Right Censoring

Right Censoring in Trial Survival Data and Economic Models

A right-censored record belongs to a patient who was still free of the event when observation stopped, so the data give a time the event had not yet reached rather than the time it happened. In health technology assessment most overall survival and progression-free survival data include such records, from patients still event-free at a trial's data cut-off or lost to follow-up. Survival methods keep each such patient in the analysis up to the last event-free time, which avoids the downward bias of averaging only the times that were observed. It is also why an economic model needs extrapolation, since mean survival beyond the end of follow-up, often a large share of the QALY gain, is never observed. Neighbouring pages treat censoring in general, administrative censoring at a planned cut-off and dependent censoring where leaving the study carries prognostic information.

How right-censored records arise in trials and cohorts

Clark and colleagues, in an introduction to survival analysis, describe a time as right censored when the event, if it occurs, lies beyond follow-up because the study closed first, the patient was lost to follow-up or a different event made further follow-up impossible. What counts as censored depends on the endpoint. In Clark and colleagues' ovarian cancer cohort, everyone who had not died was right censored for overall survival, whereas an analysis of ovarian cancer deaths alone would censor the patients who died of other causes. That second use touches competing risks, since a death from another cause is not a delayed event of interest, and unplanned exits are covered on the loss to follow-up page.

The right-censored data structure

Each patient has a potential event time and a potential censoring time, and only the earlier of the two is seen. Rodríguez, in lecture notes on survival models, describes this random censoring scheme with an observed time and an event indicator.

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

where $T_i$ is the potential event time of patient $i$, $C_i$ is that patient's potential 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 right censored.

Type I, Type II and random censoring schemes

Rodríguez distinguishes several mechanisms. Under Type I censoring a sample is followed for a fixed time, so the number of events is random. In what he calls fixed censoring, each unit has its own maximum observation time, fixed in advance. Under Type II censoring follow-up continues until a set number of units have had the event, so the study length is random. Random censoring treats censoring and event times as independent random variables.

A trial that recruits over several years and is analysed at one date gives each patient a maximum follow-up set by the date of entry, the fixed-censoring pattern, and loss to follow-up adds random censoring. Rodríguez states that all these schemes are non-informative and lead to much the same likelihood. The label Type I censoring therefore names one scheme of right censoring and is not a synonym for it.

How a right-censored record enters the likelihood

A patient who has the event at time $t_i$ contributes the density at that time, which Rodríguez writes as the survival function multiplied by the hazard. A patient still event-free at $t_i$ contributes only the probability of surviving beyond $t_i$. Both share the survival term, because both patients lived to $t_i$, and only the event adds the hazard.

$$L = \prod_{i=1}^{n} h(t_i)^{\delta_i} \, S(t_i)$$

where $L$ is the likelihood, $n$ is the number of patients, $t_i$ is the observed time for patient $i$, $\delta_i$ is the event indicator defined above, $h(t)$ is the hazard rate at time $t$ and $S(t)$ is the survival function, the probability of remaining event-free beyond $t$. The Weibull, log-normal and other curves fitted for a parametric survival model are estimated by maximum likelihood estimation of this function.

The exponential case and the events per exposure time

For the exponential distribution the hazard is a constant $\lambda$ and $S(t) = e^{-\lambda t}$. Rodríguez shows that the estimate then has a closed form.

$$\hat{\lambda} = \frac{D}{T}, \qquad \widehat{\mathrm{var}}(\hat{\lambda}) = \frac{D}{T^{2}}$$

where $\hat{\lambda}$ is the estimated hazard, $D$ is the total number of observed events, $T$ is the total observed time summed over all patients, events and censored records alike, and $\widehat{\mathrm{var}}(\hat{\lambda})$ is its large-sample variance. He notes that it is optimal only if the risk is constant. The standard error divided by the estimate equals $D^{-1/2}$, so precision rests on the number of events and not on the number of patients enrolled.

Non-informative censoring and what the data cannot show

Rodríguez gives the weakest assumption needed for the common likelihood, that censoring an observation provides no information about that unit's prospects beyond the censoring time. The event times that would reveal a violation are the ones missing, so the assumption is argued from the reasons patients left rather than tested. Clark and colleagues also warn that unequal follow-up between treatment arms may bias the analysis. Adjustments are covered under independent censoring, dependent censoring and treatment switching.

Worked example with ten patients and a 24-month cut-off

All numbers in this example are illustrative. Ten patients in one trial arm are recruited at different dates and analysed at a common cut-off, so each has a different maximum follow-up of up to 24 months. Five die, at months 3, 6, 9, 14 and 20. One patient is lost to follow-up at month 8 and four are alive at the cut-off, with 12, 16, 22 and 24 months of follow-up.

Step 1. Fit an exponential model. The observed times of the deaths sum to 3 + 6 + 9 + 14 + 20 = 52 months, and those of the censored patients to 8 + 12 + 16 + 22 + 24 = 82 months, so T is 134 months and D is 5. The estimated hazard is 5 / 134 = 0.03731 per month. Its standard error is the square root of 5 divided by 134, or 0.01669, a relative error of 45% from only five deaths.

