Selection Models for Informative Censoring

Why Model Censoring Through the Latent Outcome?

Censoring is informative when the chance that an event time remains unobserved depends on that event time, even after accounting for measured covariates. For example, people with a latent event time far in the future might be more likely to leave follow-up early. The observed data alone generally cannot establish whether this dependence exists: among people censored first, the event time that would have followed is unknown. A selection model makes the unobserved dependence explicit and asks how conclusions change over scientifically plausible values for it.

Let \(T\) be the latent event time, \(J\) its event type when competing events are possible, \(C\) the censoring time, and \(X\) the measured covariates. The observed time is \(Y=\min(T,C)\), and the event indicator is \(\Delta=I(T\le C)\). When \(\Delta=1\), the event time and cause are observed; when \(\Delta=0\), the event time and cause after censoring are not observed. A selection model describes censoring conditional on the latent outcome, such as \(P(C\mid T,J,X)\), rather than assuming that this conditional distribution can be learned completely from the observed data.

A pattern-mixture model makes the complementary factorization: it specifies outcome distributions within observed and missingness patterns, such as \(P(T,J\mid \Delta,X)\), and combines them using pattern proportions. Thus, the key distinction is which conditional distribution is modeled: selection models specify the selection mechanism conditional on the outcome, whereas pattern-mixture models specify outcome distributions conditional on the selection pattern. Neither factorization removes the need for assumptions about the unobserved outcomes.

Specify the Selection Mechanism and Its Sensitivity Parameter

One useful horizon-specific specification defines \(R=1\) when censoring has not prevented ascertainment of the endpoint by a fixed horizon \(\tau\), and \(R=0\) otherwise. A logistic selection model can be written as

Here, \(\alpha(X)\) describes how measured covariates relate to observation, \(g(T,J)\) is a specified function of the latent event time and possibly cause, and \(\delta\) is the sensitivity parameter. For a one-unit increase in \(g\), the odds of being observed are multiplied by \(\exp(\delta)\), holding other terms fixed. If \(\delta=0\), this model has no residual dependence of observation on the latent outcome through \(g\), conditional on \(X\). A positive or negative value indicates the direction of that dependence. The sign depends on how \(g\) is defined, so it must be interpreted from the actual equation rather than from the word “informative.”

In continuous-time work, a related specification can model the censoring hazard conditional on the latent event time and cause. For example, \(\lambda_C(t\mid T,J,X)=\lambda_{C0}(t)\exp\{\beta^\mathsf{T}X+\delta g(T,J,t)\}\) describes a censoring hazard that may depend on the latent outcome. The horizon-specific logistic version is convenient for illustrating sensitivity calculations; it is not a replacement for the full observed-data likelihood when exact event and censoring times are available.

The model requires a stated target, a suitable model for the observed-data distribution, and a defensible set of values for \(\delta\). The observed data can inform the covariate and baseline components only to the extent permitted by the data and model. They cannot, in general, determine the residual association between censoring and the unobserved event outcome. Consequently, \(\delta\) is varied rather than estimated as though it were identified. For competing risks, the latent outcome includes both \(T\) and \(J\); a model that varies only with time assumes no additional selection dependence on cause after its other terms are included.

A Binary-Endpoint Calculation

A simple derivation makes the role of the sensitivity parameter concrete. Let \(D=1\) indicate that a target event has occurred by \(\tau\), and suppose \(D\) is available only for individuals with \(R=1\). Write \(p_1=P(D=1\mid R=1)\), \(p_0=P(D=1\mid R=0)\), and define the selection odds ratio

By Bayes’ rule, this is also the odds ratio relating observation to event status. Given a chosen \(\Gamma\), the missing-pattern event risk follows by rearrangement:

If \(q=P(R=1)\), the overall risk is then \(p=q p_1+(1-q)p_0\). This is a sensitivity calculation: \(q\) and \(p_1\) are estimable from the observed data in this simplified setting, but \(\Gamma\), and hence \(p_0\) and \(p\), are not determined by those data alone. The calculation assumes that the binary endpoint is observed for everyone in the \(R=1\) group and that the sensitivity parameter applies comparably within the population being summarized.

