# Should one use a censored survival model when an event is only observed at death?

A colleague of mine is trying to estimate how neutron radiation exposure changes cancer incidence rates (in mice). He has autopsy data that reports whether a cancer was observed at the time of death, but we do not know what time the cancer first appeared.

I argue that he should use a survival analysis with censoring to adjust for the fact that neutron radiation exposure reduces lifespan. But he argues that he can't apply a survival analysis because we don't know when the cancer first appeared, only that it was observed at the autopsy. Instead he thinks we should use a basic logistic regression and ignore lifespan.

Who's approach is better?

• The literature on radiation carcinogenesis is large. It's a reasonable assumption that there is a lag from exposure to clinical event. It sounds as though your colleague has knowledge of the timing of exposure, so ignoring lifespan will be throwing away data. – DWin Sep 3 '15 at 3:51

Actually, what you have described is the definition of "Current Status Data". The idea being that each subject is only observed once, and all we know is that an event has occurred or has not. This results in all of our data being either left censored (i.e. at time of observation, we see that the event has occurred) or right censored (at time of observation, we see that the even has not occurred).

As such, you should use methods for interval censored data to analyze your data. Standard interval censored data allows you to define an interval for the time of event for each subject such that

$t_i \in (l_i, r_i)$

where $t_i$ is the (unobserved) event time, and $l_i$ and $r_i$ are the lower and upper limits of an interval that you know to contain $t_i$. So with current status data, if we inspect subject $i$ at time $c_i$, and the event has occurred, we record their interval as $(0, c_i)$ (i.e. we know the event occurred before $c_i$), but if it has not occurred, we say $(c_i, \infty)$ (i.e. all we know is that the event has not occurred by time $c_i$).

Extremely biased here, but I would recommend the R-package icenReg, which allows Cox-PH and proportional odds regression models for interval censored data. Another great package is interval, which allows for log-rank tests (among other functions), but not regression models. Finally, the survreg function in the standard survival package allows for fitting a fully-parametric accelerated failure time model.

The bias towards icenReg is that I am the author.

Interestingly, your dataset sounds almost identical to one of the earliest publications involving current status data. See

Hoel D. and Walburg, H.,(1972), Statistical analysis of survival experiments, The Annals of Statistics, 18, 1259-1294

In fact, this dataset can be found as an example in icenReg (see miceData).

• This is extremely helpful, Cliff. I hadn't realized that the data was censored both left and right. Critical insight for me. Appreciate your thorough answer, and I'll check out the icenReg packages. Thanks! – Ben Haley Sep 3 '15 at 13:30
• Quick followup, I used Cliff's suggestion in a simulation I wrote and it worked like a charm. Maybe someone else will find it useful. rpubs.com/benjaminhaley/survive – Ben Haley Sep 4 '15 at 20:18
• Very nice! Also, nice little trick for switching to the $(l_i, r_i)$ format, which I've historically done with much more clunky machinery. – Cliff AB Sep 4 '15 at 21:40

The literature on radiation carcinogenesis is large. It's a reasonable assumption that there is a lag from exposure to clinical event. It's likely in the extreme that there is some knowledge about this process in mice. It sounds as though your colleague has knowledge of the timing of exposure as well, .... so ignoring lifespan will be throwing away useful data.