Mediated Proportions and Their Failure Modes

Why a Mediated Proportion Needs Care

A mediation analysis can estimate how much of a genetic effect on an outcome operates through a specified mediator. Dividing the estimated indirect effect by the estimated total effect gives a concise summary. But a ratio is only informative when its numerator and denominator describe compatible effects, and it can behave badly when the total effect is small or when the two pathways act in opposite directions.

The mediated proportion is therefore not automatically a percentage between zero and one. Its value depends on the effect scale, the mediation estimand, and the relationship between the direct and indirect effects. In genetic mediation, the causal interpretation also depends on the identification assumptions for the genetic instruments and the mediation methods used; a ratio does not repair a failure of those assumptions.

Definition: Let $TE$ denote the total genetic effect of an exposure on an outcome, $IE$ the indirect effect through a specified mediator, and $DE$ the direct effect not through that mediator. When the effects are defined on the same scale and satisfy the additive decomposition $TE=DE+IE$, the mediated proportion is $$ PM=\frac{IE}{TE}=\frac{IE}{IE+DE}, $$ provided $TE\ne 0$. It is often reported as $100PM\%$, but this percentage is a ratio of effect estimates, not a proportion of people or cases.

Conditions for a Meaningful Ratio

The equation is simple; deciding whether it answers a useful question is not. The total, direct, and indirect effects must refer to the same genetic contrast, outcome, population, and effect scale. Their definitions must also support the decomposition $TE=DE+IE$. For example, in a compatible linear model without an exposure–mediator interaction, the indirect effect can be represented by the product of the genetic effect on the mediator and the mediator effect on the outcome, while the direct effect is the remaining pathway effect.

A product estimate and a difference estimate can both be used, but they should not be combined casually. If the total effect is estimated separately and the indirect effect is estimated by a product method, the corresponding difference is $TE-IE$. If direct and indirect effects are instead estimated jointly, their estimates should be compatible with the stated total-effect decomposition. Material disagreement can reflect sampling variation, different estimands, or model problems; it is not a reason to choose whichever ratio looks most plausible.

The ratio is especially difficult to interpret when the effect scale does not support additive decomposition. With an exposure–mediator interaction, a decomposition may depend on the specified intervention or contrast. For a binary outcome, effects on a non-collapsible scale need not add as they do on a linear scale. In such cases, a proportion should be reported only after defining a compatible decomposition on the chosen scale. A numerical division alone does not establish one.

Worked Example: A Positive Mediated Proportion

Suppose a hypothetical analysis estimates the total effect of a genetic contrast on a continuous inflammatory marker as $TE=0.30$ marker units per allele. The estimated indirect effect through a circulating protein is $IE=0.12$ marker units per allele. Assume the effects refer to the same contrast and scale, and that the additive decomposition is appropriate.

The direct effect implied by the decomposition is $$ DE=TE-IE=0.30-0.12=0.18 $$ marker units per allele.

The mediated proportion is $$ PM=\frac{IE}{TE}=\frac{0.12}{0.30}=0.40. $$

The estimate is 40%. Under the stated model and assumptions, the indirect pathway estimate is 40% of the total effect estimate on this scale. It does not mean that the protein accounts for 40% of people’s marker values, or that intervening on the protein would necessarily reduce the outcome by 40%. Those stronger claims require additional causal assumptions and a clearly specified intervention.

Why the Ratio Can Be Unstable

A ratio becomes unstable when its denominator is close to zero relative to its uncertainty. Small changes in the total-effect estimate can then produce large changes in $PM$, including changes in sign. If the total effect is exactly zero, the ratio is undefined. If it is merely estimated imprecisely and could plausibly be zero, a single point estimate of the mediated proportion can give a false impression of precision.

A standard error for a ratio can be approximated with the delta method when the denominator is sufficiently far from zero and the joint sampling distribution is reasonably well approximated by a normal distribution. Let $\widehat{I}$ and $\widehat{T}$ be estimates of the indirect and total effects, with estimated variances $V_I$ and $V_T$, and covariance $C_{IT}$. Since $g(I,T)=I/T$, its partial derivatives are $1/T$ and $-I/T^2$. The first-order variance approximation is therefore

\widehat{\operatorname{Var}}\!\left(\frac{\widehat{I}}{\widehat{T}}\right) \approx \frac{V_I}{\widehat{T}^{\,2}} +\frac{\widehat{I}^{\,2}V_T}{\widehat{T}^{\,4}} -\frac{2\widehat{I}C_{IT}}{\widehat{T}^{\,3}}.

This approximation makes the instability visible: the variance terms contain powers of the estimated denominator. As $\widehat{T}$ approaches zero, the approximation can become very large or cease to describe the ratio’s sampling distribution well. The covariance term matters too; treating the total and indirect effect estimates as independent when they are not can distort the uncertainty estimate.

Worked Example: An Unstable Ratio When the Total Effect Is Small

In a hypothetical analysis, the estimated indirect effect is $\widehat{I}=0.08$ outcome units per allele, with standard error $0.03$. The estimated total effect is $\widehat{T}=0.04$ outcome units per allele, with standard error $0.05$. Suppose their estimated correlation is $0.40$. The point estimate of the mediated proportion is

\widehat{PM}=\frac{0.08}{0.04}=2.00=200\%.

