Survival Curves: Tales of the Tail
While I think that when it comes to survival and survival curves, the whole picture is what counts, I also think that long term survival (which is quite literally, “the bottom line” on a survival curve) is particularly important. This means the end or the tail of a survival curve. This article is about the pitfalls interpreting the ends of real survival curves. Ideally, a survival curve is based on many patients, all of whom have long term follow-up, and in this case there is no particular difficulty interpreting the curve. If you have a cancer with well known treatments which yield good results this is probably the only kind of survival curve you should be paying attention to. But if your situation dictates that you explore new treatments which have more or less preliminary results, you will be exposed to less than ideal survival curves and there are a number of pitfalls you need to know about.
“Complete” Versus “Incomplete” Curves
Since in the end everyone dies of something, if you follow any group of people long enough, their survival curve will reach zero. At that point there is no more information to be gained about their survival. You know all there is to know. The curve is “complete”. In the real world, it’s rare to find cancer survival curves that follow people until their deaths at a ripe old age – and if you do find such curves, they obviously won’t be about the results of new treatments! But unfortunately all too many world curves for advanced cancer are complete with much less follow-up because all of the patients died of their disease. There is also a much happier reason for considering a curve to be complete.
If instead, some of the patients are cured of their cancer, the curve will ultimately end with an extended plateau – a nearly flat line – or close to that. If the plateau is long enough (and well populated enough), for you to conclude that the remaining patients have been cured, then for all practical purposes the curve is also complete – even if there is a theoretical possibility of a late death.
The first pitfall is simply not to mistake an incomplete curve which ends in a short plateau for proof that there are cures! Staircase curves are composed of plateaus and steps. The mere fact that the end of a curve is a plateau doesn’t mean there is any evidence the final plateau will happily extend indefinitely. It depends on the shape of the curve.
This incomplete curve ends with a plateau, but it’s so short in relation to the rest of the curve that there is really no way to know whether this is going to turn out to be the beginning of an extended plateau because people are being cured, or whether the curve will keep stepping down all the way to zero or anything in between. Here are two ways it could plausibly turn out:
Censored Data and Giant Steps at the End
Incomplete survival curves need not end in a plateau. They can also end in a step. A survival curve will end in a plateau if the patient with the longest follow-up was still alive at the end of his follow-up. The curve will end in a step if a patient with the longest follow-up died. In fact, if only one person had follow-up that long, the curve will plunge all the way to zero at the end just as in this example:
Every curve which reaches zero will end in a step down. But if that last step is a big one, it almost certainly means there are reliability problems with the end of the curve! It does not mean a bunch of patients suddenly died at that time – it turns out that towards the end of the curve even a single death can cause a large drop. The problem arises because patients are mathematically “removed” from the curve at the end of their follow-up time (For all the details of how this works, see my article on Kaplan Meier Estimation and Censored Data). This so-called “censoring” effectively reduces the sample size without decreasing the survival estimate. At the very end when the curve depends only on the result of a single patient with the longest follow-up, the estimate will continue to be flat if that patient is still alive, or will plunge to zero if that patient died. Either way, where the estimate of survival depends dramatically on the fate of a single patient, it is clearly unreliable, as is any other statistical estimate which depends on a single observation. With more observations at that length of follow-up, the truth would probably be found to be somewhere between the extremes of a plunge to zero or a final plateau at the level just before the end of the curve. Technically, the estimate of survival given by a big final step is still the most accurate possible estimate but the reliability of that estimate is so low that this hardly matters.
Some survival curves are not plotted beyond the time when the number of patients still in the curve (“at risk”) declines below some level just to avoid this kind of problem. Also if more than one patient has the maximum follow-up, a single death among those patients will cause the final estimate to drop only part of the way to zero. So survival curves are certainly not always unreliable at the end.
A survival curve can also have large steps near the end rather than actually at the end, and anytime you see unusually large drops toward the end of a survival curve it’s an indication of reliability problems. For instance, if the patient with the second to longest follow-up dies, then at that point the curve will drop halfway to zero. If the curve is far above zero at this point due to small sample size and/or censoring then this drop will be large. All of my comments about large final steps apply to this situation as well. Here’s an example:
This CancerGuide Page By Steve Dunn. © Steve Dunn
Page Created: 2002, Last Updated: June 4, 2002