Longitudinal TMLE for Survival Risk

Why survival requires a longitudinal target

A survival probability at a specified time is a marginal probability: it describes the proportion of a target population that would remain event-free under a treatment policy. It is not, in general, the survival probability among people whose event status happened to be observed. Follow-up can end before the target time, and treatment decisions or covariates may change over time. A longitudinal targeted maximum likelihood estimator (TMLE) represents these features as a sequence, then targets the sequential regressions to the survival parameter of interest.

We work in discrete time, with intervals \(t=1,\ldots,K\). This is natural when events are recorded at scheduled visits or when the scientific target is survival through a specified visit. The endpoint is the end of interval \(K\); other time points are handled by choosing the corresponding interval as the endpoint.

This convention makes the order explicit: an event in interval \(t\) is observed before censoring after that interval. Someone censored after remaining event-free in interval \(t\) has known event-free status through \(t\), but their later event status is unknown. A person who has an event is no longer at risk in later intervals.

Policy-specific survival and its assumptions

Let \(W\) denote baseline covariates, \(A_t\) treatment in interval \(t\), and \(L_t\) time-varying covariates measured before that interval's treatment. Write \(\mathcal H_t\) for the observed covariate and treatment history available just before \(A_t\). A longitudinal treatment policy \(g\) specifies a distribution \(g_t(a\mid\mathcal H_t)\) for treatment at each time. A deterministic regime is the special case that assigns a single treatment value at each history.

The target \(S^g(K)=P(T^g>K)\) is survival through \(K\) if treatment were assigned according to \(g\). Its complement \(1-S^g(K)\) is the policy-specific event risk. Identification from observed data requires consistency, sequential exchangeability of treatment, positivity for treatment values assigned by \(g\), and conditionally independent censoring. In particular, conditional on the observed history and treatment, censoring must not carry additional information about future event outcomes. The corresponding censoring probabilities must also be positive on histories relevant under \(g\). These assumptions are substantive; targeting does not make them true.

Define a backward sequence of conditional survival regressions under the full-data law. Set \(q_{K+1}=1\). For \(t=K,\ldots,1\), define

For continuous treatment, the sum is an integral. In the final interval \(q_{K+1}=1\), so \(m_K\) is the conditional probability of surviving interval \(K\). At earlier times, the regression averages the probability of surviving the current interval and all remaining intervals. Censoring makes some later outcomes unobserved; fitting and targeting these regressions therefore requires appropriate censoring adjustment.

Proof. Under policy \(g\), conditional on the history before treatment in interval \(t\), treatment is drawn from \(g_t\). Conditional on the treatment value \(a\), surviving the interval and all subsequent intervals has probability equal to the conditional expectation of \((1-D_t)q_{t+1}(\mathcal H_{t+1})\). Averaging first over the policy's treatment distribution gives \(q_t(\mathcal H_t)\). Starting at \(t=K\), where the probability of surviving beyond \(K\) after interval \(K\) is represented by \(q_{K+1}=1\), and iterating this argument backward yields the probability of surviving all \(K\) intervals conditional on the initial history. Averaging over the baseline history gives \(E[q_1(\mathcal H_1)]\). Exchangeability and consistency replace the counterfactual conditional distributions in this recursion by the corresponding observed-data distributions; positivity and independent censoring ensure those distributions are estimable from observed histories. \(\square\)

How longitudinal targeting enters

An initial estimate is obtained by fitting the sequence of regressions from the last interval backward, substituting each fitted \(q_{t+1}\) into the regression for interval \(t\), and integrating over the policy \(g_t\). The final initial plug-in estimate is the sample average of the fitted \(q_1\). A longitudinal TMLE then updates these regressions so that their empirical score equations target the policy-specific survival parameter.

