Why Specify a Mediator Distribution?
A mediator is a variable that may lie on a causal pathway from an exposure to an outcome. One way to describe a mediation question is to imagine setting each person’s mediator to a particular value. But a policy or scientific intervention may instead aim to shift the distribution of mediator values—for example, to make mediator values follow a distribution seen under treatment, under no treatment, or under a distribution chosen by the investigator.
A stochastic mediator intervention makes that idea precise: for each person, it generates a new mediator value by a random draw from a specified probability distribution. The distribution being used is part of the intervention’s definition. It is not, in general, the same thing as assigning everyone a fixed mediator value or giving each person their own mediator value under another exposure condition.
The draw is made for the hypothetical intervention under consideration. A common formulation makes it independent of that person’s potential outcomes and potential mediator values, conditional on the covariates used to define the draw. This distinguishes a fresh random assignment from a construction that pairs a person’s outcome under one exposure with that same person’s mediator under another exposure.
Three Choices of Mediator Distribution
The distribution \(q\) can be chosen in different ways. Two natural choices are the mediator distributions associated with treated and untreated conditions. A third is a reference distribution selected by the investigator. These choices define different interventions, even when they use the same people and the same outcome model.
Thus, \(G_1\) is a random draw from the mediator distribution under exposure level 1, and \(G_0\) is a random draw from the mediator distribution under exposure level 0. These are distributions over possible mediator values, not values attached to particular people. The user-chosen \(G_q\) need not equal either one: it might represent a target population, a feasible policy, or a deliberately selected distribution used for comparison.
In applications, the relevant distributions may be defined from the causal laws of \(M(1)\) and \(M(0)\), or they may be specified using observed data distributions. Those are conceptually different definitions. An observed conditional distribution \(P(M=m\mid A=a,C=c)\) is not automatically the causal distribution of \(M(a)\mid C=c\). Whether observed data identify a causal distribution is a separate question; defining a stochastic intervention does not by itself answer it.
Worked Example: Drawing From Treated and Untreated Binary Mediator Distributions
Suppose \(M\) is a binary mediator indicating whether a hypothetical patient reaches a specified clinical threshold. For a fixed covariate profile, suppose the causal mediator distributions are defined as \(P(M(1)=1\mid C=c)=0.70\) and \(P(M(0)=1\mid C=c)=0.30\). Describe the two stochastic interventions and calculate their mediator means.
Under the treated-condition distribution, draw \(G_1\) from a Bernoulli distribution with success probability 0.70. Thus \(G_1=1\) with probability 0.70 and \(G_1=0\) with probability \(1-0.70=0.30\). Since the mediator is binary, \(E(G_1\mid C=c)=1(0.70)+0(0.30)=0.70\).
Under the untreated-condition distribution, draw \(G_0\) from a Bernoulli distribution with success probability 0.30. Thus \(G_0=1\) with probability 0.30 and \(G_0=0\) with probability \(1-0.30=0.70\). Its mean is \(E(G_0\mid C=c)=1(0.30)+0(0.70)=0.30\).
These calculations describe the distributions from which mediator values are drawn. They do not say that every treated person receives \(M(1)\), or that every untreated person receives \(M(0)\); each \(G_a\) is a fresh draw for the intervention. Nor do the calculations determine the outcome under either intervention. That also depends on how \(Y\) responds to mediator values.
Conditional and Marginal Reference Distributions
A reference distribution can either preserve dependence on baseline covariates or ignore it. A conditional reference \(q(m\mid c)\) uses different mediator distributions for people with different covariates. A marginal reference \(q(m)\) uses the same distribution for everyone, regardless of covariates. The choice changes the intervention, not merely the notation.
Worked Example: A Conditional Reference Versus a Marginal Draw
Consider a hypothetical population divided equally between two baseline strata, \(C=L\) and \(C=H\). Suppose the probability of mediator value 1 under a reference condition is 0.20 in stratum \(L\), and 0.80 in stratum \(H\). Compare drawing from this conditional reference with drawing from its marginal mixture.
With the conditional intervention, a person in stratum \(L\) receives a Bernoulli draw with probability 0.20, while a person in stratum \(H\) receives a Bernoulli draw with probability 0.80. Because each stratum has population weight 0.50, the marginal probability of drawing mediator value 1 is \(0.50(0.20)+0.50(0.80)=0.10+0.40=0.50\).
Now define a marginal intervention that assigns a Bernoulli draw with probability 0.50 to everyone. Its overall probability of mediator value 1 is also 0.50. The two interventions therefore have the same marginal probability of mediator value 1, but they are not the same intervention: the conditional intervention uses probabilities 0.20 and 0.80 in the two strata, whereas the marginal intervention uses 0.50 in both.
If mediator values relate differently to outcomes across strata, these distinct assignments can produce different outcome distributions. The equality of the marginal mediator probabilities alone does not make the interventions equivalent.
