Kaplan Meier (June 27, 2000)

Dear Professor Mean, When I read my medical journals, I keep on coming across terms like "Kaplan-Meier Product Limit estimate" or "Kaplan-Meier survival curve." What do these terms mean and when are they used?

Often we want to measure how long it takes for something to occur. The most common (and the most morbid) example is how long it takes for someone to die. For this outcome, we want to estimate the fraction of patients who survive for at least one month, at least three months, etc. This estimate is known as a survival curve.

The term survival is sometimes misleading, because we can use it for other less severe outcomes like how long until a cancer relapse, or how long until an infection occurs. Sometimes it can even be used for a positive outcome, like how long it takes for a couple to conceive. But for the rest of this example, we'll keep things simple by assuming that the outcome is time until death.

Estimating a survival curve is often complicated by the uncooperative way in which research subjects sometimes behave. For example, some subjects decide to leave a study part of the way through. Others refuse to die before the study ends. We label these uncooperative subjects as censored observations. They survived for at least three months, but then we lost touch with them. Or they survived at least three years, but then we had to terminate the study.

Short explanation

The Kaplan-Meier estimate is a simple way of computing the survival curve in spite of all these troublesome research subjects. It involves computing the number of people who died at a certain time point, divided by the number of people who were still in the study at that time. We multiply these probabilities by any earlier computed probabilities, which is one reason this is called a "product limit estimate."

The Kaplan-Meier survival curve is often illustrated graphically. It looks like a poorly designed staircase, with vertical steps downward at the time of death of each individual subject.

Often we will compare curves for two different groups of subjects. For example, what is the survival pattern for subjects on a standard therapy compared to a newer therapy. We can look for gaps in these curves in a horizontal or vertical direction. A vertical gap means that at a specific time point, one group had a greater fraction of subjects surviving. A horizontal gap means that it took longer for one group to experience a certain fraction of deaths.

More details

To compute a survival curve, you need to note the time of occurrence of events (e.g., failures, deaths)

It is possible for two or more events to occur at the same time, in which case the number of distinct times is less than the number of deaths or failures. You need to place the t's in order from smallest to largest. That is,

You also need to define the starting point of the study,

The basic computations for the Kaplan-Meier survival curve rely on the computation of conditional survival probabilities. In particular, the probability

which can be interpreted as the probability of your surviving to a specific time, given that you survived to the previous time. This probability is easy to calculate if you know the number of deaths or failures at a specific time and if you know the number of patients at risk at that same time.

A more difficult (but more important) probability is the unconditional probability of survival,

which represents the simple probability of survival to a specific time. You can use a relationship between this unconditional probability and the conditional probability:

At first glance, this does not seem to help, because the right hand side of the equation still includes an unconditional probability. But we can apply this approach again to get

and we can continue along these lines to get

This last probability represents the probability of surviving at the start of the study. Unless we intentionally recruit dead subjects, this probability has to be 1. Therefore, the unconditional probability is equal to the cumulative product of conditional probabilities.

At each time point, you should count

You should also count

Armed with this information, you can now compute a Kaplan-Meier survival curve. First you need to calculate the number of patients at risk,

In other words, the number at risk at any specific time point is just the number at risk at the previous time point, minus the number of deaths/failures and the number of censored observations. For convenience, we define

Next you compute the conditional probability of survival:

Finally, the unconditional probability of survival is simply the cumulative product of the conditional probabilities:

Example

The following example is from Chadha et al (2000). The authors studied a sample of 36 pediatric patients undergoing acute peritoneal dialysis through Cook Catheters. They wished to examine how long these catheters performed properly. They noted the date of complication (either occlusion, leakage, exit-site infection, or peritonitis).

Half of the subjects had no complications before the catheter was removed. Reasons for removal of the catheter in this group of patients were that the patient recovered (n=4), the patient died (n=9), or the catheter was changed to a different type electively (n=5). If the catheter was removed prior to complications, that represented a censored observation, because they knew that the catheter stayed complication free at least until the time of removal.

