The Cross-World Independence Assumption

Why a Mediation Formula Needs a Cross-World Link

A natural mediation effect combines counterfactuals from different treatment settings. For example, $Y(1,M(0))$ is the outcome under treatment level 1 if the mediator were set to the value that same individual would have had under treatment level 0. The outcome and mediator in this expression belong to different hypothetical worlds. This makes natural effects scientifically interpretable for some questions, but it also creates an identification challenge: data do not ordinarily reveal how those counterfactuals are related within an individual.

The cross-world independence assumption supplies one particular link. It is not a statement that the observed outcome and observed mediator are independent, and it is not guaranteed by randomizing treatment. It concerns a counterfactual outcome under one setting and a counterfactual mediator under another.

Definition: Let $C$ denote measured baseline covariates, $M(a')$ the mediator under treatment level $a'$, and $Y(a,m)$ the outcome under treatment level $a$ with the mediator set to $m$. The cross-world independence assumption is $$ Y(a,m) \perp\!\!\!\perp M(a') \mid C $$ for the treatment levels $a,a'$ and mediator values $m$ relevant to the estimand. Conditional on the same baseline covariates, the counterfactual outcome $Y(a,m)$ is assumed independent of the counterfactual mediator $M(a')$.

The condition is often invoked for the nested counterfactual mean $E[Y(a,M(a'))]$, which is a building block of natural direct and indirect effects. The assumption concerns the joint distribution of $Y(a,m)$ and $M(a')$, not just their separate distributions. In particular, knowing the mean outcome under each fixed mediator value and the distribution of the mediator under $a'$ does not by itself determine the mean of the nested counterfactual.

How the Assumption Enters the Mediation Formula

For clarity, first consider a discrete mediator and a fixed covariate value $C=c$. Decompose the nested outcome according to the mediator value that would occur under $a'$. The cross-world assumption permits the outcome counterfactual and that mediator counterfactual to be factored:

\begin{aligned} E[Y(a,M(a'))\mid C=c] &= \sum_m E[Y(a,m)\,I\{M(a')=m\}\mid C=c]\\ &= \sum_m E[Y(a,m)\mid C=c]\, P(M(a')=m\mid C=c). \end{aligned}

The first equality follows because exactly one mediator value is realized by $M(a')$: the indicator selects the corresponding $Y(a,m)$. The second equality follows from $Y(a,m) \perp\!\!\!\perp M(a')\mid C$. Without that independence, the expectation of the product of the outcome and indicator need not equal the product of their expectations.

To turn this expression into an observed-data formula also requires the usual identification conditions. In particular, treatment assignment must be exchangeable given $C$, mediator-outcome confounding must be controlled given treatment and $C$, and the required treatment and mediator values must have positive probability in the relevant covariate strata. Consistency links observed outcomes to the corresponding counterfactuals. Under these conditions, the counterfactual outcome mean at a fixed mediator value can be represented by $E[Y\mid A=a,M=m,C=c]$, and the mediator distribution under $a'$ by $P(M=m\mid A=a',C=c)$. Averaging over the baseline covariate distribution gives the mediation formula:

E[Y(a,M(a'))] = \int\!\!\int E[Y\mid A=a,M=m,C=c]\, f(m\mid A=a',C=c)\,f(c)\,dm\,dc.

For discrete variables, the integrals are replaced by sums. The cross-world assumption is the step that connects the two counterfactual worlds; the other identification conditions connect the separate counterfactual distributions to observed data. It is important not to treat the formula as a consequence of cross-world independence alone.

Worked Example: Calculating a Covariate-Adjusted Nested Mean

Suppose a hypothetical study concerns a treatment $A$, a binary mediator $M$, and a later blood-pressure outcome $Y$, measured in mmHg. Let $C$ have two baseline strata, $c_0$ and $c_1$, with probabilities 0.40 and 0.60. The following are hypothetical conditional outcome means under $A=1$ and mediator probabilities under $A=0$:

Baseline stratum	$P(M=1\mid A=0,C)$	$E[Y\mid A=1,M=0,C]$	$E[Y\mid A=1,M=1,C]$
$c_0$	0.25	10 mmHg	18 mmHg
$c_1$	0.60	12 mmHg	20 mmHg

We calculate the mediation-formula quantity for $E[Y(1,M(0))]$. Assume, for this hypothetical calculation, consistency, the required exchangeability conditions including cross-world independence, and positivity. In stratum $c_0$, the mediator is 1 with probability 0.25 under $A=0$, so:

(1-0.25)(10)+0.25(18)=7.5+4.5=12\text{ mmHg}.

