Preserving or Breaking Mediator Dependence

Why the Dependence Between Mediators Matters

Suppose an intervention assigns two mediators, such as a patient's adherence and physical activity, according to distributions observed under a reference treatment. Each mediator can have the same marginal distribution under two intervention rules, while the relationship between the two mediators differs. One rule draws the pair together from its joint distribution; another draws each mediator separately. The second rule breaks their dependence.

If the outcome responds only additively to the two mediators, changing their dependence need not change the expected outcome. But if the outcome response includes an interaction, or is otherwise nonlinear in the mediator pair, the two intervention rules can yield different means. The difference is a feature of the specified intervention distributions and outcome response; it does not by itself establish a biological mechanism.

Definition: Let \(A\) denote treatment, \(C\) baseline covariates, \(M_1\) and \(M_2\) two mediators, and \(Y\) the outcome. A joint mediator draw samples the vector \((M_1,M_2)\) from a specified joint distribution. An independent mediator draw samples each mediator from its specified marginal distribution independently, conditional on the same \(C\). The independent rule retains the two marginal distributions but replaces their conditional joint distribution with the product of those marginals.

Two Intervention Distributions and Their Estimands

Let \(a\) be the treatment level at which the outcome is evaluated, and let \(a^*\) specify the treatment level used to define the reference mediator distribution. Write \(f(m_1,m_2\mid A=a^*,C=c)\) for the conditional joint distribution of the mediators in the reference treatment group, and \(f(m_j\mid A=a^*,C=c)\) for the corresponding marginal distribution of mediator \(M_j\).

The joint intervention draws the two mediators together according to the reference joint distribution. The independent intervention uses the reference marginal distributions but makes the draws independent within levels of \(C\). In both cases, the outcome is evaluated at treatment \(a\), and the mean is standardized over the target population distribution of \(C\).

g_{\mathrm{joint}}(m_1,m_2\mid c) = f(m_1,m_2\mid A=a^*,C=c), \qquad g_{\mathrm{ind}}(m_1,m_2\mid c) = f(m_1\mid A=a^*,C=c)f(m_2\mid A=a^*,C=c).

Let \(\mu(a,m_1,m_2,c)=E(Y\mid A=a,M_1=m_1,M_2=m_2,C=c)\) be the conditional outcome mean. The interventional mean under a mediator distribution \(g\) is the outcome mean obtained by setting treatment to \(a\), drawing the mediator vector from \(g\), and averaging over the target distribution of \(C\):

\Psi(a;g) = \sum_c P(C=c) \int \mu(a,m_1,m_2,c)\, g(m_1,m_2\mid c)\,dm_1\,dm_2.

For discrete mediators, the integral is a sum over their possible values. The difference \(\Psi(a;g_{\mathrm{joint}})-\Psi(a;g_{\mathrm{ind}})\) isolates the change in this interventional mean caused by using the reference dependence rather than breaking it, while holding the two conditional mediator marginals fixed. It is not a contrast between observed groups unless the intervention distributions and identification conditions justify that interpretation.

A causal interpretation of these means requires suitable conditions. In particular, consistency must connect observed outcomes and mediators to their corresponding potential outcomes; treatment must be exchangeable with the relevant potential outcomes and mediator distributions conditional on \(C\); and there must be no unmeasured confounding of the joint mediator-outcome relation after conditioning on treatment and \(C\). The required treatment and mediator support must also be present: the outcome regression must be informed at the mediator combinations assigned by the intervention. These conditions support identification from observed data; the formulas define the estimands whether or not those conditions hold.

Worked Example: Positive Dependence and an Outcome Interaction

Consider a hypothetical population with no baseline covariates. Under the reference treatment \(a^*=0\), the joint mediator distribution is:

\(M_1\)	\(M_2\)	Probability
0	0	0.40
0	1	0.10
1	0	0.10
1	1	0.40

Each mediator has marginal probability 0.50 of being 1. The pair is positively associated: the joint probability that both equal 1 is 0.40, greater than the 0.25 that would result from independent draws. Suppose, hypothetically, the conditional mean outcome at \(a=1\), measured in outcome points, is \( \mu(1,m_1,m_2)=10+2m_1+3m_2+4m_1m_2 \).

Joint draw. Under the reference joint distribution, \(E(M_1)=0.50\), \(E(M_2)=0.50\), and \(E(M_1M_2)=P(M_1=1,M_2=1)=0.40\). Therefore:

\Psi(1;g_{\mathrm{joint}}) = 10+2(0.50)+3(0.50)+4(0.40) = 10+1+1.5+1.6 = 14.1\text{ points}.

