The Summary-Data Wald Ratio

From Two Association Estimates to One Causal-Effect Estimate

A genetic variant can be associated with an exposure and with an outcome, with each association estimated in a different analysis. The summary-data Wald ratio uses those two estimates to form a single-variant Mendelian randomization estimate. It is useful when the available results are association estimates and standard errors rather than individual-level data.

The ratio is simple to calculate, but its interpretation depends on a causal model, and its uncertainty depends on both the exposure and outcome estimates. Treating the exposure estimate as exact can understate uncertainty, especially when it is imprecise. The calculation also becomes unstable when the estimated exposure association is close to zero.

Definition: Let $\hat{\gamma}$ be the estimated association of a genetic variant with an exposure, and let $\hat{\Gamma}$ be its estimated association with an outcome. The single-variant summary-data Wald ratio is $$ \hat{\beta}_{W}=\frac{\hat{\Gamma}}{\hat{\gamma}}. $$ Here, $\hat{\beta}_{W}$ estimates the causal effect of the exposure on the outcome under the instrumental-variable assumptions stated below. Its units are outcome units per exposure unit.

The association estimates must refer to the same variant and to a consistently oriented allele: a positive change in the allele count must refer to the same allele in both estimates. Their scales must also be clear. For example, if $\hat{\gamma}$ measures exposure standard deviations per allele and $\hat{\Gamma}$ measures outcome units per allele, the ratio is in outcome units per exposure standard deviation.

Model and Conditions for a Causal Interpretation

The ratio is a causal estimate only under substantive assumptions. In a simple linear instrumental-variable model, the variant $G$ affects the exposure $X$, and the exposure affects the outcome $Y$. For a one-unit increase in $X$, suppose the causal effect on $Y$ is $\beta$. If the variant has no effect on $Y$ except through $X$, then its population associations satisfy $\Gamma=\beta\gamma$. Dividing gives $\beta=\Gamma/\gamma$, provided $\gamma\ne 0$.

Model conditions: The Wald ratio has the intended causal interpretation when the variant is associated with the exposure; is not associated with causes of the exposure and outcome that would confound the variant–outcome relation; and affects the outcome only through the exposure. The exposure and outcome association estimates must refer to compatible populations, variant coding, and effect scales. For the ratio to represent a single effect, the exposure–outcome effect must also be adequately described by the stated linear, constant-effect model. These conditions are assumptions, not facts established by calculating the ratio.

The relevance condition is $\gamma\ne0$ in the population, but merely obtaining a nonzero estimate $\hat{\gamma}$ does not show that the ratio is precise. The exclusion condition rules out a direct variant effect on the outcome that bypasses the exposure. If that condition fails, the outcome association may include a pathway other than the exposure pathway, and the ratio need not estimate the exposure’s causal effect.

For the standard first-order uncertainty calculation, the association estimates are treated as approximately jointly normal around their population values, and the denominator is sufficiently far from zero for a local linear approximation to be useful. If the estimates come from independent samples, their estimation errors can often be treated as independent. If they share participants, their covariance may need to be included.

First-Order Standard Error

The ratio is a function of two estimates, so uncertainty in either can change the result. The delta method approximates the ratio near the estimated values by a linear function. For $f(\Gamma,\gamma)=\Gamma/\gamma$, its partial derivatives are $1/\gamma$ with respect to $\Gamma$, and $-\Gamma/\gamma^2$ with respect to $\gamma$. Applying the variance formula for a linear combination gives the first-order variance estimate below.

\widehat{\operatorname{Var}}(\hat{\beta}_{W}) \approx \frac{SE_{\Gamma}^{\,2}}{\hat{\gamma}^{\,2}} + \frac{\hat{\Gamma}^{\,2}SE_{\gamma}^{\,2}}{\hat{\gamma}^{\,4}} - \frac{2\hat{\Gamma}}{\hat{\gamma}^{\,3}} \widehat{\operatorname{Cov}}(\hat{\Gamma},\hat{\gamma}).

