Censoring Mechanism Nuisance Models

Why Model Censoring in Longitudinal TMLE?

Longitudinal TMLE uses outcome regressions to work backward through a sequence of treatment decisions. But those regressions cannot be fit from people whose outcomes and histories are no longer observed. When loss to follow-up depends on measured history, the remaining participants may not represent the full target population. A censoring nuisance model describes this observation process and helps targeting account for it.

The goal here is to define the relevant conditional probabilities and make their role concrete. They are not treatment probabilities: a treatment mechanism describes which treatment is received, whereas a censoring mechanism describes whether the next required part of a participant’s history remains observed.

Conditional Probabilities of Remaining Uncensored

Use the following timing convention. There are decision times \(t=0,\ldots,K\). At time \(t\), a participant who is still observed has history \(H_t\) and receives treatment \(A_t\). The indicator \(R_t\) equals 1 if the participant is observed through the start of time \(t\), and equals 0 otherwise; \(R_0=1\). After \(A_t\), the participant either remains observed through the next time or is censored. The terminal outcome \(Y\) is observed only if the participant remains observed through its measurement time, denoted by \(R_{K+1}=1\).

The conditioning on \(R_t=1\) matters. A participant already censored cannot newly be censored at time \(t\), and their time-\(t\) history is not available in the observed data. In practice, each \(g_t\) is estimated among participants with \(R_t=1\), using available history and treatment as predictors. The model may be parametric, such as logistic regression, or more flexible, provided its predictions are suitable for the intended estimation procedure.

For a history and treatment sequence that remains observed, the probability of remaining observed through time \(s\) is the product of the step-specific probabilities:

This factorization follows by the probability chain rule: each factor is the conditional probability of surviving the next censoring step given survival up to that step and the observed history. In applications, the fitted probabilities are evaluated along the observed trajectory. The inverse \(1/G_s\) is an inverse-probability-of-retention weight for contributions requiring observation through time \(s\).

Assumptions Behind Censoring Adjustment

A censoring model does not, by itself, make missing outcomes recoverable. Identification requires assumptions about why observation ends and whether there are enough participants who remain observed.

The first assumption says that after conditioning on the recorded history and treatment, censoring does not further select people according to their unobserved future outcomes or covariates. It is sometimes called a sequential coarsening-at-random assumption. It is not established by fitting a model: it is a substantive claim about which predictors of dropout have been measured.

Positivity requires a nonzero chance of continued observation at relevant histories. If a group with a particular history is always censored, its later outcomes cannot be learned from the observed data without additional assumptions. Very small, rather than exactly zero, retention probabilities also cause practical problems because their inverse weights are large.

Worked Examples

Two Useful Derivations

First consider one censoring step, with measured history \(H\), retention indicator \(R\), retention probability \(g(H)\), and full-data outcome \(Y\). Assume \(P(R=1\mid Y,H)=g(H)>0\). Conditional on \(H\),

Taking expectations over \(H\) gives \(E[RY/g(H)]=E[Y]\). For multiple steps, apply the same conditional-expectation argument successively at each step: conditional on the full data and the history available at a step, the expected retention indicator divided by its correct conditional retention probability is 1. Iterating over all required steps yields \(E[R_{K+1}Y/\prod_{t=0}^{K}g_t(H_t,A_t)]=E[Y]\). This identifies the full-data mean, but requires the sequential independence and positivity assumptions; it does not justify a misspecified \(g_t\). \(\square\)

For two steps, apply the chain rule: \(P(R_2=1\mid R_0=1,H_0,A_0,H_1,A_1)\) equals \(P(R_1=1\mid R_0=1,H_0,A_0)\) times \(P(R_2=1\mid R_1=1,R_0=1,H_0,A_0,H_1,A_1)\). By the definition of the step-specific probabilities, these factors are \(g_0(H_0,A_0)\) and \(g_1(H_1,A_1)\). Repeating this argument for each additional step gives \(G_s=\prod_{t=0}^{s-1}g_t(H_t,A_t)\). The conditioning histories must be the appropriate histories at each step; replacing them by a single marginal retention probability would not generally give this factorization. \(\square\)

How the Estimates Enter Longitudinal Targeting

A longitudinal TMLE begins with sequential outcome regressions, which predict outcomes conditional on histories and are evaluated backward under a specified treatment regime. The censoring nuisance estimates affect the targeting stage: observations that remain available despite a low fitted probability of retention receive more weight in the score contribution than observations with a high fitted retention probability. For a given regression step, the relevant weight reflects the probability of remaining observed through the data needed at that step.

This role is distinct from fitting the outcome regression itself. The regression describes outcomes conditional on measured history; the censoring nuisance describes which outcomes are observed. If censoring estimates are poor, the targeting update can be poorly calibrated even when the initial outcome regression appears reasonable. The targeting construction also uses treatment-mechanism estimates, which address a separate source of reweighting. The censoring component should not be mistaken for, or substituted by, the treatment component.

A useful diagnostic interpretation is that targeting score contributions should balance the outcome residuals after accounting for the estimated observation process. If large contributions come from a few participants with very small \(G_s\), the update may be numerically unstable. This is a warning about limited information in the data, not a reason to assume the extreme weights are harmless.

Diagnostics, Alternatives, and Limitations

• Check predictions by time and history. Compare observed retention frequencies with predicted probabilities across time and important covariate groups. A model that predicts nearly constant retention despite visibly different dropout patterns may be misspecified.
• Inspect the lower tail of fitted probabilities. Report minimum and quantiles of \(g_t\) and cumulative \(G_s\), along with the largest inverse weights. Tiny probabilities indicate practical positivity problems and can make targeting sensitive to a small number of observations.
• Check the risk set and timing. Each model should be fit among people who were still observed and should use predictors available before the censoring event being modeled. Including variables only observed after dropout, or using the wrong at-risk set, does not estimate the defined conditional probability.
• Consider truncation cautiously. Replacing very large weights by a maximum can reduce variance, but changes the estimating procedure and may introduce bias. It cannot restore information for histories with no chance of continued observation.

Complete-case analysis is simpler, but generally targets a selected group when retention depends on outcome-related history. Inverse-probability weighting makes the observation adjustment explicit, although it can be unstable. Outcome-regression approaches can use measured history to predict missing outcomes, but depend on their own modeling assumptions. Longitudinal TMLE combines sequential outcome regressions with targeting informed by nuisance mechanisms; it does not remove the need for adequate censoring measurement, positivity, or sensible nuisance estimation.

The principal limitation is unmeasured informative censoring. If dropout depends on an unobserved predictor of future outcome even after conditioning on recorded history, the stated sequential independence assumption fails. No observed-data censoring model can diagnose that failure conclusively. Sensitivity analysis or additional data may be needed to assess how conclusions change under plausible departures from the assumption.

Common Mistakes

• Using the probability of censoring as the retention probability. If \(P(\text{censored at }t\mid\cdot)=0.20\), then the corresponding probability of remaining observed is \(0.80\), not \(0.20\).
• Using one marginal dropout rate at every time. Retention probabilities are conditional on the participant’s risk set and history; they can vary across time and covariate patterns.
• Assuming correct prediction proves independent censoring. A model can fit observed retention well while important unmeasured predictors of future outcome remain absent.
• Ignoring small fitted retention probabilities. The resulting large inverse weights can dominate score equations, making estimates unstable and diagnostics essential.