Why a “Proportion” May Not Be a Proportion
The proportion mediated is often summarized as the share of a total effect attributed to an indirect effect through a mediator. That description can suggest a number between zero and one. But the quantity is a ratio of causal effect contrasts, not a literal partition of individuals or outcomes. If the direct and indirect effects oppose each other, or if their sum is close to zero, the ratio can be negative, greater than one, or highly unstable. Its value can also change when the outcome scale changes.
To see why, it helps to make the effect decomposition explicit. Consider a binary exposure \(A\), a mediator \(M\), baseline covariates \(C\), and an outcome \(Y\). Let \(g_{a^*}(m\mid c)\) denote the mediator distribution under exposure level \(a^*\), conditional on \(C=c\). Define \(\mu(a,a^*)\) as the mean outcome under exposure level \(a\) when the mediator is drawn from \(g_{a^*}(m\mid c)\), with the result averaged over the baseline covariate distribution.
For this tutorial, use the decomposition that changes the exposure first while holding the mediator distribution at its level under \(A=0\), then changes the mediator distribution while holding the outcome-model exposure at \(A=1\):
Here \(D\) is the interventional direct effect for this decomposition, \(I\) is the interventional indirect effect, and \(T\) is the total effect. All three contrasts must use the same outcome scale. When \(T\ne 0\), the proportion mediated is
This is a ratio of the two specified effect contrasts. It is not automatically a percentage of a meaningful “whole,” and a value outside zero to one is not, by itself, a calculation error. The definition of \(\mu(a,a^*)\) is an interventional one: under the usual identification conditions—including consistency, appropriate control of exposure and mediator-outcome confounding, positivity, and adequate estimation of the required models—these means can be identified from observed data. The numerical examples below stipulate that these conditions are adequate; they illustrate interpretation rather than establish identification from a small data set.
When the Components Point in the Same Direction
The proportion mediated is between zero and one when the direct and indirect effects have the same direction and the total effect is nonzero. This also holds if one component is zero: if \(I=0\), the ratio is zero; if \(D=0\), the ratio is one. For effects on the same scale, the ratio then describes the indirect effect relative to the total contrast. Even in this case, it is a summary of the specified estimands, not a universal measure of how much mediation exists.
The restriction to the same direction matters. When both components are nonzero and point in opposite directions, they partially cancel in \(T=D+I\). Dividing one component by the smaller remainder can produce a negative ratio or a ratio above one. The following algebra makes the possibilities precise.
For the second case, \(D+I<0\) implies \(|I|>D\), so \(I/(D+I)=|I|/(|I|-D)>1\). Reversing both signs gives the same conclusion for \(D<0,\ I>0\). If the effects have the same nonzero sign, their sum has that sign and has greater magnitude than either component, so the ratio is positive and less than one. If \(I=0\), the ratio is zero; if \(D=0\) and \(I\ne0\), it is one. These cases show why a blanket claim that the ratio must be a fraction between zero and one is incorrect.
Worked Example: Opposing Effects Produce a Negative Ratio
Suppose a hypothetical intervention is evaluated for its effect on a symptom score, measured in points, with lower scores indicating fewer symptoms. A g-computation analysis gives the following standardized mean scores:
| Mean | Hypothetical value |
|---|---|
| \(\mu(0,0)\) | 50 points |
| \(\mu(1,0)\) | 58 points |
| \(\mu(1,1)\) | 53 points |
Assume the hypothetical analysis has adequate covariate adjustment, mediator support, and model fit for these interventional means. On the point scale, the direct effect is
The indirect effect is
and the total effect is
Therefore,
The negative value does not mean that a negative fraction of people were mediated. It indicates that the indirect effect is in the opposite direction from the total effect: the \(+8\)-point direct contrast is partly offset by the \(-5\)-point indirect contrast. Reporting “\(-166.7\%\) mediated” without this explanation would invite a literal, misleading interpretation. The component effects and their directions are more informative.
Worked Example: Opposing Effects Produce a Ratio Above One
In another hypothetical analysis, consider a continuous outcome measured in points. Suppose the standardized means are \(\mu(0,0)=40\), \(\mu(1,0)=48\), and \(\mu(1,1)=38\). Assume, as in the previous example, that the identification conditions and the estimation of these means are adequate.
The direct effect is \(D=48-40=8\) points, while the indirect effect is \(I=38-48=-10\) points. Thus the total effect is \(T=38-40=-2\) points, which also equals \(8+(-10)\). The proportion mediated is
The ratio is greater than one because the indirect effect has magnitude \(10\), while the net total effect has magnitude only \(2\) after the opposing \(8\)-point direct effect is added. It does not mean that the mediator accounts for five times the outcome change in a literal or population-share sense. It is the ratio of the indirect contrast to a small net contrast.
A Near-Zero Total Effect Makes the Ratio Unstable
Even without an exactly zero total effect, a denominator close to zero can make the ratio extremely sensitive to small changes in estimated means. The component effects may each be reasonably sized while their sum is tiny. Rounding, sampling variation, or a modest change in model specification can then move the denominator across zero and change both the magnitude and the sign of the ratio.
