Sequential Interventions on Ordered Mediators

Why Specify an Order for Mediator Interventions?

When two mediators have a meaningful temporal or causal order, an intervention can be described in stages: first assign \(M_1\), then assign \(M_2\) conditional on the value of \(M_1\) that was assigned. This is more specific than saying only that the pair \((M_1,M_2)\) should follow some joint distribution. It makes explicit how the second assignment responds to the first mediator and allows a contrast to change one stage while holding the other stage's policy fixed.

For example, suppose a hypothetical intervention first assigns a measure of early treatment engagement, \(M_1\), and then assigns a later adherence measure, \(M_2\), based on the assigned engagement level. The outcome exposure can be held at a chosen value while these mediator policies are varied. The resulting mean is an interventional estimand: it averages outcomes under a specified assignment procedure, not under each person's naturally occurring mediator values.

The distributions \(g_1\) and \(g_2\) can be chosen policies or distributions taken from specified reference regimes. Superscripts such as \(r\) and \(s\) will label the policies used at the first and second stages; they do not, by themselves, imply that the policies are observed exposure levels. Let \(\mu(a_y,m_1,m_2,c)\) denote the conditional mean outcome under exposure \(a_y\) and mediator values \(m_1,m_2\), given \(C=c\).

The Sequential Mediational G-Formula

Under the identification conditions stated below, the mean outcome under first-stage policy \(g_1^r\) and second-stage policy \(g_2^s\) is

The outer expectation averages over the population distribution of baseline covariates. It applies whether \(C\) is discrete or continuous; when \(C\) is continuous, it is an integral with respect to the distribution of \(C\), not a sum over covariate values. The displayed sums over mediators apply to discrete mediators. For continuous mediators, replace the corresponding sums by integrals with respect to the intervention densities.

This formula follows the intervention's order. For each \(C\), average the outcome mean over \(M_2\) under the second-stage policy conditional on \(M_1\); then average that result over \(M_1\) under the first-stage policy; finally average over \(C\). The conditioning of \(g_2^s\) on \(m_1\) is essential: it specifies which second-stage distribution is used after each possible first-stage assignment.

Sequential Pathway Contrasts

A sequential pathway contrast changes a policy at one stage while holding the other stage's policy fixed. For example, holding the second-stage policy at \(g_2^0\), a first-stage contrast is

It compares two intervention means that differ in the distribution used to assign \(M_1\), while both use the same rule for assigning \(M_2\) conditional on the assigned \(M_1\). A second-stage contrast can instead hold \(g_1^0\) fixed and compare \(g_2^1\) with \(g_2^0\). These are contrasts between well-defined intervention regimes. Their interpretation depends on which policy is held fixed, and they should not automatically be treated as unique, context-free shares of an overall effect.

Identification Conditions

The g-formula is an identification result: it expresses the target intervention mean using observed-data distributions under assumptions. In the basic setting here, \(C\) contains the relevant baseline confounders, and there are no additional exposure-induced mediator-outcome confounders outside the ordered mediator history. The main conditions are:

• Consistency and well-defined interventions: observed outcomes agree with the corresponding potential outcomes for the exposure and mediator values actually received, and the stochastic assignment rules are sufficiently specified to define the intervention.
• No interference: one unit's outcome is not affected by other units' exposure or mediator assignments.
• Exposure exchangeability: conditional on \(C\), exposure assignment is independent of the potential outcomes relevant to the estimand. In a randomized exposure study, randomization supports this condition.
• Sequential mediator-outcome exchangeability: after conditioning on \(A,C,M_1\), there is no unmeasured confounding of the relation between \(M_2\) and the outcome; and, after conditioning on \(A,C\), there is no unmeasured confounding of the relation between \(M_1\) and the outcome that is needed for the intervention mean.
• Positivity: the exposure and mediator values needed by the target policies have support in the relevant observed-data strata. In particular, outcome regression must be identified where \(g_1\) and \(g_2\) place positive probability.

These conditions permit the outcome regression and mediator distributions in the formula to be learned from observed data. If important mediator-outcome confounders are omitted, or a policy assigns mediator values outside observed support, the formula alone does not guarantee identification. As with other interventional mediation estimands, the definition does not require assuming that a person's potential outcome under one exposure is independent of that same person's mediator value under a different exposure.

Continuous Baseline Covariates and the Outer Average

The expectation over \(C\) is part of the estimand, not a notational shortcut that can be replaced by a sum in every setting. If the baseline covariate is continuous, the inner mediator averages are first computed at each covariate value and then integrated over the covariate distribution.

How Sequential Policies Relate to Joint Interventions

A joint stochastic intervention specifies a distribution \(q(m_1,m_2\mid C)\) for the mediator pair. Its intervention mean is

Again, the outer average is an expectation over \(C\), so it applies to continuous as well as discrete baseline covariates. For continuous mediators, replace the sums over mediator values with integrals. A sequential policy induces the joint distribution \(q(m_1,m_2\mid C)=g_1(m_1\mid C)g_2(m_2\mid m_1,C)\). When the joint intervention uses exactly this distribution, both definitions give the same intervention mean. A joint distribution can also be expressed in this sequential factorization wherever the relevant conditional distributions are defined.

The distinction is therefore not that sequential and joint interventions must produce different mediator distributions. It is that the sequential specification makes the assignment order and the conditional second-stage rule explicit. That specification is useful for contrasts that change one stage's policy while retaining the other. A joint policy instead states the target distribution of the pair as a whole; it need not be presented as a procedure that assigns one mediator before the other.

Derivation and a Common Pitfall

For fixed \(C=c\), the sequential policy first gives \(M_1\) probability \(g_1^r(m_1\mid c)\), then gives \(M_2\) probability \(g_2^s(m_2\mid m_1,c)\). By the multiplication rule, the probability of the ordered pair under the intervention is their product. Averaging the outcome mean over this pair distribution yields the inner sums; averaging over baseline covariates yields the outer expectation. Explicitly, for discrete mediators:

The second line is the iterated average: first over \(M_2\) conditional on \(M_1,C\), then over \(M_1\) conditional on \(C\). The derivation also shows why the \(M_2\) policy must be evaluated at each value of \(M_1\), rather than averaged independently of it.

Common mistake: use the marginal distribution of \(M_2\) in place of \(g_2(m_2\mid m_1,C)\), then multiply separate marginal probabilities for the two mediators. That generally defines a different intervention because it removes the specified dependence of the second-stage assignment on the first-stage value. The correction is to apply the second-stage rule conditional on the realized \(M_1\), and only then average over the first-stage assignment. Another error is to write a sum over \(C\) without restricting \(C\) to be discrete; the population average must be written as \(E_C[\cdot]\), or as an integral when \(C\) is continuous.