Mendelian Randomization With Binary Outcomes

Why a Binary Outcome Changes the Interpretation

In Mendelian randomization (MR), a genetic variant’s association with a binary outcome is often estimated by logistic regression and reported as a log odds ratio. Dividing this association by the variant’s association with the exposure produces a ratio estimate, but its scale matters: the result is naturally expressed as a change in log odds per exposure unit, not as a change in probability.

This distinction is important in clinical and genetic research. An odds ratio can be converted to a risk only when a baseline risk is specified, and the same odds ratio can correspond to different risk differences in populations with different baseline risks. When an outcome is rare, odds ratios approximate risk ratios, which makes a log-odds ratio useful as an approximation to a log-relative-risk effect. That approximation is not automatic for common outcomes.

Definition: Let \(G\) be a genetic instrument, \(X\) an exposure, and \(Y\) a binary outcome. Write \(\beta_X\) for the change in the mean exposure per unit increase in \(G\), and \(\beta_Y\) for the change in the log odds of \(Y=1\) per unit increase in \(G\). The single-instrument ratio is \(\hat{\theta}=\hat{\beta}_Y/\hat{\beta}_X\). If \(\beta_X\) is measured in exposure units per allele and \(\beta_Y\) is a log odds ratio per allele, then \(\hat{\theta}\) is on the log-odds scale per exposure unit. Its exponent, \(\exp(\hat{\theta})\), is an odds ratio per exposure unit.

The ratio has a causal interpretation only under the relevant MR conditions, including instrument relevance, independence from causes of the outcome that also affect the exposure, and the exclusion restriction. For the binary-outcome interpretation, the exposure and outcome associations must also refer to compatible populations and to the same effect allele. These conditions do not turn an odds ratio into a risk difference; they support a causal interpretation on a specified effect scale under the model used.

Interpreting the Ratio on the Odds Scale

Suppose the outcome association is estimated by logistic regression. The coefficient \(\beta_Y\) is a change in log odds, so the corresponding odds ratio for a one-unit increase in \(G\) is \(\exp(\beta_Y)\). If \(G\) is coded as the number of effect alleles, one unit is one additional effect allele. Dividing \(\beta_Y\) by \(\beta_X\) expresses the outcome association per unit of exposure rather than per allele.

For example, if the exposure is standardized, one exposure unit is one standard deviation. A ratio of \(0.20\) then corresponds to an odds ratio of \(\exp(0.20)\) per standard-deviation increase in exposure. This is an odds comparison, not a statement that the probability of the outcome rises by 20%, nor that the risk rises by 20%.

Worked Example: A Ratio Estimate and Its Confidence Interval

Consider hypothetical summary associations from a two-sample analysis. The instrument is associated with a standardized exposure by \(\hat{\beta}_X=0.080\) standard deviations per effect allele, with \(SE(\hat{\beta}_X)=0.010\). Its association with a binary outcome is \(\hat{\beta}_Y=0.060\) log odds per effect allele, with \(SE(\hat{\beta}_Y)=0.020\). Assume the estimates come from independent samples, and use the first-order delta method for the ratio.

The ratio estimate is

\hat{\theta}=\frac{0.060}{0.080}=0.750

The units are log odds per one-standard-deviation increase in exposure. Exponentiating gives the odds ratio:

\exp(0.750)=2.117

Under the stated MR assumptions and the model’s scale, the estimate corresponds to odds multiplied by about 2.12 per standard-deviation increase in exposure. It does not say that the outcome probability is multiplied by 2.12.

For independent estimates, the delta-method variance of the ratio is approximated by

\widehat{\operatorname{Var}}(\hat{\theta}) \approx \frac{SE(\hat{\beta}_Y)^2}{\hat{\beta}_X^2} + \frac{\hat{\beta}_Y^2 SE(\hat{\beta}_X)^2}{\hat{\beta}_X^4}.

Substituting the values gives

\widehat{\operatorname{Var}}(\hat{\theta}) \approx \frac{0.020^2}{0.080^2} + \frac{0.060^2(0.010^2)}{0.080^4} = 0.06250+0.008789 = 0.07129, \qquad SE(\hat{\theta})\approx\sqrt{0.07129}=0.267.

A 95% Wald confidence interval on the log-odds scale is \(0.750\pm1.96(0.267)\), or approximately \((0.227,1.273)\). Exponentiating the endpoints gives an odds-ratio interval of approximately \((1.255,3.572)\) per standard deviation of exposure. This interval reflects sampling uncertainty under the approximation; it is not a 95% probability that the fixed causal parameter lies inside these particular endpoints. The normal approximation for a ratio is most credible when the instrument-exposure association is sufficiently precise and is not close to zero.

When Odds Ratios Approximate Risk Ratios

Risk is the probability of the outcome. Odds are the ratio of that probability to the probability of no outcome. Because these are different quantities, an odds ratio is not generally a risk ratio. The relationship between them shows exactly why the rare-outcome approximation can work.

Let \(p_0\) be the outcome risk in a reference group and \(p_1\) the risk in a comparison group. Their odds ratio is the odds in the comparison group divided by the odds in the reference group. Solving that definition for \(p_1\) gives the following relationship.

OR=\frac{p_1/(1-p_1)}{p_0/(1-p_0)} \quad\Longrightarrow\quad p_1=\frac{OR\,p_0}{1-p_0+OR\,p_0}.

Dividing \(p_1\) by \(p_0\) gives the risk ratio. Since \(1-p_0+OR\,p_0=1+p_0(OR-1)\), the ratio of risks is

RR=\frac{p_1}{p_0} =\frac{OR}{1+p_0(OR-1)}.

