Scale Dependence of Interventional Mediation

Why the Effect Scale Matters

An interventional mediation analysis compares outcome distributions under specified exposure levels and interventions on the mediator distribution. But a contrast between two outcome risks is not expressed in only one way. It may be a difference in risks, a ratio of risks, or a ratio of odds. These scales answer related but distinct questions, and the numerical size of a direct or indirect contrast depends on which one is used.

This matters when describing how an exposure effect is divided into interventional direct and indirect components. On the risk-difference scale, the components can add to the total risk difference. On ratio scales, the corresponding components multiply to the total ratio, or add after taking logarithms. A mediated proportion therefore depends on its scale and on the particular component contrasts used.

Thus \(\mu(0,0)\) and \(\mu(1,1)\) are the risks under the corresponding exposure and mediator-distribution settings, while \(\mu(1,0)\) combines exposure 1 with the mediator distribution under exposure 0. The risks are marginal over the target population's baseline covariate distribution. In practice, they may be identified and estimated using the mediational g-formula under the relevant consistency, exchangeability, and positivity conditions. The examples below take the risks as already identified; they do not claim that a numerical contrast by itself establishes those conditions.

Define the Contrasts Before Choosing a Scale

One useful decomposition follows a path from \(\mu(0,0)\) to \(\mu(1,0)\), then to \(\mu(1,1)\). The first step changes the exposure while holding the mediator distribution at its level under exposure 0. The second changes the mediator distribution while holding the exposure at 1. These contrasts are an interventional direct effect and an interventional indirect effect for this particular choice of settings.

Those expressions use risk differences. On a ratio scale, replace differences by ratios. For example, the total risk ratio is \(\mu(1,1)/\mu(0,0)\), the direct risk ratio along this path is \(\mu(1,0)/\mu(0,0)\), and the indirect risk ratio is \(\mu(1,1)/\mu(1,0)\). For odds ratios, first convert each risk \(\mu\) to its odds \(\mu/(1-\mu)\), then form the corresponding ratios.

Why the Decomposition Works on Each Scale

The additive decomposition follows by adding and subtracting the intermediate risk \(\mu(1,0)\). The same telescoping argument works after applying a function \(g\) to the risks. This gives additive components on the transformed scale. Choosing \(g(\mu)=\mu\) yields risk differences; choosing \(g(\mu)=\log(\mu)\) yields log risk ratios; choosing \(g(\mu)=\operatorname{logit}(\mu)=\log\{\mu/(1-\mu)\}\) yields log odds ratios.

This identity is an algebraic justification, not an additional causal assumption. For \(g(\mu)=\log(\mu)\), for example, the first component is \(\log\{\mu(1,0)/\mu(0,0)\}\), and the second is \(\log\{\mu(1,1)/\mu(1,0)\}\). Exponentiating both components gives risk ratios whose product is the total risk ratio. The same reasoning with the logit gives odds ratios whose product is the total odds ratio.

Mediated Proportions Depend on Scale

A mediated proportion is often formed by dividing an indirect contrast by a total contrast, with both contrasts defined on the same scale and along a stated decomposition. Using the first worked example, the risk-difference mediated proportion is \(0.08/0.14\approx0.571\), or 57.1%. On the risk-ratio scale, the corresponding log-scale proportion is \(\log(1.50)/\log(2.40)\approx0.463\), or 46.3%. The log-odds-scale calculation in the second example is about 48.4%, using its intermediate risk.

These proportions differ because the transformations from risks to log risks and log odds are nonlinear. Dividing the indirect risk difference by the total risk difference is not generally equivalent to dividing the log indirect ratio by the log total ratio. The scale-specific results are each mathematically coherent, but they should not be presented as though they estimate one uniquely defined fraction independent of scale.

Path Choice and Interpretation

The path through the interventional risks is another choice that must be reported. An alternative path goes from \(\mu(0,0)\) to \(\mu(0,1)\), then to \(\mu(1,1)\). It changes the mediator distribution first and the exposure second. The total contrast remains the comparison between \(\mu(1,1)\) and \(\mu(0,0)\), but the component contrasts may differ.

For example, suppose the hypothetical risks are \(\mu(0,0)=0.10\), \(\mu(1,0)=0.16\), \(\mu(0,1)=0.20\), and \(\mu(1,1)=0.24\). Along the first path, the risk-difference direct and indirect contrasts are \(0.16-0.10=0.06\) and \(0.24-0.16=0.08\). Along the alternative path, they are \(0.24-0.20=0.04\) for the direct contrast and \(0.20-0.10=0.10\) for the indirect contrast. Both paths sum to the same total risk difference, \(0.14\), but allocate it differently between components.

This path dependence can arise when exposure and mediator-distribution changes do not have constant effects across the other setting. A report should therefore name the mediator reference distribution and exposure setting used for each component, not just label a number “the indirect effect.” Comparisons across studies also require attention to whether the outcome scale, target population, and decomposition path match.

Common Mistake

• Wrong reasoning: “The mediated proportion is 57%, so 57% of the causal effect is mediated, regardless of how the effect is measured.”
• Correction: The 57% figure in the example is a ratio of risk differences along a specified path. A log-risk or log-odds calculation can give another value. State the scale, the exact numerator and denominator, and the interventional settings defining the path.
• Wrong reasoning: “The odds ratio is larger than the risk ratio, so the odds-scale indirect effect must be stronger.”
• Correction: Odds ratios and risk ratios use different denominators and cannot be ranked as though they were measurements on one common scale. Report the chosen measure with its interpretation and, when useful, provide standardized risks as well.