Survival Curves: Accrual and The Kaplan-Meier Estimate

This is a technical topic about how real survival curves are calculated using a procedure called the Kaplan-Meier method. The Kaplan-Meier method is so widely used and so well known, that in research papers survival curves are more often than not called Kaplan-Meier curves. After reading this article you will be better able to interpret certain features of survival curves and you will be alert to a important issues concerning the reliability of survival curves. Although I explain the math in detail at the end of the article, you'll get what you need even if you skip it (The calculation isn't actually too difficult though). If you aren't going to delve into the technical literature where real world survival curves live you probably don't need to read this at all.

In my basic survival curve articles, I have mostly described survival curves as if a group of patients were all followed starting at the same time so that at any later time they all have the same amount of follow-up. In the real world, it doesn't actually work that way and this complicates the construction and interpretation of survival curves.

In any realistic clinical trial it takes time to accrue the patients for the trial. Often it takes several years. This is also true of retrospective studies which review results of past patients, again because the patients will have been diagnosed or started treatment at different times, often over several years. Either way, this means that patients are followed for survival starting at different times. But the results are analyzed at one time and so at that time, the patients have varying length of follow-up. For example, suppose a trial signs up patients (and starts their treatment) during its first two years and is analyzed at 5 years, then a surviving patient who was accrued at the start has five years of follow-up, while a surviving patient who was accrued after two years has only three years of follow-up. As long as some of the patients are alive at the end of the trial, as is almost always the case, the problem of varying follow-up is sure to arise.

What is desired is to find a way to allow a patient to contribute to the survival curve for the entire length of time he was followed, but to statistically "remove" the patient from the curve after that time. For example, if a patient has been in the trial for three years and is still alive, the fact that this patient has lived three years should contribute to the survival data for the first three years of the curve, but not to the part of the curve after that. You don't want to count the patient as having died after three years, and you don't want to count him as having survived longer than that either. Of course, if a patient dies during the trial, then all information about that patient's survival is known and the survival curve will reflect that patient's death at the appropriate point with a step down.

Mathematically removing a patient from the curve at the end of their follow-up time is called "censoring" the patient. This is perhaps an unfortunate choice of words, but you should understand that in this context "censor" is a purely technical term and has nothing to do trying to limit information or prevent expression. The aim is to produce the most accurate possible survival curve taking into account all the of information available. A few conspiracy minded alternative therapy advocates accuse conventional medicine of "censoring their data, presumably to conceal something - if you ever see this argument, one thing you can conclude for sure is that the writer is totally clueless!

How Censoring Affects the Shape of the Curve

When a patient is censored the curve doesn't take a step down as it does when a patient dies. In fact, unless the curve has tick marks to show where patients were censored, there is nothing to tell you where a patient was censored. But because censoring the patient reduces the number of patients who are contributing to the curve, each death after that point represents a higher proportion of the remaining population, and so every step down afterwards will be a little bit larger than it would have been. This effect on the shape of the curve isn't usually something which you can see just by looking at the curve.

Censoring and Estimation

The part of the curve after the first patient is censored is only an estimate of survival for the group rather than the actual survival, which is not yet known since the censored patients are still alive at the time of analysis. If the analysis was done again at a future date (and often results from clinical studies are updated with increased follow-up) then the information that a previously censored patient had continued to survive or had died at some point would be incorporated into the curve. Since everyone eventually dies, eventually, hopefully in the far distant future, all patients will have died and the survival curve for the group will be precisely known. Until that point, the curve is only an estimate.

Censoring a patient in effect reduces the sample size of patients at risk after the time of censorship. Reducing sample size always reduces reliability, so the more patients are censored and the earlier they are censored the more unreliable the curve is. Because each censored patient reduces the reliability of the curve from that point forward, the end of the curve is most affected. This is unfortunate, since the end of the curve represents long term survival which is the ultimate goal. Naturally, getting data about long term survival takes a long time. With the fast pace of medical research, by the time reliable long term data is available, more promising new treatments are being tested and preliminary data on those treatments may be in. There is a tension between the uncertainty inherent in shorter term results and the promise of encouraging results from new approaches.

Many clinical trials are designed with a minimum follow-up time. This means that the results aren't reported until that amount of time after the last patient signed up for the trial. In this case, no patients will be censored until after that minimum time on the curve. But often reports of preliminary results don't include any minimum follow-up time and include patients with very short follow-up time which definitely affects the reliablity of the curve.