Expected Outcomes Under a Stochastic Draw
It is useful to express what the draw means for an outcome without yet making a claim about whether that quantity can be identified from observed data. Let \(Y(a,m)\) be the potential outcome that would occur if exposure were set to \(a\) and mediator to \(m\). Under a stochastic intervention drawing from \(q\), write \(Y(a,G_q)\) for the outcome when exposure is set to \(a\) and the mediator is assigned a random draw from \(q(m\mid C)\).
The expected outcome averages over both the population’s covariates and the intervention’s mediator randomization. In the following expression, \(F_C\) denotes the population distribution of baseline covariates, and the conditional expectation is taken over the potential outcome at the indicated exposure and mediator values.
For a discrete mediator, the inner integral is replaced by a sum over possible mediator values. This expression is an averaging identity for the specified intervention: at each covariate value, average the outcome response over the mediator distribution used by the intervention, then average over covariates. It is not, by itself, an identification formula in observed data.
The identity follows by conditioning first on \(C\) and then on the random draw \(G_q\). Given \(C=c\), the draw has distribution \(q(m\mid c)\). Averaging the conditional outcome expectation over that draw gives the inner integral. Averaging that quantity over the population distribution of \(C\) gives the outer integral. No equality to an observed-data expression has been asserted.
Worked Example: Averaging an Outcome Response Over a Reference Draw
Suppose, purely hypothetically, that for a fixed covariate profile the outcome is binary and its mean under exposure level \(a\) is 0.10 when \(m=0\), and 0.40 when \(m=1\). Let a user-chosen reference distribution assign probability 0.25 to \(m=1\) and 0.75 to \(m=0\). Calculate the expected outcome under this stochastic draw at exposure level \(a\).
Average the outcome mean at each mediator value using the specified draw probabilities: \(0.75(0.10)+0.25(0.40)=0.075+0.100=0.175\). Thus, the expected outcome under this intervention, for the stated covariate profile and exposure level, is 0.175, or 17.5% for a binary outcome.
If the reference distribution instead assigned probability 0.50 to \(m=1\), the same outcome response values would give \(0.50(0.10)+0.50(0.40)=0.050+0.200=0.250\). The expected outcome changes because the intervention changes the probabilities with which mediator values are assigned. These numbers are hypothetical and describe intervention-defined expectations; they are not estimates from a study.
What the Intervention Does—and Does Not—Specify
A stochastic mediator intervention must specify enough detail to define its draw. At minimum, this includes the mediator values the intervention can assign and the probability distribution used. If the distribution is conditional on covariates, the relevant covariates and their conditioning rule must also be stated. For an intervention to have a clear interpretation, the mediator should be well defined: the same assigned value should refer to a meaningful mediator state across the units to which it is assigned.
The intervention specifies a probability law, not a promise about the realized value for any particular person. Two people with identical \(C\) may receive different draws, and repeated implementations may assign different values to the same person. The assignment randomness is part of the intervention’s definition.
A reference distribution is a substantive choice. A treated-condition distribution asks what the intervention means when draws follow the mediator law associated with treatment; an untreated-condition distribution uses the law associated with no treatment. A user-chosen reference can instead target a distribution that neither condition naturally produces. Describing the choice clearly prevents an ambiguous statement such as “set the mediator to the treated value.”
State whether the draw follows the mediator distribution under treatment, under no treatment, or a user-chosen reference distribution.
Say whether the distribution is conditional on baseline covariates or is common to everyone, and define any covariate-dependent rule.
State that the assigned mediator is generated from the specified law, rather than identified with the person’s own potential mediator value.
Defining the intervention does not establish that its expected outcome can be recovered from observed data.
Common Mistake: Treating a Draw as a Person’s Natural Mediator
A common error is to say, “Under the treated mediator distribution, each person receives the mediator value they would have had if treated.” That describes each person’s potential mediator \(M(1)\), not a fresh stochastic draw from the population distribution of \(M(1)\). The distinction matters because the intervention uses a distribution to assign values; it does not preserve the person-specific pairing between that person’s potential mediator and other potential outcomes.
A related error is to call an observed group distribution a causal distribution without justification. For example, \(P(M=m\mid A=1,C=c)\) describes observed mediator values among exposed people with covariates \(c\). The causal distribution of \(M(1)\mid C=c\) is defined using potential outcomes. The two may coincide under suitable conditions, but the intervention’s definition alone does not supply those conditions.
Check Your Understanding
Use the definitions and distinctions in this tutorial to answer the following questions.
- In your own words, distinguish a draw \(G_0\) from the person-specific potential mediator \(M(0)\).
- A reference distribution assigns probability 0.40 to mediator value 1 for one covariate stratum and 0.70 for another. Is this a conditional or a marginal reference distribution?
- For a binary mediator with probability 0.65 of value 1, calculate the probability of value 0 and the mean of the draw.
- Explain why a marginal probability of mediator value 1 equal to 0.50 does not establish that a conditional intervention and a marginal intervention are the same.
- Why does defining a stochastic mediator intervention not, by itself, show that its expected outcome is identifiable from observed data?