In stratum $c_1$, the corresponding calculation is:

(1-0.60)(12)+0.60(20)=4.8+12=16.8\text{ mmHg}.

Now average over the baseline-stratum probabilities:

(0.40)(12)+(0.60)(16.8)=4.8+10.08=14.88\text{ mmHg}.

Thus the formula identifies a nested mean of 14.88 mmHg under the stated assumptions. This is a counterfactual mean, not automatically a causal effect by itself. A natural direct or indirect effect would involve a contrast between appropriate nested means.

Why Marginal Information Does Not Determine the Nested Mean

The cross-world assumption is substantive because it rules out some possible dependence structures. The following small hypothetical construction makes the role of that restriction explicit. Hold $C$ fixed, and suppose half the population has $M(0)=0$ and half has $M(0)=1$. Let $V$ be binary and define the potential outcomes under treatment 1 by $Y(1,0)=V$ and $Y(1,1)=1-V$. Consider two possible ways to pair $V$ with $M(0)$.

Worked Example: Two Cross-World Couplings, Two Nested Means

In coupling A, set $V=M(0)$. When $M(0)=0$, the nested outcome is $Y(1,0)=V=0$. When $M(0)=1$, it is $Y(1,1)=1-V=0$. Therefore $E[Y(1,M(0))]=0$.

In coupling B, set $V=1-M(0)$. When $M(0)=0$, the nested outcome is $Y(1,0)=V=1$. When $M(0)=1$, it is $Y(1,1)=1-V=1$. Therefore $E[Y(1,M(0))]=1$.

In both couplings, $P(M(0)=1)=0.5$, $E[Y(1,0)]=E[V]=0.5$, and $E[Y(1,1)]=E[1-V]=0.5$. The independence-based calculation would give:

$$ (0.5)(0.5)+(0.5)(0.5)=0.5. $$

That answer is neither nested mean above. In each coupling, $Y(1,m)$ and $M(0)$ are dependent, so the cross-world assumption fails. The example shows that the separate marginal distributions do not determine the nested mean; their cross-world dependence matters. It does not claim that these two constructions have identical full observed-data distributions under every study design. Rather, it isolates the unobserved joint feature that the assumption restricts.

For a general binary outcome, the uncertainty can also be expressed through bounds. If $Z=Y(1,1)$ and $M(0)$ are binary with $P(Z=1)=q$ and $P(M(0)=1)=p$, their joint probability satisfies

\max(0,p+q-1)\leq P(Z=1,M(0)=1)\leq \min(p,q).

This follows because a joint event cannot be more probable than either event, and $P(Z=1\text{ or }M(0)=1)\leq 1$ gives the lower bound. Independence selects the particular value $pq$ from this range. The joint probability can affect $E[Y(1,M(0))]$, so selecting $pq$ is an additional assumption, not a calculation forced by the marginal probabilities.

A Structural Model That Makes the Assumption Plausible

One sufficient, but not necessary, basis for cross-world independence is a structural model in which the mediator and outcome use independent sources of individual variation once baseline covariates are fixed. For example, suppose $M(a')$ is a function of $a'$, $C$, and an error $U_M$, while $Y(a,m)$ is a function of $a$, $m$, $C$, and a different error $U_Y$. If $U_M$ and $U_Y$ are independent conditional on $C$, then the two counterfactuals are independent conditional on $C$. This construction rules out a shared unmeasured cause of the mediator and outcome counterfactuals.

