Censoring Assumptions for Causal Risk Estimation

Why Censoring Needs Its Own Assumptions

A treatment-specific cumulative incidence is the probability that a particular event occurs by a specified time under a specified treatment strategy, while other event types remain possible. In a competing-risks setting, a competing event can make the event of interest impossible to experience first. Loss to follow-up is different: it stops observation, but it does not itself determine what would happen next. Treating these two situations as equivalent changes the risk being estimated.

The issue is identification. When a participant is lost before the end of follow-up, their later event type and event time are unknown. The observed data identify the treatment-specific risk only if the loss process does not selectively hide different future event experiences, after conditioning on the information used to account for censoring. This assumption is separate from the assumptions needed to compare treatments causally.

The superscript \(a\) describes a potential outcome under a treatment strategy. To interpret an observed treatment-specific risk causally, one also needs consistency, treatment exchangeability, and treatment positivity: briefly, the observed outcomes agree with the potential outcomes under the received treatment; treatment assignment is conditionally exchangeable given the measured covariates; and each treatment being compared is possible at each relevant covariate value. The earlier material on consistency and treatment versions, exchangeability, and positivity develops these requirements. This tutorial focuses on the additional requirements created by incomplete follow-up.

Independent Censoring and Censoring Positivity

A simple baseline-covariate version of the independent-censoring condition is that, within treatment and baseline-covariate groups, the censoring time carries no further information about the event time or event type. In notation, a sufficient formulation is conditional independence of \(C\) and the pair \((T,J)\) given \(A,L\). It says that among participants with the same treatment and measured baseline covariates, those still observed at a given time represent those who have been censored with respect to their future event experience.

This condition is conditional, not necessarily marginal. If people with different prognoses are more likely to leave follow-up, censoring can be informative overall even when it becomes independent of future event experience after conditioning on measured \(L\). Conversely, adjusting for a covariate does not solve the problem if relevant predictors of both loss and future events are unmeasured, poorly measured, or omitted.

A more general condition is needed when censoring depends on measured information collected during follow-up. At each time, conditional on treatment and the observed history up to that time, remaining uncensored must be independent of the subsequent event process. This is a sequential conditional independence assumption. It is not justified merely because the history is observed; the history must contain the relevant shared predictors of censoring and future events, and there must still be a positive chance of continued observation at the histories being considered.

Censoring positivity is substantive. If everyone in some treatment-covariate group is lost before a time of interest, the data contain no direct information about their event experience after that time. No amount of modeling can recover that part of the risk without additional assumptions. A very small probability of continued observation is also a warning: estimates may depend heavily on a few observed participants.

How the Assumption Connects to Cumulative Incidence

The connection is clearest in discrete time. Consider intervals \(1,\ldots,K\), conditional on a treatment and covariate group. Define \(h_j(k)\) as the probability that event type \(j\) occurs in interval \(k\), conditional on being event-free at its start. The probability of remaining event-free through interval \(k\) is denoted \(S(k)\), with \(S(0)=1\). Loss to follow-up is not an event in these definitions.

Here \(F_1(K)\) is the cumulative incidence of event type 1 by interval \(K\). The formula follows by adding the probability of a type 1 event in each interval: a person must have remained free of both event types through the preceding interval, which has probability \(S(k-1)\), and then experience type 1 in interval \(k\), with conditional probability \(h_1(k)\). The mutually exclusive first-event types imply that the probability of staying event-free over the next interval is \(1-h_1(k)-h_2(k)\).

With censoring, the hazards in this calculation are learned from participants still at risk and observed in each interval. Conditional independent censoring makes their event experience representative of the comparable participants who were censored, conditional on the variables in the assumption. Censoring positivity ensures that there are such observed participants throughout the relevant follow-up. The competing event remains in the event-free survival calculation as a real event type: it reduces the number who can subsequently experience the event of interest.

In a causal analysis, this identifies the event distribution under each treatment only when the censoring conditions are combined with the treatment-assignment assumptions and consistency. The treatment-specific risks can then be related to the target population by accounting for its covariate distribution. The particular procedure for that population aggregation is a separate step; the censoring assumption does not itself remove confounding or define the target population.

Loss to Follow-Up Is Not a Competing Event

A competing event is observed and determines that the participant has experienced a different first event. For example, if death prevents a later nonfatal event from occurring, death is an outcome type, not a missing outcome. A participant lost to follow-up has no observed event status after the loss time. The distinction matters both for what is recorded and for the probability being estimated.

The arithmetic in this example depends on the stated interval hazards and is meant to expose the distinction, not to prescribe an estimator. In real data, treating competing events as censoring in a survival calculation generally targets a net risk under a hypothetical removal of competing events, rather than the observed-world cumulative incidence. The treatment-specific cumulative incidence keeps the competing event in the event process.

What Goes Wrong When Censoring Is Informative

Diagnostics, Alternatives, and Limits

Censoring assumptions are not proven by a nonsignificant test or by a well-fitting event model. They concern unobserved outcomes after loss, so direct verification is impossible. Useful checks can reveal problems or make an assumption less plausible, but cannot establish independence.

• Describe when and for whom censoring occurs. Compare censoring proportions and times across treatment groups and measured prognostic covariates. Mark whether losses cluster around particular times or sites.
• Check follow-up support. Examine the number still observed over time within important treatment and covariate groups. Empty or nearly empty risk sets signal positivity problems and unstable inference.
• Review the information available at loss. Determine whether recorded health status, contact history, or other predictors of future events also predict loss. Their omission can make conditional independent censoring implausible.
• Separate administrative end from loss. A study's fixed end date is often set by design, whereas participants may leave early for reasons related to prognosis. The latter requires particular scrutiny. Neither circumstance licenses treating a competing event as censoring.

The most direct alternative to a conditional independent-censoring assumption is not a routine adjustment, but acknowledging that the risk may not be identified from the observed data without further assumptions. Sensitivity analyses can examine how conclusions change under departures from the assumption. If censoring is plausibly independent only after conditioning on measured covariates, an analysis must actually use those covariates in a way appropriate to its estimator; merely stating the condition is not enough. Inverse-probability-of-censoring weighting and outcome-model approaches are distinct estimation strategies, each requiring correctly specified modeling choices and adequate support. They do not make informative censoring harmless by their names.

The central limitation is that independent censoring is an assumption about the relation between observed follow-up and unobserved future outcomes. Diagnostics can identify warning signs, but no observed-data pattern alone proves that people lost to follow-up would have had the same future event distribution as comparable people who remained observed. Likewise, conditional independent censoring does not address unmeasured treatment confounding, treatment positivity failures, incorrect event classification, or a poorly chosen horizon.