For a fixed odds ratio, as \(p_0\) approaches zero, the denominator approaches one. Therefore \(RR\) approaches \(OR\), and \(\log(RR)\) approaches \(\log(OR)\). This is the rare-disease approximation: it is strongest when risks are small in the relevant population and for the exposure contrast being considered. It is an approximation, not an identity. For a common outcome, a log-odds association should not simply be relabelled a log-risk association.

Worked Example: Checking the Rare-Disease Approximation

Suppose, hypothetically, that an exposure contrast has an odds ratio of \(1.5\). Compare its implied risk ratio when the reference risk is \(0.01\) with the implied risk ratio when the reference risk is \(0.30\).

With a reference risk of \(0.01\), the comparison risk is

p_1=\frac{1.5(0.01)}{1-0.01+1.5(0.01)} =\frac{0.015}{1.005} =0.01493.

The risk ratio is \(0.01493/0.01=1.493\), close to the odds ratio of \(1.5\). Equivalently, the formula gives \(RR=1.5/[1+0.01(1.5-1)]=1.5/1.005=1.493\).

With a reference risk of \(0.30\), the comparison risk is instead

p_1=\frac{1.5(0.30)}{1-0.30+1.5(0.30)} =\frac{0.45}{1.15} =0.3913, \qquad RR=\frac{0.3913}{0.30}=1.304.

The odds ratio is still \(1.5\), but the risk ratio is about \(1.30\). Thus the same odds ratio gives a close approximation to a risk ratio in the rare-outcome example, but not in the common-outcome example.

Odds Effects Do Not Determine Absolute-Risk Effects

A ratio estimate on the log-odds scale does not, by itself, specify a change in absolute risk. To calculate a risk under an odds-ratio model, a baseline risk is needed. The conversion formula depends on that baseline risk, so an odds ratio alone cannot tell a clinician how many additional events to expect in a particular population.

This dependence is not a technical nuisance: it reflects the difference between relative comparisons and absolute outcomes. A fixed odds ratio can correspond to a small risk difference when baseline risk is low and a larger difference when baseline risk is higher. A calculated risk difference also depends on the target population’s baseline risk being appropriate for the exposure contrast and causal model under consideration.

Worked Example: The Same Odds Ratio at Two Baseline Risks

Assume hypothetically that an exposure contrast has a causal odds ratio of \(1.25\), and that the applicable baseline risk is \(0.08\) in one population and \(0.30\) in another. Use the odds-ratio conversion to calculate the corresponding risks and risk differences.

For baseline risk \(p_0=0.08\), the risk under the comparison exposure level is

p_1=\frac{1.25(0.08)}{1-0.08+1.25(0.08)} =\frac{0.100}{1.020} =0.09804.

The risk difference is \(0.09804-0.08=0.01804\), or about 1.80 percentage points. For baseline risk \(p_0=0.30\),

p_1=\frac{1.25(0.30)}{1-0.30+1.25(0.30)} =\frac{0.375}{1.075} =0.3488.

The risk difference is \(0.3488-0.30=0.04884\), or about 4.88 percentage points. Both calculations use the same odds ratio, but the absolute-risk changes differ because the baseline risks differ. These are conditional calculations under the assumed odds-ratio model, not quantities supplied by the odds ratio alone.

Scale and Model Require Careful Language

The outcome scale should be named whenever a binary-outcome MR result is reported. A ratio based on a logistic-regression outcome association is an estimate on a log-odds scale. If the outcome is rare, it may be described as approximately a log-relative-risk effect under the rare-disease approximation, with the outcome frequency and approximation made clear. For a common outcome, reporting it as a risk ratio without an additional model or calculation is misleading.

Odds ratios also depend on how the comparison is defined. In particular, conditional odds ratios from a model that adjusts for covariates need not equal marginal odds ratios from a model that does not condition on those covariates, even in the absence of confounding. This property is called non-collapsibility. Consequently, outcome association estimates should be interpreted with attention to the regression model and population used to obtain them; estimates on different odds-ratio scales should not be treated as automatically interchangeable.

A report should state the exposure unit, the outcome association scale, and whether an odds ratio is being presented or a risk-scale quantity has been derived. For an absolute-risk interpretation, it should also state the baseline risk and the assumptions used to translate the odds ratio into a risk. These details keep a mathematically correct ratio from being given an unsupported clinical meaning.

Common Mistake

Incorrect reasoning: “The MR odds ratio is 1.5, so exposure increases the probability of the outcome by 50%.” This treats an odds ratio as a risk ratio and then treats a relative effect as an absolute probability increase.

Correction: An odds ratio of \(1.5\) means that the odds are multiplied by \(1.5\) for the stated exposure contrast, under the specified model and assumptions. It does not mean the risk rises by 50%. To obtain a risk or risk difference, use an appropriate baseline risk and the odds-to-risk conversion. Only for a sufficiently rare outcome is the odds ratio approximately a risk ratio, and even then it is not an absolute risk difference.

Key takeaway: With a binary outcome, the standard ratio of a log-odds association to an exposure association is on the log-odds scale per exposure unit. Rare outcomes can make that scale approximate a log-risk scale, but neither the ratio nor its exponent alone supplies an absolute-risk effect.

Check Your Understanding

Answer each question using the scale and units explicitly.

A genetic variant is associated with an exposure by \(0.04\) units per allele and with a binary outcome by \(0.02\) log odds per allele. What is the ratio estimate, and what are its units?
Why can an odds ratio approximate a risk ratio when the outcome is rare? State the relationship that supports the approximation.
An odds ratio of \(2\) is observed for an outcome with a baseline risk of \(0.02\). Explain why it is incorrect to conclude that the risk is \(0.04\) without calculation.
Why can the same odds ratio correspond to different risk differences in two populations?
What additional information is needed to use an odds ratio to calculate an absolute risk under the odds-ratio model?