Validity and Interpretation of Two-Sample Mendelian Randomization with Binary Traits

This paper provides a formal statistical justification for using standard two-sample Mendelian randomization with binary traits by demonstrating that, under a liability-threshold framework with small genetic effects, conventional methods validly estimate scaled causal effects on the underlying continuous liability scale without requiring methodological modifications.

The Gist

The Big Problem: The "On/Off" Switch Confusion

Imagine you are trying to figure out if smoking causes heart disease. In the real world, smoking isn't just a simple "Yes" or "No." Some people smoke one cigarette a week; others smoke three packs a day. But in medical studies, we often have to simplify this into a binary switch: Smoker (1) or Non-Smoker (0).

The same goes for diseases. You either have Type 2 Diabetes or you don't. But underneath that "Yes/No" label, there is a hidden, continuous reality: your blood sugar levels. Some people are just barely over the line to be diagnosed, while others are way over the line.

The Confusion:
For years, scientists using a method called Mendelian Randomization (MR) (which uses genetics as a natural experiment to prove cause-and-effect) were worried. They thought: "If we only look at the 'Yes/No' switch, the math might break. We might get the wrong answer because we are ignoring the 'how much' part of the story."

They were afraid that using binary data (Yes/No) was like trying to measure the speed of a car by only looking at whether the speedometer says "Fast" or "Slow," without knowing the actual miles per hour.

The Solution: The "Hidden Volume Knob"

This paper, by Zixuan Wu and Jingshu Wang, says: "Don't worry! The math actually works fine, you just need to understand what the number means."

Here is the core idea using a Volume Knob analogy:

1. The Hidden Reality (The Liability): Imagine every person has a hidden "Volume Knob" inside them representing their risk for a trait (like heart disease or smoking). This knob can be turned up or down continuously.
2. The Threshold (The Light Switch): A doctor looks at this knob. If it's turned past a certain point (the threshold), they flip a light switch and say, "You have the disease!" If it's below the line, the switch stays off.
   • Binary Trait: The light switch (On/Off).
   • Liability: The actual position of the volume knob.
3. The Genetic Clue: Your genes act like tiny hands that gently nudge the volume knob up or down.

The "Magic" Discovery

The authors did some heavy math to prove a very simple thing:

Even though we can't see the volume knob, the "Yes/No" light switch still tells us exactly how the knob is moving.

They found that the genetic data we get from the "Yes/No" switch is proportional to the data we would get if we could see the volume knob directly.

• Think of it like this: If you have a blurry photo of a mountain (the binary data) and a sharp photo of the same mountain (the continuous data), the blurry photo is just a scaled-down version of the sharp one. It's not a different mountain; it's just the same mountain at a different resolution.

The "Scaling Factor": The Secret Decoder Ring

The paper explains that when you run the standard MR analysis on binary data, you aren't getting the "wrong" answer. You are getting the right answer, but it's in a different "currency."

• The Currency: Instead of measuring the effect on the "Light Switch" (Yes/No), the math is actually measuring the effect on the "Volume Knob" (the hidden risk).
• The Exchange Rate: The paper provides a simple formula (a "scaling factor") based on how common the disease is in the population (prevalence).
  • If a disease is rare (like 1% of people), the "exchange rate" is one number.
  • If a disease is common (like 50% of people), the "exchange rate" is a different number.

The Takeaway: You don't need to invent new, complicated math tools. You can keep using the standard tools everyone already uses. You just need to take the result and multiply it by this "exchange rate" to translate it back into the language of the hidden volume knob.

Why This Matters (The "Aha!" Moment)

Before this paper, researchers were hesitant to use binary data (like "Did you get cancer? Yes/No") in these genetic studies because they feared the results were unreliable.

This paper says:

1. It's Valid: You can trust the standard methods.
2. It's Interpretable: The result you get isn't about the "Yes/No" switch; it's about the underlying risk (the volume knob).
3. It's Easy: You don't need to change your software. You just need to apply a simple correction factor based on how common the disease is.

Summary Analogy

Imagine you are trying to guess how much rain fell in a city.

• The Binary Data: You only have a bucket that tells you "Wet" or "Dry."
• The Old Fear: Scientists thought, "If I only know 'Wet', I can't calculate the exact amount of rain, so my math is broken."
• This Paper's Conclusion: "Actually, if you know how often buckets get wet in this city, you can calculate the exact amount of rain that fell. The 'Wet/Dry' bucket is just a scaled-down version of the rain gauge. You don't need a new gauge; you just need to do a little math to translate 'Wet' into 'Inches of Rain'."

In short: Binary traits (Yes/No) are not a dead end for genetic research. They are just a slightly blurry version of the full picture, and this paper gives us the lens to make them crystal clear again.

Technical Summary

1. Problem Statement

Two-sample Mendelian Randomization (MR) is a dominant method for inferring causal effects using summary statistics from Genome-Wide Association Studies (GWAS). However, a significant methodological challenge arises when dealing with binary traits (e.g., disease status, smoking initiation).

• The Conflict: Standard MR methods assume linear relationships between genetic instruments and traits. Binary traits, however, are often analyzed using logistic regression (producing odds ratios) or linear regression on a 0/1 scale.
• The Interpretation Gap: It has been unclear whether standard MR estimates derived from binary traits are statistically valid or what specific causal parameter they identify.
  • Binary outcomes do not naturally support linear causal models.
  • Binary exposures often represent thresholded versions of underlying continuous processes.
  • Existing literature suggests that standard MR on binary traits might be problematic due to violations of exclusion restrictions or heterogeneity in effect sizes across individuals with the same binary status.
• The Question: Can conventional summary-data MR be formally justified for binary traits, and if so, what does the resulting estimate represent?