Worked Example: Independent Errors Imply the Factorization

Within one fixed baseline stratum, consider the hypothetical structural equations $M(0)=I(U_M<0.40)$ and $Y(1,m)=2+3m+2U_Y$, where $U_M$ is uniform on $[0,1]$, $U_Y$ is Bernoulli with success probability 0.50, and $U_M$ and $U_Y$ are independent. Let $Y$ be a score measured in points.

First, $P(M(0)=1)=0.40$. The fixed-mediator outcome means are:

E[Y(1,0)]=2+3(0)+2(0.50)=3\text{ points}, \qquad E[Y(1,1)]=2+3(1)+2(0.50)=6\text{ points}.

Because $M(0)$ depends only on $U_M$ and $Y(1,m)$ depends only on $U_Y$ once $m$ is fixed, their independence gives the mediation-formula calculation:

(1-0.40)(3)+0.40(6)=1.8+2.4=4.2\text{ points}.

Direct substitution into the nested counterfactual gives the same result:

E[Y(1,M(0))] =E[2+3M(0)+2U_Y] =2+3(0.40)+2(0.50) =4.2\text{ points}.

This model demonstrates a sufficient structural explanation for the assumption. It does not show that independent errors are necessary, or that observed data establish that the errors are independent.

Why Observed Data Generally Cannot Test It

For each individual, the data reveal at most one treatment setting and one realized mediator and outcome. They do not jointly reveal $Y(a,m)$ and $M(a')$ for different hypothetical settings. Randomization can support exchangeability of treatment assignment, but it does not reveal or force the dependence between a person’s potential outcome under one mediator setting and that same person’s potential mediator under another treatment setting.

Consequently, an observed-data analysis may estimate the separate conditional distributions used in the mediation formula while providing no direct check of their cross-world factorization. A good fit of an outcome model, or an absence of measured mediator-outcome association after adjustment, does not establish cross-world independence. Additional causal structure or scientific knowledge may make the assumption more or less credible, but the assumption is not generally testable from observed data alone.

This limitation also clarifies why measured baseline covariates matter. The assumption is usually stated conditional on $C$, allowing the distributions of counterfactuals to vary across covariate strata. Averaging across $C$ does not justify replacing conditional independence with unconditional independence, nor does unconditional independence imply conditional independence. The conditioning set and the causal role of its variables must be specified.

Common Mistake

Wrong reasoning: “Treatment was randomized, so $Y(a,m)$ must be independent of $M(a')$.” Randomization concerns assignment of the observed treatment, not the joint distribution of two counterfactuals for the same individual.
Correction: State the cross-world condition separately, conditional on the chosen baseline covariates. Then assess whether the scientific model or design makes it plausible; do not infer it from treatment randomization alone.
Wrong reasoning: “The outcome and mediator are associated in the observed data, so cross-world independence is false.” That association is not the counterfactual independence statement.
Correction: Observed association and cross-world dependence are different quantities. The assumption is generally not empirically testable, even when observed mediator and outcome distributions are available.

Key takeaway: Cross-world independence factors the joint counterfactual contribution needed for the natural mediation formula. It is a substantive assumption about counterfactuals from different treatment settings, not an implication of randomization and not a restriction that observed data can generally verify.

Check Your Understanding

Use the definitions and distinctions in this tutorial to answer the following questions.

Write the cross-world independence condition for $Y(0,m)$ and $M(1)$, conditional on baseline covariates $C$.
In the derivation of the mediation formula, which factorization step uses cross-world independence?
Why does randomizing $A$ not, by itself, establish independence between $Y(a,m)$ and $M(a')$?
In the two-coupling example, what separate probabilities and means were held fixed, and what quantity changed?
Give one structural-model condition that is sufficient for cross-world independence, and explain why it is sufficient.
Does a well-fitting observed-data model test the cross-world assumption? Explain briefly.

Baseline stratum	\(P(M=1\mid A=0,C)\)	\(E[Y\mid A=1,M=0,C]\)	\(E[Y\mid A=1,M=1,C]\)
\(c_0\)	0.25	10 mmHg	18 mmHg
\(c_1\)	0.60	12 mmHg	20 mmHg