G-Estimation With Censoring

Why Censoring Changes G-Estimation

Longitudinal g-estimation uses treatment histories and outcomes to estimate causal blip parameters. With complete follow-up, the treatment-residual equations can use the outcome and the histories needed to construct blipped-down outcomes. If some participants leave follow-up before their outcome is measured, those equations cannot simply be evaluated on the remaining complete cases: if censoring depends on observed history, the complete cases may no longer represent the full target population.

Inverse-probability-of-censoring weighting (IPCW) addresses this selection by giving more weight to observed participants who had a lower probability of remaining observed. It does not make censoring harmless by itself. Its justification depends on a correctly specified observation model, adequate probability of remaining observed at each history, and the causal assumptions required for g-estimation.

Notation and Censoring Assumptions

Suppose there are treatment times $t=0,\ldots,K$, a final outcome $Y$, and covariates $L_t$ measured before treatment $A_t$. Let $H_t=(L_0,A_0,\ldots,L_{t-1},A_{t-1},L_t)$ denote the observed covariate and treatment history just before $A_t$. Define $C_{t+1}=1$ if a person remains uncensored through the next observation point, and $C_{t+1}=0$ if censoring occurs in that interval. Let $R_0=1$ and $R_{t+1}=R_tC_{t+1}$. Thus, $R_{K+1}=1$ indicates that the final outcome is observed.

Among people still observed at time $t$, let $q_{t+1}(H_t,A_t)$ be the probability of remaining uncensored through the next observation point, conditional on the history available for the censoring decision. The notation may include additional measured variables available by that decision; the essential requirement is that the conditioning history be well defined and observed before censoring.

Definition: The cumulative probability of remaining observed through the final outcome is $$ G_{K+1}=\prod_{t=0}^{K}q_{t+1}(H_t,A_t), $$ and the unstabilized inverse-probability-of-censoring weight for a participant with an observed final outcome is $W_C=1/G_{K+1}$. Only participants with $R_{K+1}=1$ contribute the final-outcome estimating equation; their contribution is multiplied by $W_C$.

The weights address censoring, not treatment assignment. The treatment process still needs the assumptions and treatment models required by g-estimation, including sequential exchangeability and treatment positivity. For censoring, the key assumptions are:

Sequential independent censoring: conditional on the measured history available at each censoring decision, remaining uncensored is independent of the relevant future full-data quantities. In particular, after conditioning on this history, censoring does not select people according to unmeasured future outcomes or potential outcomes.
Censoring positivity: every history that occurs in the target population has a positive probability of remaining observed at each required follow-up interval. The cumulative probability $G_{K+1}$ must therefore be positive.
Correct censoring probabilities: the probabilities $q_{t+1}$, whether known or estimated, must represent the conditional observation process adequately for the weighting argument.

These assumptions are in addition to consistency, no interference, the treatment-exchangeability conditions used for the causal blip model, and adequate specification of the blip functions. IPCW cannot recover information about histories with zero chance of observation, nor can it remove bias from unmeasured causes of both censoring and the outcome.

How the Weights Enter G-Estimation

Let $\psi$ collect the structural nested mean model (SNMM) blip parameters. Blip-down transformations use $\psi$ to remove modeled treatment effects from the observed outcome. In a one-treatment-time additive model, for example, the blipped-down outcome is $Y-\psi A$. In a longitudinal model, the relevant blipped-down outcome at time $t$ removes the modeled effects of the treatment blips required by the backward g-estimation procedure.

Write $U_t(\psi)$ for the complete-data treatment-residual contribution at time $t$. In the simple one-time binary-treatment case, it is $(A-e(H))(Y-\psi A)$, where $e(H)=P(A=1\mid H)$. In the longitudinal case, the treatment residual is formed from the treatment model conditional on $H_t$, and it is paired with the appropriate blipped-down outcome. The complete-data g-estimation equations have mean zero at the true blip parameter under the causal assumptions and a correctly specified blip model. With censoring, replace each complete-data contribution by its observed contribution times the censoring weight:

\sum_{i=1}^{n} R_{i,K+1}W_{C,i}U_{it}(\psi)=0, \qquad t=0,\ldots,K.

