MendelianMR

Browser-based Mendelian Randomization analysis — 23 methods including IVW, MR-Egger, Weighted Median, MR-PRESSO, MR-RAPS, MR-Lasso, Anderson-Rubin, Fieller, Model Averaging, Sargan, MR-PEN

Data Input

Results

Scatter Plot

Forest Plot

Funnel Plot

Leave-One-Out

Radial

Clusters

Advanced

Weak Instruments

Guide

Load Example Dataset

Instrument Data

Enter SNP-exposure (β_X, SE_X) and SNP-outcome (β_Y, SE_Y) associations. One row per genetic variant. Tab/comma-separated: SNP, beta_X, se_X, beta_Y, se_Y

Exposure name

Outcome name

SNP data (SNP, beta_X, se_X, beta_Y, se_Y)

Effect scale

Significance level (α)

MR-PRESSO simulations

MR Estimates Summary

Method Comparison

Method	Estimate	SE	95% CI	P-value	Interpretation

Heterogeneity & Pleiotropy Diagnostics

Test	Statistic	P-value	Interpretation

MR-PRESSO Results

Heterogeneity-Penalized Model Averaging (Burgess et al. 2020)

Weights IVW, MR-Egger, weighted median, and mode-based estimates by their heterogeneity Q: w_method = exp(−Q/2) / ∑exp(−Q_j/2). Methods with lower heterogeneity receive more weight. More robust than choosing a single method.

Method	Estimate	Q statistic	Weight

Individual Wald Ratios

SNP	β_X	β_Y	Wald Ratio	SE	Weight (IVW)

SNP-Exposure vs SNP-Outcome Scatter Plot

Each point is a genetic variant. Lines show causal estimates from different MR methods. Slope = causal effect.

Forest Plot — Individual Wald Ratios

Funnel Plot — Asymmetry Detection

Wald ratios vs precision (1/SE). Asymmetry suggests directional pleiotropy.

Leave-One-Out Analysis (IVW)

IVW estimate when each SNP is excluded. Identifies influential variants.

Excluded SNP	IVW Estimate	SE	95% CI	P-value	Change

Radial MR (Bowden et al. 2018, Stat Med)

Transforms data to radial coordinates: X_j = √w_j, Y_j = β_Yj√w_j / β_Xj where w_j = 1/SE_Yj². Radial IVW = WLS through origin; Radial MR-Egger = WLS with intercept. Modified Cochran's Q identifies outliers more reliably than standard Q.

Radial Plot

Scatter in radial coordinates. Outliers identified by per-SNP Q contribution (|Q_j| > χ²_1,0.05 = 3.84).

Per-SNP Q Contributions (Rucker's Q')

SNP	X (radial)	Y (radial)	Q_j	P-value	Outlier

Radial Method Estimates

Method	Estimate	SE	95% CI	P-value	Q statistic

MR-Clust (Foley et al. 2021, Bioinformatics)

Clusters Wald ratios using k-means (k=2,3,4) with BIC selection. Each cluster represents a potential causal mechanism or pleiotropic pathway. Instruments with |z-residual| > 3 are assigned to a junk cluster.

Cluster Scatter Plot

SNPs color-coded by cluster assignment. Each cluster's causal estimate shown as a line through the origin.

Cluster Assignments

SNP	Wald Ratio	SE	Cluster	Cluster Estimate

Cluster-Specific Estimates

Cluster	N instruments	Estimate	SE	95% CI	Interpretation

Advanced MR Methods

Extended methods for sensitivity analysis: MR-RAPS (robust to weak instruments), Mode-based estimation (plurality valid assumption), Contamination mixture (profile likelihood), IVW prediction interval, I²_GX instrument strength, and Steiger directionality test.

Advanced Method Estimates

Method	Estimate	SE	95% CI	P-value	Interpretation

Advanced Diagnostics

Test	Statistic	P-value	Interpretation

Cochran's Q Decomposition (Bowden et al. 2019)

Decomposes total Q into Q_balanced (non-directional heterogeneity, inflates variance but does not bias) and Q_unbalanced (directional pleiotropy, biases the estimate). If Q_balanced dominates: random-effects IVW is appropriate. If Q_unbalanced is significant: need MR-Egger.

Component	Q statistic	df	P-value	Interpretation

Reverse MR / Bidirectional Test

Swaps exposure and outcome: uses β_Y as instrument-exposure and β_X as instrument-outcome, then runs IVW in the reverse direction. If both forward and reverse are significant, this may indicate a shared pathway rather than unidirectional causation.

Direction	Estimate	SE	95% CI	P-value	Interpretation

Conditional F-statistic (Sanderson et al. 2019)

Adjusts F-statistic for potential pleiotropy. F_conditional = F_overall × (1 − R²_pleiotropy). If F_cond > 10: reliable inference. Accounts for the fraction of instrument variance explained by pleiotropic pathways.

Outlier-Robust IVW (Penalized / Huber weights)

Downweights outlier SNPs using Huber weights (k=1.345). Iteratively: compute residuals, apply Huber psi function, re-weight, re-estimate. More resistant to individual pleiotropic variants than standard IVW.

Method	Estimate	SE	95% CI	P-value	Interpretation

MR-Lasso (Rees et al. 2019, Stat Med)

L1-penalized regression that simultaneously estimates the causal effect (β) and SNP-specific pleiotropy parameters (γ_j). Instruments with γ_j = 0 are classified as valid (no detectable pleiotropy). The penalty λ is chosen by BIC. Soft-thresholding via coordinate descent.

SNP	Wald Ratio	γ_j (pleiotropy)	Status

Sargan/Hansen Overidentification Test

Under the null of all instruments being valid, Q ~ χ²(k − 1). Sequential removal drops the SNP with the largest Q contribution until the Sargan P > 0.05, identifying a valid instrument subset.

