Censoring

Censoring and Incomplete Follow-Up in Survival Data and Economic Models

A censored record shows that a patient's event, such as death or disease progression, had not happened by a certain date, or happened inside a window between visits, without showing exactly when. Survival methods keep that partial information instead of discarding the patient or treating the event as one that never happens. In health economics the distinction matters because cost-effectiveness depends on mean survival, and censored trial data rarely reveal the mean directly, so the amount and pattern of censoring decide how much of the QALY gain rests on observation and how much on extrapolation. How a censored record enters the likelihood and the Kaplan-Meier estimator rests on the non-informative censoring assumption, and censoring also bears on costs, treatment switching, truncation and competing events.

Three ways an event time can be incomplete

Clark and colleagues, in an introduction to survival analysis, list three routes to censoring. A patient has not yet had the outcome when the study closes, a patient is lost to follow-up, or a patient has a different event that makes further follow-up impossible. In each case the recorded time understates the true, unknown time to the event. Which side of the record is missing defines the form.

The left-censoring example follows Clark and colleagues, who note that most survival data include right-censored observations and that separate methods exist for left- and interval-censored data. Right censoring has several causes with different implications. Administrative censoring comes from a planned data cut-off, whereas loss to follow-up or withdrawal may be linked to how the patient was faring.

How a censored record enters the analysis

NICE Decision Support Unit Technical Support Document (TSD) 14 states that censored data remain useful because they provide a lower bound for the unobserved survival time, and survival techniques allow them to be used rather than excluded. Each patient therefore contributes a follow-up time and an event indicator. The standard notation for right-censored data is

$$Y_i = \min(T_i, C_i), \qquad \delta_i = I(T_i \le C_i)$$

where $T_i$ is the true event time for patient $i$, $C_i$ is that patient's censoring time, $Y_i$ is the time actually observed, $I(\cdot)$ is the indicator function and $\delta_i$ equals 1 when the event is observed and 0 when the record is censored. Under non-informative censoring a parametric model's likelihood contribution for patient $i$ is

$$L_i = f(Y_i)^{\delta_i}\, S(Y_i)^{1-\delta_i}$$

where $f(t)$ is the probability density of the event time and $S(t)$ is the survival function, the probability of remaining event-free beyond time $t$. An observed event contributes the density at its time; a censored record contributes only the probability of having survived to its censoring time. This is how censored patients inform the Weibull, log-normal and other curves fitted for economic models.

The Kaplan-Meier estimator uses censored records through the risk set, as follows.

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

where $t_j$ are the distinct event times up to $t$, $d_j$ is the number of events at $t_j$ and $n_j$ is the number of patients still at risk just before $t_j$. A censored patient stays in $n_j$ for every event time up to the censoring time and then leaves without causing a step in the curve, which is why censoring appears as tick marks on a plotted curve rather than drops.

The non-informative censoring assumption

Censored records are only as good as the assumption that justifies using them. Clark and colleagues state that standard methods for censored survival data are valid only if censoring is noninformative, meaning that those censored at a given time should be as likely to have a later event as those who remain in the study. TSD 14 makes the same point for health technology assessment, noting that its standard methods are suitable only if censoring is uninformative.

The assumption is most plausible for a planned data cut-off, where censoring depends on calendar date of entry rather than on prognosis. It is least plausible when withdrawal is driven by the patient's condition; Clark and colleagues give withdrawal because of drug toxicity or worsening clinical condition as examples of informative censoring, and add that little bias is likely when few patients are lost. In regression models the requirement can be relaxed to independence conditional on the covariates in the model, as in standard texts such as Kalbfleisch and Prentice. Whether the assumption holds cannot be checked from the censored outcomes themselves, because the event times that would reveal a violation are the ones that are missing. Independent censoring and dependent censoring have their own pages, including adjustment by inverse probability of censoring weighting.

Worked example with six patients and three ways to handle censoring

The following numbers are illustrative only. Six patients in a single trial arm are followed for progression-free survival over 12 months.

Step 1 (Kaplan-Meier). At month 2, six patients are at risk and one progresses, so $\hat{S}(2) = 1 - 1/6 = 0.8333$. Patient B leaves the risk set at month 4 with no step. At month 5, four patients (C, D, E, F) are at risk, so $\hat{S}(5) = 0.8333 \times (1 - 1/4) = 0.6250$. Patient D leaves at month 8. At month 9, two patients (E, F) are at risk, so $\hat{S}(9) = 0.6250 \times (1 - 1/2) = 0.3125$. Patient F is censored at month 12, and the estimate at 12 months stays at 0.3125.

Step 2 (restricted mean). The area under the curve to 12 months is the restricted mean survival time, here $\mathrm{RMST}(12) = 1 \times 2 + 0.8333 \times 3 + 0.6250 \times 4 + 0.3125 \times 3 = 2 + 2.5 + 2.5 + 0.9375 = 7.94$ months of progression-free survival.

Step 3 (two naive alternatives). Dropping the three censored patients leaves A, C and E, all of whom progressed, giving 0% progression-free at 12 months and a mean of $(2 + 5 + 9)/3 = 5.33$ months. Treating the censoring times as if they were progression times gives a mean of $(2 + 4 + 5 + 8 + 9 + 12)/6 = 40/6 = 6.67$ months. Both understate the 7.94 months the Kaplan-Meier curve supports. Treating every censored patient as progression-free to month 12 goes the other way, giving 3 of 6, or 50%, progression-free at 12 months, against the Kaplan-Meier estimate of 31%.