How Censoring is Shown in Real World Survival Curves

It is useful to see when patients are censored so as to get a feeling for the reliability of the curve over time. Survival curves often have a tick mark at each point where a patient was censored. They may show the number remaining "at risk" at several points instead. The number at risk at any point in the survival curve is the number of patients who are still alive and whose follow- up extends at least that far into the curve. Unfortunately, all too many survival curves show neither tick marks nor the number at risk, and so it may be impossible to get a feel for the reliability of the curve, especially towards the end. If the paper does give a minimum follow-up time, at least you know no patients were censored earlier than that.

Last, and probably least, it's time for the math! The good news is that the Kaplan-Meier calculation is relatively straightforward. First, suppose that at after two years, the survival curve has reached 60%. Suppose that during the third year 10% of the surviving patients die. Then, at the start of the fourth year you can calculate that 90% of 60% = 54% of patients are still alive. If during the fourth year, again 10% of the patients who made it to the start of the year die, then at the end of the fourth year 90% of 54% = 48.6% will be alive. To generalize, if the time that the curve covers is broken up into intervals, then the percentage surviving at the start of any interval is equal to the probability of surviving each of the preceding intervals all multiplied together. In the example, the probability of survival at the start of the fifth year (the end of the fourth) is 0.6 * 0.9 * 0.9 = 0.48 (or 48%).

Of course, if at the start of the trial there are 100 patients and 48 are still alive at the start of the 5th year, you can obviously calculate the survival at that point in time as simply 48/100 * 100% = 48%! The above calculation using the probability of surviving all preceding time intervals seems to be rather the long way around! Rest assured there is a method in the madness.

Remember, that the aim here is to find a way to account for censored patients, that is to "remove" them from the curve at the time their follow-up ends. With the product of intervals method it turns out to be simple. When a patient is censored, the number of patients "at risk" is reduced by one. So when a patient dies, the survival for the interval ending with his death is calculated according to the number remaining at risk at the time of death. So for example, if at the start of the interval 25 patients are alive, and during the interval, two patients are censored then at the time of death 23 patients were at risk (25-2 censored patients). So the chance of surviving the interval is 22/23. If two patients died at the same time, the chance of surviving the interval would be 21/23.

Now that we can calculate the probability of surviving each of the intervals taking the censoring into account, the calculation of the entire survival curve using the product of intervals method I started with is simple. To get the level of the curve at the end of each interval, multiply the chance of surviving all the preceding intervals together. Connect these points with like a staircase and you have your typical survival curve.

Now that you've seen how the calculation works, you can see why censoring patients results in slightly larger steps from then on. In the example, we reduced the number of patients at risk during an interval from 25 to 23 to account for two censored patients. If we had not had censored patients in the interval the death would have caused the curve to step down by 1/25. But because of the censoring, the curve stepped down by 1/23 instead. 1/23 is a little bigger than 1/25 so the step is a little bigger than it would have been. You can also see that if the number of patients still at risk is reasonably large, the effect on the step size will be quite small as in this case, and you won't be able to pick it up just by looking at the curve.

An Example

To keep things simple, this example will concern only seven patients - most any real world survival curve would have more. To begin, suppose the survivals of these seven patients (sorted by length) are:

Note that the "+" signs mean that the patient was still alive at the end of his or her follow-up. In other words, these are the censored patients.

Here is the actual curve plotted from this computation:

Here's a more detailed explanation of the calculation for one interval, the interval from 1 to 4 years. When the interval started, 6 patients were alive and still in the curve (at risk). During the interval two patients were censored (2+ and 3+) so that at the end of the interval four patients were still at risk. Since the interval ends with the death of one of those, the chance of surviving the interval is estimated as 3/4. Also notice that at the start of the next interval (4 through 10 years), only three patients were at risk due to the death at the end of the interval.

The first and last intervals are special since they are bounded by the time zero or the end of the curve whether or not there was a death at these times. It is very rare for there to be a death at time zero, but fortunately it is not rare for the patient with the longest follow-up to be alive at the end of the curve, as in my example. When that happens, the curve doesn't step down at the end but rather is horizontal - the estimated survival probablity doesn't change. Note that if only one patient has the longest follow-up as was the case here, but unlike this example, that patient died, then the curve would step all the way down to zero at the end. This dramatic result would be based on only one patient, and it would be a big mistake to assume that the chance for similar patients to survive longer than this is actually zero! I've seen this in more than one curve in the literature. It's good to know that this sudden plunge to zero by no means forecloses a better future for the next patient, for instance you.