Here, $i$ indexes participants. In practice, the equations may be conditional or stacked estimating equations, and treatment models or censoring probabilities may also be estimated. The principle is the same: the censoring-weighted equation targets the corresponding complete-data moment, rather than the moment among the selected complete cases alone.

Why Inverse-Probability Weighting Recovers the Moment

The basic justification is an expectation identity. Let $f$ be a full-data quantity used in an estimating equation, and suppose sequential independent censoring holds given the histories used to construct $G_{K+1}$. Then the product of observation indicators divided by the product of conditional observation probabilities has expectation one in the required conditional sense. As a result, weighting the observed $f$ recovers its full-data expectation:

E\left[\frac{R_{K+1}}{G_{K+1}}f\right]=E[f].

The identity can be justified one censoring interval at a time. At an interval with history $H_t,A_t$, the conditional probability of being observed through that interval is $q_{t+1}(H_t,A_t)$. Under independent censoring, the conditional expectation of the observation indicator times the relevant future quantity, divided by this probability, equals the conditional expectation of that quantity:

E\left[\left.\frac{C_{t+1}}{q_{t+1}(H_t,A_t)}f_{\mathrm{future}}\right|H_t,A_t,R_t=1\right] = E\left[f_{\mathrm{future}}\mid H_t,A_t,R_t=1\right].

Applying this step successively across follow-up intervals yields the cumulative-weight identity. Taking $f$ to be a complete-data g-estimation contribution gives $E[R_{K+1}W_CU_t(\psi)]=E[U_t(\psi)]$. Thus, at the true parameter, the weighted equation has expectation zero whenever the corresponding complete-data equation does.

This is an identification and estimating-equation argument, not a guarantee that a finite-sample weighted estimate will equal the estimate from complete data. Estimated probabilities, random variation, model misspecification, and large weights can all affect the result.

Worked Examples

Worked Example: Constructing a Cumulative Censoring Weight

Consider a hypothetical participant who receives treatment at three times. Given that participant’s measured histories, the estimated probabilities of remaining uncensored through the next observation points are $0.90$, $0.80$, and $0.75$. Assume the participant remains observed at every point, including measurement of the final outcome.

The cumulative probability of complete observation is the product of the interval-specific probabilities:

\widehat{G}_{4}=0.90\times0.80\times0.75=0.54.

The participant’s inverse-probability-of-censoring weight is therefore

\widehat{W}_C=\frac{1}{0.54}=1.852.

In a weighted estimating equation, this observed participant contributes $1.852$ times their unweighted contribution. The interpretation is not that the participant literally represents 1.852 people. Rather, under the censoring assumptions, the weight compensates for the participant’s measured-history-specific probability of being observed through the outcome.

For comparison, if another observed participant had interval probabilities $0.90$, $0.50$, and $0.80$, their cumulative probability would be $0.36$ and their weight would be $1/0.36=2.778$. The larger weight reflects the lower probability of complete observation along that history.

Worked Example: Solving a Weighted One-Time G-Estimation Equation

Suppose a hypothetical one-time binary-treatment SNMM has blip $\psi A$, so the blipped-down outcome is $Y-\psi A$. Assume, for this illustration, that the treatment probability conditional on the measured pre-treatment history is $e(H)=0.50$ for each record. Four participants have observed final outcomes; their censoring probabilities and data are shown below. The displayed weights are the inverse of their estimated probabilities of remaining observed through the outcome.

Participant	$A$	$Y$	$G$	$W_C=1/G$
1	1	8	1.00	1
2	1	12	0.50	2
3	0	4	0.50	2
4	0	8	1.00	1

The weighted treatment-residual equation is

\sum_i W_{C,i}(A_i-0.50)(Y_i-\psi A_i)=0.

