# Analysis of Interval Censored Data

I am working with survival data. I have 2 groups, one for radiation and the other group for radiation/chemotherapy as adjuvant treatment for cancer. I am given an interval for every participant $$(x^{(start)}_1, x^{(end)}_1), \dots, (x^{(start)}_n, x^{(end)}_n)$$. So the data is both left and right censored. Some of the $$x^{(end)}_n$$ are not given.

I want to compare the survival times for both groups. I am supposed to do this in 2 different ways.

One way is through the log rank test, which I did through the ictest function in R.

For the second way, I am supposed to use an adaptive method. I started off by finding the survival curves using the NPMLE, but I am not sure how to go about after that.

• I'm guessing the issue has to do with treatment for cancer. Is the issue that you're performing periodic radiologic scans of the disease to determine if progression occurred, and that, if a progression is identified, you can't identify the time point at which the progression happened? Dec 13, 2021 at 23:48
• Yes that's right! A patient is monitored at every month for cosmetic effect of the therapy and may miss some visits and return in a possibly changed state. The survival time of interest is the time until the appearance of breast retraction. Dec 13, 2021 at 23:52

In clinical oncology, confirmatory trials may track disease response endpoints using clinical response criteria such as RECIST. This algorithm reflects the clinical practice of surveilling the disease status with periodic follow-up scans, and sum of longest diameters drawn on lesions in 3D space that grow by a certain percentage are declared "progressive disease" and are considered an event in time to event analyses.

Standard statistical analysis plans specify censoring criteria in these cases that roughly require that subjects

1. Attend scans at a desired frequency.
2. Are censored at the last time point of an evaluable scan
3. Are considered an "event" at the moment it's known that event criteria are met (i.e. death, or radiographic progression)

Example of an accepted, publicly available protocol and statistical analysis plan for these cases can be found on the US government clinical trials database: https://clinicaltrials.gov/ct2/show/NCT04516746

Typically scans occur every 2 months, and if 2 scans are missed, the subject is censored at the last evaluable scan. Even if PD is known to occur at 4 months and a few days, the patient is censored due to the long period with unknown clinical status.

The idea of finding the exact time point at which a disease meets criteria for PD is a theoretical issue that has no bearing on clinical practice. In practice, patients monitor the disease status in exactly the same way: attending periodic scans, and (more or less) presenting with symptoms of unequivocal disease progression. As such there's no misleading information or bias in using the actual date of radiologic progression as the event time, even though the subject may have met those criteria days or weeks prior to the scan.