Step 2. Split mean survival at the end of follow-up. The exponential mean is 1 / 0.03731 = 26.80 months. Fitted survival at 24 months is $e^{-0.03731 \times 24} = e^{-0.8955} = 0.4084$. The mean restricted to 24 months is 0.5916 (one minus 0.4084) multiplied by 26.80, or 15.86 months. That leaves 10.94 months (26.80 less 15.86), or 40.8% of the mean, beyond the last observed time. Under a constant hazard that share always equals fitted survival at the end of follow-up.

Step 3. Check against the Kaplan-Meier curve. The risk set is 10 at month 3, 9 at month 6, 7 at month 9 after the loss at month 8, 5 at month 14 and 3 at month 20. The estimates are 0.900, 0.900 × 8/9 = 0.800, 0.800 × 6/7 = 0.6857, 0.6857 × 4/5 = 0.5486 and 0.5486 × 2/3 = 0.3657. The area under the steps to month 24 is 3 × 1 + 3 × 0.900 + 3 × 0.800 + 5 × 0.6857 + 6 × 0.5486 + 4 × 0.3657 = 16.28 months, close to the fitted 15.86. Kaplan-Meier survival at 24 months is 0.366 against the fitted 0.408, a gap ten patients cannot resolve.

Step 4. Compare two shortcuts. Dropping the five censored patients gives a mean of 52 / 5 = 10.40 months. Treating every censoring time as a death gives a hazard of 10 / 134 = 0.07463 and a mean of 13.40 months, which equals the plain average of all ten observed times. Both fall below the restricted mean alone.

Step 5. Translate into QALYs. At an illustrative utility of 0.70, undiscounted, the exponential mean of 26.80 months (2.233 years) gives 1.563 QALYs per patient. The observed period contributes 15.86 months (1.321 years), or 0.925 QALYs, and the extrapolated period 10.94 months (0.912 years), or 0.638 QALYs.

Interpretation. Right-censored records raise the estimate above both shortcuts because each adds event-free time without adding an event. About two fifths of the modelled QALYs rest on the constant-hazard assumption beyond month 24, which the five deaths can barely inform. A Weibull or other fitted curve with a hazard that rises over time would usually put less of the mean beyond the cut-off, so the choice of curve matters most where censoring is heaviest.

How much of a model's mean survival lies beyond follow-up

In the Figure 1 example of NICE Decision Support Unit Technical Support Document (TSD) 14, follow-up ends at about 40 weeks with about 45% of the treatment group still event-free, so only a restricted mean can be estimated directly, and parametric models are fitted to estimate the whole area under the curve. The restricted mean survival time is the directly estimable part, and in a partitioned survival model the extrapolated parts of each curve set state occupancy for the rest of the time horizon.

Heavy right censoring in the tail also affects how curves are fitted. TSD 14 records that in TA86 and TA121 exponential models were fitted only up to certain time points, leaving out heavily censored final data, an approach it judges highly questionable because excluding data increases uncertainty.

Reconstructing right-censored data from published curves

Economic models are often built from published Kaplan-Meier curves rather than patient-level data. Guyot and colleagues note that the usual reported results, median times and Cox hazard ratios, are not sufficient statistics for meta-analysis or cost-effectiveness analysis, and that the average survival time is a biased estimate when censored observations are present. Their algorithm inverts the Kaplan-Meier equations from digitised curves. The published numbers at risk approximate how many patients were censored in each interval, and the method assumes that censoring occurs at a constant rate within each interval, which the authors judge reasonable if censoring is non-informative. With neither numbers at risk nor total events the method assumes no censoring, and reconstructed hazard ratios became, in their words, frankly unusable. They recommend that all trials report numbers at risk, a practice TSD 14 describes as relatively rare.

Right censoring beside left censoring, interval censoring and truncation

The forms of censoring differ in which side of the event time is missing. A right-censored record bounds the event time from below. Left censoring bounds it from above, as with a recurrence already present at the first scan, and interval censoring places it between two assessments, with right censoring as the special case whose window never closes. Truncation differs again because it concerns who enters the data at all, and under left truncation a patient whose event came before entry never appears.

Misreadings of right-censored data

A censored patient is known to be event-free only up to the censoring time. A tick mark on a Kaplan-Meier plot is an exit from the risk set, and a flat stretch at the end of a curve may reflect a handful of patients still at risk rather than an absence of events. Clark and colleagues note that confidence limits are often wide at the tail of the curve, making meaningful interpretation difficult.

Median follow-up is easy to understate. In Clark and colleagues' ovarian cohort the median follow-up of all patients was 1.7 years, while the reverse Kaplan-Meier estimate, in which being censored is treated as the outcome, was 5.3 years.

Sources

• 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/
• Guyot P, Ades AE, Ouwens MJNM, Welton NJ. Enhanced secondary analysis of survival data: reconstructing the data from published Kaplan-Meier survival curves. BMC Medical Research Methodology. 2012;12:9. https://doi.org/10.1186/1471-2288-12-9
• 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; June 2011, last updated March 2013. https://doi.org/10.15131/shef.data.33745843
• Rodríguez G. Lecture Notes on Generalized Linear Models. Princeton University; 2007. Chapter 7, Survival models. https://grodri.github.io/glms/notes/c7.pdf