Separate the terms involving the blipped-down outcome. Because $A_i=1$ for the treated participants and $A_i=0$ for the untreated participants, the coefficient of $\psi$ is $\sum_i W_{C,i}(A_i-0.50)A_i=0.50(1+2)=1.5$. The term not involving $\psi$ is

\sum_i W_{C,i}(A_i-0.50)Y_i =0.50(1\times8+2\times12)-0.50(2\times4+1\times8) =16-8=8.

Therefore, $8-1.5\psi=0$, giving

\widehat{\psi}=\frac{8}{1.5}=5.333.

The weighted g-estimation equation yields an estimated additive blip of $5.333$ outcome units for treatment versus no treatment under this model and its assumptions. This small calculation is illustrative: with real data, treatment probabilities and censoring probabilities may be estimated, and multiple time-specific equations may need to be solved jointly.

Worked Example: Why Complete-Case G-Estimation Can Differ

Use the same hypothetical records. If censoring weights are ignored, the unweighted treatment-residual equation with $e(H)=0.50$ gives the difference between the observed treated and untreated means. The treated mean is $(8+12)/2=10$, while the untreated mean is $(4+8)/2=6$. Thus the unweighted estimate is $10-6=4$ outcome units.

The weighted estimate was $5.333$ outcome units. The difference occurs because the observed treated and untreated records receive different relative weights: among treated participants the weights are $1$ and $2$, while among untreated participants they are $2$ and $1$. The unweighted estimate describes the estimating equation in the selected observed sample; the weighted equation aims, under the stated assumptions, at the complete-data moment for the target population.

The numerical difference does not establish that weighting is always preferable in a particular data set. If censoring is independent of the relevant full-data quantities without conditioning on additional history, complete-case analysis may not have this selection problem. If the censoring model or assumptions are wrong, weighting may instead introduce bias.

Practical Checks and Common Mistakes

Before interpreting a censoring-weighted g-estimate, verify what each probability in the censoring model represents. It must be the conditional probability of remaining observed over the specified interval, among those still at risk, given variables measured in time to inform that censoring decision. Multiplying probabilities from incompatible risk sets or conditioning on information that would not have been observed before censoring does not produce the intended cumulative probability.

Using only the final censoring probability. If follow-up involves several intervals, the probability of observing the final outcome is generally the product of the conditional interval probabilities, not just the probability for the last interval.
Treating weighting as a substitute for causal assumptions. IPCW addresses selection due to measured predictors of censoring under sequential independent censoring. It does not eliminate confounding of treatment and outcome or repair a misspecified blip model.
Ignoring positivity problems. If a history has a very small probability of continued observation, its inverse weight is large. A few such records can dominate the estimating equation and make estimates unstable. Positivity and the distribution of weights should be examined.
Claiming weights create observed outcomes. A censored participant’s missing outcome is not filled in by the weight. Instead, observed participants with similar measured histories represent the full-data contributions under the assumptions.

Weight truncation or other weight modifications may reduce the influence of extreme values, but they change the estimating procedure and can introduce bias. They should not be presented as a remedy that preserves the original identification argument automatically. Report the approach and assess how conclusions depend on it.

Key takeaway: Censoring-adjusted g-estimation multiplies observed complete-data treatment-residual contributions by the inverse of the modeled cumulative probability of remaining observed. The weighted equations recover their complete-data targets only when the censoring, positivity, treatment, and blip-model assumptions are credible.

Check Your Understanding

Use the assumptions and equations above to answer the following questions.

A participant’s interval-specific probabilities of remaining uncensored are $0.8$, $0.9$, and $0.5$. What is the cumulative probability of observing the final outcome, and what is the corresponding inverse-probability weight?
State the sequential independent-censoring assumption in words. Why is conditioning on measured history important?
In the one-time binary-treatment model, what is the blipped-down outcome when the blip is $\psi A$?
Why does a small cumulative probability of remaining observed produce a large weight, and what practical concern does that raise?
Does censoring weighting by itself address unmeasured confounding of treatment and outcome? Explain.

What You'll Learn