Dependent Censoring

Dependent Censoring in Survival Curves and Censored Costs

Survival curves used in economic models treat patients who leave follow-up early as if they would have fared like those who stayed. Dependent censoring, more often called informative censoring in health technology assessment, is the failure of that premise. When patients who are deteriorating are more likely to be withdrawn, switched or lost, those left at risk are a selected group, and the Kaplan-Meier estimator and everything built on it are biased. A consequential case in appraisals is treatment switching in oncology trials, where censoring control patients when they switch to the new drug is a common but discouraged shortcut. Adjustment methods for switching are set out in NICE Decision Support Unit TSD 16 and its 2024 update TSD 24, censored costs raise a parallel problem, and an illustrative control arm worked by hand shows how large the bias can become.

When stopping follow-up carries information about the event

Survival data record either the time of the event or the time at which observation stopped, a censoring time. Whether standard methods remain valid depends on why observations stopped; the number lost limits how large the bias can be. With $T$ the event time and $C$ the time at which observation would stop, standard methods need the hazard among patients still observed to equal the hazard among everyone still event free, as follows.

$$\lim_{\Delta t \to 0} \frac{P(t \le T < t + \Delta t \mid T \ge t, C \ge t)}{\Delta t} = \lim_{\Delta t \to 0} \frac{P(t \le T < t + \Delta t \mid T \ge t)}{\Delta t}$$

where the left side is the hazard at time $t$ among patients still under observation and the right side is the hazard in the whole population still event free. Independence of $T$ and $C$ guarantees the equality; censoring is dependent when it fails. A weaker condition, independence given measured covariates, underlies the adjustment methods below.

The condition cannot be confirmed from the data. Tsiatis showed in 1975, for competing causes of death, that without the not directly verifiable assumption of independence the observed data do not identify the quantities of interest. Clark and colleagues note that informative censoring may occur when patients withdraw because of drug toxicity or a worsening condition, although little bias is likely when few are lost. Dependent and informative censoring are used interchangeably in HTA; the assumption they violate is independent censoring, also called non-informative censoring.

Why Kaplan-Meier curves and Cox models go wrong

The product-limit estimator only sees the patients still at risk. A censored patient simply leaves the denominator at the next event time.

$$\hat{S}(t) = \prod_{j:\, t_j \le t} \left(1 - \frac{d_j}{n_j}\right)$$

where $\hat{S}(t)$ is the estimated probability of being event free beyond $t$, $t_j$ are the distinct event times, $d_j$ is the number of events at $t_j$ and $n_j$ is the number at risk just before $t_j$. If poor-prognosis patients are censored more often, the risk set over-represents good prognosis, each ratio $d_j / n_j$ understates the hazard and the curve sits too high; the reverse pattern pulls it too low. The restricted mean survival time and any extrapolation inherit the error.

The Cox proportional hazards model uses the same risk sets. Different censoring rates between arms do not bias the hazard ratio if censoring is independent; bias arises when censoring depends on prognosis in one or both arms, as when only control patients can switch and switching follows prognosis. Censoring that depends only on baseline covariates in the model leaves the conditional estimates valid but not an unadjusted curve. TSD 16 explains that, with the relevant prognostic factors measured, the Kaplan-Meier estimator, log-rank test and Cox estimator can be replaced by inverse probability of censoring weighted versions.

Treatment switching in oncology appraisals

TSD 16, by Latimer and Abrams (2014), concerns control patients moving onto the experimental treatment, common in cancer trials once interim analyses show benefit. If switchers benefit, an intention-to-treat analysis underestimates the overall survival gain, and the trial no longer compares a world with the technology against one without it.

Censoring switchers at the switch turns this into dependent censoring. TSD 16 states that such analyses are prone to selection bias through informative censoring, because randomisation balance is broken if switching is associated with prognosis, which is highly likely when clinicians decide who switches. The NICE manual (PMG36, section 4.6.26) says to avoid censoring or excluding patients who cross over, because these methods are very susceptible to selection bias, and asks for the uncertainty from any adjustment to be explored and quantified where appropriate.

The choice moves the ICER. Of 45 NICE appraisals in advanced or metastatic cancer completed by December 2009 and reviewed in TSD 16, 25 involved switching, and 11 of those used censoring or exclusion. For sunitinib in gastrointestinal stromal tumours (TA179) an incremental cost-effectiveness ratio of about £77,000 per QALY under intention-to-treat fell to about £27,000 with a rank preserving structural failure time model. In a later appraisal, of pazopanib in renal cell carcinoma (TA215), inverse probability of censoring weighting gave about £49,000 and the rank preserving model about £33,000.

