Cross-World Assumptions for Natural Effects

Why Cross-World Assumptions Matter

Natural direct and indirect effects compare outcomes under a treatment value with mediator values that would arise under another treatment value. That combination is the defining feature of a natural effect—and its main identification difficulty. The outcome and mediator values in such a contrast generally cannot both be observed for the same person.

With competing events, the outcome must also preserve the actual event structure. Let $T(a,m)$ be the event time and $J(a,m)$ the event cause if treatment were set to $a$ and the mediator to $m$. For cause $k$ by a fixed time $t$, define $Y_k(t;a,m)=I\{T(a,m)\le t,\ J(a,m)=k\}$. This indicator is one only if cause $k$ occurs by $t$ before any competing event. It is not an outcome in a hypothetical world where competing events have been removed.

The introductory material on natural direct and indirect effects with competing events defines the contrasts. Here the focus is the extra assumption that permits their identification, why that assumption is stronger than ordinary observed-data comparisons, and how competing events sharpen the interpretation.

Natural Effects and the Cross-World Condition

Definition: $M(a')$ is the mediator value that would arise under treatment $a'$. The nested potential outcome $Y_k(t;a,M(a'))$ is the cause-$k$ event indicator by time $t$ under outcome treatment $a$, with the mediator set to the value it would naturally take under treatment $a'$. For binary treatments $0$ and $1$, one reference convention defines the natural direct effect on the risk-difference scale as $E[Y_k(t;1,M(0))]-E[Y_k(t;0,M(0))]$, and the natural indirect effect as $E[Y_k(t;1,M(1))]-E[Y_k(t;1,M(0))]$. Their sum is the total effect $E[Y_k(t;1,M(1))]-E[Y_k(t;0,M(0))]$.

These are contrasts between population risks, not statements that an individual’s event was caused by a particular pathway. The chosen reference treatment matters: a direct effect holding the mediator distribution to $M(0)$ need not equal one holding it to $M(1)$, especially when treatment and mediator interact in their effects on risk.

Cross-world independence assumption: Conditional on measured baseline covariates $C$, the cause-$k$ potential outcome under treatment $a$ and mediator value $m$ is independent of the mediator value under treatment $a'$: $$ Y_k(t;a,m)\ \perp\ M(a')\mid C $$ for the treatment values, mediator values, causes, and times involved in the estimand. A stronger process-level version asserts conditional independence between the full event-time-and-cause outcome under $(a,m)$ and $M(a')$.

The condition pairs counterfactuals from different treatment worlds. It says that, within baseline-covariate strata, knowing a person’s mediator value under $a'$ gives no further information about that person’s cause-$k$ outcome under the intervention $(a,m)$. It does not say that the mediator and outcome are unrelated in observed data. Nor does it follow merely from random assignment of treatment.

Identification Requires More Than Cross-World Independence

The cross-world condition is part of an assumption package. One also needs well-defined interventions and consistency: a person who actually receives $A=a$ and has $M=m$ has the corresponding potential outcome, and the recorded treatment and mediator represent the interventions being compared. There must be no interference between individuals. The mediator must be meaningfully defined under each treatment regime for the target population; if a competing event makes it undefined, the natural effect needs a different, explicitly defined estimand.

For identification from observed data, assume treatment is randomized or conditionally exchangeable given $C$, so treatment assignment carries no residual information about the relevant potential outcomes and mediators after conditioning on $C$. Also assume mediator-outcome exchangeability: within treatment and baseline-covariate strata, there is no unmeasured common cause of the mediator and the potential outcome. In particular, there must not be an unaddressed mediator-outcome confounder caused by treatment. These assumptions are distinct from cross-world independence; none should be silently substituted for another.

Under these assumptions, define the observed conditional risk $\mu_k(t,a,m,c)=P(Y_k(t)=1\mid A=a,M=m,C=c)$. For discrete mediators, the mediation formula is

E\{Y_k(t;a,M(a'))\} = \int \sum_m \mu_k(t,a,m,c)\, P(M=m\mid A=a',C=c)\,dF_C(c).

For a continuous mediator, replace the sum by an integral and the conditional probability by a density:

E\{Y_k(t;a,M(a'))\} = \int\!\int \mu_k(t,a,m,c)\, f(m\mid A=a',C=c)\,dm\,dF_C(c).

The formula averages the outcome risk under treatment $a$ over the mediator distribution under treatment $a'$, then averages over the baseline-covariate distribution of the target population. It identifies the natural-effect risks only if the assumptions and support conditions hold; fitting the two observed-data models alone does not establish them.

