A practical overview and statistical analysis of interval‐censored data in cancer

In many clinical, biomedical, and epidemiologic cancer studies, the outcome of interest is the time until an event (or failure) occurs, and “survival analysis” is performed to estimate the effect of covariates on survival. Most clinical investigators are familiar with the term “right censoring”. The type of censoring we focus on in this paper is “interval-censoring” [1-3]. Interval-censoring occurs when the event of interest cannot be easily observed but is determined by imaging and/or blood biomarker tests at clinic visits. For example, suppose a patient comes in for imaging every 3 months, to detect the event (cancer recurrence). If recurrence is found at a visit, it occurred sometime between that visit and the previous visit where no recurrence was detected—this is “interval-censoring”. Researchers often code the event date as the visit when recurrence is detected, but this overestimates time to recurrence since it occurred earlier within the interval. Modern statistical software supports proper interval-censoring methods, making it unnecessary to use visit dates as event dates. Since most cancer recurrences are identified through imaging or biomarker tests at clinic visits, interval-censoring methods should be used, at least as a sensitivity analysis to assess the impact of coding the event date as the visit date when recurrence is detected . Consider a study of biochemical recurrence in 260 patients with localized prostate cancer who underwent radical prostatectomy [4]. After surgery, prostate-specific antigen (PSA) levels drop to 0 ng/mL, and recurrence is defined as the time PSA exceeds 0.1 ng/mL. Since PSA is measured only at regular intervals (every 3 or 6 months), the recurrence time is unknown and interval-censored between visits. Clinicians often use the first elevated PSA test as the recurrence date, though the true event occurred between the current and previous visit. An interval-censored Cox model assessed the impact of Gleason score, surgical margin, preoperative PSA, and pathologic stage on recurrence time. Parameters of survival models are estimated by maximizing a “likelihood” (ML). A patient with an exact failure time t contributes the probability of failure at t to the likelihood. A patient right-censored at time t contributes the probability of surviving beyond t. An interval-censored patient whose event occurs in the interval (L, U), contributes the probability of failure within that interval. After assuming a survival model (e.g., proportional hazards), the probabilities of failure at a given time (no censoring), between two time points (interval-censoring), or greater than a time point (right-censoring), are easily calculated and used in ML. Despite the long history of interval-censoring statistical methodology [1-3], until recently, few software packages were available to analyze interval-censored data in a statistically valid way. Consequently, the event date was often assigned to the clinic visit date when the event was ascertained, allowing use of survival software that only supported right-censoring. Interval-censoring approaches are now readily available in SAS version 9.4 (SAS Institute Inc), R, and Stata Version 19 (StataCorp). In SAS, proc iclifetest is designed for Kaplan-Meier analysis for interval-censored data, and proc icphreg for interval-censored proportional hazards models. Kaplan-Meier curves for interval-censored data can be fit using the R package Icens (version 1.81.0). The Cox proportional hazards model for the interval-censored data can be estimated in the R package icenReg (version 2.0.16). In Stata, the intcens and stintreg modules support estimation. In our simulation below, we used the SAS procedures. When using any software program to analyze interval-censored data, the dataset needs to have two variables corresponding to the lower and upper endpoints, (L, U), described above. For example, if the event is found on imaging at 6 months with the previous visit at 3 months with no event, you give the software program the endpoints (3, 6). We performed a small-scale simulation to show possible bias in estimating the hazard ratio and survival percentages using a Cox model with different approaches to account for interval-censoring. We simulated datasets based on the model and parameter estimates reported in a Bladder Cancer Recurrence Randomized Clinical Trial (Southwest Oncology Group [SWOG] S0337 [5]. Patients with low-grade non-muscle-invasive bladder cancer typically undergo transurethral resection of bladder tumor (TURBT). In the referenced clinical trial, the intervention group received Gemcitabine bladder instillation immediately after TURBT, while the control group received saline. Recurrence rates by year 4 were significantly lower in the Gemcitabine group (35%) compared to the control group (47%). Patients were monitored with cystoscopy and cytology every 3 months for the first 2 years and every 6 months thereafter, resulting in interval-censored recurrence times. For simplicity, we simulated time-to-recurrence data using a piecewise-exponential distribution, calibrated to approximately match the 1-, 2-, 3-, and 4-year recurrence percentages and a hazard ratio of 0.68 between arms, consistent with the trial findings. Additional simulation details are provided in Table 1’s caption. 1-year recurrence intervention (20.0%) 1-year recurrence control (28.0%) 4-year recurrence intervention (35%) 4-year recurrence control (47%) Hazard ratio (0.68) The results from the simulation study in Table 1 demonstrated that using the interval-censoring Cox regression produced unbiased estimates for all quantities. However, using the upper endpoint of the interval as the event time can overestimate the recurrence time, and accordingly, underestimate the percentage with recurrence at a given time. This was especially true for estimating the percent recurring at an earlier time point (1 year). Underestimation worsened as the interval between visits increased. Although unrealistic in practice, if the follow-up visits were only scheduled annually, the 1-year percent recurrence was severely underestimated (approximately 70% biased downward in both arms) using the upper endpoint of the interval as the recurrence time. Even with a 6-month follow-up interval (more realistic), the 1-year percent recurrence was considerably underestimated (approximately 25% biased downward in both arms) using the upper endpoint. Notably, the estimated 4-year recurrence percentages were nearly unbiased across all methods. Similarly, hazard ratio estimates remained unbiased across all methods. This was expected, as the Cox model relies only on the ordering (ranks) of event times, not their exact values. Thus, the hazard ratio estimate in the Cox model was less sensitive to using the upper endpoint than the survival percentage estimate, which also required estimation of the baseline survival function on the correct time scale. Thus, if estimating the hazard ratio, using the upper endpoint with the Cox model appeared to have little bias. Additionally, examining later time points—after most follow-up data was collected (e.g., year 4)—also showed minimal bias. Because of the broad range of possible interval-censoring configurations (e.g., time between follow-up visits) and underlying survival distributions generating the data, it was difficult to draw definitive conclusions from a small-scale simulation study. We can only make general suggestions. Based on our simulation, the interval-censored Cox model performed best. While using the upper limit of the follow-up interval may be acceptable with frequent follow-up visits, it can introduce bias when intervals are longer, especially at earlier time points. Given that interval-censoring methods are widely implemented in statistical software, we recommend their use, to at least perform a sensitivity analysis to assess the impact of using simplified approaches (e.g., using the upper endpoint of the interval), especially if early failure is of interest. This simulation focuses on bias in estimation, not on determining optimal follow-up schedules, which should be guided by the recurrence risk and disease severity. Tanujit Dey: conducted statistical data analysis, developed the research concept and focus, led manuscript editing, directed the project, and served as project co-lead. Stuart Lipsitz: conducted statistical data analysis, developed the research concept and focus, led manuscript editing, directed the project, and served as project co-lead. Zara Cooper: contributed to manuscript editing and provided consultation. Debajyoti Sinha: contributed to manuscript editing and provided consultation. Quoc-Dien Trinh: contributed to manuscript editing, provided consultation, and directed the project. Alexander Cole: contributed to manuscript editing, provided consultation, and directed the project. Tim Clinton: involved in manuscript editing, offered consultation, and guided project direction. All authors read and approved the final manuscript. Not applicable. The authors declare that they have no conflicts of interest relevant to this work. The authors declare that they have no funding associated with this work Not applicable. This study was conducted entirely using simulated data; no patient data were used, generated, or analyzed. The simulated datasets can be reproduced based on the methods described in the article. The simulation code used to generate the data is available upon reasonable request from the corresponding author. 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