Understanding survival analysis: Kaplan-Meier estimate
Manish Kumar Goel, Pardeep Khanna, Jugal Kishore1
Department of Community Medicine, Post Graduate Institute of
Medical Science, Rohtak, Haryana, 1Maulana Azad Medical College, New Delhi, India.
A B S T R A C T
Kaplan-Meier estimate
is one of the best options to be used to measure the fraction of subjects living for a certain
amount of time after treatment. In clinical trials
or community trials,
the effect of an intervention is assessed by measuring the number of subjects survived or saved after that intervention over a period of time. The
time starting from a defined point to the occurrence
of a given event, for example death is called as survival time and the analysis of group data
as survival analysis. This can be affected
by subjects under study that are uncooperative and refused to be remained in the study or when some of the subjects may not experience the event or death before
the end of the study, although they would have experienced or died if observation continued, or we lose touch with them midway in the study. We
label these situations as censored
observations. The Kaplan-Meier estimate
is the simplest way of computing the survival over time in spite of all these difficulties associated with subjects or situations. The survival curve can be created assuming various situations. It involves computing
of probabilities of occurrence of event at a certain
point of time and multiplying these successive probabilities by any earlier
computed probabilities to get the final estimate. This can be calculated for two groups
of subjects and also their
statistical difference
in the survivals. This can be used in
Ayurveda research when they
are comparing two drugs and looking for survival of subjects.
Key words: Survival
analysis, Kaplan-Meier estimate
IntroductIon
For human subjects, to compare efficacy and safety, controlled
experiments are conducted which are called as clinical trials.[1] In clinical or community trials, the effect of an intervention is assessed by measuring the
number of subjects survived or saved after that intervention over a period of time. Sometime
it is interesting to compare the survival of subjects in two or more
interventions. In situations where survival is the issue
then the variable of interest would be the length of time that elapses
Address
for correspondence:
Prof. Jugal Kishore,
Department of Community Medicine,
|
|||||||||||
Submission Date: 08-10-10 Accepted Date: 15-01-11
before some event to occur. In many of the situations this length
of time is very long for example in
cancer therapy; in such case per unit duration of time number of events such as death
can be assessed. In other
situations, the duration
for how long until a cancer relapses or how long
until an infection occurs can be assessed.
Sometimes it can even be used for a specific outcome, like how long it takes
for a couple to conceive. The time starting from a defined point to the occurrence of a given event is called as the survival
time[2] and the analysis of group
data as the survival analysis.[3]
These analyses are often complicated when subjects
under study are uncooperative and refused to be remained in the study or
when some of the subjects may not experience the event or death before
the end
of the
study,
although they would have experience or died, or we lose touch with them midway in the study. We label these situations as right-censored
observations.[2] For these subjects we have partial
information. We know that the event occurred (or will
occur) sometime after the date of last follow-up.
We
do not want to ignore
these subjects, because they provide some information about survival. We will know that they survived beyond a certain point, but we do not know the exact
date of death.
Sometimes we have subjects that become a part of the study
later, i.e. a significant time has elapsed from the start. We
have a shorter observation time for those subjects and these subjects may or
may not experience the event in that short stipulated time. However, we cannot exclude those subjects since
otherwise sample size of the study may become small. The Kaplan-Meier estimate is the simplest
way of computing the survival over time in spite of all these difficulties
associated
with subjects or situations.
The Kaplan-Meier survival curve
is defined as the probability
of surviving in a given length of time while considering time in many small intervals.[3] There are three
assumptions used in this
analysis. Firstly, we assume that at any time patients who are
censored have the same survival
prospects as those who
continue to be followed. Secondly, we assume that the survival
probabilities are the same for subjects recruited
early and late in
the study. Thirdly, we assume that the event
happens at the time
specified. This creates problem
in some conditions when the event would be detected
at a regular examination. All we
know is that the event happened between two examinations. Estimated survival can be
more accurately calculated by carrying out follow-up of the individuals frequently at shorter time intervals; as short as
accuracy of recording permits i.e. for one day (maximum).
The Kaplan-Meier estimate is also called as “product limit estimate”. It
involves computing of probabilities of occurrence of event at a certain
point of time. We
multiply these successive probabilities by any earlier computed probabilities to get the final estimate. The survival
probability at any particular time is calculated by the formula given below:
Number of subjects
– Number of subjects
|
= living
at the start died
t Number of subjects living at the start
For each time interval, survival probability is calculated as
the number of subjects surviving
divided by the number of
patients at risk. Subjects who have died, dropped out, or move
out are
not counted
as “at
risk” i.e., subjects who are
lost are considered “censored” and are not counted in the denominator. Total
probability of survival
till that time
interval is calculated by multiplying all the
probabilities of survival at all
time intervals preceding that time (by applying law of multiplication
of
probability to calculate cumulative probability). For example,
the probability
of a patient surviving two days
after a kidney transplant can be considered to be probability of surviving the
one
day multiplied by the probability surviving the second day
given that patient survived the first day. This second probability
is
called as a conditional probability. Although
the probability calculated
at any given interval is not very accurate because of
the
small number of events, the overall probability of surviving
to each
point is more accurate. Let
us take
a hypothetical data of a group of patients
receiving standard anti-retroviral therapy. The data shows the time of survival (in days) among the patients
entered in a clinical trial - (E.g. 1)- 6, 12, 21, 27, 32, 39, 43, 43,
46*, 89, 115*, 139*, 181*, 211*, 217*, 261, 263, 270, 295*,
311, 335*, 346*, 365* (* means these patients
are still surviving
after mentioned
days in the trial.)
