Building Conditional Censoring Hazard Models

Why model the censoring hazard?

Inverse-probability-of-censoring weights require predictions of who remains observed at each follow-up time, conditional on the history available before that time’s censoring decision. A censoring hazard model supplies those predictions. Its output is not itself the weight: the denominator for a person observed through time \(t\) is the product of the predicted probabilities of remaining uncensored at each preceding decision.

The distinction between a censoring probability and a retention probability matters. A model may predict a high probability of censoring in one interval, while the corresponding contribution to the weight denominator is the lower probability of not being censored. The model must also use only history known before the decision being modeled; information recorded after censoring cannot be used to predict that decision.

Definition and assumptions

Once censored, an individual is no longer at risk for later censoring decisions in this follow-up process. Equivalently, define \(R_{it}=R_{i,t-1}(1-C_{it})\); then only person-times with \(R_{i,t-1}=1\) contribute a censoring decision. \(H_{it}\) can include baseline variables, treatment, and prior observed time-varying covariates or outcomes, provided each is available before the decision in interval \(t\).

The conditional model is used under the sequential independent censoring and positivity assumptions established in the earlier material on those subjects. Informally, after conditioning on the recorded history, censoring must not convey further information about the relevant future outcome, and the probability of remaining observed must be positive for histories relevant to the target. For a fitted model to provide valid denominator predictions, its conditional probabilities must also be adequately specified or estimated. Correct specification cannot be guaranteed by a good-looking fit alone.

From hazards to denominator predictions

Suppose a model indexed by parameter \(\beta\) specifies \(\lambda_t(h;\beta)\). For an individual observed through time \(t\), calculate the predicted retention probability at each decision using that individual’s history at that time. The censoring-weight denominator is \(G_i(t;\beta)=\prod_{k=1}^{t}\{1-\lambda_k(H_{ik};\beta)\}\). The inverse-denominator weight is \(1/G_i(t;\beta)\). If a stabilized weight is being used, its numerator is chosen separately; the denominator here is the product of the conditional retention predictions.

For example, a model may use different time-specific intercepts, or allow history variables to modify the hazard over time. A binary regression model is one possible specification, but the modeling task is broader than choosing a particular regression family: the model must define a probability between zero and one for each at-risk history and follow-up time. The later treatment of pooled logistic regression develops one particular way to estimate such hazards.

Conditional model likelihood

For a specified hazard model, each at-risk decision contributes a Bernoulli conditional likelihood term. Conditional on the histories used by the censoring model, an individual’s contribution through the last observed decision is \(\prod_{t:R_{i,t-1}=1}\lambda_t(H_{it};\beta)^{C_{it}}\{1-\lambda_t(H_{it};\beta)\}^{1-C_{it}}\). When \(C_{it}=1\), the term is the probability of censoring at that decision; when \(C_{it}=0\), it is the probability of remaining uncensored.

Within one individual, multiplying these terms follows the chain rule for the sequence of censoring decisions conditional on the histories available at those decisions. Multiplying individual contributions to obtain a sample likelihood requires an additional assumption: individuals’ trajectories are independent, conditional on the modeled histories, or else a specified joint model must account for their dependence.

Proof. For an individual, apply the chain rule in time order. At each time \(t\) for which \(R_{i,t-1}=1\), the conditional probability of the observed censoring decision is \(\lambda_t(H_{it};\beta)\) if \(C_{it}=1\), and \(1-\lambda_t(H_{it};\beta)\) if \(C_{it}=0\). These two cases are written together as \(\lambda_t(H_{it};\beta)^{C_{it}}\{1-\lambda_t(H_{it};\beta)\}^{1-C_{it}}\). Multiplying across that individual’s at-risk decisions gives \(L_i(\beta)\). Independence of the individual trajectories then makes the conditional joint likelihood the product of these contributions across \(i\). If trajectories are dependent, the chain rule for decisions within each person still holds, but the across-person product need not equal the joint likelihood. \(\square\)

This likelihood describes the censoring-process component conditional on the histories supplied to the model. It does not assert that the histories themselves are independent over time or that their distribution has been modeled. If dependence between people is plausible, such as within clinics or households, an independence-based likelihood should not be presented as the joint likelihood without a suitable dependence model.

Checking the model and recognizing failure

Diagnostics should examine predicted hazards and the resulting denominator probabilities. Useful checks include comparing observed and predicted censoring frequencies across follow-up times and across important history groups, examining whether fitted probabilities are close to zero or one, and inspecting the distribution of \(G_i(t)\) and its inverse among those still observed. Large inverse weights often indicate that some observed histories were assigned very low probabilities of remaining observed.

These checks have limits. Agreement in an overall censoring rate does not establish conditional calibration, as the previous example shows. Conversely, a sparse history group may show unstable observed rates even when the model is reasonable. Extreme predictions can signal a positivity problem, model misspecification, sparse data, or some combination. A model cannot create information in histories where nearly everyone is censored; truncating extreme weights may change the bias-variance tradeoff rather than repair the underlying lack of support.

A common error is to fit or interpret a model as if censoring were independent without conditioning on the relevant observed history. The incorrect reasoning is: “The overall censoring fraction is modest, so censoring is harmless.” The correction is that a modest marginal fraction can conceal near-certain censoring in a particular history group. Assess conditional hazards and retention predictions for histories relevant to the target, not just the overall fraction censored.

Model choices and limitations

The appropriate choice depends on the data’s support and the structure of the histories. A simple model may be too restrictive; a highly detailed model may produce unstable predictions. Whatever the approach, denominator predictions should correspond to the conditional probability of remaining uncensored at each at-risk decision, under the time ordering and assumptions of the analysis.

The model’s role is limited: it estimates the censoring mechanism, not the causal outcome contrast. Valid inverse-probability weighting also relies on the censoring assumptions, adequate measurement of relevant history, and positivity. A well-fitted hazard model cannot compensate for unmeasured predictors of both censoring and future outcomes, nor can it justify extrapolation where retention probabilities are effectively zero.