Adjustment methods and the assumptions they trade

Inverse probability of censoring weighting (IPCW) censors switchers and up-weights similar patients who have not switched; Robins and Finkelstein applied it to adjust for dependent censoring at stopping or switching therapy in an AIDS trial. The rank preserving structural failure time model (RPSFTM) relies on randomisation instead, and two-stage methods on a secondary baseline such as progression. The unstabilised IPCW weight is the inverse of the probability of having stayed unswitched, given covariate history, and is written below.

$$w_i(t) = \prod_{k=0}^{t} \frac{1}{P\left[C_i(k) = 0 \mid \bar{C}_i(k-1) = 0, \bar{L}_i(k), T_i > k\right]}$$

where $w_i(t)$ is the weight for patient $i$ in interval $t$, $C_i(k)$ equals 1 if the patient has switched (and been artificially censored) by the end of interval $k$, $\bar{C}_i(k-1) = 0$ means no earlier switch, $\bar{L}_i(k)$ is the baseline and time-dependent covariate history up to the start of interval $k$, and $T_i > k$ means the patient is still alive. Stabilised weights, which TSD 16 reports Robins and Finkelstein recommend as more efficient, replace each numerator of 1 with the same probability conditioned on baseline covariates only (TSD 16 equation A1). Past treatment drops out of the conditioning for switching, because a patient is censored as soon as they switch.

IPCW needs data on every factor that predicts both switching and survival, which TSD 16 notes cannot be tested; with all confounders unmeasured it reduces to simple censoring, and it fails if some covariate level makes switching certain. In simulations with 500 patients its bias became extremely high when about 90% of control patients switched, the relevant proportion being among those eligible to switch. TSD 24 adds that truncating extreme weights, often at the 99th or 95th percentile, trades bias for precision and needs its own sensitivity analysis. Two-stage methods assume no unmeasured confounders at the secondary baseline and no time-dependent confounding between the secondary baseline and the time of switch.

The RPSFTM estimates switchers' survival without the new drug, assuming they gain the same effect, relative to time on treatment, as patients randomised to it. Adjusting censoring times only in switchers links censoring to prognosis, so dependent censoring reappears in the counterfactual data. Re-censoring breaks that link but discards long-term data, more so for larger effects; TSD 24 reports that analyses with and without re-censoring were both prone to bias, often in opposite directions, and asks for both.

Censored costs and quality-adjusted survival

Cost data can be dependently censored even when survival time is not. Lin and colleagues showed in 1997 that naive averages of observed costs can be severely biased, and Wijeysundera and colleagues explain why survival methods fail when cost replaces time. Cost at censoring and cost at death are both driven by the same patient's spending rate, so with a constant rate $R$ they are $R$ times the censoring time and $R$ times the death time, and are not independent.

Bang and Tsiatis introduced weighted estimators that account for censoring. The simplest weights each complete case by its inverse probability of remaining uncensored, as follows.

$$\hat{\mu} = \frac{1}{N} \sum_{i=1}^{N} \frac{\Delta_i A_i(t_i)}{\hat{S}_c(t_i)}$$

where $\hat{\mu}$ is the mean cost over a restricted period, $N$ is the number of patients, $\Delta_i$ is 1 for a complete case (death or full follow-up) and 0 if censored, $A_i(t_i)$ is the cost observed by time $t_i$, and $\hat{S}_c(t_i)$ is the Kaplan-Meier probability of remaining uncensored beyond $t_i$, with censoring as the event. In heart failure data with 32% censored over 1,080 days, Wijeysundera and colleagues found a full-sample mean of 30,420 against 36,490 reweighted, in 2008 Canadian dollars. In their simulations the simple estimator overestimated costs with 53% censoring.

Quality-adjusted survival has the same flaw. TSD 19 traces partitioned survival models to 1989 and 1990 work showing that standard methods induced informative censoring, as people with worse quality of life were censored earlier.

Worked example with switchers censored in a control arm

The figures are illustrative. A control arm of 200 patients loses 50 to death in year 1, leaving 60 progressed and 90 progression free. Only progressed patients may switch, and 30 do so at the start of year 2. Without switching, year-2 mortality would be 40% after progression and 10% without it; switching depends only on progression, so no confounder is unmeasured.

1. Target. With no switching, year-2 deaths would be 0.40 × 60 + 0.10 × 90 = 33, so the target is as follows.

$$S_{true}(2) = \frac{150}{200} \times \left(1 - \frac{33}{150}\right) = 0.75 \times 0.78 = 0.585$$

where $S_{true}(2)$ is two-year survival had nobody switched.

2. Censoring switchers. The risk set falls to 30 progressed and 90 progression-free patients, with 12 and 9 deaths, giving the estimate below.

$$S_{naive}(2) = 0.75 \times \left(1 - \frac{21}{120}\right) = 0.75 \times 0.825 = 0.61875$$

where $S_{naive}(2)$ is the Kaplan-Meier estimate with switchers censored, 3.4 percentage points too high because switchers came from the poorer-prognosis group.

3. Weighting. Using unstabilised weights, half the progressed patients stayed unswitched, so each progressed non-switcher gets weight 1/0.5 = 2, and progression-free patients, who cannot switch, get 1. The weighted risk set is 30 × 2 + 90 × 1 = 150 and weighted deaths are 12 × 2 + 9 × 1 = 33, so year-2 survival is 117/150 = 0.78 and the estimate returns to 0.585. Stabilised weights (120 of 150 unswitched overall) are 0.8/0.5 = 1.6 and 0.8/1 = 0.8, giving 26.4 weighted deaths in a weighted risk set of 120; 26.4/120 = 0.22, the same 0.78.