Here, $SE_{\Gamma}$ and $SE_{\gamma}$ are the standard errors of the outcome and exposure association estimates. If those estimates are independent, the covariance is zero, giving the commonly used independent-estimates formula:

SE(\hat{\beta}_{W}) \approx \sqrt{ \frac{SE_{\Gamma}^{\,2}}{\hat{\gamma}^{\,2}} + \frac{\hat{\Gamma}^{\,2}SE_{\gamma}^{\,2}}{\hat{\gamma}^{\,4}} }.

The first variance term reflects uncertainty in the outcome association. The second reflects uncertainty in the exposure association. The minus sign on the covariance term follows from the negative derivative with respect to the denominator; the term’s numerical contribution can therefore be positive or negative, depending on the signs of the estimates and covariance. The total variance remains nonnegative when the covariance and variances are valid.

First-order Wald interval: Under the normal approximation, an approximate $100(1-\alpha)\%$ confidence interval is $\hat{\beta}_{W}\pm z_{1-\alpha/2}SE(\hat{\beta}_{W})$, where $z_{1-\alpha/2}$ is the corresponding standard normal quantile. This interval is an approximation; it can be poor when the denominator is imprecise or close to zero.

Worked Calculations

Worked Example: Ratio and Uncertainty From Independent Estimates

Suppose, hypothetically, a variant’s exposure association is estimated as $\hat{\gamma}=0.20$ exposure standard deviations per allele, with $SE_{\gamma}=0.04$. Its outcome association is $\hat{\Gamma}=0.06$ outcome units per allele, with $SE_{\Gamma}=0.015$. Assume the estimates are independent, the allele coding is consistent, and the causal model conditions are appropriate.

The ratio estimate is $$ \hat{\beta}_{W}=\frac{0.06}{0.20}=0.30 \quad\text{outcome units per exposure standard deviation}. $$

The first variance contribution is $0.015^2/0.20^2=0.000225/0.04=0.005625$. The second is $0.06^2(0.04^2)/0.20^4=0.0036(0.0016)/0.0016=0.003600$. Thus, $$ SE(\hat{\beta}_{W})\approx\sqrt{0.005625+0.003600} =\sqrt{0.009225}=0.0960. $$

Using $1.96$ for an approximate 95% interval gives $0.30\pm1.96(0.0960)=0.30\pm0.188$, or approximately $[0.112,0.488]$ outcome units per exposure standard deviation. Under the stated assumptions and approximation, the interval describes values compatible with the data and model at this confidence level; it is not a probability statement about the fixed causal effect.

Worked Example: Why the Denominator’s Uncertainty Matters

In another hypothetical analysis, let $\hat{\gamma}=0.10$ exposure units per allele with $SE_{\gamma}=0.02$, and $\hat{\Gamma}=0.04$ outcome units per allele with $SE_{\Gamma}=0.01$. Suppose the association estimates are independent and the causal assumptions hold.

The ratio is $0.04/0.10=0.40$ outcome units per exposure unit. The outcome-association contribution to variance is $0.01^2/0.10^2=0.01$. The exposure-association contribution is $0.04^2(0.02^2)/0.10^4=0.0016(0.0004)/0.0001=0.0064$. Therefore, $$ SE(\hat{\beta}_{W})\approx\sqrt{0.01+0.0064} =\sqrt{0.0164}=0.128. $$

If the exposure association were incorrectly treated as known without error, the calculated standard error would be only $\sqrt{0.01}=0.100$. The first-order standard error that includes both estimates is larger, $0.128$, and the corresponding approximate 95% interval is $0.40\pm1.96(0.128)=[0.149,0.651]$ outcome units per exposure unit. Omitting denominator uncertainty would make this interval too narrow under the stated model.

Worked Example: Including Covariance

Suppose hypothetical estimates are $\hat{\gamma}=0.25$ exposure units per allele, $SE_{\gamma}=0.05$, $\hat{\Gamma}=0.10$ outcome units per allele, and $SE_{\Gamma}=0.02$. Assume they have correlation $0.40$, so their covariance is $0.40(0.05)(0.02)=0.0004$. Such dependence can arise when estimates share participants. Assume consistent coding and the causal model conditions.

