Sequential Independent Censoring Given Observed History

Why censoring must be considered sequentially

At each follow-up time, some participants may stop contributing observed data. If the decision to remain observed carries information about their future outcomes that is not contained in the data already observed, the remaining participants need not represent the full group with the same history. The key question is therefore not simply whether censoring is associated with outcome overall. It is whether, at each time, censoring is independent of future potential outcomes after conditioning on the treatment and covariate history observed before that censoring decision.

This assumption is sequential because it is stated at every follow-up step, using the history available at that step. A single statement about whether participants are ever censored can conceal important changes in treatment, covariates, and censoring over time. The earlier material on censoring indicators and at-risk histories distinguishes being at risk of censoring from having already been censored; here that distinction lets us state the assumption precisely.

Notation and the assumption

Consider discrete follow-up times \(k=0,1,\ldots,K\). Let \(L_k\) be the covariate information measured at time \(k\), and let \(A_k\) be the treatment at that time. Write \(\bar L_k=(L_0,\ldots,L_k)\) and \(\bar A_k=(A_0,\ldots,A_k)\). Let \(R_k=1\) mean that the participant remains uncensored through time \(k\). For a participant with \(R_k=1\), define \(C_{k+1}=1\) if censoring occurs after the history at time \(k\) is recorded and before the next follow-up measurement; otherwise \(C_{k+1}=0\). Thus \(R_{k+1}=R_k(1-C_{k+1})\).

For each treatment regime \(\bar a=(a_0,\ldots,a_K)\), let \(Y^{\bar a}\) be the potential outcome under that regime. Let \(W^{\bar a}_{k+1:K}\) denote the vector of potential future outcomes from time \(k+1\) onward under the regime, including \(Y^{\bar a}\). Including the future outcome vector makes the claim relevant to any outcome feature of interest, not just its mean.

In words, once that observed history is fixed, whether a participant is censored at the next step provides no additional information about their future potential outcomes under the specified regime. The treatment history is part of the condition, not an optional addition: censoring can depend on past treatment, and past treatment can predict future outcomes. The covariate history must likewise include the measured variables that jointly predict censoring and future outcomes.

This is a conditional, regime-specific assumption. It does not say that censoring is independent of outcomes without conditioning, that everyone follows the same treatment, or that future outcomes are observed for censored participants. It also does not by itself guarantee that each relevant history has uncensored participants; that is a separate support condition.

What the assumption implies within a history stratum

Proof. Fix \(k,\bar a,h\), and abbreviate \(W=W^{\bar a}_{k+1:K}\), \(C=C_{k+1}\), and \(Q=\{R_k=1,h\}\). Sequential independent censoring says that \(W\) and \(C\) are conditionally independent given \(Q\). For any measurable \(B\),

The first equality is the definition of conditional probability, the second is conditional independence, and the last uses \(P(C=0\mid Q)>0\). Thus, within a fixed observed history stratum, the future potential-outcome distribution among those not censored at this step equals the distribution in the stratum before that censoring decision. This conclusion is conditional on the stated history; it is not a claim that the retained sample has the same overall outcome distribution as the original sample. \(\square\)

For an integrable scalar potential outcome, the same argument gives equality of conditional means. Apply the event result to the conditional distributions, or use the defining property of conditional expectation, to obtain \(E[Y^{\bar a}\mid C_{k+1}=0,R_k=1,h]=E[Y^{\bar a}\mid R_k=1,h]\). The equality is meaningful only when the relevant conditional expectations exist and the retained-history stratum has positive probability.

Worked examples

A useful decomposition and a common mistake

The previous example illustrates a general fact. Let \(Y\) and \(C\) be square-integrable variables and \(H\) an observed history. The law of total covariance gives

If conditional independence holds, then \(\operatorname{Cov}(Y,C\mid H)=0\) in every history stratum. It follows that \(\operatorname{Cov}(Y,C)=\operatorname{Cov}(E[Y\mid H],E[C\mid H])\), which need not be zero.

Derivation. Write \(Y-E[Y]=(Y-E[Y\mid H])+(E[Y\mid H]-E[Y])\), and use the corresponding decomposition for \(C-E[C]\). On multiplying and taking expectations, the cross terms vanish because \(E[Y-E[Y\mid H]\mid H]=0\) and \(E[C-E[C\mid H]\mid H]=0\). The remaining terms are the expected conditional covariance and the covariance of the conditional means, yielding the displayed identity. Thus independence within history strata does not generally imply marginal independence. \(\square\)

Common mistake: “Censoring and outcome are associated in the full sample, so independent censoring must be false.” The conclusion does not follow. As the decomposition shows, that association can arise because history predicts both censoring and outcome. The relevant question is whether censoring still carries information about future potential outcomes after conditioning on the observed history.

Timing, diagnostics, and limits

The conditioning variables must be observed before the censoring decision being modeled. A covariate measured only after dropout cannot be treated as part of the history available at the decision. Likewise, omitting an observed treatment or covariate history that predicts both censoring and future outcome leaves open a conditional association the assumption was meant to rule out. For artificial censoring used to represent adherence to a treatment regime, the observed treatment history is especially important: the assumption concerns censoring after conditioning on that history, not after ignoring the reason a participant left the regime.

The assumption is not generally testable from observed data alone, because the future potential outcomes of censored participants are not observed. Diagnostics can still reveal problems with a proposed specification: summarize censoring across measured history strata, check whether important predictors of censoring are recorded before censoring, examine whether censoring patterns change over time, and investigate whether conclusions are sensitive to plausible additions or functional forms for history. These checks can identify model or data concerns, but they cannot verify conditional independence.

Sequential independent censoring is also distinct from positivity. It concerns whether censoring is independent of future potential outcomes after conditioning; positivity concerns whether participants with a given relevant history have a nonzero chance of remaining uncensored. One can hold while the other fails. For example, a group may have no censoring at all (so the censoring decision conveys no outcome information) yet provide no observed uncensored follow-up for a different history stratum. The separate support condition is needed for that stratum to contribute information about later outcomes.

Finally, complete-case analysis is not a substitute for stating this assumption. It discards participants with incomplete follow-up and compares the resulting sample without necessarily accounting for changes in history composition. The sequential assumption instead describes the conditional relation required at each step; subsequent methods use that structure to address informative loss of follow-up. The assumption remains a substantive, untestable claim, not a consequence of fitting a censoring model or observing similar crude outcomes.