MR-Lasso for Detecting Pleiotropic Instruments

Why Penalize Direct Effects?

In Mendelian randomization (MR), a genetic variant can affect an outcome through the exposure of interest and also through another pathway. The latter is horizontal pleiotropy: it violates the exclusion restriction for that variant. Earlier material on MR has distinguished this from vertical pleiotropy, in which effects on the outcome operate through the exposure. MR-Lasso addresses a particular working model: most variants have no direct effect on the outcome, while a smaller number may have such effects.

The method makes this sparsity idea explicit by fitting direct effects and penalizing their absolute sizes. A variant’s direct-effect estimate is shrunk toward zero; sufficiently large residual effects can remain nonzero and be flagged for investigation. This is a selection procedure under a model, not a test that proves a variant is invalid or a guarantee that the remaining variants satisfy the instrumental-variable assumptions.

Definition: Let $\hat{\beta}_j$ be the estimated association of variant $j$ with the exposure, and let $\hat{\Gamma}_j$ be its estimated association with the outcome. The summary-association model is $$ \hat{\Gamma}_j = \theta \hat{\beta}_j + \alpha_j + \varepsilon_j, $$ where $\theta$ is the causal-effect parameter, $\alpha_j$ is the variant’s direct effect on the outcome not operating through the exposure, and $\varepsilon_j$ represents estimation error. MR-Lasso estimates $\theta$ and the $\alpha_j$ values by minimizing a weighted residual sum of squares plus an L1 penalty on the direct effects.

This formulation treats the exposure associations as fixed in the basic version of the procedure. It is most defensible when their estimation error is small relative to the outcome-association uncertainty (the NOME approximation). If that approximation is poor, uncertainty in $\hat{\beta}_j$ needs to be addressed rather than ignored. In the simple independent-variant setting, a common fixed-weight objective uses $w_j=1/SE(\hat{\Gamma}_j)^2$. The weights give less influence to less precise outcome associations.

(\hat{\theta}_{L},\hat{\alpha}_{1,L},\ldots,\hat{\alpha}_{m,L}) = \underset{\theta,\alpha_1,\ldots,\alpha_m}{\operatorname{argmin}} \left\{ \frac{1}{2}\sum_{j=1}^{m} w_j (\hat{\Gamma}_j-\theta\hat{\beta}_j-\alpha_j)^2 +\lambda\sum_{j=1}^{m}|\alpha_j| \right\}.

Here $m$ is the number of variants and $\lambda\geq 0$ controls the strength of penalization. The factor $1/2$ is a convenient scaling convention; other presentations may absorb it into $\lambda$. With $\lambda=0$, the direct effects are not shrunk. As $\lambda$ increases, nonzero direct-effect estimates are pulled toward zero, and some may become exactly zero. For a fixed $\lambda$, a variant is selected by this procedure when $\hat{\alpha}_{j,L}\ne 0$. In this context, “selected” means flagged as potentially pleiotropic, not selected as a preferred instrument.

How the Penalty Produces Selection

The L1 penalty has a thresholding effect. To see its role, temporarily hold $\theta$ fixed and define the residual before fitting a direct effect as $r_j=\hat{\Gamma}_j-\theta\hat{\beta}_j$. The part of the objective involving $\alpha_j$ is $\frac{1}{2}w_j(r_j-\alpha_j)^2+\lambda|\alpha_j|$. Its minimizer is

\hat{\alpha}_j = \operatorname{sign}(r_j) \max\left(|r_j|-\frac{\lambda}{w_j},\,0\right).

Thus, if $|r_j|\leq\lambda/w_j$, the fitted direct effect is zero; a larger residual leaves a nonzero estimate, reduced in magnitude by $\lambda/w_j$. This formula is a coordinate-wise result with $\theta$ held fixed. In the full MR-Lasso fit, $\theta$ and all direct effects are estimated together, so changing $\theta$ can change the residuals and which variants are selected.

The penalty does not decide which causal pathways produced a residual. A nonzero $\hat{\alpha}_{j,L}$ can reflect horizontal pleiotropy, but it can also arise from sampling variability, a poorly specified model, or dependence between variants not handled by the fit. Conversely, a true direct effect can be shrunk to zero. The interpretation is therefore “potentially pleiotropic under this model,” rather than “known to violate the exclusion restriction.”

