Worked Example of Survival Estimation with Informative Dropout

From Dropout to a Weighted Survival Estimate

When dropout depends on characteristics related to survival, the people still observed may no longer represent the original cohort. A survival estimate based only on those still followed can therefore be distorted. Inverse-probability-of-censoring weighting (IPCW) addresses this by giving more weight to observed people whose histories make continued observation less likely.

This worked example uses two discrete follow-up intervals and estimates survival through interval 2. The event and dropout ordering matters: at the start of each interval, eligible participants either drop out or remain observed; events during that interval are then ascertained among those retained. Thus, a participant who drops out before an interval’s event assessment contributes no event information for that interval.

The target is the survival probability in the original cohort, not merely among participants who complete follow-up. The weighted event hazard for interval \(t\) is the weighted fraction of the event-free risk set experiencing an event:

Here, \(R_{it}\) includes remaining observed through all required assessments up to interval \(t\); participants who dropped out earlier do not enter that interval’s sums. The weights use the estimated probability of remaining observed through the interval being analyzed.

Worked Dataset and Censoring-Model Fit

Consider ten participants, five in each baseline risk stratum. The binary covariate \(Z\) is 0 for the lower-risk stratum and 1 for the higher-risk stratum. The table records status in each interval: E means an event, S means observed and event-free, and C means dropout before event ascertainment. A dash means the participant had already experienced an event or dropped out and was no longer eligible.

Fit a saturated retention model with separate probabilities for each combination of interval and \(Z\). Each probability is estimated among participants event-free at the interval’s start and still observed up to that point. For a group with \(m\) eligible participants, of whom \(r\) remain observed, the Bernoulli likelihood is \(p^r(1-p)^{m-r}\). Its log-likelihood is \(r\log(p)+(m-r)\log(1-p)\); setting its derivative \(r/p-(m-r)/(1-p)\) to zero gives \(\hat p=r/m\).

In interval 1, the lower-risk stratum retains four of five participants, while the higher-risk stratum retains three of five. In interval 2, three lower-risk participants are event-free and eligible, and two remain observed; two higher-risk participants are eligible, and one remains observed. Therefore:

These are model estimates of conditional retention, not event probabilities. The fitted model uses \(Z\) because retention differs across its strata. In a larger study, the model could condition on other measured history; the required condition is that the history used adequately captures predictors of both dropout and subsequent event risk.

Compute Survival Through Interval 2

In interval 1, the observed event-free risk set contains four lower-risk and three higher-risk participants. The weighted denominator is \(4(5/4)+3(5/3)=5+5=10\). There is one event in each stratum, so the weighted event total is \(5/4+5/3=35/12\). Thus:

At interval 2, three lower-risk and two higher-risk participants were event-free at the start, but only two and one, respectively, remain observed for event ascertainment. The weighted risk denominator is \(2(15/8)+1(10/3)=15/4+10/3=85/12\). There is one observed event, in the lower-risk stratum, with weight \(15/8\). Hence:

The estimated probability of surviving both intervals is the product of the two estimated conditional survival probabilities:

This is a point estimate for survival through the prespecified horizon, interval 2. It is not a claim that the estimated retention probabilities are known without error; in practice, uncertainty from estimating the censoring model must also be accounted for.

Why the Weighted Hazard Targets the Cohort Hazard

To see the identity, consider a person who is event-free at the start of interval \(t\). Let \(H_{ij}\) be the observed history before the retention decision at interval \(j\), and write \(G_{it}=\prod_{j=1}^{t}p_{ij}\). Under the stated sequential independence condition, weighting by \(R_{it}/G_{it}\) removes the selection induced by retention. In one interval, conditional on the history, the event indicator \(D_{it}\) that would be ascertained under continued observation is independent of \(R_{it}\), and \(E[R_{it}/p_{it}\mid H_{it}]=1\). Therefore:

For multiple intervals, apply the same conditional-expectation argument successively, beginning with the last retention decision and then moving backward through the earlier decisions. Each factor \(R_{ij}/p_{ij}\) has conditional mean one and is conditionally independent of the future event information given the relevant history. Thus the weighted event total has expectation equal to the full-cohort event total. The same argument with the event indicator replaced by 1 gives the corresponding identity for the weighted event-free risk total. Their population ratio is the full-cohort interval hazard.

The sample weighted ratio is not generally exactly unbiased: it is a ratio of random totals. Under independent sampling, correct model specification, positivity, finite weight moments, and a positive population risk denominator, the law of large numbers makes the two sample totals converge to their expectations. The continuous mapping theorem then gives convergence of the weighted hazard to the cohort hazard. With a finite number of intervals and positive survival through each interval, multiplying the estimated conditional survival probabilities yields a consistent horizon survival estimate.

Comparison, Diagnostics, and Limits

For comparison, a naive discrete-time estimate that ignores dropout uses the observed risk sets. In this dataset it estimates the interval 1 hazard as \(2/7\). At interval 2, three people remain observed and event-free at the start, with one event, giving a hazard of \(1/3\). Its horizon survival estimate is therefore \((5/7)(2/3)=10/21\approx0.476\), compared with the IPCW estimate \(25/48\approx0.521\). The difference illustrates how accounting for observed predictors of dropout can change an estimate; this tiny dataset cannot establish which estimate is closer to the unknown truth.

A complete-case analysis would restrict attention to people observed through the horizon, discarding additional event information and generally changing the population represented. The ordinary Kaplan–Meier approach handles censoring under its corresponding independent-censoring condition; when censoring is independent only after conditioning on measured history, IPCW can instead reweight the observed risk sets. Neither method repairs unmeasured predictors of dropout, an incorrect censoring model, or lack of positivity.

A practical check is to trace the counts used to fit each retention probability: only participants eligible for that interval belong in its denominator. Also verify that cumulative weights multiply the conditional retention probabilities across intervals, and that each event numerator uses the same interval-specific weight as its risk-set denominator. Large or unstable weights can signal near-zero fitted retention probabilities or sparse histories. Such signs warrant investigation, but a reasonable-looking weight distribution cannot prove that the censoring model or its assumptions are correct.

Common mistake: Treating a dropout before interval 2 as though the participant had survived interval 2, or using only the interval 2 retention probability in the weight. Neither is justified. The event status after dropout is unknown, and the correct cumulative weight is the reciprocal of the product of retention probabilities along the observed path. IPCW adjusts selection under stated assumptions; it does not impute an individual’s unobserved event time.