Interval Censoring

Interval Censoring in Trial Follow-Up and Survival Models

Interval censoring is the form of censoring in survival analysis where a patient's event, such as cancer progression, is recorded only as happening somewhere after one check and before the next. Each patient therefore contributes a window, bounded by the last visit without the event and the first visit with it. It arises whenever an outcome is detected at scheduled scans or clinic visits, as in most progression-free survival data used in cancer appraisals. Treating the detection date as the event date pushes every event later than it really happened, and that error carries through to mean survival, state occupancy and QALYs in the economic model.

How assessment schedules create intervals

Zhang and Sun describe the common case as studies with periodic follow-up, where the true event time is greater than the last observation time at which the change had not occurred and no later than the first observation time at which it had. The interval is written $(L_i, R_i]$ for patient $i$. A patient who never shows the event has $R_i = \infty$, which is ordinary right censoring, and a patient who already has the event at the first visit has $L_i = 0$, which is left censoring.

Zhang and Sun distinguish current status data, where each person is examined once and the only information is whether the event had happened by then, from the general case of repeated visits at times that differ between patients. They also separate interval-censored data from grouped survival data. In grouped data every patient's intervals are identical or do not overlap, as in a fixed-interval life table, whereas interval-censored windows may overlap in any way.

The likelihood for a window rather than a date

Each patient contributes the probability that the event fell inside the observed window. An exact event time would contribute a density instead. For $n$ independent patients the likelihood is

$$\mathcal{L} = \prod_{i=1}^{n} \left[ S(L_i) - S(R_i) \right]$$

where $S(t)$ is the survival function, the probability of remaining event-free beyond time $t$, $L_i$ is the last assessment time at which patient $i$ was known to be event-free, $R_i$ is the first assessment time at which the event was found, and $S(R_i) = 0$ when $R_i = \infty$, so a right-censored patient contributes $S(L_i)$. Parametric models are fitted to this function by maximum likelihood estimation.

The likelihood relies on noninformative interval censoring. Zhang and Sun state the assumption as saying that, apart from the fact that the event lies between the two observed times, the interval and its endpoints carry no extra information about the event time. It fails if, for example, visits are brought forward because a patient feels worse, so that the timing of the visit says something about the timing of progression.

Turnbull's estimate and its blind spots

With exact or right-censored times the nonparametric maximum likelihood estimate of survival is the Kaplan-Meier estimator, which has a closed form. For interval-censored data Zhang and Sun note that there is no closed form, and that the simplest algorithm is Turnbull's self-consistency algorithm, an application of the EM algorithm that converges slowly. Fay and Shaw, describing the R package interval, explain that Turnbull showed the estimate is unique only up to a set of innermost intervals, often called Turnbull intervals, inside which the likelihood says nothing about where the probability mass sits. Zhang and Sun add that the estimate converges at rate $n^{1/3}$ rather than the usual $n^{1/2}$, because interval censoring gives much less information than right censoring, although they note that linear functionals of the estimate still converge at rate $n^{1/2}$.

Regression and parametric models

Covariate effects need models fitted to the same window likelihood. Zhang and Sun describe the proportional hazards model for general interval-censored data, with Finkelstein's approach estimating the coefficients and baseline cumulative hazard together. For economic models the parametric route matters most, since a parametric survival model fitted to the windows can be extrapolated in the usual way.

Shortcuts that replace the window with one date

The commonest shortcut uses the visit at which the event was first seen as the event time. Panageas and colleagues state that an analysis that ignores interval censoring and uses the detection date as the date of progression unavoidably overestimates median progression-free survival. A longer median may then reflect only the length of the surveillance interval.

The second shortcut imputes the midpoint of each window and then applies right-censored methods. Law and Brookmeyer found that the performance of midpoint imputation depends strongly on the underlying distributions and on the width of the interval between tests. Fay and Shaw report from that work that, when the censoring mechanism differed between two groups, a logrank test with a nominal size of 0.05 had an estimated size as high as 0.19 in one case. Vandormael and colleagues, simulating HIV incidence cohorts, found that midpoint estimates fell spuriously towards the end of follow-up once testing rates dropped below 80%.