Worked Example: A Variant With a Large Residual

Suppose a hypothetical variant has an outcome-association residual $r_j=0.12$ outcome units after fixing $\theta$. Its outcome-association standard error is $0.04$ outcome units, so $w_j=1/(0.04)^2=625$ per squared outcome unit. Let $\lambda=50$ in the corresponding penalty scale. Find the direct-effect update for this coordinate.

The threshold is $$ \frac{\lambda}{w_j}=\frac{50}{625}=0.08 $$ outcome units. Since $r_j=0.12>0.08$, the update is positive and nonzero: $$ \hat{\alpha}_j=0.12-0.08=0.04 $$ outcome units.

The penalty has reduced the residual-based direct-effect estimate from $0.12$ to $0.04$ outcome units; it has not left it unchanged. If instead the residual had been $0.06$ outcome units, its magnitude would be below the $0.08$-unit threshold and the coordinate update would be zero. These are conditional updates, not by themselves the full joint MR-Lasso estimate.

A Joint Fit and the Meaning of Post-Selection Estimation

MR-Lasso produces two quantities that should not be conflated. The joint penalized estimate $\hat{\theta}_{L}$ is obtained while estimating $\theta$ and the direct effects together. A post-selection estimate is calculated after excluding the variants flagged by that fit, often by applying inverse-variance weighted (IVW) estimation to the remaining variants. The first estimate reflects the penalty and all variants in the joint optimization; the second is a refit on a data-selected subset.

For independent variants, the fixed-weight IVW estimate on a specified set $S$ of variants is $$ \hat{\theta}_{IVW,S} = \frac{\sum_{j\in S}w_j\hat{\beta}_j\hat{\Gamma}_j} {\sum_{j\in S}w_j\hat{\beta}_j^2}. $$ This refit can be useful if the selected variants are genuinely invalid and the remaining variants satisfy the relevant MR assumptions. But selection and estimation use the same data, so a post-selection estimate is not automatically free of selection bias, and its usual standard error may not account for the selection step. Selection also cannot repair confounding, population structure, or other violations affecting the variants that remain.

Worked Example: Checking a Joint Penalized Solution

Consider three hypothetical, independent variants with equal weights $w_j=1$, exposure associations $(1,1,1)$, outcome associations $(0.75,0.75,3.00)$, and $\lambda=0.50$. Treat the associations as being in the same arbitrary units for this simplified calculation. A proposed joint solution is $\hat{\theta}_{L}=1.00$, $\hat{\alpha}_{1,L}=0$, $\hat{\alpha}_{2,L}=0$, and $\hat{\alpha}_{3,L}=1.50$. Its residuals are $$ (0.75-1,\;0.75-1,\;3-1-1.50)=(-0.25,-0.25,0.50). $$

To verify the proposed solution, the derivative condition for $\theta$ is that the exposure associations multiplied by the residuals sum to zero. Here $1(-0.25)+1(-0.25)+1(0.50)=0$. For a nonzero direct effect, the residual must equal $\lambda$ times its sign; the third residual is $0.50$, matching the positive $\hat{\alpha}_{3,L}$. For each direct effect estimated as zero, the residual magnitude must be no larger than $\lambda$. The first two magnitudes are $0.25\leq0.50$. These conditions establish a minimizer of this convex objective.

The objective value at the solution is $$ \frac{1}{2}\{(-0.25)^2+(-0.25)^2+(0.50)^2\} +0.50(1.50) =0.1875+0.75=0.9375. $$ The third variant is flagged by the penalized fit. For comparison, IVW using all three variants is $(0.75+0.75+3.00)/3=1.50$. Excluding the flagged variant and refitting gives $(0.75+0.75)/2=0.75$. These are different estimators; neither arithmetic nor the selection alone establishes which value is the true causal effect.

Worked Example: A Clean Proportional Pattern

In a second hypothetical data set, three independent variants have exposure associations $(0.10,0.20,0.30)$ and outcome associations $(0.08,0.16,0.24)$, with equal weights. Every outcome association is $0.80$ times its exposure association. At $\theta=0.80$, all residuals are zero, so choosing all $\alpha_j=0$ gives an objective of zero for any $\lambda\geq0$.

Because the objective cannot be less than zero, this is a global minimum. The penalized fit selects no direct effects, and IVW using all variants gives $$ \hat{\theta}_{IVW} = \frac{(0.10)(0.08)+(0.20)(0.16)+(0.30)(0.24)} {(0.10)^2+(0.20)^2+(0.30)^2} = \frac{0.112}{0.14}=0.80. $$ In this constructed example, the estimate is $0.80$ outcome-association units per exposure-association unit. A lack of selected direct effects is consistent with the proportional pattern, but does not prove that every exclusion restriction holds in a real analysis.

