Causal Survival Targets for Weighted Marginal Structural Models

Why Specify a Survival Target?

A causal question about survival is not fully defined by saying that one treatment “improves survival.” The answer depends on the treatment strategies being compared, the event of interest, and the time scale. It also depends on what summary is wanted: the probability of remaining event-free at a particular time, the risk of an event by that time, the average event-free time over a fixed horizon, or a comparison of instantaneous event rates.

In a longitudinal study, a treatment regime specifies what treatment is assigned at each decision time, possibly as a function of measured history. The survival quantities below are defined for the counterfactual event time that would occur if everyone followed a particular regime. These are population-level targets; they are not, by themselves, claims about how a particular weighted estimator is computed.

Here \(d\) can be a static strategy, such as receiving treatment at every decision time, or a dynamic strategy that assigns treatment according to measured history. The event must be specified: for example, death from any cause, recurrence, or a composite endpoint. If death is a competing event for a nonfatal outcome, the target must state whether it concerns the cumulative incidence of that event or a different survival outcome. A target also requires a time origin and a horizon. The same pair of regimes can have different risk contrasts at different times.

Identification: When Do the Counterfactual Targets Have an Observed-Data Interpretation?

The causal definitions above do not alone tell us whether the targets can be learned from observed data. For a standard longitudinal treatment setting, let \(A_k\) be treatment at decision time \(k\), and let \(L_k\) denote measured covariate information available before that decision. Write \(H_k\) for the observed history available just before \(A_k\). For a deterministic regime \(d\), consistency requires that a person whose observed treatment history follows \(d\) has observed event time equal to \(T^d\). The usual identification conditions also include no interference between individuals, sequential exchangeability, and treatment positivity.

Sequential exchangeability means that, at each decision time, conditional on \(H_k\), treatment assignment is independent of the counterfactual outcomes under the regimes of interest. Positivity means that every treatment prescribed by a regime has positive conditional probability for histories that can arise under that regime. These are substantive assumptions, not properties established merely by fitting a treatment model. For outcomes affected by loss to follow-up, identification additionally requires an appropriate conditional independent-censoring assumption and positive probability of remaining observed over the relevant follow-up, given measured history.

The following derivation gives the treatment-weighting identity for a deterministic regime. Let \(Y^d\) be any bounded counterfactual outcome under \(d\), such as \(I(T^d>t)\). Let \(g_k(d_k(H_k)\mid H_k)\) denote the conditional probability of receiving the treatment specified by \(d\) at time \(k\), given the observed history. Under consistency, sequential exchangeability, and positivity,

To see why, sum over the possible covariate histories in sequence. At the final decision time, conditional on \(H_K\), the indicator that the prescribed treatment is received has expectation \(g_K(d_K(H_K)\mid H_K)\). It cancels the final denominator factor. Exchangeability and consistency then identify the outcome contribution among those receiving that treatment with the corresponding counterfactual outcome contribution under \(d_K\). Repeating this cancellation at times \(K-1,\ldots,0\) removes every treatment probability factor. The remaining distribution is the distribution of covariate histories generated while following \(d\), and its outcome expectation is \(E[Y^d]\). Positivity ensures that all denominators used in this argument are nonzero. Applying the identity to \(Y=I(T>t)\) identifies \(S^d(t)\); applying it to \(Y=I(T\leq t)\) identifies \(F^d(t)\). If censoring occurs, treatment weights alone do not repair informative loss to follow-up; censoring must also be addressed under its own assumptions. \(\square\)

Survival, Risk, and Restricted Mean Survival Time

The survival function \(S^d(t)\) answers a direct question: under regime \(d\), what proportion of the target population would remain event-free beyond time \(t\)? The cumulative risk \(F^d(t)\) answers the complementary question: what proportion would experience the event by \(t\)? A risk difference at a chosen time is \(F^{d_1}(t)-F^{d_0}(t)\); a risk ratio is \(F^{d_1}(t)/F^{d_0}(t)\), when the denominator is positive. These contrasts are often readily interpreted in absolute terms.

Restricted mean survival time (RMST) is the expected event-free time accumulated before \(\tau\). It has units of time and does not require a proportional-hazards model. For any nonnegative event time, its integral representation follows from the tail-integral formula. Since \(\min(T^d,\tau)=\int_0^\tau I(T^d>t)\,dt\), and the integrand is nonnegative, Tonelli’s theorem permits the expectation and integral to be interchanged:

This identity is useful for interpretation and calculation: the area under a regime-specific survival curve through \(\tau\) is the RMST. The contrast \(\mu^{d_1}(\tau)-\mu^{d_0}(\tau)\) is the difference in mean event-free time accumulated through that horizon. It is not the difference in unrestricted mean event times unless the horizon and tail behavior justify that interpretation.

Marginal Hazards and Marginal Hazard Ratios

When \(T^d\) has a continuous distribution with density \(f^d(t)\), its marginal hazard at time \(t\), where \(S^d(t)>0\), is

The hazard is an instantaneous event rate among people who have remained event-free up to \(t\) under regime \(d\). It is not the probability of an event in a fixed interval, and it is not a risk ratio. The marginal hazard ratio comparing two regimes is

when the denominator is positive. A single hazard ratio is a meaningful summary only under a proportional-hazards restriction, \(\lambda^{d_1}(t)=\exp(\beta)\lambda^{d_0}(t)\) for all times in the stated range, or under an explicitly chosen summary of a time-varying ratio. Even if treatment has a simple effect within baseline subgroups, the marginal hazard ratio can change over time: the surviving populations under the two regimes can have different compositions.

Comparing Targets and Checking Their Meaning

These alternatives are not interchangeable. A risk difference is an absolute contrast at a fixed time; an RMST difference summarizes area under the curves over a fixed interval; a hazard ratio compares instantaneous rates in regime-specific survivor populations. In particular, a hazard ratio below one does not specify the absolute risk reduction without additional information about baseline hazards and time. The introductory material on marginal structural models supplies the broader weighting context; here the key task is to decide which survival estimand answers the scientific question.

Common Pitfalls and Diagnostics

• Leaving the horizon implicit. RMST and risk contrasts depend on \(\tau\) or \(t\). State the horizon and units in the estimand and report.
• Calling a hazard a probability. A hazard is a rate conditional on survival to that time; it is not a risk over a fixed interval.
• Assuming a marginal hazard ratio is constant. Proportional hazards is a substantive model restriction, not a consequence of using a marginal structural model. Inspect time-specific survival patterns and consider whether one number adequately represents the contrast.
• Ignoring support and censoring. Sparse treatment histories or near-zero treatment probabilities can make weighted targets unstable. If follow-up is incomplete, check whether censoring is plausibly independent conditional on measured history and whether adequate follow-up support exists through the chosen horizon.
• Overinterpreting truncated-weight estimates. The earlier material on truncation explains that clipping can change the target or introduce bias. State the truncation rule and distinguish the target defined by the causal regime from the behavior of a particular clipped estimator.
• Reporting a single summary without the survival curves or risks. A hazard ratio can conceal crossing or non-proportional curves. When scientifically useful, report regime-specific survival probabilities at clinically relevant times alongside a time-based summary such as RMST.

A final limitation is that no target repairs unmeasured confounding, inconsistency, or lack of positivity. Likewise, a well-defined estimand does not guarantee precise estimation: limited events and heavy censoring can leave late-time survival or hazard comparisons weakly supported. Restricting interpretation to horizons with adequate follow-up is often more defensible than extrapolating beyond the observed information.