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Course 2: Instrumental Variables and MR Assumptions

Course02-InstrumentalVariablesAndMRAssumptions.Rmd

What you’ll learn

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

  • define an instrumental variable in plain language,
  • state the three core MR assumptions,
  • explain weak instruments, pleiotropy, and population stratification,
  • recognize common threats to a valid MR design.

Why this matters

MR is persuasive only when the genetic variants really behave like useful instruments. That is why serious MR work spends a lot of time checking assumptions, not just calculating estimates.

What is an instrumental variable?

An instrumental variable (IV) is a variable that changes the exposure in a way that can help us learn about the causal effect of the exposure on the outcome.

For MR, the instrument is usually one or more genetic variants.

To be useful, the variant needs to do three things:

  1. predict the exposure,
  2. avoid tracking major confounders,
  3. affect the outcome mainly through the exposure.

Those are the three classic IV assumptions.

Assumption 1: Relevance

The instrument must genuinely affect the exposure.

If a variant barely changes LDL-C, it contains very little information about the LDL-C to outcome pathway. In practice, such variants are called weak instruments.

Weak instruments matter because they can make MR estimates unstable and biased.

A simple strength check

MR studies often summarize instrument strength using the approximate F-statistic:

[ F ()^2 ]

Here:

  • (_{ZX}) is the SNP-exposure association,
  • ((_{ZX})) is its standard error.

A common rule of thumb is that (F < 10) suggests a weak instrument.

beta_zx <- c(0.20, 0.06, 0.30)
se_zx <- c(0.04, 0.05, 0.06)
data.frame(
  snp = c("rsA", "rsB", "rsC"),
  beta_zx = beta_zx,
  se_zx = se_zx,
  approx_F = (beta_zx / se_zx)^2
)
#>   snp beta_zx se_zx approx_F
#> 1 rsA    0.20  0.04    25.00
#> 2 rsB    0.06  0.05     1.44
#> 3 rsC    0.30  0.06    25.00

Key takeaway: if the instrument does not move the exposure, it cannot tell you much about the exposure’s causal effect.

Assumption 2: Independence

The instrument should be independent of major confounders of the exposure-outcome relationship.

Why might this fail?

  • Population stratification: ancestry differences can create associations between variants and environment.
  • Selection bias: the analytic sample may depend on both genotype and outcome-related factors.
  • Dynastic or family effects: parental genotype can shape the environment of offspring.

In practice, this assumption is motivated by the biology of inheritance, but it is not automatically guaranteed in every dataset.

Assumption 3: Exclusion restriction

The instrument should affect the outcome mainly through the exposure of interest.

This is usually the hardest assumption to defend.

The main threat is pleiotropy, which means a genetic variant affects more than one trait.

There are two broad cases:

  • Vertical pleiotropy: the variant affects an upstream trait, which then affects downstream traits in the same causal pathway. This is usually not a problem for MR.
  • Horizontal pleiotropy: the variant affects the outcome through a separate pathway outside the exposure of interest. This can bias MR estimates.

Example

Suppose a variant changes LDL-C and also independently changes inflammation. If inflammation affects CAD risk, then the variant has a second route to the outcome. That violates exclusion restriction for an LDL-C MR analysis.

Key takeaway: the instrument must not have an important back door into the outcome outside the intended exposure pathway.

A picture in words

For a valid MR design, we want:

  • genetic variant -> exposure -> outcome

and we do not want:

  • confounder -> genetic variant
  • genetic variant -> other pathway -> outcome

That verbal graph is often enough to orient your thinking, even before drawing a formal DAG.

A small simulation: strong vs weak instruments 

set.seed(1002)
n <- 5000
strong_snp <- rbinom(n, 2, 0.3)
weak_snp <- rbinom(n, 2, 0.3)

exposure_strong <- 0.40 * strong_snp + rnorm(n)
exposure_weak <- 0.04 * weak_snp + rnorm(n)

summary(lm(exposure_strong ~ strong_snp))$coefficients["strong_snp", ]
#>     Estimate   Std. Error      t value     Pr(>|t|) 
#> 4.025216e-01 2.185009e-02 1.842196e+01 2.242399e-73
summary(lm(exposure_weak ~ weak_snp))$coefficients["weak_snp", ]
#>    Estimate  Std. Error     t value    Pr(>|t|) 
#> 0.003518649 0.022374208 0.157263618 0.875043445

The strong variant produces a clearer exposure shift. The weak variant produces a noisier, less informative relationship.

Exercise set 1

Exercise 1

Which MR assumption is threatened most directly by a variant that barely changes the exposure?

Show solution

The relevance assumption is threatened. If the variant barely affects the exposure, it is a weak instrument.

Exercise 2

A variant affects body mass index and, independently, smoking behavior. You are using it to study body mass index and lung disease. Which assumption is at risk?

Show solution

The exclusion restriction is at risk, because the variant may affect lung disease through smoking behavior, not only through body mass index.

Exercise 3

Why can population stratification threaten MR, even though genes are inherited at conception?

Show solution

If ancestry groups differ in both allele frequencies and environmental or health-related factors, genotype can become associated with confounders at the sample level. That weakens the independence assumption.

How Medusa helps with assumptions

Medusa cannot prove the assumptions, but it gives you tools that support assumption checking:

  • runInstrumentDiagnostics() reports approximate F-statistics,
  • the instrument PheWAS looks for associations with measured covariates,
  • allele-frequency comparisons can flag harmonization or population issues,
  • genotype missingness summaries can reveal data-quality concerns.

Those diagnostics do not replace scientific judgment, but they improve transparency.

Common misunderstanding

It is tempting to say “because genes are randomized, MR assumptions are automatically true.” That is too strong. Inheritance helps, but weak instruments, pleiotropy, population structure, and sample selection can still cause problems.

Chapter summary

  • MR is an instrumental variable method using genetic variants as instruments.
  • The three core assumptions are relevance, independence, and exclusion restriction.
  • Weak instruments threaten relevance.
  • Pleiotropy most often threatens exclusion restriction.
  • Population stratification and selection can threaten independence.

References

  • Angrist, J.D., Imbens, G.W., & Rubin, D.B. (1996). Identification of causal effects using instrumental variables. Journal of the American Statistical Association, 91(434), 444–455.
  • Staiger, D. & Stock, J.H. (1997). Instrumental variables regression with weak instruments. Econometrica, 65(3), 557–586.
  • Davey Smith, G. & Hemani, G. (2014). Mendelian randomization: genetic anchors for causal inference in epidemiological studies. Human Molecular Genetics, 23(R1), R89–R98.
  • 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.

Next chapter

Next: Course 3: How two-sample MR works in practice

For a package-oriented preview of diagnostics, see Getting Started with Medusa and IL-6 Signaling and Colorectal Cancer.