Scale Dependence in Mediation Decompositions

Why the Effect Scale Matters

A mediation analysis separates a treatment’s effect into a component associated with changing treatment while holding the mediator’s counterfactual source fixed, and a component associated with changing that source while holding treatment fixed. The numbers assigned to those components depend on the scale used to compare outcomes. A risk difference measures changes in probability points; a risk ratio compares probabilities multiplicatively; and an odds ratio compares odds. These are different summaries of the same counterfactual outcome distribution, not interchangeable labels for one numerical answer.

For a binary outcome, the chosen scale can change the size of each component substantially, especially when the outcome is common. The overall direction of a contrast, however, cannot change among these three scales when the risks are strictly between zero and one: a higher risk also means a risk ratio and an odds ratio greater than one. The scale changes magnitude and decomposition, not the direction relative to each scale’s null value.

Counterfactual Effects and Three Scales

Let treatment \(A\) be binary, let \(M\) be a mediator, and let \(Y\) be a binary outcome. Write \(M(a)\) for the mediator value that would occur under treatment \(a\), and \(Y(a,m)\) for the outcome under treatment \(a\) if the mediator were set to \(m\). The nested counterfactual \(Y(a,M(b))\) combines treatment level \(a\) with the mediator value that would arise under treatment level \(b\). Define its population risk as \(q_{ab}=E[Y(a,M(b))]\), with the first subscript denoting the outcome treatment and the second the mediator’s treatment source. Thus \(q_{00}\) and \(q_{11}\) are the risks under the two aligned treatment regimes, while \(q_{10}\) and \(q_{01}\) are mixed counterfactual risks.

The definitions below use one particular decomposition path: start at \(q_{00}\), change treatment while retaining the mediator source \(0\), then change the mediator source from \(0\) to \(1\) while retaining treatment \(1\). These are population counterfactual quantities. A causal interpretation requires meaningful, well-defined interventions and appropriate conditions for interpreting the counterfactuals. The examples stipulate their counterfactual risks; they do not claim that those risks can be identified from observed data without further assumptions.

Definition: On the additive scale, the natural direct effect along this path is \(q_{10}-q_{00}\), and the natural indirect effect is \(q_{11}-q_{10}\). The total effect is \(q_{11}-q_{00}\). On the risk-ratio scale, the corresponding effects are \(q_{10}/q_{00}\), \(q_{11}/q_{10}\), and \(q_{11}/q_{00}\), when the denominators are positive. For the odds-ratio scale, define \(o(q)=q/(1-q)\) for \(0<q<1\). The corresponding direct, indirect, and total effects are \(o(q_{10})/o(q_{00})\), \(o(q_{11})/o(q_{10})\), and \(o(q_{11})/o(q_{00})\).

The risk-difference components are measured in probability points. Risk ratios and odds ratios are dimensionless; their null value is one, whereas the null value for a risk difference is zero. For example, a risk difference of \(0.10\) is a 10-percentage-point increase, not a 10% increase. A risk ratio of \(1.10\) is a 10% relative increase in risk.

Why Decomposition Holds on Each Scale

On the additive scale, the decomposition follows by inserting the intermediate risk \(q_{10}\). On the ratio scale, the corresponding step is multiplication: the intermediate risk cancels from the product. The same cancellation works for odds ratios when each risk lies strictly between zero and one.

(q_{10}-q_{00})+(q_{11}-q_{10})=q_{11}-q_{00}

\frac{q_{10}}{q_{00}}\times\frac{q_{11}}{q_{10}} =\frac{q_{11}}{q_{00}}, \qquad \frac{o(q_{10})}{o(q_{00})}\times\frac{o(q_{11})}{o(q_{10})} =\frac{o(q_{11})}{o(q_{00})}.

Equivalently, logarithms turn the two multiplicative decompositions into sums: the log risk-ratio components add to the log total risk ratio, and the log odds-ratio components add to the log total odds ratio. These are algebraic identities for the defined counterfactual risks. They do not say that the components have the same substantive meaning or numerical size across scales.

Worked Example: One Counterfactual Table, Three Decompositions

Suppose a hypothetical population has the following nested counterfactual risks: \(q_{00}=0.20\), \(q_{10}=0.32\), \(q_{01}=0.30\), and \(q_{11}=0.50\). Treat these as stipulated risks, not estimates from a real study. First use the path \(q_{00}\to q_{10}\to q_{11}\).

