Interpreting Mendelian Randomization With Binary Outcomes

What a Binary-Outcome MR Ratio Estimates

When the outcome is binary, its association with a genetic variant is commonly reported as a log odds ratio. A Mendelian randomization (MR) ratio divides that association by the variant’s association with the exposure. The result is on a log-odds scale per unit of exposure, provided that the exposure association and outcome association have compatible definitions and scales. Exponentiating the ratio gives an odds ratio, not a risk ratio.

That calculation alone does not establish a causal interpretation. As in the earlier material on pleiotropy and interpretation in multivariable MR, causal interpretation depends on the instrumental-variable assumptions and on what causal contrast the model represents. With a binary outcome, it is especially important to distinguish the odds ratio’s mathematical scale from a risk change, and a genetic liability effect from the effect of assigning an intervention.

A common working relation for the genetic associations is \(\beta_{Yj}=\theta\beta_{Xj}+\alpha_j\), where \(\alpha_j\) represents a variant’s outcome association not operating through the exposure. The ratio has the intended causal interpretation when the instrument is relevant, independent of causes of the outcome (subject to the study design and adjustment), and has no pathway to the outcome outside the exposure, so that \(\alpha_j=0\). In addition, the exposure and outcome associations must be suitably aligned, and the causal effect must be representable by the chosen model and scale. For multiple variants, the same scale and interpretation issues apply to the combined estimate; this tutorial focuses on interpreting the ratio rather than choosing an estimator.

From Log Odds to Odds and Risk

For an odds ratio \(R\), the log-odds effect is \(\theta=\log(R)\), so \(R=\exp(\theta)\). An odds ratio compares odds, \(p/(1-p)\), rather than probabilities \(p\). To translate an odds ratio into a probability at a specified baseline risk \(p_0\), multiply the baseline odds by the odds ratio and convert the resulting odds back to risk:

The resulting probability is an illustration for the specified baseline risk and a contrast that supports this calculation. It is not a universal risk change: the same odds ratio yields different absolute risk differences at different baseline risks. This conversion also does not, by itself, establish that a genetic contrast corresponds to an intervention that could be assigned.

Uncertainty on the Odds-Ratio Scale

A ratio estimate is uncertain because both association estimates are uncertain. One approximate standard error can be obtained by the delta method when the estimates are approximately normal, the denominator is sufficiently far from zero, and the two estimates are independent. If their standard errors are \(SE_Y\) and \(SE_X\), the approximation is \(SE(\hat{\theta})\approx\sqrt{SE_Y^2/\hat{\beta}_X^2+\hat{\beta}_Y^2SE_X^2/\hat{\beta}_X^4}\). With non-independent estimates, the covariance term must also be included; the independence assumption should not be used automatically.

Why Odds Ratios Are Non-Collapsible

An odds ratio is non-collapsible: a conditional odds ratio and a marginal odds ratio can differ even when the conditioning variable is not a confounder. Conditional odds compare outcome odds between exposure groups within levels of another variable. Marginal odds compare odds after averaging risks across that variable’s distribution. Averaging probabilities and then taking odds is not equivalent to taking odds within each group and averaging those odds.

The following hypothetical calculation demonstrates the distinction. Let \(Z\) define two equally common strata. In each stratum, suppose exposure doubles the outcome odds. In stratum A, the unexposed risk is 0.10, so the exposed risk is \(2/11\). In stratum B, the unexposed risk is 0.50, so the exposed risk is \(2/3\). Each stratum-specific odds ratio is exactly 2, but the risks averaged across the equally sized strata give a different marginal odds ratio.

Genetic Liability Is Not Automatically an Intervention

A genetic variant generally shifts exposure over a lifetime, often by a small amount. A ratio estimate consequently describes a contrast associated with genetically induced differences in exposure under the MR assumptions. It does not automatically describe the effect of starting a drug, changing a behaviour at a particular age, or assigning an exposure value in a clinical trial. Such an intervention may differ in timing, duration, magnitude, and affected biological pathways.

The distinction is especially important when the exposure itself is binary or when its genetic association represents a change in liability to being exposed. A ratio involving a genetic association with an exposure does not turn a liability contrast into the effect of assigning everyone from unexposed to exposed. The estimand depends on how exposure is defined and modeled, the size and nature of the genetic shift, and the assumptions linking that shift to the outcome. Interpret the estimate using the actual exposure scale, not as a generic treatment effect.

Diagnostics, Alternatives, and Common Mistakes

Before interpreting a binary-outcome ratio, check that the reported outcome association is a log odds ratio, the exposure association has a clear unit, and both estimates refer to the same allele and compatible population. Verify that the exposure association is not close to zero and that standard errors and covariance assumptions suit the data sources. Examine whether the causal model supports a constant effect on the chosen scale; if effects vary, one ratio may not summarize all relevant contrasts.

The odds ratio is only one effect measure. A risk ratio compares probabilities, and a risk difference subtracts them. These measures answer different questions and do not generally equal a transformed odds ratio without a baseline risk and additional assumptions. For common outcomes, the odds ratio can be farther from 1 than the risk ratio for the same comparison. When absolute clinical impact is the target, reporting an odds ratio alone may be insufficient; a risk translation requires a relevant baseline risk and a defensible target population.

• Mistake: “An odds ratio of 1.5 means 50% more people will have the outcome.” It means 50% higher odds. The probability change depends on baseline risk.
• Mistake: “A causal MR ratio is automatically the effect of treatment.” The estimate pertains to a genetically induced exposure contrast under its assumptions; an intervention effect may be a different estimand.
• Mistake: “Different marginal and conditional odds ratios prove confounding.” They can differ through non-collapsibility alone. Confounding and non-collapsibility are distinct issues.
• Mistake: “Exponentiating a ratio makes its causal interpretation valid.” Exponentiation changes the scale, not the assumptions. Instrument validity, compatible association scales, and an appropriate causal model remain necessary.

A practical interpretation should name the outcome, exposure unit, scale, and causal caveat together. For example: “Under the MR assumptions and a compatible log-odds model, a one-standard-deviation genetically induced increase in the exposure is associated with an estimated odds ratio of 1.49 for the binary outcome.” Avoid replacing “genetically induced” with a direct intervention claim unless the design and estimand justify that step.