The Wald Ratio For A Single Variant

Why Divide Two Genetic Associations?

A genetic variant can provide an instrumental-variable estimate of the causal effect of an exposure on an outcome. With one variant, the basic calculation is a ratio: the variant’s association with the outcome divided by its association with the exposure. The numerator describes how much the outcome changes per allele; the denominator describes how much the exposure changes per allele. Dividing cancels the allele scale and expresses the result as an outcome change per unit of exposure.

The arithmetic is simple, but its interpretation is conditional. The variant must be relevant to the exposure, independent of causes of the outcome, and affect the outcome only through the exposure. The estimated associations must also refer to the same effect allele and compatible populations and scales. The earlier material on Harmonizing Exposure And Outcome Alleles covers aligning effect alleles before comparing association estimates.

Definition: Let $\beta_{XG}$ denote the association of a genetic variant $G$ with an exposure $X$, and let $\beta_{YG}$ denote its association with an outcome $Y$. Both coefficients are defined per copy of the same effect allele, with compatible adjustment and association models. If $\beta_{XG}\ne 0$, the single-variant Wald ratio estimate is $$ \hat{\theta}_{W}=\frac{\hat{\beta}_{YG}}{\hat{\beta}_{XG}}. $$ Here $\hat{\beta}_{XG}$ and $\hat{\beta}_{YG}$ are estimated variant-association coefficients, and $\hat{\theta}_{W}$ estimates the causal effect of $X$ on $Y$ under the instrumental-variable assumptions.

For example, if $X$ is measured in kilograms per square metre and $Y$ in millimetres of mercury, then $\hat{\beta}_{XG}$ has units of kilograms per square metre per allele and $\hat{\beta}_{YG}$ has units of millimetres of mercury per allele. Their ratio has units of millimetres of mercury per kilogram per square metre. The allele unit cancels; the exposure unit does not.

Assumptions and Derivation

The ratio is a causal estimate only when the variant functions as a valid instrument for the exposure. In this setting, the key assumptions are the three instrumental-variable assumptions:

Relevance: the variant is associated with the exposure in the target population, so $\beta_{XG}\ne 0$. A very small association may make the ratio highly unstable even if it is not exactly zero.
Independence: the variant is independent of causes of the outcome, including relevant confounders of the exposure–outcome relationship. Population structure, relatedness, or selection can threaten this assumption if not adequately addressed.
Exclusion restriction: the variant affects the outcome only through the exposure. A direct genetic effect on $Y$, or a pathway through another trait that affects $Y$, violates this assumption.

A standard derivation uses a linear structural model with a constant causal effect $\theta$. Let $G$ denote allele dosage, such as 0, 1, or 2 copies of the effect allele. Suppose the outcome follows $Y=\alpha+\theta X+\varepsilon$, where $\varepsilon$ collects other outcome determinants. Under the assumptions, $G$ is associated with $Y$ through its effect on $X$, and it is uncorrelated with $\varepsilon$. This gives the following justification.

Take the association with the outcome.
Taking covariance with $G$ in the structural model gives $\operatorname{Cov}(G,Y)=\theta\operatorname{Cov}(G,X)+\operatorname{Cov}(G,\varepsilon)$.

Use instrument validity.
Independence and exclusion imply $\operatorname{Cov}(G,\varepsilon)=0$, so $\operatorname{Cov}(G,Y)=\theta\operatorname{Cov}(G,X)$.

Divide the two variant associations.
When $\operatorname{Cov}(G,X)\ne 0$, division yields $\theta=\operatorname{Cov}(G,Y)/\operatorname{Cov}(G,X)$. In linear association models, the exposure and outcome coefficients per allele share the same genotype scale, so their ratio estimates this causal slope.

This argument requires more than a nonzero denominator. It uses a linear causal relationship with a constant effect and compatible association coefficients. In practice, the coefficients may come from a single sample or from separate exposure and outcome samples. Either way, the effect allele, ancestry and target population, phenotype definitions, and relevant covariate adjustments should be sufficiently compatible for the ratio to have the intended interpretation. In two-sample analyses, estimates from different populations or substantially different measurement scales can make the quotient difficult to interpret.

Worked Example: A Positive Estimate With Physical Units

Suppose a hypothetical study reports an effect-allele association of $0.20$ kilograms per square metre per allele with body mass index and $0.08$ millimetres of mercury per allele with systolic blood pressure. Assume the allele has been harmonized across the two association estimates and, for interpreting the ratio causally, that the instrument assumptions and linear model are appropriate.

The Wald estimate is $$ \hat{\theta}_{W}=\frac{\hat{\beta}_{YG}}{\hat{\beta}_{XG}} =\frac{0.08\ \text{mm Hg per allele}}{0.20\ \text{kg/m}^2\text{ per allele}} =0.40\ \frac{\text{mm Hg}}{\text{kg/m}^2}. $$

Thus, the ratio estimates an increase of $0.40$ mm Hg in systolic blood pressure per $1$ kg/m² higher body mass index, under the stated assumptions. The per-allele unit cancels. This is a causal interpretation only if the assumptions are credible; the arithmetic alone does not establish them.

Reading the Sign and Checking the Allele

The sign of the ratio depends on the signs of both associations. If the variant increases the exposure and increases the outcome, the ratio is positive. If it decreases both, the ratio is also positive. If its associations have opposite signs, the ratio is negative. Therefore, the numerator’s sign alone does not determine whether the estimated causal effect is positive or negative.

