Why Take a Ratio of Genetic Associations?
A genetic variant can be associated with an exposure and with an outcome. The single-variant Wald estimator uses those two associations to estimate the causal effect of the exposure on the outcome. The key idea is that, if the variant influences the outcome only through the exposure, the variant’s outcome association is the exposure association multiplied by the causal effect. Dividing by the exposure association then recovers that effect.
The ratio is simple, but its causal interpretation is not automatic. It depends on a model and on assumptions about how the variant relates to the exposure and outcome. We first state those conditions, then derive the ratio and work through hypothetical examples.
Model and Assumptions
Use the following linear model to express the relationships that produce the ratio:
Here \(\mu_X\) and \(\mu_Y\) are intercepts, \(\gamma\) is the change in mean exposure per one-unit increase in the coded variant dosage, and \(\beta\) is the constant causal change in the outcome per one-unit increase in the exposure. The error terms represent other influences on the exposure and outcome. They may be related to one another, so the exposure need not be unconfounded with the outcome.
For the ratio to identify \(\beta\), the following conditions are needed. They describe an idealized instrumental-variable setting; observing the two associations alone does not establish that they hold.
- Relevance: the variant changes the exposure, so \(\gamma\ne 0\). Without this, there is no nonzero exposure association to divide by.
- Independence: the variant is not associated with causes of the outcome that could create an outcome association apart from its effect through \(X\). In the displayed model, a sufficient mean-level condition for the derivation is that \(G\) is uncorrelated with \(\varepsilon_X\) and \(\varepsilon_Y\).
- Exclusion: the variant affects the outcome only through the exposure. The outcome equation therefore has no additional direct term involving \(G\).
- Constant linear effect: a one-unit increase in \(X\) has the same additive causal effect \(\beta\) on \(Y\) throughout the model. This is what permits one ratio to represent a single causal effect.
The model also presumes that the recorded \(X\), \(Y\), and \(G\) refer to the same relevant units and that the effect units are well-defined. For example, if \(X\) is measured in kilograms and \(Y\) in millimetres of mercury, \(\beta\) has units of millimetres of mercury per kilogram. The variant associations must use the same variant coding: \(\gamma\) and \(\delta\) are both effects per one-unit increase in \(G\).
Deriving the Wald Ratio
The derivation follows by comparing the variant association with the exposure to its association with the outcome. Let \(\operatorname{Cov}(G,Z)\) denote the covariance between \(G\) and a variable \(Z\), and let \(\operatorname{Var}(G)\) denote the variance of \(G\). In the linear model, the slopes from regressing \(X\) and \(Y\) on \(G\) are respectively \(\gamma=\operatorname{Cov}(G,X)/\operatorname{Var}(G)\) and \(\delta=\operatorname{Cov}(G,Y)/\operatorname{Var}(G)\), when \(\operatorname{Var}(G)>0\).
The second equality uses the outcome model. The constant \(\mu_Y\) has zero covariance with \(G\). The fourth equality uses the independence condition \(\operatorname{Cov}(G,\varepsilon_Y)=0\). Since \(\gamma\ne0\), rearranging gives the population identity:
The same result can be seen from conditional means when the model specifies linear mean relationships in \(G\): each unit increase in \(G\) changes mean \(X\) by \(\gamma\), and, through \(X\), changes mean \(Y\) by \(\beta\gamma\). Thus \(\delta=\beta\gamma\). Independence and exclusion are essential to this reasoning: an additional route from \(G\) to \(Y\) could contribute to \(\delta\) without being part of \(\beta\gamma\).
In data, the population associations are unknown. Replacing them with estimates gives \(\hat{\beta}_{W}=\hat{\delta}/\hat{\gamma}\). This plug-in calculation defines the single-variant Wald estimate. The derivation explains its causal target under the stated model; it does not by itself guarantee that the estimate is precise or that the assumptions hold.
Worked Example: A Binary Variant and Two Group Means
Suppose a hypothetical study codes a variant as \(G=0\) or \(G=1\). The exposure is LDL cholesterol in mmol/L, and the outcome is systolic blood pressure in mmHg. The hypothetical group means are:
| Variant dosage \(G\) | Mean LDL cholesterol (mmol/L) | Mean systolic blood pressure (mmHg) |
|---|---|---|
| 0 | 2.8 | 118 |
| 1 | 4.3 | 121 |
Exposure association. For a binary \(G\), the linear slope is the difference in mean exposure between \(G=1\) and \(G=0\): \(\hat{\gamma}=4.3-2.8=1.5\) mmol/L per variant unit.
Outcome association. Using the same comparison, \(\hat{\delta}=121-118=3\) mmHg per variant unit.
Wald estimate. The ratio is \(\hat{\beta}_{W}=\hat{\delta}/\hat{\gamma}=3/1.5=2.00\) mmHg per mmol/L. Under the instrumental-variable assumptions and the constant-effect model, this estimates an increase of 2.00 mmHg in systolic blood pressure per 1 mmol/L increase in LDL cholesterol.