The ratio is $0.10/0.25=0.40$ outcome units per exposure unit. The two variance contributions are $0.02^2/0.25^2=0.0004/0.0625=0.0064$ and $0.10^2(0.05^2)/0.25^4=0.01(0.0025)/0.00390625=0.0064$. The covariance contribution is $$ -\frac{2(0.10)}{0.25^3}(0.0004) =-\frac{0.00008}{0.015625} =-0.00512. $$

Consequently, the estimated variance is $0.0064+0.0064-0.00512=0.00768$, and the standard error is $\sqrt{0.00768}=0.0876$ outcome units per exposure unit. For comparison, ignoring the covariance would give $\sqrt{0.0128}=0.113$. The difference shows why independence should not be assumed when overlapping data induce dependence.

Derivation of the Approximation

For a small estimation error around $(\Gamma,\gamma)$, a first-order Taylor expansion of $f(\hat{\Gamma},\hat{\gamma})=\hat{\Gamma}/\hat{\gamma}$ gives

\frac{\hat{\Gamma}}{\hat{\gamma}} \approx \frac{\Gamma}{\gamma} + \frac{1}{\gamma}(\hat{\Gamma}-\Gamma) - \frac{\Gamma}{\gamma^2}(\hat{\gamma}-\gamma).

For random quantities $A$ and $B$, the variance of $aA+bB$ is $a^2\operatorname{Var}(A)+b^2\operatorname{Var}(B)+2ab\operatorname{Cov}(A,B)$. Applying this identity to the two error terms above gives

\operatorname{Var}(\hat{\beta}_{W}) \approx \frac{\operatorname{Var}(\hat{\Gamma})}{\gamma^2} + \frac{\Gamma^2\operatorname{Var}(\hat{\gamma})}{\gamma^4} - \frac{2\Gamma}{\gamma^3}\operatorname{Cov}(\hat{\Gamma},\hat{\gamma}).

Replacing the unknown quantities and variances with their estimates yields the first-order formula. This derivation also explains its limitation: the approximation uses local slopes of the ratio function, which change sharply near $\gamma=0$. It can therefore misrepresent the ratio’s sampling distribution when the denominator is close to zero.

Interpretation and Common Pitfall

Report the ratio with its units, standard error or interval, and the assumptions that support its causal interpretation. A positive ratio indicates a positive estimated causal effect under the model; it does not establish that the genetic variant acts only through the exposure. Association estimates and a precise ratio do not, by themselves, verify the instrumental-variable assumptions.

Pitfall: treating the denominator as fixed. The reasoning “the outcome association has a standard error, so that is all the ratio needs” ignores variation in the estimated exposure association. The correction is to include the denominator term in the delta-method variance unless a justified approximation permits neglecting it.
Pitfall: assuming separate estimates are independent. Different association analyses can still share participants. The correction is to use the covariance term when it is nonzero and available, rather than automatically setting it to zero.
Pitfall: interpreting any computed ratio as causal. Division does not establish relevance, independence from confounding, or the exclusion restriction. The correction is to state the causal assumptions and interpret the ratio conditionally on them.
Pitfall: relying on a narrow-looking Wald interval near a weak denominator. A first-order normal interval can be misleading when the denominator is near zero. The correction is to recognize that the approximation may fail, rather than treating the interval as dependable solely because it is easy to calculate.

Key takeaway: The summary-data Wald ratio is the outcome association divided by the exposure association. Its first-order uncertainty generally includes both association standard errors and, when relevant, their covariance. The estimate’s causal meaning depends on the instrumental-variable assumptions, not on the arithmetic alone.

Check Your Understanding

Answer each question using the ratio definition, its uncertainty formula, and the stated model conditions.

A variant has an outcome association of 0.08 outcome units per allele and an exposure association of 0.20 exposure units per allele. What is the Wald ratio and its unit?
Under independence, which two sources of uncertainty appear in the first-order variance of the ratio?
Why can setting the covariance to zero be inappropriate when exposure and outcome association estimates share participants?
State the exclusion condition required for the ratio to represent the exposure’s causal effect on the outcome.
Why can a first-order Wald confidence interval be unreliable when the estimated exposure association is close to zero?

What You'll Learn