Interpretation. The censored patients add information without being counted as progressions, and the correct estimate lies between the pessimistic and optimistic shortcuts. The figure of 7.94 months still covers only the first year. With 31% estimated to be progression-free at month 12, the remainder of mean progression-free survival, and the QALYs that go with it, can only come from a fitted model extended beyond the data. If patient B had withdrawn because of toxicity that foreshadowed early progression, the non-informative assumption in Step 1 would fail and 0.3125 would be too high.

Why censoring matters in economic evaluation

TSD 14 notes that economic evaluation needs mean rather than median effects, and that survival data are commonly censored, so standard statistical methods cannot be used. When a proportion of patients are still alive at the end of follow-up, only a mean restricted to that time can be estimated directly. Unless the data are complete or very close to complete, TSD 14 states that extrapolation is required before survival data can be used in an economic model, typically through a parametric survival model. The NICE manual (PMG36, section 4.2.24) observes that a lifetime time horizon often requires extrapolation beyond the duration of the available studies and consideration of the associated uncertainty. Where censoring is substantial, TSD 14 suggests reporting a scenario based only on observed data to show how much the extrapolated period drives the incremental cost-effectiveness ratio.

The effects run through the whole model. In a partitioned survival model, censoring in overall survival and progression-free survival shapes the areas that become state occupancy, and so how costs and utilities divide between pre- and post-progression states. Costs bring a further problem. Lin and colleagues showed that the naive average of observed costs, taken over all patients or over uncensored cases only, can be severely biased, and that standard survival techniques do not apply to costs directly; their estimators instead combine Kaplan-Meier probabilities with costs within short time intervals. TSD 19 records that partitioned survival analysis itself grew out of work by Goldhirsch, Glasziou and colleagues around 1989 to 1990, who noted that applying standard survival methods to quality-adjusted survival induced informative censoring, because patients with worse health-related quality of life were more likely to be censored earlier.

Treatment switching shows how censoring can be misapplied. When control-arm patients cross over to the new treatment, censoring them at the switch makes censoring depend on prognosis. Section 4.6.26 of PMG36 advises against simple adjustment methods such as censoring or excluding data from patients who cross over, because they are very susceptible to selection bias.

Censoring, truncation and competing events

Three nearby ideas are often mistaken for censoring. Truncation concerns who enters the data at all. Cain and colleagues state the contrast. An event that occurred before study entry is left censored if the person is enrolled and left truncated if they are not. A censored patient appears in the data with partial timing; a truncated one is absent, and their simulations show that naive analyses ignoring left truncation can be substantially biased, with standard errors severely underestimated.

Competing risks arise from the third route in Clark and colleagues' list, an event that prevents the event of interest. Austin, Lee and Fine explain that treating competing events as censored is legitimate when estimating cause-specific hazards, but that one minus the Kaplan-Meier curve then overstates the cumulative incidence of the event of interest, regardless of whether the competing events are independent. They recommend the cumulative incidence function for absolute risk, which matters for a model that needs, for example, the probability of cardiovascular death in a population that can also die of other causes.

Missing data is the broader category; censoring is a structured kind in which the observed follow-up still carries information, which is why deleting censored records, as in Step 3, discards valid evidence.

Common misreadings of censored data

A censored patient is known to be event-free only up to the censoring time, not beyond it, and a tick mark on a Kaplan-Meier plot marks an exit from the risk set rather than an event. Using the same statistical method in both arms does not make censoring non-informative in either.

The proportion censored is an incomplete guide to maturity. The timing of censoring, the number still at risk in the tail and the reasons for leaving all matter, and Clark and colleagues warn that unequal follow-up between treatment arms may bias a comparison. For summarising follow-up, they prefer the reverse Kaplan-Meier method, in which the event indicator is reversed so that being censored becomes the outcome, because simpler medians of follow-up tend to understate it.

Sources

• Austin PC, Lee DS, Fine JP. Introduction to the analysis of survival data in the presence of competing risks. Circulation. 2016;133(6):601-609. https://doi.org/10.1161/CIRCULATIONAHA.115.017719
• Cain KC, Harlow SD, Little RJ, Nan B, Yosef M, Taffe JR, Elliott MR. Bias due to left truncation and left censoring in longitudinal studies of developmental and disease processes. American Journal of Epidemiology. 2011;173(9):1078-1084. https://pubmed.ncbi.nlm.nih.gov/21422059/ (abstract checked)
• 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 N. NICE DSU Technical Support Document 14: Survival analysis for economic evaluations alongside clinical trials, extrapolation with patient-level data. Sheffield: Decision Support Unit, ScHARR; 2011, last updated March 2013. https://doi.org/10.15131/shef.data.33745843
• 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. Sections 4.2.24 and 4.6.26. https://www.nice.org.uk/process/pmg36/chapter/economic-evaluation-2
• 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. https://orda.shef.ac.uk/articles/report/TSD_19_Partitioned_survival_analysis_as_a_decision_modelling_tool/33848653