Censoring-Adjusted Survival Estimation Under Dropout

Why Dropout Must Be Distinguished From an Event

In longitudinal survival data, an individual may experience the event of interest or may stop being observed before the event status is known. These are different outcomes. Coding a person who drops out as event-free through the end of follow-up changes the data; coding dropout as an event changes the scientific question. Censoring-adjusted estimation instead uses the observed history and assumptions about why follow-up ended to recover the survival quantity of interest.

We use discrete intervals \(t=1,\ldots,K\), with a fixed treatment regime understood throughout. The focus here is dropout, so treatment assignment is treated as already handled. Let \(T\) denote the event time, and define \(\Delta_t=I(T=t)\), the event indicator in interval \(t\), among people event-free at its start. Let \(S_t=I(T>t)\) denote survival beyond interval \(t\). Define \(R_t=1\) if the person is observed and event-free at the start of interval \(t\), with \(R_1=1\). After surviving interval \(t\), \(R_{t+1}=1\) means follow-up continues and the next history can be observed. Because dropout is monotone, once \(R_{t+1}=0\), later \(R_j=0\) for \(j>t+1\). A person who has the event is no longer at risk, which is not the same as being censored.

The timing convention matters: an event during interval \(t\) is recorded before the dropout decision for a person who survives that interval. Thus \(R_{t+1}\) is needed to observe subsequent follow-up, not to decide whether an event in interval \(t\) occurred. At the final horizon, dropout after the last event assessment does not affect whether \(S_K\) was observed.

Identification and the Sequential Survival Recursion

Let \(H_t\) contain the information available at the start of interval \(t\), including relevant past covariates. Under a fixed regime, define \(Q_{K+1}=1\), and, for a person alive and observed at the start of \(t\), define the full-data conditional survival function recursively by

When \(T=t\), the factor \(1-\Delta_t\) is zero, so later survival contributes nothing. If the person survives interval \(t\), that factor is one and the recursion averages future survival over the next history. Since \(Q_{K+1}=1\), \(Q_K\) is simply the conditional probability of surviving the last interval. The target marginal survival probability is \(E[Q_1(H_1)]\) under the target baseline population and regime.

Dropout makes some terms in this full-data recursion unobserved. Identification requires sequential conditional independent censoring: among people alive and observed at the start of an interval, conditional on the recorded history used in the analysis, continuing observation is independent of the subsequent full-data event and covariate outcomes relevant to the target. It also requires retention positivity: every history with positive probability in the target population has a positive probability of continued observation when that observation is needed. These conditions do not say that dropout is unrelated to prognosis unconditionally. They say that the recorded history is sufficient to account for the relationship.

Proof. Write \(Z_t=(1-\Delta_t)Q_{t+1}(H_{t+1})\). By the stated conditional independent-censoring assumption, \(Z_t\) and \(R_{t+1}\) are conditionally independent given \(H_t\), survival to the start of interval \(t\), and the regime. Positivity ensures that the conditional distribution among \(R_{t+1}=1\) is defined for the target histories. Therefore,

The left side is estimable from retained individuals; substituting this equality at each time in the backward recursion identifies \(Q_t\), and hence its target-population average. \(\square\)

Worked Estimates and Sequential Updates

In longitudinal targeted maximum likelihood estimation (TMLE), the backward regressions estimate the \(Q_t\) functions, and sequential targeting updates these estimates so the final plug-in survival estimate reflects the relevant censoring-adjusted score. At interval \(t\), the regression outcome is zero for an event in that interval and otherwise is the next-time survival prediction. It is observed for the histories needed in that update only when follow-up continues. A standard targeting score therefore uses an inverse cumulative-retention factor: a retained observation is weighted in proportion to \(1/G_t\), with the precise index determined by whether the update is made before or after the interval-\(t\) retention decision. Under a treatment regime with treatment adherence also to be accounted for, the corresponding treatment factors enter as well; they are not censoring probabilities.

Why the Assumptions Matter: Diagnostics and Alternatives

The identification result is conditional, not automatic. A model for retention can be estimated from observed histories and used to calculate \(G_t\), but a fitted model cannot verify that dropout is independent of future outcomes after conditioning on those histories. Investigators should assess whether measured predictors of both dropout and survival are included, compare retention across important histories, and report how small estimated retention probabilities become. When some \(G_t\) values are near zero, inverse weights become large and estimates can be unstable; when a target history has zero chance of continued observation, positivity fails and the data do not identify its later survival without additional assumptions.

The main approaches have different strengths. Risk-set or product-limit estimation is direct and efficient in simple settings with independent censoring, but does not by itself adjust for complex measured histories. Inverse-probability-of-censoring weighting makes the correction explicit and can handle measured predictors of dropout, but extreme weights can inflate variability. Sequential regression models future survival conditional on history and can use covariates to predict outcomes; longitudinal TMLE adds targeted updates to these regressions while incorporating the censoring mechanism. Complete-case analysis is generally not a substitute: it selects people who stayed observed and is justified only under stronger conditions that make this selection harmless for the target.

A common mistake is to use one indicator for both event occurrence and dropout. For instance, setting \(\Delta_t=1\) whenever \(R_{t+1}=0\) makes dropout look like an event and generally lowers estimated survival. The correction is to record the event indicator only when event status was observed, retain a separate observation indicator, and use the retention assumption and probability to address missing future outcomes. Another mistake is to multiply the target survival by the probability of remaining observed. That estimates the chance of being both alive and observed, not survival in the target population.