Risk differences. The direct effect is \(0.32-0.20=0.12\), or a 12-percentage-point increase. The indirect effect is \(0.50-0.32=0.18\), or an 18-percentage-point increase. Their sum is \(0.12+0.18=0.30\), the total risk difference \(0.50-0.20=0.30\), or 30 percentage points.

Risk ratios. The direct risk ratio is \(0.32/0.20=1.60\). The indirect risk ratio is \(0.50/0.32=1.5625\), and the total risk ratio is \(0.50/0.20=2.50\). The displayed component ratios multiply exactly: \(1.60\times1.5625=2.50\). In context, the total risk is 2.50 times the baseline counterfactual risk.

Odds ratios. The relevant odds are \(o(0.20)=0.20/0.80=0.25\), \(o(0.32)=0.32/0.68=0.470588\), and \(o(0.50)=1.00\). Thus the direct odds ratio is \(0.470588/0.25=1.88235\); the indirect odds ratio is \(1.00/0.470588=2.125\); and the total odds ratio is \(1.00/0.25=4.00\). Their product is \(1.88235\times2.125\approx4.00\), with the unrounded values giving exactly 4.00. The total odds are four times the baseline odds; this is not a statement that the risk is four times as large.

The values \(q_{01}\) and \(q_{10}\) also show that the decomposition depends on the chosen path. The alternative path \(q_{00}\to q_{01}\to q_{11}\) gives additive components \(0.30-0.20=0.10\) and \(0.50-0.30=0.20\). On the risk-ratio scale it gives \(0.30/0.20=1.50\) and \(0.50/0.30=1.6667\); on the odds scale it gives \((0.30/0.70)/(0.20/0.80)=1.7143\) and \((0.50/0.50)/(0.30/0.70)=2.3333\). Each pair still combines to the same total on its scale, but the component sizes differ from those on the first path.

Baseline Risk Changes the Meaning of an Odds Ratio

A risk ratio has a direct relative-risk interpretation, but the same risk ratio can correspond to very different odds ratios at different baseline risks. Odds become increasingly larger than risks as risks rise toward one. Consequently, odds-ratio mediation components can change markedly with the outcome distribution, even when the relative-risk components stay fixed.

Worked Example: The Same Risk-Ratio Components at Different Baseline Risks

Consider two hypothetical sequences of nested counterfactual risks. In the first, the risks are \(0.05\), \(0.075\), and \(0.1125\) for \(q_{00}\), \(q_{10}\), and \(q_{11}\), respectively. The two risk ratios are \(0.075/0.05=1.50\) and \(0.1125/0.075=1.50\), so the total risk ratio is \(0.1125/0.05=2.25\). The additive components are \(0.025\) and \(0.0375\), summing to a total risk difference of \(0.0625\).

For this first sequence, the odds at the three risks are \(1/19\), \(3/37\), and \(9/71\). The direct odds ratio is \((3/37)/(1/19)=57/37=1.54054\). The indirect odds ratio is \((9/71)/(3/37)=111/71=1.56338\). The total odds ratio is \((9/71)/(1/19)=171/71=2.40845\). The unrounded component odds ratios multiply to the total odds ratio.

Now consider risks \(0.40\), \(0.60\), and \(0.90\). Again, each step has risk ratio \(1.50\), and the total risk ratio is \(2.25\). But the additive components are now \(0.20\) and \(0.30\), with total \(0.50\). The odds are \(2/3\), \(3/2\), and \(9\), so the direct odds ratio is \((3/2)/(2/3)=2.25\), the indirect odds ratio is \(9/(3/2)=6.00\), and the total odds ratio is \(9/(2/3)=13.5\). The component odds ratios multiply to \(2.25\times6.00=13.5\).

Both sequences have identical risk-ratio components, yet their odds-ratio components differ substantially. This is a consequence of their different baseline risks and intermediate outcome distributions, not a contradiction. When the outcome is rare, odds and risks are close, so odds ratios may approximate risk ratios. As the outcome becomes common, that approximation can be poor.

Marginal Effects and Conditional Odds Ratios

A mediation effect is often summarized using population-average counterfactual risks. It is therefore important not to substitute a conditional odds ratio, such as a coefficient from a logistic regression, for a marginal odds ratio without justification. Odds ratios are non-collapsible: averaging risks across strata and then taking their odds does not generally preserve a common stratum-specific odds ratio. This distinction is separate from confounding; it can occur simply because odds are a nonlinear transformation of risk.