Figure 3.1 Failures and censored observations for catheter study.

The table above lists the days at which failures and/or censored observations occurred.

Figure 3.2 Computation of number of patients at risk

To compute a Kaplan-Meier survival curve, you first need to compute the number of catheters at risk on each day. This is just the number of catheters that were not previously censored or failures. These calculations appear in the table shown above.

Figure 3.3 Compuation of conditional probability of survival.

Next you need to compute the conditional probability of survival. This is the probabilty that a catheter will survive at a specific time point, given that it survived (and was not censored) at any previous time point. These calculations appear in the table shown above.

Figure 3.4 Computation of unconditional survival probabilities.

Finally, you need to compute the cumulative product: the product of each conditional probability with all previous conditional probabilities. This provides the estimates of survival probability used in the Kaplan-Meier curve. These calculations appear in the table shown above.

Figure 3.5 Graph of unconditional survival probabilities (Kaplan-Meier curve).

The graph you see above is the Kaplan-Meier curve as computed by SPSS. Select ANALYZE | SURVIVAL | KAPLAN-MEIER from the menu to get this graph.

Figure 3.6 SPSS dialog box for Kaplan-Meier procedure. [Image is already full size]

The figure above shows the SPSS dialog box. The date of the event (either failure or censoring) goes in the TIME field. In the STATUS field, you should place the variable which indicates whether the event was a failure or a censored observation. Click on the DEFINE EVENT button to tell SPSS what codes you used.

Figure 3.7 SPSS dialog box for defining events. [Image is already full size]

The figure shown above is the SPSS dialog box where you distinguish between failures and censoring. In this data set, a value of 1 indicates a failure and 0 represents censoring.

Figure 3.8 SPSS dialog box for Kaplan-Meier options. [Image is already full size]

Also be sure to click on the OPTIONS button in the main dialog box. The figure above shows you the dialog box you see when you click on the OPTIONS button. Be sure that the SURVIVAL PLOTS option is checked.

Reference

Tenckhoff Catheters Prove Superior to Cook Catheters in Pediatric Acute Peritoneal Dialysis.
Chada V, Warady BA, Blowey DL, Simckes AM, Alon US.
American Journal of Kidney Diseases (2000), 35(6):1111-1116.

Further reading

There are many beginning level books on biostatistics that discuss the Kaplan-Meier curve, such as Woolson's book. You can find a more advanced and detailed approach in Collett's book.

1. Modelling Survival Data in Medical Research.
   Collett D.
   London England: Chapman and Hall (1994).
   ISBN: 0-412-44890-4.
2. Statistical Methods for the Analysis of Biomedical Data
   Woolson RF.
   New York NY: John Wiley & Sons, Inc. (1987).
   ISBN: 0-471-80615-3.

Here is some extra material that I need to integrate into the above description.

Survival probabilities involve the estimation of the time to some event. Usually, the event involves death or failure of some sort. Some of the patients may not experience the event, because the study ends before they die, or we lose touch with them partway through the study. For these patients we have partial information, we know that the event occurred (or will occur) sometime after the date of last follow-up. We refer to these patients as censored observations. We don't want to ignore these patients, because they provide some information about survival, but we need to handle them differently.

The first step in a survival data analysis is to estimate survival probabilities for each group. When we know the exact date of death (or failure) for each patient, this computation is trivial. In most situations, however, we will have partial information on some of the patients. We will know that they survived beyond a certain point, but because the study ended before all the patients died, or because we lost touch with some of the patients, or because they withdrew from the study, we do not know the exact date of death. These patients represent censored observations, observations that you have to account for differently than others.

A simple example of censored data involves failure of a device, and not the death of a person. In a study of catheters for peritoneal dialysis, these catheters can fail due to occlusion, leakage, or infection. Some catheters are removed prior to failure, usually either because the patient completed dialysis or the patient died. If the catheter is removed prior to failure, that is considered a censored observation.