Independent draw. Independent Bernoulli draws with the same two marginal probabilities have \(E(M_1M_2)=0.50(0.50)=0.25\). Thus:

\Psi(1;g_{\mathrm{ind}}) = 10+2(0.50)+3(0.50)+4(0.25) = 10+1+1.5+1 = 13.5\text{ points}.

The joint-draw mean exceeds the independent-draw mean by \(14.1-13.5=0.60\) outcome points. The mediator marginals are identical in the two calculations, so the difference comes from the interaction term and the different expected product \(E(M_1M_2)\).

When Does Changing Dependence Change the Mean?

A useful exact result is available when the conditional outcome mean has an additive form plus a product interaction. Suppose, at the treatment level being evaluated, it can be written as:

\mu(a,m_1,m_2,c) = \beta_0(a,c) +\beta_1(a,c)m_1 +\beta_2(a,c)m_2 +\beta_{12}(a,c)m_1m_2.

Within each \(C=c\), the joint and independent interventions have the same marginal distributions for \(M_1\) and \(M_2\). Consequently, their expected contributions from the intercept and the two main-effect terms agree. Only the product term can differ. The following derivation makes that claim explicit.

\begin{aligned} & E\{\mu(a,M_1,M_2,c)\mid g_{\mathrm{joint}}\} - E\{\mu(a,M_1,M_2,c)\mid g_{\mathrm{ind}}\} \\ &= \beta_{12}(a,c) \left[ E(M_1M_2\mid A=a^*,C=c) - E(M_1\mid A=a^*,C=c)E(M_2\mid A=a^*,C=c) \right] \\ &= \beta_{12}(a,c)\operatorname{Cov}(M_1,M_2\mid A=a^*,C=c). \end{aligned}

The last equality follows from the definition of conditional covariance: it is the conditional expected product minus the product of the conditional means. Standardizing over \(C\) gives:

\Psi(a;g_{\mathrm{joint}})-\Psi(a;g_{\mathrm{ind}}) = \sum_c P(C=c)\, \beta_{12}(a,c) \operatorname{Cov}(M_1,M_2\mid A=a^*,C=c).

Thus, under this outcome-mean model, the dependence contrast is zero if the interaction coefficient is zero in every stratum, or if the conditional mediator covariance is zero in every stratum. Otherwise, its direction depends on both the sign of the interaction and the sign of the dependence. The formula relies on the stated model; for a general nonlinear outcome mean, dependence can matter in ways not summarized by a single covariance.

Worked Example: Negative Dependence Reverses the Contrast

Now consider a different hypothetical reference population, again with no baseline covariates. Suppose the mediator probabilities are \(P(0,0)=0.10\), \(P(0,1)=0.40\), \(P(1,0)=0.40\), and \(P(1,1)=0.10\), where each ordered pair lists \((M_1,M_2)\). Both marginals are still 0.50, but the mediators are negatively associated because \(P(M_1=1,M_2=1)=0.10\), below the independent value 0.25.

Use the same hypothetical outcome mean as before: \( \mu(1,m_1,m_2)=10+2m_1+3m_2+4m_1m_2 \). Under the joint draw, the expected product is 0.10, giving:

\Psi(1;g_{\mathrm{joint}}) = 10+2(0.50)+3(0.50)+4(0.10) = 10+1+1.5+0.4 = 12.9\text{ points}.

Under independent draws, the marginals are unchanged and the expected product is \(0.50(0.50)=0.25\):

\Psi(1;g_{\mathrm{ind}}) = 10+2(0.50)+3(0.50)+4(0.25) = 13.5\text{ points}.

Here the joint-draw mean is \(12.9-13.5=-0.60\) points lower than the independent-draw mean. The interaction coefficient is positive, but the joint reference distribution has negative covariance. The direction therefore reverses relative to the positively associated example.

Dependence Must Be Defined Within Covariate Strata

When baseline covariates are present, the independent intervention is ordinarily defined by multiplying the mediator marginals conditional on the same covariate value. This matters because the covariates can affect both mediators. If one instead pools people across covariate strata before forming marginal distributions, the resulting intervention can change mediator distributions across \(C\) as well as their within-stratum dependence.