Progression-free survival and the scan schedule

The European Medicines Agency appendix on progression-free and disease-free survival endpoints (EMA/CHMP/27994/2008/Rev.1) notes that the exact time of progression will generally not be known and that, for the primary analysis, interval censoring is generally ignored and the times of detected progression are analysed. It asks for supportive analyses based on scheduled evaluation times and using interval censoring, and for a description of compliance with the visit schedule by treatment arm.

The appendix also explains why the schedule matters for bias. Progressions found between scheduled visits, mixed with those found at scheduled visits, can introduce detection bias, for example when one arm has more toxicity or symptoms and is examined earlier. It states that the methods and frequency of tumour assessment should be the same across study arms, and that more frequent assessment does little to increase statistical power unless progression is very rapid.

Worked example with three-monthly scans

All numbers here are illustrative. Ten patients in one trial arm are scanned every 3 months for 12 months. Progression is first seen at month 3 in two patients, at month 6 in three, at month 9 in one and at month 12 in one. Three patients have a clear scan at month 12 and are censored there.

Step 1. Nonparametric estimate. With shared visit times the windows are grouped data, so the Turnbull estimate reduces to a life-table product built interval by interval. In the first interval 10 patients are at risk and 2 progress, so $S(3) = 1 - 2/10 = 0.80$. Then $S(6) = 0.80 \times (1 - 3/8) = 0.50$, $S(9) = 0.50 \times (1 - 1/5) = 0.40$ and $S(12) = 0.40 \times (1 - 1/4) = 0.30$. These values are identified only at the scan times.

Step 2. Restricted mean to 12 months. A Kaplan-Meier curve drawn at detection dates keeps each step until the scan, giving an area of 3 × (1 + 0.80 + 0.50 + 0.40) = 8.10 months. If every progression had happened just after the previous scan, the area would be 3 × (0.80 + 0.50 + 0.40 + 0.30) = 6.00 months. The data cannot separate values in that range, and the detection-date figure is its upper limit. Linear interpolation within each interval gives the midpoint, 7.05 months of restricted mean survival time.

Step 3. Interval-censored exponential model. For an exponential distribution with visits every $k$ months, the window likelihood has the closed-form solution

$$\hat{p} = \frac{d}{\sum_{i} m_i}, \qquad \hat{\lambda} = \frac{-\ln(1 - \hat{p})}{k}$$

where $d$ is the number of progressions, $m_i$ is the number of scan intervals for which patient $i$ was at risk (including the interval in which progression was found), $\hat{p}$ is the estimated probability of progressing within one interval, $k$ is the interval length and $\hat{\lambda}$ is the estimated monthly hazard. Here the intervals at risk sum to 2 × 1 + 3 × 2 + 3 + 4 + 3 × 4 = 27 and there are 7 progressions, so $\hat{p} = 7/27 = 0.2593$ and $\hat{\lambda} = -\ln(0.7407)/3 = 0.3001/3 = 0.1000$ per month. Mean progression-free survival is 1/0.1000 = 10.0 months, the median is 0.6931/0.1000 = 6.93 months and the fitted 12-month survival is $e^{-1.200} = 0.301$, close to the nonparametric 0.30.

Step 4. Detection dates and midpoints. Treating detection dates as exact gives total time 45 + 36 = 81 months, a hazard of 7/81 = 0.0864 and a mean of 81/7 = 11.57 months. Midpoint imputation gives 34.5 + 36 = 70.5 months, a hazard of 7/70.5 = 0.0993 and a mean of 10.07 months.

Step 5. Interpretation. The detection-date analysis adds 1.57 months (11.57 less 10.00), or 0.131 years, to mean progression-free survival. At an illustrative progression-free utility of 0.75, that is 0.75 × 0.131 = 0.098 undiscounted QALYs credited to time that patients did not spend progression-free. The midpoint comes close here only because every scan happened on schedule, the hazard is constant and the 3-month interval is short relative to the 10-month mean, and the simulations cited above show that agreement weakening once scans are missed.