Day	Catheters removed prior to failure	Catheters failed
1	8	2
2	2	2
3	1	1
4	1	1
5	5	3
6		2
7		1
10		2
12		2
13		1

If you wanted to estimate the probability that a catheter will survive its first day, that's easy. There were 34 catheters, 2 did not survive the first day, 15 failed on days 2-13. For 17 of the catheters, we did not know when they would have failed, but we do know that they all survived at least one day.

So the probability of surviving the first day is 32/34 = 94%.

But how would we estimate the probability of surviving two days? four days? ten days?

This is tricky, because the censored observations provide information up to the day of censoring, but cannot tell us anything more about surviving beyond that day. What we need to do is compute the number of catheters at risk on each day. This is the number of catheters that would be at risk for failure on that day. It would exclude any catheters that failed on previous days and it would exclude any catheters that were censored on previous days.

Day	Catheters removed prior to failure	Catheters failed	Catheters at risk
1	8	2	34
2	2	2	34-8-2=24
3	1	1	24-2-2=20
4	1	1	20-1-1=18
5	5	3	18-1-1=16
6		2	16-5-3=8
7		1	8-2=6
10		2	6-1=5
12		2	5-2=3
13		1	3-2=1

We then need to compute the conditional probability of surviving at each time point given that the catheter survived the previous time point. This conditional probability would be

(number at risk - number of failures)/(number at risk)

Day	Catheters removed prior to failure	Catheters failed	Catheters at risk	Conditional probability
1	8	2	34	32/34 =0.94
2	2	2	34-8-2=24	22/24 =0.92
3	1	1	24-2-2=20	19/20 =0.95
4	1	1	20-1-1=18	17/18 =0.94
5	5	3	18-1-1=16	13/16 =0.81
6		2	16-5-3=8	6/8 =0.75
7		1	8-2=6	5/6 =0.83
10		2	6-1=5	3/5 =0.60
12		2	5-2=3	1/3 =0.33
13		1	3-2=1	0/1 =0.00

Then we compute the cumulative product of these probabilities. This represents the Kaplan-Meier estimate of the survival probability.

Day	Catheters removed prior to failure	Catheters failed	Catheters at risk	Conditional probability	Cumulative product
1	8	2	34	32/34 =0.94	0.94
2	2	2	34-8-2=24	22/24 =0.92	0.94*0.92 =0.86
3	1	1	24-2-2=20	19/20 =0.95	0.86*0.95 =0.82
4	1	1	20-1-1=18	17/18 =0.94	0.82*0.94 =0.77
5	5	3	18-1-1=16	13/16 =0.81	0.77*0.81 =0.62
6		2	16-5-3=8	6/8 =0.75	0.62*0.75 =0.46
7		1	8-2=6	5/6 =0.83	0.46*0.83 =0.38
10		2	6-1=5	3/5 =0.60	0.38*0.60 =0.23
12		2	5-2=3	1/3 =0.33	0.23*0.33 =0.08
13		1	3-2=1	0/1 =0.00	0.08*0.00 =0.00

Here is a graph of these survival probabilities. 

The plot has a "stair step" pattern, because we don't know the survival probability at fractional days (such as 2.5 days) and at some integer days (such as 9 days). By convention, we estimate the survival probability for these values as equaling the survival probability of the closest value that is still smaller (the 2 day survival probability for 2.5 days, and the 7 day survival probability at 9 days).

Notice that the estimated median survival time (the time at which 50% of the catheters survived) is six days.

Tenckhoff Catheters Prove Superior to Cook Catheters in Pediatric Acute Peritoneal Dialysis. Chadha V. American Journal of Kidney Diseases 2000:35(6);1111-1116.

This webpage was written by Steve Simon on 200-06-27 and was last modified on 07/14/2008. Category: Ask Professor Mean, Category: Survival analysis