For a binary survival outcome, a standard fluctuation is logistic. At step \(t\), let \(Z_t=(1-D_t)q_{t+1}(\mathcal H_{t+1})\), using the updated later regression. The conditional mean model is fluctuated as

Here \(\mathsf H_t\) is the clever covariate: it incorporates the likelihood ratio for following \(g\) rather than the observed treatment mechanism and the inverse probability of remaining uncensored for the required outcome. A common schematic form on the relevant observed risk set is

where \(p_s\) is the observed treatment probability and \(G_{t+1}\) is the conditional probability of remaining uncensored long enough to observe the outcome needed at this step. Exact risk-set indicators and the censoring survival term must match the data's event/censoring timing convention. The fluctuation parameter is chosen to solve the weighted score equation, schematically \(P_n[\mathsf H_t(Z_t-m_t^{\epsilon})]=0\) over records contributing an observed outcome to that step. If an event makes the continuation outcome known to be zero, that known outcome can be used without requiring later follow-up.

After updating \(m_t\), integrate it under \(g_t\) to obtain the updated \(q_t\); then use that updated regression in the preceding step. The final estimate is the average of the updated baseline predictions. This backward procedure ensures that changes at a later step are propagated into earlier predictions rather than being added only to the terminal estimate.

Derivation. For one contribution with outcome \(Z\), fitted mean \(\mu_\epsilon=\operatorname{expit}(\operatorname{logit}\widehat m+\epsilon\mathsf H)\), and weight \(w\), the Bernoulli log-likelihood contribution is \(w\{Z\log\mu_\epsilon+(1-Z)\log(1-\mu_\epsilon)\}\). Since \(\partial\mu_\epsilon/\partial\epsilon=\mathsf H\mu_\epsilon(1-\mu_\epsilon)\), differentiating gives

Summing over observations gives the stated score. If \(Z\) is a fitted continuation probability rather than a binary value, the same equation is the score of the corresponding weighted Bernoulli quasi-likelihood. This derivation explains the score being solved; it does not by itself establish asymptotic normality or cure misspecified nuisance models. \(\square\)

Diagnostics, alternatives, and limitations

Several checks are especially important for survival targeting. Inspect estimated treatment and censoring probabilities over the histories that contribute to the policy target. Very small probabilities create large clever covariates, so a few people can dominate the update. Check the number at risk and the number censored in each interval; sparse late risk sets can make conditional regressions unstable. Also verify that fitted conditional survival probabilities lie in \([0,1]\), that the backward recursion uses the updated later regressions, and that the final risk is exactly one minus the final survival estimate.

A useful comparison is with the parametric g-formula, which also models sequential conditional distributions and averages under a policy, but does not by itself perform the TMLE targeting step. Inverse-probability-weighted estimation instead relies directly on treatment and censoring weights. Kaplan–Meier estimation is a standard option for an unadjusted survival curve under its censoring assumptions, but it does not generally identify survival under a time-varying treatment policy with confounding. These approaches answer different questions and rely on different modeling or design conditions; TMLE is not automatically superior when its assumptions fail.

• Common mistake: treating observed event statuses as a random sample. The observed-status event proportion need not estimate the full-cohort risk under independent censoring alone. For example, if half the cohort has an event at \(\tau/4\) and half has no event by \(\tau\), and independent censoring occurs at \(\tau/2\) for half the cohort, all events are observed but only half of event-free people have known status through \(\tau\). The observed-status event proportion is \(0.5/(0.5+0.25)=2/3\), while the full-cohort risk is \(1/2\). Use a suitable survival or censoring-adjusted procedure; a crude observed-status fraction needs additional assumptions, such as complete follow-up independent of event time.
• Confusing survival among survivors with survival in the original population. A conditional interval survival probability must be multiplied through the recursion and averaged over the appropriate policy-induced histories.
• Assuming targeting repairs poor overlap. If the policy assigns treatments rarely observed at relevant histories, the likelihood ratios can be unstable or undefined. Targeting cannot recover information absent from the data.

The method also depends on a correctly specified event and censoring time convention, sufficient measurements of confounders, and nuisance estimates adequate for the target. A discrete-time analysis may be inappropriate if visit intervals are too coarse for the scientific question. TMLE's bounded plug-in form is useful, but it does not eliminate bias from unmeasured confounding, informative censoring left unaccounted for, or severe positivity violations.