Deriving the Product Estimator and Its Uncertainty

Why the Product Estimate Needs Its Own Uncertainty Calculation

In a linear mediation analysis, the indirect effect is often estimated by multiplying two component effects. In genetic mediation, these may be estimated in two steps: one estimate describes the effect of an exposure on a mediator, and another describes the effect of the mediator on an outcome. Multiplying them gives an estimate on the outcome-per-exposure scale. Its uncertainty is not found by simply multiplying the two standard errors: both component estimates are uncertain, and they may also be correlated.

This tutorial focuses on the product estimator and on two ways to quantify its sampling uncertainty: the delta method and bootstrap intervals. These calculations address uncertainty in the estimated product. They do not establish that the product has a causal interpretation; that interpretation still depends on the mediation identification assumptions, valid genetic instruments for the relevant steps, and a compatible effect scale. The simple product also presumes a linear additive mediation model without an exposure–mediator interaction.

Definition and Working Model

Write the estimated component effects as \(\hat{a}=a+\varepsilon_a\) and \(\hat{b}=b+\varepsilon_b\), where the estimation errors have mean zero and variances \(V_a\) and \(V_b\). Their covariance is \(C_{ab}=\operatorname{Cov}(\hat{a},\hat{b})\). The usual delta-method calculation relies on component estimates having sufficiently well-behaved sampling distributions for a first-order approximation to be informative, and on estimates of their variances and covariance being appropriate.

When the estimates come from independent, non-overlapping samples, their covariance may be negligible. That is not automatic: shared participants, shared data, or a joint estimation procedure can induce dependence. The general variance formula includes this covariance rather than assuming it away.

Deriving the Delta-Method Standard Error

The delta method approximates the sampling variation of a smooth function of estimated quantities by linearizing that function near the true values. Here the function is \(g(a,b)=ab\). Its first-order Taylor expansion around \((a,b)\) is

The gradient is \((b,a)\). Taking the variance of the linear part gives

In practice, the unknown component effects and covariance are replaced by their estimates. Since \(V_a=SE(\hat{a})^2\), \(V_b=SE(\hat{b})^2\), and \(C_{ab}\) is estimated from the joint sampling information, the estimated delta-method standard error is

If the component estimates are independent, \(\widehat{C}_{ab}=0\), so the covariance term drops out. This is a first-order approximation, not an exact expression for the variance of a product. In particular, the Taylor expansion omits the second-order term \((\hat{a}-a)(\hat{b}-b)\); its influence may matter when component estimates are imprecise or their sampling distributions depart substantially from the approximation.

Bootstrap Intervals for the Product

A bootstrap forms repeated estimates from resampled data and uses the resulting distribution of the target statistic—in this case, the product—to describe uncertainty. With individual-level data, a bootstrap replicate should resample the relevant observational units and re-estimate both component effects in that replicate. Resampling the components as a pair retains dependence between them. For summary-data analyses without individual-level records, a parametric bootstrap can instead draw paired component estimates from an estimated joint sampling distribution, using the estimated covariance.

A common interval is the percentile interval: for a nominal 95% interval, take the 2.5th and 97.5th percentiles of the bootstrap product estimates. Its endpoints need not be symmetric around \(\hat{a}\hat{b}\). This can be useful because products may have asymmetric sampling distributions, especially when a component is close to zero or imprecisely estimated. However, bootstrap intervals depend on the resampling scheme and on whether it represents the actual sampling process. They are not automatically better calibrated than delta-method intervals.

Why the Covariance Term Matters

The covariance term can either increase or decrease the estimated uncertainty, depending on the signs of the component estimates and their covariance. The following example shows why setting it to zero without justification can materially change the result.

Interpreting the Two Approaches

The delta method is quick, transparent, and requires component standard errors plus their covariance. A bootstrap can show the empirical shape of the product estimates and produce an interval that is not forced to be symmetric. The choice between them should reflect the design, the availability of data, and how reliable their approximations are—not a rule that one method is always preferable.

In genetic mediation, an imprecise component may arise, for example, when the genetic evidence for one step is weak. A bootstrap can represent the product’s skewness under a suitable resampling procedure, but it does not repair weak identification, invalid instruments, or a misspecified mediation model. Nor does an interval that excludes zero establish a causal pathway unless the assumptions supporting the indirect-effect interpretation are credible. Interpret the interval in the stated units and in the context of those assumptions.

Common Mistakes

• Multiplying standard errors. The standard error of a product is not \(SE(\hat{a})SE(\hat{b})\). Use the delta-method variance, including covariance when relevant, or obtain uncertainty from a suitable bootstrap.
• Assuming the estimates are independent. Shared samples or joint estimation can induce dependence. Omitting a nonzero covariance can either understate or overstate uncertainty.
• Treating a bootstrap as a cure for weak evidence. Resampling quantifies uncertainty under the sampling scheme; it does not create information or solve identification problems.
• Calling an interval a probability statement about the fixed effect. A 95% confidence procedure has 95% coverage in repeated sampling under its assumptions; it does not mean there is a 95% probability that this particular interval contains the fixed indirect effect.