Why the Formula Follows

For a fixed $C=c$, first average the nested outcome over mediator values that would occur under $a'$. The cross-world independence condition permits the outcome risk at each $m$ to be separated from the distribution of $M(a')$. Exchangeability and consistency then replace these counterfactual distributions with the corresponding observed conditional distributions.

\begin{aligned} E\{Y_k(t;a,M(a'))\mid C=c\} &=\int E\{Y_k(t;a,m)\mid M(a')=m,C=c\} \,dF_{M(a')\mid C=c}(m)\\ &=\int E\{Y_k(t;a,m)\mid C=c\} \,dF_{M(a')\mid C=c}(m)\\ &=\int \mu_k(t,a,m,c)\, dF_{M\mid A=a',C=c}(m). \end{aligned}

The first equality is averaging over the natural mediator value. The second uses cross-world independence. The last uses the treatment, mediator-outcome, and consistency assumptions to connect potential quantities to observed data. Finally, averaging over $F_C$ gives the population formula. Without the second equality, the conditional outcome may depend on which people would have mediator value $m$ under $a'$; the product-form mediation formula need not be valid.

Worked Example: Discrete Mediator and Competing-Risk Risks

Suppose a hypothetical randomized study has no baseline covariates. The mediator $M$ is binary. At a fixed follow-up time, the observed conditional cause-1 risks are $0.10$ and $0.20$ under treatment $A=0$ for $M=0$ and $M=1$, respectively; under $A=1$, they are $0.18$ and $0.30$. Suppose $P(M=1\mid A=0)=0.20$ and $P(M=1\mid A=1)=0.50$. For coherence, the corresponding competing-event risks are $0.15, 0.20, 0.22,$ and $0.25$, respectively. Each pair of cause-specific cumulative incidences sums to at most one.

Write $Q(a,a')=E\{Y_1(t;a,M(a'))\}$. Under the identification assumptions, the mediator distribution under $a'$ supplies the weights. Thus

\begin{aligned} Q(0,0)&=(0.10)(0.80)+(0.20)(0.20)=0.120,\\ Q(1,0)&=(0.18)(0.80)+(0.30)(0.20)=0.204,\\ Q(1,1)&=(0.18)(0.50)+(0.30)(0.50)=0.240. \end{aligned}

The natural direct risk difference using $M(0)$ as the reference mediator distribution is $Q(1,0)-Q(0,0)=0.204-0.120=0.084$, or 8.4 percentage points by the stated follow-up time. The natural indirect risk difference under treatment $1$ is $Q(1,1)-Q(1,0)=0.240-0.204=0.036$, or 3.6 percentage points. Their sum is $0.120$, the total cause-1 risk difference $Q(1,1)-Q(0,0)$. These are hypothetical calculations; the causal interpretations require the assumptions above. The competing-event risks remain part of the outcome process rather than being treated as censoring.

Positivity and Continuous Mediators

The formula also requires support. Treatment positivity requires positive probability of each treatment level used in the contrast within the relevant $C$ strata. For a discrete mediator, mediator values assigned positive probability by the distribution being averaged over must also have positive probability under the outcome treatment regime, within those strata. Otherwise the observed conditional risk $\mu_k(t,a,m,c)$ is unavailable for part of the required average.

For a continuous mediator, an individual value ordinarily has probability zero, so positivity is not a requirement that each value have positive probability. Instead, there must be appropriate support or density overlap. For the formula involving outcome treatment $a$ and mediator source treatment $a'$, mediator values with positive density under $f(m\mid A=a',C=c)$ must be supported under the outcome-regime mediator distribution $f(m\mid A=a,C=c)$, within relevant covariate strata. Estimating a conditional outcome mean at unsupported values is extrapolation, not a consequence of the identification formula.

Worked Example: Continuous Mediator With Support Overlap

In a hypothetical setting, let $M\mid A=0$ be uniform on $[0,0.8]$, and let $M\mid A=1$ be uniform on $[0,1]$. Suppose the conditional cause-1 risk by time $t$ is $\mu_1(t,a,m)=0.10+0.04a+0.03m$. Suppose the competing-event risk is $0.18+0.02a+0.01m$. Throughout the stated ranges, both risks are nonnegative and their sum is below one.

The mediator means are $E(M\mid A=0)=0.4$ and $E(M\mid A=1)=0.5$. Because the risk is linear in $m$, averaging it over each mediator distribution gives

\begin{aligned} Q(0,0)&=0.10+0.03(0.4)=0.112,\\ Q(1,0)&=0.10+0.04+0.03(0.4)=0.152,\\ Q(1,1)&=0.10+0.04+0.03(0.5)=0.155. \end{aligned}

The natural direct risk difference is $0.152-0.112=0.040$, or 4.0 percentage points; the natural indirect risk difference is $0.155-0.152=0.003$, or 0.3 percentage points. The total difference is $0.043$, or 4.3 percentage points. For the required contrasts, the source distribution $M(0)$ is supported throughout $[0,0.8]$, which is contained in the support of $M\mid A=1$; the $M(1)$ distribution is also supported under outcome treatment $1$. Thus the mediator support needed for these contrasts is present. Reversing the relevant treatment roles could require support not available here.

Why Competing Events Make the Assumption Consequential

For a cause-specific cumulative incidence, $Y_k(t;a,m)$ is determined not only by the occurrence of cause $k$, but also by whether a competing event occurs first. A mediator can alter either event process. Consequently, a natural effect on cause $k$ may reflect changes in both the cause-$k$ process and the competing-event process. It is not generally an effect on a cause-$k$ event time with competitors removed.

