What Natural Effects Compare
A treatment can affect an outcome through more than one route. It may change the outcome while the mediator is held at a specified level, and it may also change the mediator, which in turn changes the outcome. Natural direct and indirect effects express these routes using counterfactual outcomes. Their definitions specify two separate treatment values: one for the outcome and one that determines the mediator value.
Use \(M(a)\) for the mediator value that would occur under treatment \(a\), and \(Y(a,m)\) for the outcome that would occur if treatment were set to \(a\) and the mediator to \(m\). The nested counterfactual \(Y(a,M(a'))\) is the outcome under treatment \(a\) when the mediator is set to the value it would naturally take under treatment \(a'\). The two treatment values need not be the same.
For population-average effects, take expectations of the individual-level contrasts. For example, \(E[Y(a,M(a'))-Y(a',M(a'))]\) compares the two outcome-treatment values while using the mediator value under \(a'\) in both counterfactual outcomes. These are causal estimands defined in terms of potential outcomes; their definitions alone do not tell us whether they can be identified or estimated from observed data.
Two Reference-Treatment Conventions
Let \(a\) be the treatment level of interest and \(a'\) the comparison level. The reference convention determines which natural mediator value is held fixed for the direct effect. Equivalently, it determines the treatment at which the indirect effect compares mediator values. The two conventions can be laid out explicitly:
| Convention | Natural direct effect | Natural indirect effect | What is held fixed? |
|---|---|---|---|
| Reference mediator under \(a'\) | \(E[Y(a,M(a'))-Y(a',M(a'))]\) | \(E[Y(a,M(a))-Y(a,M(a'))]\) | Direct: \(M(a')\). Indirect: outcome treatment \(a\). |
| Reference mediator under \(a\) | \(E[Y(a,M(a))-Y(a',M(a))]\) | \(E[Y(a',M(a))-Y(a',M(a'))]\) | Direct: \(M(a)\). Indirect: outcome treatment \(a'\). |
In words, the first convention asks for a direct contrast with the mediator set to the value it would have under the comparison treatment, and an indirect contrast evaluated at the treatment of interest. The second asks for a direct contrast with the mediator set to its value under the treatment of interest, and an indirect contrast evaluated at the comparison treatment.
When comparing treatment \(a=1\) with \(a'=0\), the first direct effect is often called the pure natural direct effect, and the first indirect effect the total natural indirect effect. The second direct effect is often called the total natural direct effect, and the second indirect effect the pure natural indirect effect. The “pure” and “total” labels are conventional; writing the counterfactual contrast is the clearest way to make the chosen reference explicit.
Worked Example: Calculating Both Conventions
Consider a hypothetical finite population of three patients. Let the outcome be symptom-free days during a follow-up period. The treatment is either \(0\) or \(1\), and the mediator is binary. The table gives each patient's mediator values under each treatment and the outcome potential outcomes for each treatment–mediator combination.
| Patient | \(M(0)\) | \(M(1)\) | \(Y(0,0)\) | \(Y(0,1)\) | \(Y(1,0)\) | \(Y(1,1)\) |
|---|---|---|---|---|---|---|
| A | 0 | 1 | 10 | 14 | 12 | 18 |
| B | 1 | 1 | 8 | 11 | 9 | 13 |
| C | 0 | 0 | 6 | 10 | 7 | 12 |
First use the convention that references the mediator to \(M(0)\). The nested outcomes under treatment \(1\) with each patient's \(M(0)\) are \(12,13,7\); under treatment \(0\) with the same mediator values, they are \(10,11,6\). Thus the natural direct effect is
For the natural indirect effect under this convention, keep the outcome treatment at \(1\) and change the mediator from \(M(0)\) to \(M(1)\). The relevant outcomes are \(Y(1,M(1))=(18,13,7)\) and \(Y(1,M(0))=(12,13,7)\), so
Now use the convention that references the mediator to \(M(1)\). The direct-effect contrast is \(Y(1,M(1))-Y(0,M(1))\). Its patient-level differences are \(18-14=4\), \(13-11=2\), and \(7-6=1\), giving
The corresponding indirect effect keeps the outcome treatment at \(0\). Its patient-level differences are \(Y(0,M(1))-Y(0,M(0))=(14-10,\;11-11,\;6-6)=(4,0,0)\), so the average is \(4/3=1.333\) symptom-free days. Each calculation uses the mediator value generated by the specified reference treatment; it does not substitute one common mediator value for all patients.
How to Read the Nested Counterfactual
The expression \(Y(a,M(a'))\) contains two instructions. The outer treatment value \(a\) specifies the treatment under which the outcome is evaluated. The inner value \(M(a')\) specifies which mediator value is supplied to that outcome. Changing the outer value while leaving the inner one unchanged defines a natural direct contrast. Changing the inner value while leaving the outer one unchanged defines a natural indirect contrast.