Interpreting and Varying the Parameter

The binary selection odds ratio is a special case of the selection-model idea. To see how a continuous-time specification is interpreted, suppose a hypothetical model for the odds of remaining observed through \(\tau\) is

where \(t-\tau\) is measured in years. Holding \(X\) fixed, compare latent event times of \(\tau+3\) years and \(\tau+1\) year. Their difference in the linear predictor is \(-0.50(3-1)=-1\), so the ratio of observation odds is \(\exp(-1)=0.3679\). Under this assumed model, a person whose latent event time is two years later has 0.3679 times the odds of remaining observed through \(\tau\), compared with a person whose latent event time is one year later. This is an odds ratio, not a probability ratio, and the coefficient does not by itself give either person’s observation probability; that also depends on \(\alpha(X)\).

The parameter should be presented on an interpretable scale, with the unit of \(g\) stated. For competing risks, analysts can vary a parameter for latent event time, cause, or both, if the scientific question justifies those terms. A cause-specific dependence term needs a clear reference cause and coding. Plausible values should be grounded where possible in follow-up practice, subject-matter knowledge, or external information, not chosen because they produce a preferred answer.

Fitting, Diagnostics, and Alternatives

In a full survival or competing-risks analysis, a practical approach is to specify the outcome and censoring components, fix \(\delta\) over a prespecified grid, and recompute the target estimate for each value. Depending on the method, this can involve likelihood integration over latent event times, imputation under the selection model, or a compatible estimating procedure. For every grid value, report the parameterization, target estimand, estimate, and uncertainty interval. A plot of the estimate against \(\delta\) often makes the sensitivity of the substantive conclusion easier to assess than a single “adjusted” result.

Diagnostics can assess whether the observed-data parts of the model are behaving plausibly, but they cannot verify the unobserved dependence parameter. Check whether fitted censoring or observation probabilities match observed patterns over time and covariates; inspect convergence, sparse strata, and predicted probabilities near zero or one; and check whether the chosen \(g(T,J)\) creates plausible dependence over the relevant event-time range. If a model implies nearly deterministic observation for some covariate and outcome combinations, results may be driven by extrapolation. Refit with reasonable alternative functional forms and parameter grids. A good fit to observed censoring patterns does not establish that \(\delta\) is correct.

This approach differs from inverse-probability-of-censoring weighting, which ordinarily relies on censoring being independent of the event process conditional on included measured covariates. It also differs from a pattern-mixture analysis in its factorization, although both approaches address dependence that cannot be learned fully from the observed data. A joint model may use shared latent effects to connect event and censoring processes; its conclusions likewise depend on assumptions about that connection. These methods are not interchangeable without considering the estimand, data structure, and substantive assumptions.

Important limitations remain. Sensitivity results may depend strongly on the chosen function \(g\), on the range of \(\delta\), and on modeling assumptions for event times and causes. A single parameter can be too simple if the association varies by time, covariate, or event type. If no scientifically plausible range can be defended, the analysis should say so rather than presenting one value as known. The purpose is not to identify the unobserved censoring relationship from unavailable information; it is to make its consequences for inference transparent.

Common Mistakes

• Reversing the factorization. A selection model specifies observation or censoring conditional on the latent outcome, such as \(P(R\mid T,J,X)\). A pattern-mixture model specifies outcome distributions conditional on observation pattern, such as \(P(T,J\mid R,X)\).
• Calling the sensitivity parameter estimated. The observed data generally do not identify the residual association with an unobserved event time. Fix and vary the parameter unless additional identifying information or assumptions are supplied.
• Misreading an odds ratio as a probability ratio. \(\exp(\delta)\) multiplies odds for a one-unit change in the modeled function; it does not directly multiply probabilities.
• Reporting sensitivity variation as sampling uncertainty. Differences across fixed parameter values reflect different assumptions. Confidence intervals or other uncertainty summaries at each value address sampling variability separately.
• Assuming diagnostics can validate missing outcomes. Observed-data checks can reveal misspecification, but cannot test the relationship between censoring and event outcomes that were never observed.