The group differences supply the two associations, but they do not verify the assumptions. In particular, the calculation alone cannot show that the variant has no route to blood pressure other than through LDL cholesterol.
Worked Example: A Per-Allele Ratio From Three Dosage Means
In a second hypothetical setting, \(G\) is coded as 0, 1, or 2 copies of an allele. The exposure is a biomarker measured in arbitrary units, and the outcome is measured in units of a clinical scale. Suppose the group means are:
| Dosage \(G\) | Mean exposure | Mean outcome |
|---|---|---|
| 0 | 4.0 | 10.0 |
| 1 | 4.8 | 11.2 |
| 2 | 5.6 | 12.4 |
The means increase linearly across dosage groups. The exposure changes by \(0.8\) units for each additional allele, so the per-allele exposure slope is \(\hat{\gamma}=0.8\). The outcome changes by \(1.2\) units per additional allele, so \(\hat{\delta}=1.2\). For example, the slopes across the full dosage range are \((5.6-4.0)/(2-0)=0.8\) and \((12.4-10.0)/(2-0)=1.2\), consistent with the adjacent-group differences.
Therefore, \(\hat{\beta}_{W}=1.2/0.8=1.50\) outcome units per exposure unit. With the assumptions in force, the estimate represents a 1.50-unit increase in the outcome per one-unit increase in the exposure. If the dosage-group associations were not linear, one slope might not adequately summarize the variant association; the linear model would need to be reconsidered rather than silently assumed.
Worked Example: Negative Associations and Allele Recoding
Consider a third hypothetical study with \(G=0,1,2\). The exposure is weekly intake in servings, and the outcome is a concentration measured in mg/L. Suppose the exposure means are \(0.0\), \(0.4\), and \(0.8\) servings, while the outcome means are \(100.0\), \(98.8\), and \(97.6\) mg/L. These means imply \(\hat{\gamma}=0.4\) servings per allele and \(\hat{\delta}=-1.2\) mg/L per allele.
The Wald estimate is \(\hat{\beta}_{W}=-1.2/0.4=-3.00\) mg/L per serving. Under the model, one additional serving of the exposure corresponds to a 3.00 mg/L decrease in the outcome. The negative sign follows from the outcome association being negative while the exposure association is positive.
Now reverse the allele coding by setting \(G^*=2-G\). A one-unit increase in \(G^*\) is a one-unit decrease in \(G\), so both estimated associations change sign: \(\hat{\gamma}^*=-0.4\) and \(\hat{\delta}^*=1.2\). Their ratio is still \(1.2/(-0.4)=-3.00\) mg/L per serving. Changing the allele orientation reverses both associations, not their ratio.
Interpreting the Ratio and Its Limits
The Wald ratio is a ratio of two variant-level associations. Its units are outcome units divided by exposure units because the variant units cancel:
It is not the ratio \(Y/X\) for an individual, nor is it the outcome difference divided by the exposure difference for an arbitrary pair of people. It is a ratio of slopes or corresponding mean differences associated with the same genetic variant. Its interpretation is causal only when the model and instrumental-variable assumptions support that interpretation.
The denominator matters. If \(\gamma=0\), then the population ratio is undefined, and the variant provides no change in exposure under this model. If \(\gamma\) is close to zero, the ratio may be highly sensitive to small changes in either estimated association. Relevance is therefore not just a technical detail: a nonzero denominator is required for the algebra, and a meaningful exposure association is needed for the instrument to be informative. This tutorial’s ratio derivation does not, by itself, provide a procedure for evaluating instrument strength or quantifying uncertainty.
The exclusion condition also has a direct implication for the derivation. If a variant changes the outcome through an additional pathway, then the outcome association may contain a component beyond \(\beta\gamma\). Dividing that total association by \(\gamma\) need not recover the causal effect of the exposure. Likewise, if the causal effect is not constant, a single ratio need not correspond to one common effect for every individual. The ratio remains a calculable number, but its target then requires a more careful model.
Common Mistake
Wrong reasoning: “The variant is associated with both the exposure and outcome, so their ratio must be the causal effect.”
Correction: Association alone establishes only that \(\gamma\) and \(\delta\) can be calculated. The identity \(\delta=\beta\gamma\), which justifies the ratio, follows here because the outcome depends linearly on the exposure with constant effect, the variant has no direct effect on the outcome, and the variant is unassociated with the outcome error. If these conditions fail, the ratio need not estimate the causal effect. State the model and assumptions before giving a causal interpretation.
Check Your Understanding
Answer each question by referring to the model and assumptions in this tutorial.
- In the model, what do \(\gamma\) and \(\delta\) measure, and why must they use the same variant coding?
- Suppose a variant changes mean exposure by \(0.25\) units per allele and mean outcome by \(0.75\) units per allele. Calculate the Wald ratio and state its units if the outcome is measured in points.
- Which step in the derivation uses the condition that the variant has no association with the outcome error?
- If the effect of an additional allele on the exposure is zero, can the Wald ratio identify the causal effect? Explain.
- Why does reversing the coded allele change the signs of both associations but leave their ratio unchanged?