This reading follows directly from the definition of \(Y(a,m)\): substitute the treatment-specific mediator value \(M(a')\) for \(m\). For instance, comparing \(Y(a,M(a'))\) with \(Y(a',M(a'))\) changes the first argument from \(a\) to \(a'\) and leaves the mediator argument \(M(a')\) unchanged. In contrast, comparing \(Y(a,M(a))\) with \(Y(a,M(a'))\) holds the first argument at \(a\) and changes the mediator argument. This index-by-index check is a useful safeguard against reversing the two conventions.
Worked Example: Choosing the Intended Reference
Suppose a hypothetical study compares a new treatment, \(a=1\), with usual care, \(a'=0\). The question is: “What would the treatment contrast be if each person's mediator were set to the value that person would have had under usual care?” Because the question fixes the mediator to \(M(0)\), it calls for
The first outcome uses treatment \(1\), and the second uses treatment \(0\); both use the mediator value under treatment \(0\). This is the natural direct effect under the reference-mediator-under-\(a'\) convention.
If instead the question asks how changing the mediator from its usual-care value to its treatment value would affect the outcome while the outcome treatment is set to \(1\), the contrast is
Here treatment \(1\) stays fixed in both outcomes, and only the mediator's natural treatment assignment changes. The contrast is the natural indirect effect paired with the first convention. Naming the fixed quantity and the changing quantity before writing the formula makes the intended estimand unambiguous.
Why the Reference Choice Matters
The two conventions need not produce the same numerical direct effects, nor the same numerical indirect effects. Their contrasts evaluate outcome changes at different mediator values or different outcome-treatment levels. If the effect of the mediator on the outcome varies across treatment levels, the indirect contrasts can differ; likewise, the direct contrasts can differ when the treatment effect varies across mediator values. The convention is therefore part of the estimand, not a cosmetic choice of notation.
Worked Example: Different Reference Values, Different Contrasts
For a second hypothetical population, suppose every person has \(M(0)=0\) and \(M(1)=1\), and the average potential outcomes are described by the following simple model, in symptom-free days:
Under the reference-mediator-under-\(0\) convention, the direct effect compares \(Y(1,0)=23\) with \(Y(0,0)=20\), a difference of \(3\) days. The indirect effect at outcome treatment \(1\) compares \(Y(1,1)=29\) with \(Y(1,0)=23\), a difference of \(6\) days.
Under the reference-mediator-under-\(1\) convention, the direct effect compares \(Y(1,1)=29\) with \(Y(0,1)=22\), a difference of \(7\) days. The indirect effect at outcome treatment \(0\) compares \(Y(0,1)=22\) with \(Y(0,0)=20\), a difference of \(2\) days. The arithmetic follows by substitution into the stated model. The two conventions answer different counterfactual questions, so the different values are not a contradiction.
Natural effects should also be distinguished from controlled direct effects. A controlled direct effect compares \(Y(a,m)\) with \(Y(a',m)\) at a specified mediator value \(m\). A natural direct effect instead uses a treatment-specific mediator value, such as \(M(a')\), which may differ from person to person. The two quantities coincide only under conditions that make their mediator settings equivalent for the comparison at hand; they are not interchangeable by definition.
Common Mistakes
- Swapping the two treatment roles. In \(Y(a,M(a'))\), \(a\) governs the outcome and \(a'\) governs the mediator value. Read the nested expression from the outside in.
- Changing both arguments in a direct-effect contrast. A natural direct effect changes the outcome-treatment value while keeping the mediator value at the chosen reference, such as \(M(a')\).
- Calling any fixed-mediator contrast a natural direct effect. \(Y(a,m)-Y(a',m)\) uses a specified \(m\); a natural direct effect uses a mediator value such as \(M(a')\) or \(M(a)\).
- Assuming the two reference conventions are numerically identical. They are different contrasts and can have different values. State the reference treatment when reporting an effect.
- Treating a counterfactual definition as an estimation procedure. Defining the estimand does not establish that observed data identify it or specify how to estimate it.
Check Your Understanding
For each item, identify the treatment component held fixed and the component changed.
- Write the natural direct effect that uses \(M(a')\) as the reference mediator.
- In \(E[Y(a,M(a))-Y(a,M(a'))]\), which treatment value governs the outcome, and what changes between the two terms?
- For a comparison of treatment \(1\) with treatment \(0\), what is the natural indirect effect under the reference-mediator-under-\(1\) convention?
- Explain why \(E[Y(a,m)-Y(a',m)]\) is not, by definition, a natural direct effect.
- Can the two reference conventions give different numerical estimates of natural direct and indirect effects? Explain what is different between their counterfactual contrasts.