Worked Example: A Common Conditional Odds Ratio Need Not Be the Marginal Odds Ratio

Suppose two equally common baseline strata have counterfactual risks \(0.10\) and \(0.50\) in one nested state, denoted \(q_{00}\) within each stratum. In the other state, suppose the conditional odds ratio is 2 in both strata. For the lower-risk stratum, the initial odds are \(0.10/0.90=1/9\). Doubling the odds gives \(2/9\), corresponding to risk \((2/9)/(1+2/9)=2/11\). For the higher-risk stratum, the initial odds are \(0.50/0.50=1\); doubling them gives odds 2 and risk \(2/3\). Thus the conditional odds ratio is 2 in each stratum.

The marginal baseline risk is \((0.10+0.50)/2=0.30\), with odds \(0.30/0.70=3/7\). The marginal risk in the other state is \((2/11+2/3)/2=(6/33+22/33)/2=14/33=0.424242\ldots\). Its odds are \(14/19\). Therefore the marginal odds ratio is \((14/19)/(3/7)=98/57=1.7193\), not 2. The difference arises from marginalizing over strata before computing the odds ratio. For a marginal mediation estimand, use the marginal counterfactual risks specified by the estimand rather than interpreting a conditional logistic coefficient as though it were that marginal effect.

Choosing and Reporting a Decomposition

The scale should follow the scientific question and be stated explicitly. A risk difference communicates absolute changes in outcome probability. A risk ratio communicates relative changes in risk. An odds ratio communicates relative changes in odds, which can diverge from relative changes in risk when the outcome is common. In all cases, report the counterfactual contrasts and their units or interpretation, rather than presenting unlabeled component numbers.

The path also matters. The first path defined the direct effect at mediator source \(0\) and the indirect effect at treatment \(1\). The alternative path defined the indirect effect at treatment \(0\) and the direct effect at mediator source \(1\). Each path telescopes to the same total on a chosen scale, but its component values need not match the other path. This is especially relevant when treatment and mediator contributions interact on that scale. A decomposition is not a unique partition of the total effect unless the scale and reference path are specified.

Key takeaway: Additive components sum; risk-ratio and odds-ratio components multiply. The chosen scale and decomposition path determine component magnitudes. For binary outcomes, odds-ratio components also depend strongly on baseline risk, and marginal odds ratios need not equal conditional odds ratios.

Common Mistakes

Comparing magnitudes without naming the scale. A difference of \(0.10\), a risk ratio of \(1.10\), and an odds ratio of \(1.10\) are different quantities. State whether each component is a risk difference, risk ratio, or odds ratio.
Claiming that scale changes can reverse the direction. For risks strictly between zero and one, \(q_b>q_a\) implies \(q_b-q_a>0\), \(q_b/q_a>1\), and \(o(q_b)/o(q_a)>1\). The numerical magnitudes differ; the direction relative to the corresponding null does not.
Calling an odds ratio a risk ratio. An odds ratio of 2 does not generally mean the risk doubled. When the baseline risk is \(0.40\), for example, doubling the odds gives risk \(4/7\approx0.571\), not \(0.80\).
Assuming there is one path-independent pair of components. Both paths can sum or multiply to the same total while assigning different amounts to the direct and indirect components. Report which path defines the effects.
Treating an algebraic decomposition as an identification result. Telescoping identities hold for specified counterfactual risks. They do not establish that those risks are identified from observed data.

Check Your Understanding

Use the definitions and examples above to answer the following questions.

If \(q_{00}=0.10\), \(q_{10}=0.20\), and \(q_{11}=0.40\), calculate the additive direct, indirect, and total effects along the path \(q_{00}\to q_{10}\to q_{11}\). What are the corresponding risk ratios?
Why do risk-ratio components multiply to the total risk ratio, while risk-difference components add to the total risk difference?
Two counterfactual risk sequences have the same risk-ratio components but different baseline risks. Why can their odds-ratio components differ?
Can a positive risk difference correspond to a risk ratio below one when both risks are positive? Explain.
In your own words, why might a conditional odds ratio from a regression differ from a marginal odds ratio used to summarize population counterfactual risks?