Course 4: Estimation and Sensitivity Analysis • Medusa

What you’ll learn

By the end of this chapter, you should be able to:

explain what IVW, MR-Egger, weighted median, leave-one-out, and Steiger filtering do,
describe what each method protects against,
explain why sensitivity analyses are necessary in MR,
run Medusa’s summarized-data sensitivity analysis workflow on synthetic data.

Why this matters

A single MR estimate is rarely enough. If your result changes dramatically when one SNP is removed, or if methods with different assumptions disagree sharply, you need to know that before interpreting the analysis as causal evidence.

Sensitivity analyses exist because MR assumptions can fail in different ways.

The main summarized-data estimators

IVW: inverse-variance weighted MR

IVW combines SNP-specific information into one overall estimate, giving more weight to precise SNP-outcome associations.

It is often treated as the default summarized-data MR estimator when most or all instruments are believed to be valid.

What it is good at: efficient estimation when the instruments are broadly valid.

What it is vulnerable to: directional pleiotropy and influential invalid instruments.

MR-Egger

MR-Egger allows a non-zero intercept in the weighted regression of beta_ZY on beta_ZX.

Intuition:

if the intercept is far from zero, the SNPs may be pushing the outcome in a systematic way outside the exposure pathway.

What it is good at: detecting and sometimes adjusting for directional pleiotropy under additional assumptions.

What it is vulnerable to: low precision and sensitivity to weak-instrument problems.

Weighted median

The weighted median estimator takes the weighted median of the SNP-specific ratio estimates.

What it is good at: giving a consistent estimate when less than half of the total weight comes from invalid instruments.

What it is vulnerable to: strong violations concentrated in many SNPs or complex dependence structures.

Leave-one-out

Leave-one-out removes one SNP at a time and re-runs the IVW estimate.

What it is good at: detecting whether one SNP dominates the result.

Steiger filtering

Steiger filtering asks whether a SNP appears to explain more variance in the exposure than the outcome. If not, the proposed causal direction becomes less convincing.

In Medusa:

the internal Steiger implementation is for continuous outcomes,
binary outcomes require different assumptions and are not handled by the internal version.

A small synthetic summary dataset

library(Medusa)

set.seed(1004)
n_snps <- 8
beta_zx <- c(0.22, 0.18, 0.25, 0.20, 0.16, 0.24, 0.19, 0.21)
true_effect <- 0.45
beta_zy <- true_effect * beta_zx + rnorm(n_snps, sd = 0.015)

per_snp <- data.frame(
  snp_id = paste0("rs", seq_len(n_snps)),
  effect_allele = c("A", "C", "G", "T", "A", "C", "G", "A"),
  other_allele = c("G", "T", "A", "C", "C", "T", "T", "G"),
  eaf = seq(0.15, 0.50, length.out = n_snps),
  beta_ZY = beta_zy,
  se_ZY = rep(0.03, n_snps),
  beta_ZX = beta_zx,
  se_ZX = rep(0.04, n_snps),
  stringsAsFactors = FALSE
)

per_snp
#>   snp_id effect_allele other_allele  eaf    beta_ZY se_ZY beta_ZX se_ZX
#> 1    rs1             A            G 0.15 0.08988239  0.03    0.22  0.04
#> 2    rs2             C            T 0.20 0.09251444  0.03    0.18  0.04
#> 3    rs3             G            A 0.25 0.11003592  0.03    0.25  0.04
#> 4    rs4             T            C 0.30 0.08956835  0.03    0.20  0.04
#> 5    rs5             A            C 0.35 0.07220337  0.03    0.16  0.04
#> 6    rs6             C            T 0.40 0.09726425  0.03    0.24  0.04
#> 7    rs7             G            T 0.45 0.06333035  0.03    0.19  0.04
#> 8    rs8             A            G 0.50 0.10692022  0.03    0.21  0.04

