Joint Interventions on Multiple Mediators

Why Intervene on the Mediator Vector Jointly?

A study may measure several mediators, such as symptom severity and treatment adherence, or a biomarker and a downstream clinical measure. These mediators can be statistically dependent: for example, individuals with one mediator value may be more likely to have a particular value of the other. A stochastic intervention can account for that dependence by drawing the mediator values as a vector from a specified joint distribution.

This matters because an intervention defined by the joint distribution is not generally the same as one that draws each mediator separately from its own marginal distribution. The latter can assign combinations of mediator values that are uncommon, or absent, in the joint distribution. The target of this tutorial is the mean outcome when the mediator vector is drawn jointly, and the assumptions under which that mean can be identified from observed data.

Definition: Let \(A\) be a treatment or exposure, \(C\) baseline covariates, and \(M=(M_1,\ldots,M_K)\) a vector of mediators. For a treatment level \(a^*\), define \(G_{a^*}\) as a random draw of the entire vector \(M\), conditional on \(C=c\), from the joint distribution \(f(m\mid A=a^*,C=c)\). The draw is made according to this distribution, not by assigning each person their own mediator values under a different treatment. Write \(Y_{a,G_{a^*}}\) for the outcome under treatment level \(a\) and this joint stochastic mediator intervention.

The draw is defined at the vector level. It retains the dependence present in the specified joint distribution. It does not, by itself, assert that the mediators are causally ordered or that one mediator causes another. Nor does it require the mediators to be independent.

Definition: The joint interventional mean is \(\psi(a,a^*)=E[Y_{a,G_{a^*}}]\). The first argument specifies the treatment level used for the outcome; the second specifies the treatment level whose conditional joint mediator distribution supplies the stochastic draw.

The two arguments can differ. For instance, \(\psi(1,0)\) uses the outcome under treatment level 1 while drawing the mediator vector from its joint distribution under treatment level 0, conditional on baseline covariates. This is a population-level stochastic intervention, not a claim about the mediators an individual would naturally have had under both treatment levels.

Identification of the Joint Interventional Mean

For a concise identification result, consider a setting in which all relevant confounders of treatment and the mediator–outcome relationship are measured before treatment in \(C\). Let \(\mu(a,m,c)=E[Y\mid A=a,M=m,C=c]\). Under the conditions below, the joint interventional mean is identified by the mediational g-formula.

Theorem: Suppose consistency holds, there is no interference between individuals, treatment is conditionally exchangeable given \(C\), and the mediator vector is conditionally exchangeable with the outcome given \(A\) and \(C\). Suppose also that treatment and mediator positivity hold. Then

\psi(a,a^*)= \int\!\!\int \mu(a,m,c)\, f(m\mid A=a^*,C=c)\, f(c)\,dm\,dc.

Here \(f(m\mid A=a^*,C=c)\) is the joint conditional distribution of the full vector \(M\). For discrete mediators, the integrals over \(m\) are sums over the possible mediator vectors. The formula averages the outcome regression at treatment level \(a\) over the mediator vectors drawn from the reference distribution and then averages over the baseline covariate distribution in the target population.

The assumptions have specific roles:

Consistency and well-defined interventions. Observed outcomes agree with the corresponding potential outcomes under the treatment and mediator values actually received. The treatment and joint mediator intervention must be sufficiently well-defined for those potential outcomes to be meaningful.
No interference. One individual’s treatment or mediator values do not alter another individual’s outcome.
Treatment exchangeability. Conditional on \(C\), treatment assignment is independent of the relevant potential outcomes and mediator vectors. This allows observed treatment groups to identify the corresponding conditional distributions and outcome means.
Mediator–outcome exchangeability. Conditional on \(A\) and \(C\), the mediator vector has no unmeasured common causes with the outcome. In particular, the outcome regression for a mediator vector can identify the corresponding potential-outcome mean.
Positivity. Each treatment level used in the formula must occur with positive probability at relevant values of \(C\). Also, every mediator vector receiving positive probability under \(f(m\mid A=a^*,C=c)\) must be supported among individuals with \(A=a,C=c\), so that \(\mu(a,m,c)\) can be identified there.

These are substantive conditions, not consequences of the formula. In particular, measuring several mediators does not by itself control mediator–outcome confounding. If a relevant mediator–outcome confounder is unmeasured, the observed conditional mean need not identify the required potential-outcome mean.

Why the Formula Has This Form

The derivation follows from averaging over the stochastic intervention. Conditional on \(C=c\), the intervention draws a mediator vector with density \(f(m\mid A=a^*,C=c)\). For any such draw, the mean outcome under treatment \(a\) is \(E[Y(a,m)\mid C=c]\). Averaging first over the draw and then over \(C\) gives

E[Y_{a,G_{a^*}}] = \int\!\!\int E[Y(a,m)\mid C=c]\, f(m\mid A=a^*,C=c)f(c)\,dm\,dc.

Under treatment exchangeability, mediator–outcome exchangeability, and consistency, the potential-outcome mean in this expression equals the observed conditional mean: \(E[Y(a,m)\mid C=c]=E[Y\mid A=a,M=m,C=c]=\mu(a,m,c)\). Substitution yields the stated formula. The mediator distribution remains the joint distribution throughout; the derivation does not replace it with separate distributions for its components.

Worked Examples

Worked Example: A Joint Draw With Two Dependent Binary Mediators

Suppose two binary mediators, \(M_1\) and \(M_2\), have the following hypothetical joint distribution among individuals with \(A=0\). The table also gives hypothetical conditional mean outcomes under \(A=1\).

