valid cluster can identify an effect even when some variants are invalid and the valid group is not a majority. But “robust” is conditional: if invalid variants form the dominant cluster, the method can follow them. ZEMPA is not directly verifiable from the observed ratios, because those ratios do not reveal which variants have direct effects. A prominent cluster is evidence about the shape of the estimates, not proof of instrument validity.

The choice of bandwidth matters. If it is too small, sampling noise can split one genuine cluster into several narrow peaks. If it is too large, distinct valid and invalid clusters can blur together or the estimated mode can shift. Weighted estimates add a second consideration: a small number of highly precise variants can dominate the density. Analysts should examine whether the selected mode is stable across reasonable bandwidth choices and compare simple and weighted modes. Agreement is reassuring about sensitivity to weighting, but it does not establish ZEMPA.

Interpretation also depends on the causal model. The ratio identity \(\Gamma_j/\gamma_j=\beta+\alpha_j/\gamma_j\) assumes a common causal effect \(\beta\). When causal effects vary across variants, the ratios need not share a single causal target; the modal estimate then requires an explicit interpretation in terms of the effect distribution rather than an automatic claim to an average effect. As with other MR estimators, relevance and independence assumptions still matter, and a mode estimate does not repair population structure or other violations of instrument validity.

Common Mistakes

• Assuming a majority of instruments must be valid. ZEMPA is about the modal cluster, not necessarily more than half of the variants. A smaller valid group can be modal if it is sufficiently concentrated.
• Assuming the mode must be valid because the method is called robust. If invalid variants form the strongest cluster, the estimated mode can be biased, as the worked example shows.
• Ignoring precision weights. A weighted mode can be dominated by a few precise ratios. Check which variants carry the weight, and do not interpret a weighted result as a simple count of instruments.
• Treating the bandwidth as irrelevant. The bandwidth controls smoothing and can alter the number and location of apparent clusters. Report and assess sensitivity to reasonable choices.
• Claiming that a stable mode proves causality. Stability across settings is a sensitivity check, not a test of ZEMPA, exclusion restriction, or all other IV assumptions.