Changing which allele is labelled as the effect allele reverses the signs of both association coefficients. If $G$ is recoded to count the other allele, then $\hat{\beta}_{XG}$ and $\hat{\beta}_{YG}$ both change sign, but their ratio does not:

\frac{-\hat{\beta}_{YG}}{-\hat{\beta}_{XG}} = \frac{\hat{\beta}_{YG}}{\hat{\beta}_{XG}}.

That invariance is not a reason to skip harmonization. If one coefficient refers to an allele that increases dosage while the other is inadvertently aligned to the opposite allele, only one sign may be reversed. The resulting ratio then has the wrong direction. The exposure and outcome estimates must refer to the same allele before the quotient is formed.

Worked Example: Two Negative Associations, Positive Ratio

In a hypothetical analysis, the same effect allele is associated with a $0.15$ milligram per decilitre decrease in an exposure and a $0.06$ millimetre of mercury decrease in an outcome. Suppose the exposure is measured in milligrams per decilitre, the outcome in millimetres of mercury, and the required causal assumptions are taken as plausible for illustration.

Here $\hat{\beta}_{XG}=-0.15$ mg/dL per allele and $\hat{\beta}_{YG}=-0.06$ mm Hg per allele. Then $$ \hat{\theta}_{W} =\frac{-0.06\ \text{mm Hg per allele}}{-0.15\ \text{mg/dL per allele}} =0.40\ \frac{\text{mm Hg}}{\text{mg/dL}}. $$

The positive estimate is consistent with both variant associations pointing in the same direction: the allele that lowers the exposure also lowers the outcome. The estimate is $0.40$ mm Hg per 1 mg/dL higher exposure, subject to the instrumental-variable assumptions. It would be incorrect to call the estimated causal effect negative merely because both associations are negative.

Worked Example: Opposite Signs and a Negative Estimate

Consider a separate hypothetical example with an exposure association of $0.25$ units per allele and an outcome association of $-0.10$ outcome units per allele. Assume the exposure and outcome coefficients are harmonized to the same effect allele, and suppose the linear instrumental-variable model is appropriate.

The calculation is $$ \hat{\theta}_{W} =\frac{-0.10\ \text{outcome units per allele}}{0.25\ \text{exposure units per allele}} =-0.40\ \frac{\text{outcome units}}{\text{exposure unit}}. $$

Under the assumptions, the estimate means that a one-unit increase in the exposure corresponds to a $0.40$-unit decrease in the outcome. It is the contrast between the signs—not simply whether either coefficient is negative—that produces the negative ratio.

When the Ratio Is Uninformative

A ratio can be calculated whenever its denominator is nonzero, but not every calculated ratio is useful. If $\hat{\beta}_{XG}$ is close to zero, even a modest change in either association estimate can produce a large change in their quotient. A zero denominator makes the ratio undefined. A weak exposure association also raises the substantive concern that the variant provides little information about how the exposure relates to the outcome. The size and reliability of uncertainty for the ratio require separate assessment; this tutorial concerns the point estimate, not its standard error or confidence interval.

The assumptions also deserve scrutiny independent of the arithmetic. A variant associated with the exposure does not thereby satisfy independence or exclusion. Population structure can induce variant–outcome association, while horizontal pleiotropy can create a pathway to the outcome that does not pass through the exposure. Either can make the ratio differ from the causal effect. The one-variant design cannot use disagreement among several variant-specific estimates as a diagnostic, because it supplies only one ratio.

The ratio is useful because it gives a transparent estimate from one instrument, with an immediate interpretation in outcome units per exposure unit. It is not a substitute for other designs or for checking the instrument assumptions. A randomized trial estimates effects through treatment assignment; a conventional exposure–outcome regression estimates an association that can be confounded; and multi-variant Mendelian randomization combines information from several instruments under its own assumptions. Those approaches answer related questions but are not interchangeable with a single-variant Wald ratio.

Common Mistakes and Practical Checks

Dividing in the wrong order. The Wald ratio is the outcome-variant association divided by the exposure-variant association. Reversing the order gives units of exposure per outcome, not the desired causal effect of exposure on outcome.
Ignoring allele alignment. A sign mismatch can reverse the ratio’s direction. Confirm that both coefficients use the same effect allele before dividing.
Forgetting the units. The per-allele units cancel, but the ratio retains outcome units per exposure unit. State that quantity in context.
Treating an association ratio as proof of causality. The quotient is a causal estimate only under instrument validity and the model conditions. A precise calculation cannot repair a violated exclusion restriction.
Overinterpreting a small denominator. A very weak exposure association can make the ratio unstable. Do not interpret a large quotient as evidence of a large causal effect without assessing instrument strength and uncertainty.

Key takeaway: For a single harmonized variant, divide its outcome association by its exposure association. The result has units of outcome per exposure unit and estimates a causal effect only when the instrumental-variable assumptions and the modeling conditions are defensible.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

A variant is associated with an exposure at $0.30$ units per allele and an outcome at $0.12$ units per allele. What is the Wald estimate, including units?
If both harmonized associations are negative, must the Wald estimate be negative? Explain.
What happens to the ratio if the effect allele is switched consistently in both association estimates?
Why is a nonzero exposure-variant association not sufficient to establish a causal interpretation?

Solutions. (1) The ratio is $0.12/0.30=0.40$ outcome units per exposure unit. (2) No. If both coefficients are negative, their quotient is positive; for example, $(-0.12)/(-0.30)=0.40$. (3) Both coefficients change sign, so the ratio is unchanged: $(-\hat{\beta}_{YG})/(-\hat{\beta}_{XG})=\hat{\beta}_{YG}/\hat{\beta}_{XG}$. (4) Relevance addresses only the variant’s association with the exposure. Independence from outcome causes and the exclusion restriction are also required, along with appropriate modeling and compatible association estimates.