\(M_1\)	\(M_2\)	\(P(M_1,M_2\mid A=0)\)	\(\mu(1,M_1,M_2)\)
0	0	0.40	0.10
0	1	0.10	0.30
1	0	0.10	0.40
1	1	0.40	0.80

The four probabilities sum to 1. The mediators are dependent: each has marginal probability 0.50 of being 1, but \(P(M_1=1,M_2=1\mid A=0)=0.40\), not \(0.50\times0.50=0.25\). The formula for the mean outcome under treatment 1 and the joint mediator distribution under treatment 0 is

\begin{aligned} \psi(1,0) &= \sum_{m_1,m_2} \mu(1,m_1,m_2)P(m_1,m_2\mid A=0)\\ &=(0.10)(0.40)+(0.30)(0.10)+(0.40)(0.10)+(0.80)(0.40)\\ &=0.04+0.03+0.04+0.32\\ &=0.43. \end{aligned}

Thus, under the stated assumptions, the identified mean outcome is 0.43 outcome units when treatment is fixed at 1 and the mediator vector is drawn jointly from its distribution under treatment 0. If the outcome is, for example, a risk indicator, the result is a risk of 0.43 under this hypothetical intervention.

Worked Example: Averaging Over Baseline Covariate Strata

Now suppose \(C\) is a binary baseline covariate, with \(P(C=0)=0.60\) and \(P(C=1)=0.40\). For each stratum, the following hypothetical table gives the joint mediator distribution under \(A=0\) and the mean outcome under \(A=1\).

\(C\)	\((M_1,M_2)\)	\(P(M_1,M_2\mid A=0,C)\)	\(\mu(1,M_1,M_2,C)\)
0	(0,0)	0.50	0.10
0	(0,1)	0.10	0.30
0	(1,0)	0.20	0.50
0	(1,1)	0.20	0.70
1	(0,0)	0.10	0.20
1	(0,1)	0.20	0.40
1	(1,0)	0.20	0.60
1	(1,1)	0.50	0.90

Within \(C=0\), the weighted mean is \(0.50(0.10)+0.10(0.30)+0.20(0.50)+0.20(0.70)=0.32\). Within \(C=1\), it is \(0.10(0.20)+0.20(0.40)+0.20(0.60)+0.50(0.90)=0.67\). The population mean is then

\psi(1,0)=0.60(0.32)+0.40(0.67)=0.192+0.268=0.460.

The calculation uses the mediator distribution and outcome regression appropriate to each covariate stratum, then averages using the covariate distribution in the target population. Averaging the conditional mediator distributions first and ignoring \(C\) would generally give a different result.

Worked Example: Comparing Two Joint Interventional Means

Consider a separate hypothetical setting with no baseline covariates. Suppose the joint mediator distribution under \(A=0\) assigns probabilities 0.20, 0.30, 0.10, and 0.40 to \((0,0),(0,1),(1,0),(1,1)\), respectively. Suppose the outcome means at these four vectors are as follows under each treatment level.

\((M_1,M_2)\)	Probability under \(A=0\)	\(\mu(0,M_1,M_2)\)	\(\mu(1,M_1,M_2)\)
(0,0)	0.20	0.08	0.12
(0,1)	0.30	0.18	0.22
(1,0)	0.10	0.28	0.32
(1,1)	0.40	0.48	0.62

Both interventional means use the same joint mediator distribution under \(A=0\). For treatment 0,

\begin{aligned} \psi(0,0) &=(0.20)(0.08)+(0.30)(0.18)+(0.10)(0.28)+(0.40)(0.48)\\ &=0.016+0.054+0.028+0.192=0.290. \end{aligned}

For treatment 1,

\begin{aligned} \psi(1,0) &=(0.20)(0.12)+(0.30)(0.22)+(0.10)(0.32)+(0.40)(0.62)\\ &=0.024+0.066+0.032+0.248=0.370. \end{aligned}

The contrast is \(\psi(1,0)-\psi(0,0)=0.370-0.290=0.080\) outcome units. It compares treatment levels while holding the stochastic mediator regime fixed at the joint distribution under \(A=0\). It is not, by itself, a comparison of the observed treatment-group means.

Common Pitfall: Replacing a Joint Distribution With Marginals

A tempting shortcut is to estimate each mediator’s distribution separately and multiply the marginal probabilities. That calculation silently defines a different joint draw unless the mediators are conditionally independent under the reference distribution. The first example shows the consequence: the actual joint probabilities under \(A=0\) are \(0.40,0.10,0.10,0.40\). Multiplying the marginals, both of which assign probability 0.50 to 1, would instead give probability 0.25 to each of the four vectors. Averaging the example’s outcome means under that product distribution gives

\frac{0.10+0.30+0.40+0.80}{4}=0.40,

rather than the joint-intervention mean of 0.43. The mistake is not arithmetic: it is changing the intervention being evaluated. Use the product of marginal distributions only when that product is the intended intervention distribution, or when conditional independence justifies it.

Key takeaway: For a joint stochastic intervention on multiple mediators, draw and average over the full mediator vector’s joint distribution. Identification requires support for the relevant joint mediator values and assumptions that identify both the reference distribution and the outcome regression.

Check Your Understanding

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

In \(\psi(a,a^*)\), what does each argument specify?
Why can multiplying the marginal mediator distributions produce a different interventional mean from using the joint distribution?
In the identification formula, which distribution is averaged over first, and which population distribution is used for the final average?
What positivity condition is needed for a mediator vector given positive probability by the reference distribution?
If \(\psi(1,0)=0.46\) and \(\psi(0,0)=0.39\), what is their contrast, and what mediator regime is held fixed in that comparison?