Step	Removed SNP	Q Contribution	Remaining k	Sargan Q	Sargan P

Pleiotropy-Robust Penalized IVW (MR-PEN)

Penalizes the IVW objective with adaptive Lasso weights: w_pen,j = 1/|residual_j|^γ. Downweights SNPs with large residuals more aggressively than Huber, specifically targeting pleiotropy. More principled than simple outlier removal.

Method	Estimate	SE	95% CI	P-value	Interpretation

Method Details

Weak-Instrument Robust Inference

Standard Wald-type CIs rely on strong instruments (F > 10). When instruments are weak, the ratio estimator has a non-standard (Cauchy-like) distribution and standard CIs can be misleading. The Anderson-Rubin test and Fieller’s theorem provide valid inference regardless of instrument strength.

Anderson-Rubin Test (Weak-Instrument Robust CI)

The AR statistic AR(β) = ∑ w_j(b_Y,j − β b_X,j)² follows χ²(k) under the null. Inverting this test yields a CI that is valid even when F < 10. The CI is found by grid search: all β values where AR(β) < χ²_k,α.

Method	Estimate	95% CI	P-value	Interpretation

Fieller’s Theorem CI (Ratio Estimator)

The Wald ratio β_Y/β_X has a non-standard distribution when β_X is imprecisely estimated. Fieller’s theorem solves the quadratic: (β_Y − β·β_X)² ≤ z²_α(SE²_Y + β²·SE²_X). This can yield bounded, unbounded, or empty CIs depending on instrument strength. For IVW: aggregated across instruments.

Method	CI Lower	CI Upper	Type	Interpretation

Weak IV Method Comparison

Comparison of standard IVW CI vs weak-instrument-robust CIs. When instruments are strong (F > 10), all intervals should be similar. Discrepancies indicate that weak instruments may be distorting standard inference.

Method	CI Lower	CI Upper	Width	Covers Null

Mendelian Randomization — Method Guide

What is MR? Mendelian Randomization uses genetic variants as instrumental variables to estimate causal effects of modifiable exposures on outcomes. Because genotypes are randomly allocated at conception (Mendel's second law), MR mimics a natural randomized trial.

Methods Implemented

IVW	Inverse-variance weighted regression through origin. Assumes all instruments are valid (no pleiotropy). The primary estimate.
MR-Egger	Regression with intercept. Intercept ≠ 0 indicates directional pleiotropy. Relies on InSIDE assumption (instrument strength independent of direct effect).
Weighted Median	Consistent when ≥50% of weight comes from valid instruments. Robust to outliers.
Simple Median	Unweighted median of Wald ratios. Consistent when ≥50% of instruments are valid.
MR-PRESSO	Detects and corrects for horizontal pleiotropy outliers. Global test + outlier removal + distortion test.
MR-RAPS	Robust Adjusted Profile Score (Zhao et al. 2020). Iterative profile score with overdispersion. More robust than IVW under weak instruments.
Mode-based	Hartwig et al. 2017. Weighted kernel density mode of Wald ratios. Consistent if plurality (not majority) of instruments are valid.
Contamination Mixture	Profile likelihood classifying instruments as valid/invalid. Robust to mixed validity.
Radial MR	Bowden et al. 2018. Transforms to radial coordinates for more reliable outlier detection via modified Cochran's Q. Radial IVW (through origin) and Radial MR-Egger (with intercept).
MR-Clust	Foley et al. 2021. Clusters Wald ratios via k-means with BIC selection to identify distinct causal mechanisms or pleiotropic pathways.
Q Decomposition	Bowden et al. 2019. Splits Cochran's Q into balanced (non-directional) and unbalanced (directional pleiotropy) components.
Reverse MR	Bidirectional test swapping exposure and outcome to check for reverse causation or shared pathways.
Conditional F	Sanderson et al. 2019. Adjusts F-statistic for pleiotropy. F_cond > 10 for reliable inference.
Outlier-Robust IVW	Penalized IVW using Huber weights (k=1.345) to downweight outlier SNPs iteratively.
MR-Lasso	Rees et al. 2019. L1-penalized regression estimating SNP-specific pleiotropy. Classifies instruments as valid (gamma=0) or pleiotropic. Lambda by BIC.
Anderson-Rubin	Weak-instrument robust CI by inverting the AR chi-squared test. Valid even when F < 10, unlike standard Wald CIs.
Fieller's theorem	Solves the quadratic for the ratio estimator CI. Can yield bounded, unbounded, or empty CIs depending on instrument strength.
Model Averaging	Burgess et al. 2020. Weights IVW, Egger, median, mode by exp(-Q/2). More robust than choosing a single method.
Sargan/Hansen	Overidentification test. Sequential removal of worst-fitting SNP until Q passes. Identifies valid instrument subset.
MR-PEN	Pleiotropy-robust penalized IVW with adaptive Lasso weights targeting pleiotropic outliers more aggressively than Huber.

Key Assumptions

Relevance	Genetic variants must be strongly associated with the exposure (F-statistic > 10).
Independence	Variants must not be associated with confounders of the exposure-outcome relationship.
Exclusion restriction	Variants affect the outcome only through the exposure (no horizontal pleiotropy).

Interpretation Guide

All methods agree	Strong evidence for causal effect. Report IVW as primary.
IVW ≠ MR-Egger	Possible directional pleiotropy. Check Egger intercept p-value.
High Q / I²	Heterogeneity among Wald ratios. Consider outlier analysis (MR-PRESSO).
MR-PRESSO outliers	Remove outlier SNPs and compare corrected vs. raw estimates (distortion test).