2. Methodology: The Liability-Threshold Framework

The authors propose a formal statistical justification based on the liability-threshold model.

• Latent Variables: They posit that observed binary traits ($X$ and $Y$) are dichotomizations of unobserved, underlying continuous variables called liabilities ($X^*$ and $Y^*$).

  • $X = 1$ if $X^* > t_X$ (where $t_X$ is a threshold).
  • $Y = 1$ if $Y^* > t_Y$.

• Causal Definition: The true causal effect of interest ($\beta$) is defined as the effect of the exposure liability ($X^*$) on the outcome liability ($Y^*$), rather than the effect of the binary status $X$ on $Y$.

• Derivation of Scaling Factors:

  • The authors derive the mathematical relationship between GWAS coefficients obtained from observed binary traits (via logistic or linear regression) and the true marginal genetic associations on the liability scale.
  • Key Assumption: Genetic effects on complex traits are typically small. Under this assumption, the change in the probability of a binary outcome induced by a single SNP can be approximated by a local linear shift in the underlying liability.
  • Resulting Proportionality: The observed GWAS coefficient ($\gamma_j$) is approximately proportional to the liability-scale coefficient ($\gamma^*_j$):
    $$\gamma_j \approx s_X \cdot \gamma^*_j$$
    Where $s_X$ is a scaling factor dependent on:
    1. Trait Prevalence ($p_X$): The proportion of cases in the population.
    2. Regression Model: Logistic vs. Linear.
    3. Sampling Design: Cohort vs. Case-Control.

• Specific Scaling Formulas:

  • Logistic Regression (Cohort): $s_X \approx \frac{\phi(t_X)}{p_X(1-p_X)}$
  • Linear Regression (Cohort): $s_X \approx \frac{1}{\sqrt{p_X(1-p_X)}} \cdot \frac{1}{\phi(t_X)}$ (Note: The paper provides specific derivations for linear regression on standardized binary outcomes).
  • Case-Control Design: Logistic regression coefficients remain approximately invariant to retrospective sampling, whereas linear regression coefficients require adjustment based on the case fraction in the sample.

3. Key Contributions

1. Formal Justification: The paper provides the first formal statistical derivation proving that standard two-sample MR methods applied to binary traits are valid under the liability framework, provided genetic effects are small.
2. Identification of the Causal Parameter: The authors clarify that standard MR on binary traits estimates a scaled causal effect between the underlying liabilities ($\beta \cdot \frac{s_Y}{s_X}$), not an effect on the observed binary scale.
3. No Methodological Modification Required: Crucially, the authors demonstrate that existing MR methods (e.g., IVW, MR-Egger, GRAPPLE) do not need to be modified. The validity holds because the scaling factors cancel out or are constant across SNPs within a study.
4. Rescaling Procedure: They provide a practical method to "rescale" MR estimates to recover the true liability-scale causal effect ($\beta$) using known prevalence and study design parameters.
5. Extension to Ordinal Traits: The logic extends to ordered categorical traits, viewing them as discretizations of a continuous liability with multiple thresholds.

4. Results

The authors validated their theoretical findings through simulations and real-world data analysis.

• Simulation Studies:

  • Proportionality: Simulations confirmed that GWAS coefficients from binary traits are approximately proportional to liability-scale effects. Dividing observed coefficients by the theoretical scaling factor ($s_X$) aligned them perfectly with the identity line.
  • MR Recovery: In MR simulations involving horizontal pleiotropy, unscaled estimates varied significantly depending on trait definitions (binary vs. continuous). However, rescaled estimates consistently recovered the true liability-scale causal effect, matching the benchmark of continuous-trait analyses.
  • Accuracy: The approximation was most accurate for logistic regression and moderate prevalences. Accuracy decreased slightly for extremely rare traits or linear regression, but remained robust.

• UK Biobank Application:

  • Setup: Analyzed the causal effect of Body Mass Index (BMI) on Systolic Blood Pressure (SBP).
  • Comparisons: Compared four scenarios: Continuous BMI $\to$ Continuous SBP, Continuous BMI $\to$ Binary Hypertension, Binary Obesity $\to$ Continuous SBP, and Binary Obesity $\to$ Binary Hypertension.
  • Finding: Observed-scale estimates differed widely across these definitions. After applying the prevalence-based rescaling, all estimates converged to nearly identical confidence intervals, demonstrating that the apparent discrepancies were purely due to scale transformations, not violations of causal assumptions.

5. Significance and Implications

• Resolving Long-standing Uncertainty: The paper resolves the ambiguity regarding the use of binary traits in MR. It confirms that researchers do not need to discard binary data or develop entirely new estimators.
• Practical Utility: It enables the routine use of binary exposures and outcomes (which are abundant in GWAS) in MR studies while ensuring the results are interpretable.
• Interpretation Shift: Researchers must interpret MR estimates from binary traits as effects on the underlying liability (e.g., underlying risk of disease) rather than the probability of the binary event itself.
• Broad Applicability: The findings support the use of binary traits in complex MR frameworks, including multivariable MR, within-family MR, and life-course MR.
• Limitations: The authors note that the proportionality relies on the assumption of small genetic effects. Variants with very large effects or traits with extreme prevalence (near 0 or 1) may show deviations from the linear approximation. Additionally, the model assumes a single threshold, whereas real-world diagnostic criteria may vary across subgroups.

In conclusion, this work provides a rigorous theoretical foundation that validates the widespread practice of using summary-data MR with binary traits, provided the results are interpreted as scaled effects on latent liabilities.