The target population distribution of \(C\) is also part of the estimand. Each stratum's joint and independent means are calculated under the mediator distributions conditional on that stratum, then averaged using the target stratum weights. The next example makes the distinction numerical.

Worked Example: Standardizing Stratum-Specific Dependence

Suppose a hypothetical target population has two equally common baseline strata, \(C=0\) and \(C=1\). In the reference treatment group, both mediator marginals equal 0.50 when \(C=0\), and the joint probability \(P(M_1=1,M_2=1\mid C=0)\) is 0.40. When \(C=1\), the marginals are \(P(M_1=1\mid C=1)=0.20\) and \(P(M_2=1\mid C=1)=0.80\), and the joint probability is 0.16. The latter equals the product \(0.20(0.80)\), so the mediators are conditionally independent in that stratum.

Use the same outcome mean in each stratum, \( \mu(1,m_1,m_2,c)=10+2m_1+3m_2+4m_1m_2 \). In \(C=0\), the joint-draw mean is \(14.1\) points, as calculated above, and the independent-draw mean is \(13.5\) points. In \(C=1\), both rules give the same expected product, 0.16, and therefore:

E\{\mu\mid C=1,g\} = 10+2(0.20)+3(0.80)+4(0.16) = 10+0.4+2.4+0.64 = 13.44\text{ points}

Because the strata each have weight 0.50, the standardized means are:

\begin{aligned} \Psi(1;g_{\mathrm{joint}})&=0.50(14.1)+0.50(13.44)=13.77\text{ points},\\ \Psi(1;g_{\mathrm{ind}})&=0.50(13.5)+0.50(13.44)=13.47\text{ points}. \end{aligned}

The standardized contrast is \(13.77-13.47=0.30\) points. Only the \(C=0\) stratum contributes a difference, and its 0.60-point difference is weighted by 0.50.

To see why conditioning matters, pool the mediator marginals across strata first. The pooled marginals are \(P(M_1=1)=0.50(0.50)+0.50(0.20)=0.35\) and \(P(M_2=1)=0.50(0.50)+0.50(0.80)=0.65\). Their product is 0.2275. But using that pooled product would not reproduce the specified within-stratum independent intervention: it would draw mediators without respecting each person's \(C\)-specific marginal distributions. It defines a different intervention and can yield a different estimand.

Practical Interpretation and Common Pitfalls

A joint draw is appropriate when the intervention is intended to retain the reference distribution's observed conditional pairing of mediators. An independent draw is a different hypothetical intervention: it keeps each mediator's conditional distribution but removes their conditional association. Neither is automatically the uniquely correct choice. The scientific question must specify which mediator-generating rule is meaningful, and results should name the rule used.

Pitfall: “The marginal distributions match, so the interventions are equivalent.” Matching marginals does not imply matching joint distributions. With an outcome interaction, the expected mediator product can differ, changing the mean outcome.
Pitfall: “Independence means no association after pooling everyone.” The intervention defined here makes draws independent conditional on \(C\). Marginal association can remain because the covariates influence both mediator distributions.
Pitfall: “The larger mean proves that dependence causes a better outcome.” The contrast compares two specified stochastic intervention distributions under a causal identification model. Its value alone does not establish a mechanism, nor does it imply that either intervention is feasible or clinically desirable.
Pitfall: “Every mediator pair can be assigned because the marginals are observed.” A product distribution may assign probability to combinations that are rare or absent in the data. Outcome predictions for those combinations require support and modeling scrutiny; marginal support alone does not guarantee joint support.

For a transparent analysis, state the reference treatment, the target population for \(C\), and whether the mediator draw is joint or independent. Then report the two interventional means, their contrast, and the assumptions and support needed to identify them. This makes clear whether a difference reflects altered mediator marginals or, as in the comparisons here, a deliberate change in their dependence.

Check Your Understanding

Use the definitions and calculations in this tutorial to answer the following.

If two binary mediators each have probability 0.40 of equaling 1, what is \(E(M_1M_2)\) under independent draws?
In the additive-plus-interaction outcome model, which term makes the joint and independent intervention means differ when the conditional mediator marginals match?
If the interaction coefficient is positive and the reference mediators have negative conditional covariance, what is the sign of the joint-minus-independent mean contrast under that model?
Why should the independent intervention typically multiply mediator marginals conditional on \(C\), rather than multiply marginals pooled across \(C\)?
Give one reason that matching both mediator marginals does not guarantee that the two interventions produce the same expected outcome.