Multi-state Markov models fitted to visit data

The same problem appears when transition rates for a Markov model are estimated from cohort or registry data in which disease stage is recorded only at visits. The manual for the R package msm, by Jackson, notes that in such multi-state models the times of changes of state are usually interval censored. Its likelihood uses, for each pair of successive observed states, the entry of the transition probability matrix for the time elapsed between the two visits, computed from a matrix of transition intensities.

Consequences for cost-effectiveness estimates

NICE Decision Support Unit Technical Support Document (TSD) 14 states that mean rather than median effects matter for economic evaluation, so the whole survival distribution is needed, and it treats parametric models as the likely preferred method for extrapolation. It also notes that Kaplan-Meier curves are stepped when follow-up happens only at set intervals, and that interval censoring methods should be considered where those intervals are relatively long. Interval censoring affects that mean twice. Within the trial period, curves built at detection dates place each event at the end of its window and overstate the area, as Step 2 shows. A parametric model fitted to detection dates carries the same shift into the extrapolated tail.

In a partitioned survival model, progression-free survival sets the split between pre- and post-progression time, while the date of death behind overall survival is usually known exactly, as the msm manual notes for chronic disease studies. A detection-date curve therefore moves time out of the progressed state and into the progression-free state, which changes QALYs and the costs attached to each state. When both arms share one schedule much of the shift appears in both, but absolute occupancy is still affected. Where schedules or visit compliance differ between arms, or between a trial and the source of a comparator curve, the incremental estimate itself can be biased.

Boundaries and common misreadings

Interval censoring concerns when an event happened, whereas truncation concerns whether a patient enters the data at all, and Zhang and Sun note that the two can occur together. In their terms, doubly censored data arise when both the starting event and the end event are right or interval censored, as with the incubation period between HIV infection and AIDS diagnosis.

A Kaplan-Meier curve with steps at scan dates can look like exact data, but its steps record when progression was found. A hazard rate estimated at detection dates describes time to detection under one schedule and is not the same quantity as time to the event. Because the EMA appendix prefers detection dates for the primary regulatory analysis, the headline progression-free curve is usually of that kind, and the supportive interval-censored results, where provided, are usually the better basis for the progression-free curve in an economic model.

Sources

• European Medicines Agency. Appendix 1 to the guideline on the evaluation of anticancer medicinal products in man. Methodological consideration for using progression-free survival (PFS) or disease-free survival (DFS) in confirmatory trials (EMA/CHMP/27994/2008/Rev.1). London: EMA; adopted 13 December 2012, effective 1 July 2013. https://www.ema.europa.eu/en/documents/scientific-guideline/appendix-1-guideline-evaluation-anticancer-medicinal-products-man-methodological-consideration-using_en.pdf
• Fay MP, Shaw PA. Exact and asymptotic weighted logrank tests for interval censored data: the interval R package. Journal of Statistical Software. 2010;36(2):1-34. https://doi.org/10.18637/jss.v036.i02
• Jackson C. Multi-state modelling with R: the msm package. Version 1.8.2. Cambridge: MRC Biostatistics Unit; 2024. https://cran.r-project.org/web/packages/msm/vignettes/msm-manual.pdf
• 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
• Law CG, Brookmeyer R. Effects of mid-point imputation on the analysis of doubly censored data. Statistics in Medicine. 1992;11(12):1569-1578. https://doi.org/10.1002/sim.4780111204 (abstract checked)
• Panageas KS, Ben-Porat L, Dickler MN, Chapman PB, Schrag D. When you look matters: the effect of assessment schedule on progression-free survival. Journal of the National Cancer Institute. 2007;99(6):428-432. https://doi.org/10.1093/jnci/djk091 (abstract checked)
• Vandormael A, Dobra A, Bärnighausen T, de Oliveira T, Tanser F. Incidence rate estimation, periodic testing and the limitations of the mid-point imputation approach. International Journal of Epidemiology. 2018;47(1):236-245. https://doi.org/10.1093/ije/dyx134
• Zhang Z, Sun J. Interval censoring. Statistical Methods in Medical Research. 2010;19(1):53-70. https://doi.org/10.1177/0962280209105023