Worked Example: Increasing the Penalty Can Remove a Flag

Return to the three-variant data from the joint-fit example, but consider how the solution changes as $\lambda$ varies. For a solution with only the third direct effect nonzero, let its residual equal $\lambda$. The first two residuals are both $0.75-\theta$. The slope condition requires their sum plus the third residual to be zero: $$ 2(0.75-\theta)+\lambda=0, \qquad \theta=0.75+\frac{\lambda}{2}. $$

The third direct effect is then $$ \alpha_3=3-\theta-\lambda =2.25-1.5\lambda. $$ At $\lambda=0.50$, this gives $\theta=1.00$ and $\alpha_3=1.50$, as before. The other two residuals are each $-0.25$, so their magnitudes are below $0.50$, as required for their direct effects to remain zero. At $\lambda=1.50$, $\alpha_3=0$; the all-zero direct-effect fit has $\theta=1.50$, with residuals $(-0.75,-0.75,1.50)$, all no larger in magnitude than the penalty threshold $1.50$. Thus, at this penalty the third effect need no longer be selected. Increasing $\lambda$ can reduce the number of selected variants, but does not show that the remaining variants are valid.

Choosing the Penalty and Stating Assumptions

In practice, $\lambda$ is usually chosen from a sequence of candidate values using a model-selection criterion such as the Bayesian information criterion (BIC), or another specified tuning procedure. Under fixed known weights, a BIC comparison can be based on the weighted residual sum of squares at each candidate fit, with a complexity cost for the number of fitted parameters. The exact implementation and scaling matter: the penalty convention, weight definition, and treatment of the exposure-association uncertainty should be reported. A larger penalty generally favors fewer nonzero direct effects; it does not necessarily give a more credible causal estimate.

The central modeling assumption is sparsity: most variants used in the fit have $\alpha_j=0$, or direct effects small enough to be treated as zero under the chosen penalty. Identification also requires enough information in the exposure associations to estimate $\theta$, and valid variants must provide the signal anchoring the exposure-mediated relationship. In addition, the basic independent-variant weighted objective assumes variants are independent, or that the relevant covariance is otherwise handled. If variants are in linkage disequilibrium and correlations are ignored, the stated objective and its uncertainty calculations may be inappropriate.

The fixed-weight model also treats $\hat{\beta}_j$ as sufficiently precise. When exposure-association uncertainty is material, methods that account for it are needed; simply applying this version of MR-Lasso does not remove weak-instrument bias. Inference after selecting variants deserves particular care because the selection step adds uncertainty not captured by treating the selected set as fixed. MR-Lasso is therefore best understood as a robustness and outlier-selection strategy under explicit assumptions, not as a replacement for considering instrument strength, study design, or alternative explanations for heterogeneity.

Common Mistake

Wrong reasoning: “MR-Lasso selected a variant, so that variant is definitely horizontally pleiotropic, and the estimate after removing it is unbiased.”
Correction: Selection means its fitted direct effect is nonzero at the chosen penalty under the specified model. Sampling error or model misspecification can produce this result, and real direct effects can be shrunk to zero. The refit also uses a data-selected subset; its validity depends on the remaining instruments and its uncertainty should not be interpreted as if selection had been fixed in advance.

Key takeaway: MR-Lasso fits a causal slope and sparse direct effects together. The L1 penalty can flag variants with residual outcome associations, but selected effects are evidence for sensitivity analysis, not proof of pleiotropy. Always distinguish the penalized estimate from any post-selection refit.

Check Your Understanding

Answer each question using the model and assumptions described above.

In the summary-association model, what does $\alpha_j$ represent, and how does it differ from an effect mediated by the exposure?
For a fixed $\theta$, a variant has residual $r_j=0.05$, weight $w_j=100$, and penalty $\lambda=4$. What is its coordinate-wise direct-effect update?
Why can increasing $\lambda$ reduce the number of variants flagged by MR-Lasso?
How does the joint penalized estimate differ from an IVW estimate refitted after removing selected variants?
Give two reasons that a nonzero fitted direct effect does not prove horizontal pleiotropy.
Why can the usual standard error from a post-selection refit understate uncertainty?