Running Medusa sensitivity analyses

sensitivity <- runSensitivityAnalyses(
  per_snp,
  methods = c("IVW", "MREgger", "WeightedMedian", "Steiger", "LeaveOneOut"),
  outcomeSampleSize = 15000,
  exposureSampleSize = 15000,
  outcomeType = "continuous",
  engine = "internal"
)
#> Running sensitivity analyses with 8 SNPs...
#>   Engine: internal
#>   IVW...
#>   MR-Egger...
#>   Weighted Median...
#>   Steiger filtering...
#> Warning in computeSteiger(perSnpEstimates, outcomeSampleSize,
#> exposureSampleSize, : All SNPs failed Steiger filter. Steiger-filtered estimate
#> not available. Investigate instrument validity.
#>   Leave-One-Out...
#> Sensitivity analyses complete.

sensitivity$summary
#>            method   beta_MR      se_MR   ci_lower  ci_upper         pval
#> 1             IVW 0.4362023 0.05095002 0.33634029 0.5360644 1.115270e-17
#> 2        MR-Egger 0.3720929 0.15241266 0.07336409 0.6708217 5.037909e-02
#> 3 Weighted Median 0.4401437 0.06266758 0.31731524 0.5629721 2.164288e-12

How to read the results

Questions to ask:

Are the IVW and weighted median estimates in the same rough direction?
Is the MR-Egger slope wildly different or very imprecise?
Is the MR-Egger intercept suggestive of directional pleiotropy?
Does leave-one-out show a single highly influential SNP?
Does Steiger filtering remove many SNPs?

Agreement across methods does not prove causality, but it strengthens the case. Strong disagreement tells you to slow down and inspect the assumptions.

Key takeaway: sensitivity analyses are not optional add-ons. They are part of the causal argument.

Common misunderstanding

There is no single “bias-proof” MR estimator. Each method protects against some problems under specific assumptions. The purpose of running several methods is to see whether your inference is robust across different failure scenarios.

Exercise set 1

Exercise 1

Which method is most directly designed to show whether one SNP is dominating the overall result?

Show solution

Leave-one-out analysis, because it removes each SNP in turn and recomputes the estimate.

Exercise 2

If the MR-Egger intercept is clearly non-zero, what concern does that raise?

Show solution

It raises concern about directional pleiotropy, meaning the SNPs may be affecting the outcome through pathways other than the exposure of interest.

Exercise 3

Why might IVW and weighted median differ, even when both are computed correctly?

Show solution

They make different assumptions about invalid instruments. Weighted median can remain reliable when a minority of the weight comes from invalid SNPs, whereas IVW can be more sensitive to those SNPs.

Exercise set 2: inspect a result table

Use the summary table above and answer:

Which methods produce point estimates in the same general direction?
Is there evidence that one method is much less precise than the others?

Show solution

You should usually see IVW, MR-Egger, and weighted median pointing in the same general direction in this synthetic example. MR-Egger is often less precise than IVW because it estimates both an intercept and a slope.

Chapter summary

IVW is the standard efficient summarized-data estimator under broadly valid instruments.
MR-Egger checks for and can adjust for some directional pleiotropy.
Weighted median is robust when less than half of the total weight is invalid.
Leave-one-out identifies influential SNPs.
Steiger filtering checks whether the proposed causal direction looks plausible.

References

Burgess, S., Butterworth, A., & Thompson, S.G. (2013). Mendelian randomization analysis with multiple genetic variants using summarized data. Genetic Epidemiology, 37(7), 658–665.
Bowden, J., Davey Smith, G., & Burgess, S. (2015). Mendelian randomization with invalid instruments: effect estimation and bias detection through Egger regression. International Journal of Epidemiology, 44(2), 512–525.
Bowden, J., Davey Smith, G., Haycock, P.C., & Burgess, S. (2016). Consistent estimation in Mendelian randomization with some invalid instruments using a weighted median estimator. Genetic Epidemiology, 40(4), 304–314.
Hemani, G., Tilling, K., & Davey Smith, G. (2017). Orienting the causal relationship between imprecisely measured traits using GWAS summary data. PLOS Genetics, 13(11), e1007081.

Next chapter

Next: Course 5: From textbook two-sample MR to the Medusa workflow

For a more complete package example after this chapter, see Getting Started with Medusa.