Right-Censoring Adjustments in Failure-Time G-Estimation

Why ordinary right censoring changes the estimating equation

Failure-time g-estimation uses a candidate blip parameter to transform a treated subject’s failure time to the time that would have occurred without treatment. With complete failure-time data, the transformed time can be evaluated for every subject. With ordinary right censoring, however, some failure times are only known to exceed the end of follow-up. Treating the censoring time as if it were the failure time changes the data and generally biases the estimating equation.

This tutorial considers one treatment decision at baseline. Let \(A\) be a binary treatment, \(L\) measured baseline covariates, \(T\) the factual failure time, and \(C\) the censoring time. The observed follow-up and event indicator are \(U=\min(T,C)\) and \(\Delta=I(T\le C)\). For an accelerated failure-time blip with parameter \(\psi\), define the candidate no-treatment failure time \(T^0_\psi=T\exp(-\psi A)\). At the true parameter, the structural model says that this transformation recovers the relevant no-treatment failure time.

Write \(p(L)=P(A=1\mid L)\), and let \(G(t\mid A,L)=P(C\ge t\mid A,L)\) be the censoring survival probability. The left-limit notation \(G(t-\mid A,L)\) denotes the probability of remaining uncensored through an event at time \(t\), including the event-time boundary under the convention \(\Delta=I(T\le C)\). In continuous-time settings without ties, this is the usual censor survival probability evaluated at the event time.

The censoring-weighted g-estimation equation

Suppose \(q(t,L)\) is a chosen scalar function, or a vector of functions, used to form an estimating equation. At a candidate \(\psi\), an observed failure contributes its transformed time \(U\exp(-\psi A)\). Its inverse-probability-of-censoring weight is the reciprocal of the probability that censoring would allow an event at the observed time to be seen.

The equation is motivated by two distinct requirements. First, treatment must be conditionally unconfounded for the no-treatment failure time: \(A\) and \(T^0\) are independent given \(L\), where \(T^0=T^0_{\psi_0}\) at the true parameter \(\psi_0\). Second, censoring must be independent of the failure time conditional on the variables in the censoring model, here \(A,L\). Treatment positivity requires \(0<p(L)<1\) on the covariate support of interest. Censoring positivity requires \(G(t-\mid A,L)>0\) wherever the failure-time moments are to be identified. The blip model and the chosen moments must also identify the target parameter; a valid weighted equation need not have a unique root.

Why inverse censoring weights recover the target moment

The key weighting identity explains why the event indicator and censoring weight appear together. Assume that, conditional on \(A,L\), \(C\) is independent of \(T\), and that \(G(T-\mid A,L)>0\) on the event-time support. For any integrable function \(h(T,A,L)\), the observed-data contribution \(\Delta h(U,A,L)/G(U-\mid A,L)\) has the same expectation as the full-data quantity \(h(T,A,L)\).

Proof. Conditional on \(T,A,L\), an event is observed exactly when \(C\ge T\). On that event, \(U=T\); otherwise \(\Delta=0\). Conditional independence gives \(P(C\ge T\mid T,A,L)=G(T-\mid A,L)\). Hence

Taking expectations on both sides gives equality of the unconditional moments, provided the quantities are integrable. The positive-survival condition is needed for the division to be defined. \(\square\)

Applying this identity with \(h(T,A,L)=\{A-p(L)\}q(T\exp(-\psi_0 A),L)\) recovers the complete-data g-estimation moment. At \(\psi_0\), write \(T^0=T\exp(-\psi_0 A)\). Conditional on \(L\), treatment unconfoundedness gives independence of \(A\) and \(T^0\), so

The first factor is zero because \(E[A\mid L]=p(L)\). Thus the censoring-weighted moment has expectation zero at the true blip parameter. This argument establishes the population moment under the stated assumptions; consistency of a sample root additionally requires suitable regularity, correct nuisance estimation, and an identifying moment.

Actual censoring is not artificial censoring

Ordinary right censoring is a feature of the observed data: follow-up ends before the event, and the subject’s failure time is unknown beyond that point. In contrast, artificial censoring is introduced by an estimator at a candidate blip value. It is a construction used by certain procedures to define comparable transformed risk sets or to avoid relying on unsupported transformed follow-up. Its role depends on the procedure; it is not evidence that the subject’s actual follow-up ended.

For example, if \(A=1\), \(\psi=\log(2)\), and actual follow-up ends at \(C=5\), the corresponding censoring point on the candidate transformed-time scale is \(5\exp[-\log(2)]=2.5\). This transformed point does not turn the censored subject into a failure at 2.5. If an actual failure is observed at factual time \(T=4\), its candidate transformed failure time is 2, but its censoring weight is based on the probability of being uncensored through factual time 4, namely \(G(4-\mid A,L)\), not simply on \(G(2.5\mid A,L)\). The time scale and censoring model must match the estimator’s definition.

A procedure that also imposes candidate-dependent artificial censoring needs to follow its own risk-set and weighting construction. An ordinary censoring weight corrects for actual loss to follow-up; it does not automatically undo artificial censoring. Nor does multiplying two weights automatically give a valid estimator: that step requires the relevant conditional independence assumptions and a derivation for the combined observation process.

Diagnostics, alternatives, and limitations

Diagnostics should examine both the censoring model and the consequences of its weights. Inspect estimated censoring survival across treatment and important covariate groups, the distribution and maximum of the inverse weights, and how much the result depends on a small number of heavily weighted failures. Check whether event times needed by the analysis lie in regions with positive estimated censoring survival. A model that omits predictors of censoring can leave residual dependence between censoring and failure, invalidating the weighting identity.

Complete-case g-estimation discards censored subjects and is generally not justified merely because the equation is otherwise correctly specified: event observation is selected by follow-up. A correctly specified likelihood for the joint event and censoring data is a different alternative, but it entails its own modeling assumptions. IPCW retains an estimating-equation approach and makes the censoring assumption explicit, at the cost of potentially high variance and sensitivity to the censoring model. If censoring depends on observed predictors beyond \(A,L\), the censoring probability must condition on an adequate set of those predictors; conditioning only on baseline variables is not sufficient by default.

The weights address missing failure times under the stated censoring assumptions. They do not solve unmeasured confounding, a misspecified failure-time blip, a misspecified treatment model, or weak identification of \(\psi\). Estimated propensities and censoring probabilities also add uncertainty; treating them as known can give incorrect standard errors. The earlier material on g-estimation and failure-time blips supplies the complete-data moment and transformation; the essential change here is to recover that moment from observed failures using censoring weights.