The approximate 95% confidence interval for the total effect is $$ 0.04\pm 1.96(0.05)=[-0.058,\ 0.138] $$ outcome units per allele. Because this interval includes zero, the data do not pin down the denominator away from zero. The 200% point estimate is consequently not a stable summary, even though it is arithmetically correct.

The covariance is $C_{IT}=0.40(0.03)(0.05)=0.0006$. Using the delta-method formula gives $$ \widehat{\operatorname{Var}}(\widehat{PM}) \approx \frac{0.0009}{0.04^2} +\frac{0.08^2(0.0025)}{0.04^4} -\frac{2(0.08)(0.0006)}{0.04^3} =0.5625+6.25-1.50=5.3125. $$

Thus the approximate standard error is $\sqrt{5.3125}=2.305$. A symmetric normal interval would be $2.00\pm1.96(2.305)=[-2.52,\ 6.52]$, or approximately $-252\%$ to $652\%$. That interval is a warning, not a useful precise estimate: the denominator can be near zero, and the ratio’s distribution may be strongly non-normal. A ratio-specific method, such as a Fieller confidence set, can reflect this by producing an unbounded or disjoint set when the denominator is insufficiently separated from zero. The appropriate conclusion is that the mediated proportion is poorly determined, not that the true proportion is likely to be an extreme value in the displayed range.

Opposing Effects and Values Outside Zero to One

If the direct and indirect effects have the same sign, and the total effect is their sum, the ratio will generally lie between zero and one. When they have opposite signs, they partially cancel in the total effect. The indirect effect can then be larger in magnitude than the total, producing a ratio above one; if the indirect effect is negative while the total is positive, the ratio can be negative. These values are not arithmetic errors. They indicate that the signed pathway effects do not behave like non-overlapping pieces of a positive whole.

Worked Example: An Indirect Effect Larger Than the Total

Consider hypothetical estimates, all measured in outcome units per allele: $IE=0.12$ through a mediator and $DE=-0.10$ through other pathways. Under the assumed additive decomposition, the total effect is $$ TE=IE+DE=0.12+(-0.10)=0.02. $$

The mediated proportion is $$ PM=\frac{0.12}{0.02}=6.00=600\%. $$

The estimate does not mean that 600% of an outcome, or 600% of individuals, is mediated. It means that the positive indirect effect is six times the small positive net total effect because the negative direct effect offsets most of it. Reporting only “600% mediated” hides the cancellation. A clear report gives the direct, indirect, and total effects, their units, their uncertainty, and the fact that the component effects have opposite signs.

A Practical Interpretation and Reporting Check

Before interpreting a mediated proportion, check the estimand and the component estimates. In particular, confirm that the effects use compatible units and scales, that the decomposition is justified for the model, and that the denominator is not so small or uncertain that the ratio is unstable. Then report the signed direct and indirect effects alongside the total. This allows readers to distinguish a substantial pathway effect from a small net effect created by cancellation.

For uncertainty, use a method appropriate to a ratio rather than assuming that a symmetric interval is always adequate. The delta method can be useful when the total effect is well separated from zero and the approximation is reasonable. When that condition is doubtful, a Fieller-type confidence set or a suitable resampling approach may better reveal the uncertainty. Resampling does not, however, fix an incoherent estimand or failed identification assumptions.

Check compatibility.
Confirm that the direct, indirect, and total effects concern the same genetic contrast, outcome, population, and effect scale, and that the proposed decomposition is appropriate.

Calculate and inspect the components.
Compute $PM=IE/TE$, while also checking whether $TE$ agrees with $DE+IE$ within the estimates’ uncertainty and reporting all three signed effects.

Assess stability before translating the ratio.
Examine uncertainty in the total effect and use a ratio-appropriate interval method if the denominator may be near zero. Explain cancellation when the direct and indirect effects have opposite signs.

Common Mistakes

Calling every ratio a percentage between zero and 100%. This assumes same-direction components and a positive total. Opposing effects can make the ratio negative or greater than one.
Interpreting the ratio as a fraction of people or cases. It is a ratio of effect estimates on a specified scale, not a population fraction.
Ignoring a small denominator. A large point estimate can result from a total effect close to zero, with very high uncertainty. Report the total-effect uncertainty and avoid a confident percentage interpretation.
Combining effects from incompatible scales or definitions. If the effects do not support the same additive decomposition, their ratio has no straightforward mediated-proportion interpretation.
Reporting the ratio without its components. A value such as 600% conceals whether it arose from a large indirect effect, a small total effect, or cancellation. Give the signed component estimates and their units.

Key takeaway: A mediated proportion is interpretable only as a ratio of compatible effects under a justified decomposition. When the total effect is small or the direct and indirect effects oppose, the ratio can be unstable or fall outside zero to one; show the components and their uncertainty rather than treating it as a literal fraction.

Check Your Understanding

Use the effect scale and decomposition stated in each question.

If $IE=0.09$ and $TE=0.30$, what is the mediated proportion, and what does it mean on that scale?
If $IE=0.14$ and $DE=-0.12$, calculate $TE$ and $PM$. Why would “the percentage of cases mediated” be an inappropriate interpretation?
Why does a confidence interval for the total effect that includes zero raise concern about the mediated proportion?
What must be checked before dividing an indirect effect estimate by a total effect estimate from a different mediation method?
Why can a delta-method interval be unreliable when the total effect is close to zero?