4. Mean survival. Assuming, for illustration, a constant hazard in year 1 and the year-2 hazard thereafter, mean survival is given by the expression below.

$$\bar{T} = \frac{1 - S_1}{h_1} + \frac{S_1}{h_2}, \qquad h_1 = -\ln S_1, \qquad h_2 = -\ln q_2$$

where $\bar{T}$ is mean survival in years, $S_1$ is one-year survival (0.75), $q_2$ is conditional survival through year 2, and $h_1$ and $h_2$ are annual hazards. The first term is 0.25/0.2877 = 0.869 years.

A 3.4-point gap at two years becomes 0.88 life-years once the biased hazard is extrapolated, inflating control survival and shrinking the incremental QALY gain.

5. Extreme weights. Had 57 of 60 progressed patients switched, each of the 3 non-switchers would weigh 60/3 = 20, and two-year survival would be $S(2) = 0.75 \times (1 - 29/150) = 0.605$ with one death among them but $S(2) = 0.75 \times (1 - 49/150) = 0.505$ with two, the instability TSD 16 describes when nearly all eligible patients switch.

Sensitivity analyses when the mechanism cannot be checked

Because the censoring assumptions cannot be verified, an analysis should show how far the decision depends on them. TSD 16 states that the intention-to-treat analysis should always be presented, calls for sensitivity analysis where the preferable method is unclear, and describes equal overall survival in both arms as a worst-case analysis giving a maximum ICER. TSD 24 recommends several adjustments resting on different assumptions, with and without re-censoring, and reporting of the range of IPCW weights.

Outside switching, a scenario analysis can give censored patients a higher hazard than those still observed, as for data missing not at random, and report how high it must go to change the decision.

Dependent censoring and neighbouring concepts

Administrative censoring at a planned data cut-off is usually independent unless prognosis changes with entry date, and with competing risks, treating the other event as censoring assumes an independence the data cannot confirm. Left truncation, or delayed entry, concerns when observation starts, not why it ends.

Errors that recur in submissions and models

One error is to treat censoring at switch as neutral or conservative. Its direction depends on who switches. If poorer-prognosis patients switch, as in the example, the benefit is understated; if fitter patients switch, the bias reverses.

Similar baseline profiles in censored and uncensored patients do not prove independence, since unmeasured factors may differ. Running a Kaplan-Meier analysis on cumulative cost is another error, as is accepting IPCW results without inspecting the weights; TSD 16 warns that very high weights for a few patients lead to erroneous results.

Sources

• Bang H, Tsiatis AA. Estimating medical costs with censored data. Biometrika. 2000;87(2):329-343. https://doi.org/10.1093/biomet/87.2.329 (abstract checked)
• Bell Gorrod H, Latimer NR, Abrams KR. NICE DSU Technical Support Document 24: Adjusting survival time estimates in the presence of treatment switching: an update to TSD 16. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2024. Sections 4.2 and 6. https://doi.org/10.15131/shef.data.33850897
• Clark TG, Bradburn MJ, Love SB, Altman DG. Survival analysis part I: basic concepts and first analyses. British Journal of Cancer. 2003;89(2):232-238. https://pmc.ncbi.nlm.nih.gov/articles/PMC2394262/
• Kalbfleisch JD, Prentice RL. The Statistical Analysis of Failure Time Data. 2nd ed. Hoboken, NJ: Wiley; 2002.
• Latimer NR, Abrams KR. NICE DSU Technical Support Document 16: Adjusting survival time estimates in the presence of treatment switching. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2014. Sections 1 to 7 and Appendix A. https://doi.org/10.15131/shef.data.33847621
• Lin DY, Feuer EJ, Etzioni R, Wax Y. Estimating medical costs from incomplete follow-up data. Biometrics. 1997;53(2):419-434. https://pubmed.ncbi.nlm.nih.gov/9192444/ (abstract checked)
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). London: NICE; 2022, last updated 13 October 2026. Section 4.6.26.
• Robins JM, Finkelstein DM. Correcting for noncompliance and dependent censoring in an AIDS clinical trial with inverse probability of censoring weighted (IPCW) log-rank tests. Biometrics. 2000;56(3):779-788. https://pubmed.ncbi.nlm.nih.gov/10985216/ (abstract checked)
• Tsiatis A. A nonidentifiability aspect of the problem of competing risks. Proceedings of the National Academy of Sciences of the USA. 1975;72(1):20-22. https://pubmed.ncbi.nlm.nih.gov/1054494/ (abstract checked)
• Wijeysundera HC, Wang X, Tomlinson G, Ko DT, Krahn MD. Techniques for estimating health care costs with censored data: an overview for the health services researcher. ClinicoEconomics and Outcomes Research. 2012;4:145-155. https://pmc.ncbi.nlm.nih.gov/articles/PMC3377439/
• Woods B, Sideris E, Palmer S, Latimer N, Soares M. NICE DSU Technical Support Document 19: Partitioned survival analysis for decision modelling in health care: a critical review. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2017. Section 2. https://orda.shef.ac.uk/articles/report/TSD_19_Partitioned_survival_analysis_as_a_decision_modelling_tool/33848653