Product and Difference Methods for Mediation

Two Routes to a Linear Indirect Effect

In genetic mediation, the scientific question is often whether an exposure affects an outcome partly through a proposed mediator. For example, genetic variation associated with an exposure such as circulating lipid concentration might affect a cardiovascular outcome partly through a second biomarker. Two familiar coefficient-based approaches summarize that mediated pathway: multiply the exposure-to-mediator and mediator-to-outcome coefficients, or subtract the direct-effect coefficient from the total-effect coefficient.

These calculations are simple, but their interpretation is not automatic. They concern the same indirect effect only when the coefficients belong to a coherent linear model on a common scale. And even when the arithmetic agrees exactly, causal interpretation still depends on the identification assumptions for genetic mediation. The formulas do not establish those assumptions by themselves.

The Linear Mediation Model and Its Identity

Consider the following additive models, with the same set of pre-exposure covariates \(C\) in each equation:

Here \(a\) is the change in the mediator per unit increase in the exposure, and \(b\) is the change in the outcome per unit increase in the mediator while holding the exposure and covariates fixed. The coefficient \(c'\) is the exposure coefficient conditional on the mediator. A corresponding total-effect model regresses \(Y\) on \(X\) and the same covariates, without including \(M\); its exposure coefficient is \(c\).

The path product is \(\hat a\hat b\). Its units are the mediator units per exposure unit, multiplied by outcome units per mediator unit, giving outcome units per exposure unit. The difference \(\hat c-\hat c'\) has the same units because both coefficients measure outcome change per exposure unit. Matching units are necessary, but do not alone guarantee that the two calculations refer to the same estimand.

Why the Two Methods Agree in the Matched Linear Model

The central identity follows by substituting the mediator equation into the outcome equation. For the systematic part of the model, the substitution gives:

Thus the coefficient of \(X\) in the total-effect model is \(c=c'+ab\), so rearranging gives \(c-c'=ab\). This identity describes the linear decomposition: the direct path contributes \(c'\), and the path through \(M\) contributes \(ab\).

For ordinary least-squares regressions, the corresponding coefficient identity also holds exactly in a finite sample when the mediator model, adjusted outcome model, and total-effect model use the same observations, the same covariates and intercept, and the specified linear forms. In that matched setting, the fitted total coefficient equals the fitted direct coefficient plus the product of the fitted path coefficients. Different samples, covariate sets, scales, or estimation procedures can break this exact equality.

Worked Example: Calculating the Product and Checking the Decomposition

Worked Example: Starting From the Difference Method

When the Coefficients Have a Causal Interpretation

In genetic mediation, genetic variation may help identify causal effects of the exposure, mediator, or both, but the product and difference formulas do not perform that identification. Their causal interpretation relies on the relevant identification assumptions being satisfied, as well as on a suitable linear mediation model. In particular, the analyst needs a defensible account of how genetic variation relates to the exposure, mediator, outcome, and any confounders.

For a causal reading of the linear decomposition, the assumptions must support the component effects represented by \(a\), \(b\), and \(c'\). They include consistency and a well-defined exposure contrast; adequate control of confounding of the exposure–mediator and exposure–outcome relations; and adequate control of mediator–outcome confounding. The latter is especially demanding if an exposure-induced variable confounds the mediator–outcome relation. The genetic identification assumptions discussed in genetic mediation are also required where genetic variation is used to support causal claims. Neither estimator can repair a violation of those assumptions.

The model must also be appropriate for the target effect. In particular, the decomposition above assumes a linear outcome relation without an \(X\)-by-\(M\) interaction on the analyzed scale, and compatible adjustment and measurement across the models. If these conditions fail, \(c-c'\) need not represent the same pathway effect as \(ab\). A numerical match is not evidence that the assumptions hold, and a numerical mismatch does not by itself identify which assumption failed.

Common Mistakes and Practical Checks

• Calling the product causal by definition. Multiplying two fitted coefficients gives a statistical estimate. Causal interpretation requires the identification assumptions and model conditions to be credible.
• Subtracting coefficients from incompatible models. The difference \(\hat c-\hat c'\) is meaningful as the same decomposition only when both coefficients refer to compatible exposure contrasts, outcome scales, covariates, and target populations.
• Ignoring units. Check that the mediator units cancel in \(\hat a\hat b\), leaving outcome units per exposure unit, as for \(\hat c-\hat c'\).
• Assuming agreement proves mediation. The algebraic identity can hold for regression coefficients even when the causal assumptions fail. Agreement is a model-consistency check, not a test of causal identification.
• Treating any discrepancy as a biological contradiction. Different samples, covariate adjustments, scales, or estimation methods can produce different estimates. First determine whether the two calculations are comparable.

In practice, report which coefficient definitions and scales were used, whether the same observations and adjustment set were used, and whether the models include an exposure–mediator interaction. State the indirect-effect estimate with its units, and keep its statistical calculation distinct from the justification for a causal interpretation.