The cross-world condition is especially consequential because it links a mediator value generated in one treatment world to an event outcome in another. Randomization can support comparisons of treatment regimes, but it does not reveal whether people with a particular $M(a')$ would have had the same distribution of competing-risk outcomes under $(a,m)$ as other people with the same $C$. This dependence concerns joint counterfactuals and is not directly testable from observed data.

A further issue arises if a competing event prevents mediator measurement or makes the mediator conceptually undefined. For example, a mediator defined as a post-treatment biomarker may not exist for someone who has already experienced a fatal competing event. One cannot automatically assign such a person a hypothetical mediator value and apply the natural-effect formula. The mediator and intervention must be defined for the target population and timing, or a different estimand is needed.

Worked Example: Randomization Does Not Ensure Cross-World Independence

Consider a hypothetical population with no measured baseline covariates and a binary latent factor $U$, with $P(U=1)=0.5$. Treatment is randomized. Under either treatment, let the mediator be generated so that $P(M(a')=1\mid U=1)=0.8$ and $P(M(a')=1\mid U=0)=0.2$. Under any fixed outcome intervention $(a,m)$, let the cause-1 risk be $0.05+0.30U+0.15m$, and the competing-event risk be $0.10+0.10U$. These risks are nonnegative and sum to no more than $0.70$.

The mediator has marginal probability $P(M(a')=1)=(0.5)(0.8)+(0.5)(0.2)=0.5$. Among people with $M(a')=1$, Bayes’ rule gives $P(U=1\mid M(a')=1)=0.8$. Therefore, for any fixed $a,m$,

\begin{aligned} E\{Y_1(t;a,m)\mid M(a')=1\} &=0.05+0.30(0.8)+0.15m\\ &=0.29+0.15m,\\ E\{Y_1(t;a,m)\} &=0.05+0.30(0.5)+0.15m\\ &=0.20+0.15m. \end{aligned}

The conditional and marginal risks differ by $0.09$, so $Y_1(t;a,m)$ is not independent of $M(a')$. Randomization of treatment has not removed their common dependence on $U$. The same latent factor also affects competing-event risk. Without sufficient measurement and adjustment for such common causes—or a defensible alternative identifying argument—the cross-world condition and the resulting natural-effect identification are not justified.

Assumptions, Diagnostics, and Alternatives

No observed-data diagnostic can prove cross-world independence. Useful checks can still reveal problems with the rest of the identification strategy: examine treatment and mediator overlap within covariate strata; flag estimates that rely on extreme extrapolation; assess whether the mediator is defined before competing events can preclude its measurement; and investigate plausible unmeasured mediator-outcome common causes. Sensitivity analyses can show how natural-effect estimates change under departures from the assumed cross-world relationship, but they do not turn the assumption into an empirically verified fact.

Natural effects answer a more demanding question than a total treatment effect on observed cumulative incidence. A total effect compares the risks under treatment regimes without splitting the contrast according to a mediator’s natural value in another world. A controlled direct effect instead compares outcomes when the mediator is set to a specified common value; it does not use $M(a')$. These alternatives can be more directly aligned with an intervention or require different assumptions, but they answer different causal questions. The natural-effect decomposition is useful when its pathway interpretation is substantively meaningful and its assumptions are credible; it is not automatically the preferred analysis simply because the total effect can be decomposed algebraically.

Common pitfall: “Treatment was randomized, so the natural direct and indirect effects are identified.” Randomization addresses treatment assignment, not the dependence between $Y_k(t;a,m)$ and $M(a')$, nor unmeasured mediator-outcome confounding. The correction is to state and justify the full identification assumptions, including cross-world independence, and to check the relevant support conditions before applying the mediation formula.

Check Your Understanding

Answer each question, then compare with the concise solutions below.

In words, what does $Y_k(t;a,m)\perp M(a')\mid C$ assert, and why does randomization of $A$ alone not establish it?
A binary mediator value has positive probability under $M(a')$ but zero probability under treatment $a$ in a covariate stratum. What is the problem for the mediation formula?
For a continuous mediator, why is “every mediator value must have positive probability” not the right positivity condition?
Can a natural effect on cause-$k$ cumulative incidence be interpreted as an effect with competing events eliminated? Explain.

Solutions. (1) Within each $C$ stratum, the outcome under $(a,m)$ carries no further information about the mediator value under $a'$; randomization concerns $A$, not dependence between these two potential quantities. (2) The required outcome risk under $(a,m)$ is unsupported for a mediator value receiving weight under $a'$, so the formula requires extrapolation and is not identified from the observed data there. (3) Individual values have probability zero for continuous distributions; instead, mediator values carrying density under the source treatment distribution must lie in supported regions of the outcome-regime mediator distribution, within relevant covariate strata. (4) No. The outcome indicator records cause $k$ occurring by $t$ before a competing event, so its natural effect includes how interventions affect the competing-event process as well.