Worked Example: A Small Change Reverses the Ratio
Suppose a hypothetical analysis of a score in points estimates \(\mu(0,0)=40\), \(\mu(1,0)=44\), and \(\mu(1,1)=40.2\). Assume adequate support and confounding control, and treat these as illustrative estimated means. Then \(D=44-40=4\) points, \(I=40.2-44=-3.8\) points, and \(T=40.2-40=0.2\) points. The ratio is
Now suppose a small change in the estimated last mean gives \(\mu(1,1)=39.8\) rather than \(40.2\), with the other two means unchanged. The direct effect remains \(4\) points, the indirect effect becomes \(39.8-44=-4.2\) points, and the total becomes \(39.8-40=-0.2\) points. The ratio is now
A change of \(0.4\) points in one mean has changed the ratio from \(-19\) to \(21\), even though the component effects remain close to their earlier values. At exactly \(\mu(1,1)=40\), the total effect would be zero and the ratio would be undefined. In such situations, emphasize the estimated direct, indirect, and total effects with their uncertainty; do not treat the ratio as a stable summary.
The same issue applies to uncertainty intervals. Because the ratio has a denominator that may be close to zero, ordinary approximations that assume a well-behaved denominator can give unreliable intervals. A bounded, tidy-looking interval is not guaranteed to represent the uncertainty well. If an interval for the total effect includes values near zero, that is a warning that the ratio may be poorly determined; the component estimates and their joint uncertainty should remain central to the report.
The Outcome Scale Changes the Ratio
A proportion mediated is defined relative to an outcome scale. For example, a binary outcome can be analyzed using risk differences, or one can compare marginal risks on a log-odds scale. These contrasts need not yield the same direct-to-total ratio. A value calculated on one scale should not be presented as though it were scale-free.
Worked Example: Risk-Difference and Log-Odds Ratios Differ
Consider a hypothetical binary outcome and three standardized risks: \(\mu(0,0)=0.10\), \(\mu(1,0)=0.20\), and \(\mu(1,1)=0.30\). Suppose the required identification assumptions and model adequacy hold. On the risk-difference scale, the direct effect is \(0.20-0.10=0.10\), the indirect effect is \(0.30-0.20=0.10\), and the total effect is \(0.30-0.10=0.20\). Thus,
For comparison, define the marginal log-odds contrast by applying the logit function to each standardized risk, where \(\operatorname{logit}(p)=\log\{p/(1-p)\}\). The three transformed values are approximately \(-2.1972\), \(-1.3863\), and \(-0.8473\). The direct log-odds contrast is \(-1.3863-(-2.1972)=0.8109\); the indirect contrast is \(-0.8473-(-1.3863)=0.5390\); and the total is \(-0.8473-(-2.1972)=1.3499\). Therefore,
The two ratios, \(0.50\) and approximately \(0.3993\), summarize contrasts on different scales. The transformed calculation uses contrasts between logits of the standardized risks; it is not a risk-difference proportion. Neither value is inherently the uniquely correct proportion. The scale should be chosen for the scientific question, stated explicitly, and used consistently for all components.
Reporting and Common Pitfalls
The proportion mediated can be a compact descriptive summary when the total effect is clearly nonzero, the component effects point in the same direction (or one is zero), and the scale is appropriate to the question. It is much less informative when effects oppose, when the total is near zero, or when different scales are being compared. In those settings, readers need to see what the ratio alone conceals.
- Pitfall: Assuming every “proportion” lies between zero and one. That range applies when the components point in the same direction, or one component is zero, and the total is nonzero. Opposing nonzero components can yield a negative value or a value above one.
- Pitfall: Interpreting a negative or greater-than-one value as a literal share. Such values reflect cancellation in the total-effect denominator. Report the signed direct and indirect contrasts and explain their directions.
- Pitfall: Treating an undefined ratio as evidence that the effects do not exist. If \(T=0\), the ratio is undefined, but \(D\) and \(I\) can each be nonzero and cancel exactly.
- Pitfall: Comparing ratios without naming the scale. State whether effects are, for example, risk differences or marginal log-odds contrasts, and compute the numerator and denominator on that same scale.
- Pitfall: Reporting a highly unstable ratio as the main result. When \(T\) is near zero, give the component estimates, total effect, units, and uncertainty. Explain that the ratio is sensitive to the denominator.
Check Your Understanding
Use the definitions and sign reasoning above to answer the following.
- If \(D=6\) points and \(I=2\) points, calculate \(T\) and the proportion mediated. Why is the result between zero and one?
- If \(D=7\) points and \(I=-3\) points, calculate the ratio and explain its sign in context.
- Give values of nonzero, opposing \(D\) and \(I\) that produce a proportion mediated greater than one. Explain why the ratio exceeds one.
- What happens to the proportion mediated when \(D=-I\), and can either component still be nonzero?
- Why can a change in the estimated total effect from \(0.2\) to \(-0.2\) cause a large change in the ratio even if the component effects change only slightly?
- In a binary-outcome analysis, why must the scale be stated when reporting a proportion mediated?