Inverse Censoring Intensity Weights in Continuous Time

From a Censoring Intensity to a Weight

In continuous time, censoring can occur at any moment during follow-up. The probability of remaining uncensored therefore changes continuously, and the corresponding inverse probability weight depends on the entire censoring hazard accumulated up to the time of interest. The central calculation is simple: integrate the conditional censoring intensity, convert that cumulative hazard to a survival probability, and take its reciprocal for people still observed.

Let \(C\) be a censoring time and let \(H(t-)\) denote the measured history available immediately before time \(t\). Write \(\lambda_C(t\mid H(t-))\) for the conditional censoring intensity among individuals still uncensored. An intensity is a conditional instantaneous rate: over a short interval of length \(dt\), the conditional probability of censoring is approximately \(\lambda_C(t\mid H(t-))dt\), given the history and continued observation up to \(t\). The history may include baseline covariates and time-varying measurements.

The notation \(h\) here represents the history information relevant to the censoring law through \(t\). For a time-varying history, the integral uses the values of that history at each time, not just its value at the end of follow-up. In applications, the intensity is specified by a model conditional on the observed history; the validity of using that model for weighting also requires the independent-censoring and positivity conditions discussed below.

Deriving the Survival Probability

The exponential formula follows from the definition of a hazard, rather than being an extra modeling convention. Fix a history path \(h\), and abbreviate \(G(t\mid h)\) to \(G(t)\) and \(\lambda_C(t\mid h(t-))\) to \(\lambda(t)\). In the absence of censoring atoms, the conditional probability of censoring in a short interval, given survival to its start, is \(\lambda(t)dt+o(dt)\). Consequently,

for almost every \(t\), with \(G(0)=1\). To solve this differential equation, define \(A(t)=\int_0^t\lambda(s)\,ds\). The product \(e^{A(t)}G(t)\) is absolutely continuous, and its derivative is

almost everywhere. An absolutely continuous function whose derivative is zero almost everywhere is constant. Its value at time zero is one, so \(e^{A(t)}G(t)=1\), which gives \(G(t)=e^{-A(t)}\). This argument requires an integrable intensity and an absolutely continuous survival function; if censoring has point masses, the exponential-of-integral formula alone does not represent its survival law.

Worked Calculations

When Does Inverse Survival Weighting Work?

The preceding recovery calculation depends on assumptions, not merely on the algebra of taking a reciprocal. Let \(\mathcal F^*\) denote the information in the full measured history relevant to the target quantity through time \(t\), including information that would have been observed had follow-up continued. For a full-history quantity \(Q\), a key condition is that, given \(\mathcal F^*\), remaining uncensored through \(t\) has probability \(G(t\mid\mathcal F^*)\). This is the continuous-time form of independent censoring conditional on measured history. It rules out residual dependence of censoring on the target quantity after conditioning on the specified history.

Also require consistency between the recorded history and its full-history counterpart while observed, a correctly specified or consistently estimated censoring survival law, and censoring positivity: \(G(t\mid\mathcal F^*)>0\) for histories and times in the target population. For stable population expectations, the relevant weighted quantity must be integrable; for standard large-sample inference, additional moment and regularity conditions are typically needed. Positivity is not secured just because a fitted model returns positive numbers: the true survival probability must not be zero in a target stratum.

To prove the result, take the conditional expectation given \(\mathcal F^*\). Since \(Q\) and \(G(t\mid\mathcal F^*)\) are measurable with respect to that information, they can be taken outside the conditional expectation:

Taking expectations of both sides gives the result by the law of iterated expectations. Integrability ensures these expectations are defined; positivity ensures the denominator is nonzero. If \(Q\) can be negative, the argument applies to its positive and negative parts when both have finite expectation. \(\square\)

Diagnostics, Alternatives, and Limitations

In practice, the conditional survival probability is estimated, and its inverse may be highly variable. Inspect estimated survival probabilities over time and across important histories, along with the distribution of the resulting weights. Very small estimated \(G\) values, unusually large weights, or a few observations dominating weighted summaries are warnings of weak positivity or a poorly supported model. Comparing observed history patterns among those still followed with the patterns represented after weighting can reveal discrepancies, but no diagnostic can establish conditional independent censoring from the observed data alone.

The continuous-time expression is closely related to discrete-time IPCW: in discrete time, one multiplies inverse conditional probabilities of surviving each interval; in continuous time, the cumulative hazard integrates the instantaneous rates and the survival probability is its exponential. A discrete-time approximation can be useful when observation times naturally form intervals, but its accuracy depends on interval width and the model for within-interval censoring. Complete-case analysis is simpler, but it generally does not recover the full-history distribution when censoring depends on measured prognostic history.

Inverse censoring weights cannot repair censoring that depends on unmeasured predictors of the outcome after conditioning on the modeled history. Nor do they create information in strata with zero probability of remaining observed. Even with correct assumptions, extreme weights can make finite-sample estimates unstable; truncating weights may reduce variance but changes the estimating procedure and can introduce bias. These are reasons to report the censoring model, the support of estimated survival probabilities, and sensitivity to influential weights, not reasons to treat a fitted inverse survival probability as automatically valid.

Common Pitfall

A frequent error is to use the reciprocal of the censoring intensity at time \(t\) as the weight. An intensity has units of inverse time and describes an instantaneous rate; it is not the probability of remaining uncensored through \(t\). The correction is to integrate the intensity from the start of follow-up, form \(G(t)=\exp\{-\Lambda_C(t)\}\) under the stated continuous-hazard conditions, and then use \(1/G(t)\) for an individual still uncensored. A second error is to interpret the conditional mean-one identity as proof that the weight model is correct. That identity follows only when the conditional survival probability and the independent-censoring assumptions hold.