Weighted Kaplan-Meier Estimation Under Informative Dropout

Why Weight a Kaplan–Meier Curve?

The Kaplan–Meier estimator builds a survival curve by multiplying estimated conditional event-free probabilities across event times. Its usual risk sets represent the people still event-free and observed at each time. If dropout depends on observed prognosis, however, later observed risk sets can overrepresent people with a different event risk from the full population. The ordinary Kaplan–Meier curve may then describe the selected observed group rather than the survival distribution that would have been seen with complete follow-up.

Inverse-probability-of-censoring weighting (IPCW) addresses this selection by giving more weight to people whose observed histories made continued follow-up less likely. The weighted Kaplan–Meier estimator uses these weights in both the event count and the risk-set count. Its target is the marginal survival distribution under complete follow-up, provided censoring is conditionally independent of the event process given the recorded history and the required follow-up probabilities are positive.

Setup and Estimator

Consider a cohort followed over discrete event times \(k=1,\ldots,K\). Let \(T_i\) be person \(i\)'s event time and let \(C_i\) denote the time of censoring, using this timing convention: event status is observed during interval \(k\), and censoring at the end of interval \(k\) affects observation at later intervals. Define the full-follow-up risk and event indicators

Let \(R_i(k)=1\) if person \(i\) is still observed and event-free when entering interval \(k\), and \(R_i(k)=0\) otherwise. Thus an observed event at \(k\) has \(R_i(k)=1\). Let \(H_i(j)\) be the observed history available by the end of interval \(j\), for a person still under follow-up and event-free then. Write

This is the conditional probability of remaining observed and event-free for entry into the next interval, among people eligible to do so. The probability of being observed when entering interval \(k\), conditional on the relevant history along the way, is the product of the preceding retention probabilities. The corresponding IPCW is

In a sample of \(n\) people, define the weighted risk-set and event counts by

The weighted event hazard estimate is \(\hat h_w(k)=D_w(k)/Y_w(k)\) when \(Y_w(k)>0\). The weighted Kaplan–Meier estimate is

These are weights for entering the interval: a person’s weight at \(k\) corrects for the probability of remaining observed through the earlier intervals, not for censoring after the event opportunity at \(k\).

Why the Weighted Hazard Targets the Marginal Hazard

Define the full-follow-up marginal event hazard at time \(k\) as

provided \(P(T\ge k)>0\). The key idea is that inverse-probability weighting reconstructs the full risk set as well as its events. A weight applied only to events, or only to people who remain under observation, would not give this hazard.

Proof. Conditional on the relevant full event and history information, sequential independent censoring gives the probability of being observed when entering interval \(k\) as the product of the conditional retention probabilities along the eligible, event-free history. Denote that probability by \(\pi_i(k)\). By construction, \(W_i(k)=1/\pi_i(k)\). Thus the conditional expectation of the observation indicator times the weight is

Both \(Y_i(k)\) and \(D_i(k)\) are determined by the full event information, and \(D_i(k)=1\) implies \(Y_i(k)=1\). Multiplying the conditional identity first by \(Y_i(k)\), then by \(D_i(k)\), and taking expectations gives the two stated equalities. Dividing the event expectation by the risk expectation gives \(E\{D_i(k)\}/E\{Y_i(k)\}=P(T=k\mid T\ge k)\), as required. \(\square\)

For a finite number of times, this identity also yields a large-sample result. If the true weights are known, the weighted contributions have finite expectations and sample averages converge to those expectations, and each target risk probability \(P(T\ge k)\) is positive, then the law of large numbers and the continuous mapping theorem give \(\hat h_w(k)\xrightarrow{p}h(k)\). Applying continuity to the finite product gives \(\hat S_w(k)\xrightarrow{p}S(k)\), where \(S(k)=P(T>k)\).

Indeed, the event \(T>k\) requires surviving each interval \(1,\ldots,k\), and \(1-h(j)=P(T>j\mid T\ge j)\). Multiplying these conditional probabilities gives \(P(T>k)\). If retention probabilities are estimated, consistency additionally requires adequate estimation of those probabilities and control of the resulting weights; the known-weight argument alone does not guarantee that fitted weights behave well.

Assumptions, Diagnostics, and Alternatives

The estimator needs more than a censoring model that predicts dropout well. Its interpretation depends on the following conditions:

• Sequential independent censoring: conditional on the recorded history, continued observation does not convey additional information about the event process. If important predictors of both dropout and future events are unrecorded, weighting cannot correct that selection.
• Positivity: each relevant history among people at risk has a positive probability of remaining observed. If a history has retention probability zero, the full-follow-up risk set for that history cannot be reconstructed from observed data.
• Correct retention probabilities: the conditional probabilities used in the weights must be known or adequately estimated. Model misspecification can leave residual selection bias.
• Well-behaved weights and risk sets: very small retention probabilities produce large weights, and the estimator is undefined at a time when its weighted risk-set total is zero.

In practice, inspect the distribution of weights by time, the largest weights, and the weighted risk-set totals. Also assess whether fitted retention probabilities approach zero in particular histories. A rising weight distribution, a few observations dominating the weighted counts, or a nearly empty weighted risk set are warnings that the estimate may be unstable. Truncating weights can reduce variance, but changes the estimating procedure and can introduce bias; it does not repair a genuine positivity violation.

When censoring is independent of the event time without conditioning on history, ordinary Kaplan–Meier estimation is the simpler approach and generally avoids the extra variability from estimating weights. When censoring is independent only after conditioning on observed history, IPCW can recover the marginal curve without requiring the observed risk set to have the same history composition as the full risk set. Direct IPCW estimation of a fixed-horizon mean answers a different, single-time summary question; the weighted Kaplan–Meier estimator assembles hazards across event times into a survival curve. Stratification or standardization can also address observed prognostic differences, but require specifying how the stratum-specific results are combined. None of these methods can recover information about histories with no chance of continued observation.

Common Pitfall