We know about the time of the event, i.e. death in each subject, after he/she had entered
the trial, may it be at different times. There are also a few subjects
who are still
surviving i.e. at the
end of the trial. Even in these
conditions we can calculate the Kaplan-Meier estimates as summarized
in Table
1.
The time ‘t’ for which the value of ‘L’ i.e. total probability of survival at the end of a
particular time is 0.50 is called as median survival time. The
estimates obtained are invariably
expressed in graphical form.
The
graph plotted between
estimated survival probabilities/estimated survival percentages
(on
Y
axis) and time
past after entry into the study (on X axis)
consists of horizontal and vertical lines.[4] The survival curve is drawn as a step
function: the proportion surviving remains unchanged between
the events, even if there
are some intermediate
censored observations. It is incorrect to join the calculated points by sloping
lines [Figure 1].
We can compare curves for two different groups of subjects. For example,
compare the survival pattern for subjects on a standard therapy with 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.
Let us take another hypothetical
data for example of a group of patients receiving new Ayurvedic
therapy for HIV infection.
The data
shows the time of survival
(in days)
among the patients entered in a clinical trial (as in e.g. 1) 9, 13, 27, 38,
45*, 49, 49, 79*, 93, 118*, 118*, 126, 159*, 211*, 218, 229*,
Table 1: Kaplan-Meier estimate for patients mentioned in
e.g. 1
|
Time
of event
|
No.
of Pt. died
|
Live
at the start of the
|
Estimated
|
probability
|
Probability
of survivors at the end
|
|
(t)
|
(d)
|
day (n)
|
death
(d/n)
|
survival
(1 - d/n)
|
of time (L)
|
|
6
|
1
|
23
|
0.0435
|
0.9565
|
0.9565
|
|
12
|
1
|
22
|
0.0455
|
0.9545
|
0.9565 × 0.9545 = 0.9130
|
|
21
|
1
|
21
|
0.0476
|
0.9524
|
0.9130 × 0.9523 =
0.8695
|
|
27
|
1
|
20
|
0.0500
|
0.9500
|
0.8695 × 0.9500 = 0.8260
|
|
32
|
1
|
19
|
0.0526
|
0.9474
|
0.7826
|
|
39
|
1
|
18
|
0.0556
|
0.9444
|
0.7391
|
|
43
|
2
|
17
|
0.1176
|
0.8824
|
0.6522
|
|
89
|
1
|
14
|
0.0714
|
0.9286
|
0.6056
|
|
261
|
1
|
8
|
0.125
|
0.875
|
0.5299
|
|
263
|
1
|
7
|
0.1429
|
0.8571
|
0.4542
|
|
270
|
1
|
6
|
0.1667
|
0.8333
|
0.3785
|
|
311
|
1
|
4
|
0.25
|
0.75
|
0.2839
|
The
time ‘t’ for which the value of ‘L’ i.e. total probability of survival at
the end of a particular time is 0.50
is called as median survival time. The
estimates obtained are invariably expressed in graphical form. The graph plotted between estimated
survival probabilities/estimated survival percentages (on Y axis) and time
past after entry into the study (on X axis) consists of horizontal and
vertical lines.[4] The survival curve is drawn
as a step function: the proportion surviving remains unchanged between the events, even
if there are some intermediate censored observations. It is incorrect to join
the calculated points by sloping lines (Figure 1).
263*, 298*, 301, 333, 346*, 353*, 362* (* means these patients
are still surviving after mentioned days in
the trial.)
The Kaplan-Meier estimate for the above example is
summarized in
Table 2.
The two survival
curves can be compared statistically
by testing the null hypothesis i.e. there is no difference regarding survival among two interventions. This null hypothesis
is statistically tested by
another test known as log-rank test and Cox proportion hazard test.[5] In log-rank test we calculate
the expected number of events in each group i.e. E and E while O
1 2 1
|
and O
are the total number of observed events
in each group,
respectively [Figure 2]. The
test statistic is
Figure 2: Plots of Kaplan-Meier product
limit estimates of survival of a
(0 – E )2
(0 – E )2
group of patients
(as in e.g. 1 and 2) receiving ART and new Ayurvedic
|
|
Log-rank test statistic = 1 1 + 2 2
therapy for HIV Infection.
1 2
|
The total number of expected events in a group (e.g. E ) is the sum of expected number of events, at
the time of each event in any of the group, taking both groups together. At the time
of event in any group the expected number of events is
Considering the above example the log-rank
test can be applied
as shown in Table 3.
(0 – E ) (0
– E )
|
|
|
Log-rank test
statistic = 1 1
+ 2 2
2 2
the product of risk of event at that time with the total number
of subjects alive at the start of the time of event in that very
group (e.g. at day 6, 46 patients
were alive at the start of the
(13 – 11.78)2
= +
(11 – 12.22)2
day and one died, so the risk of event was 1/46 = 0.021739.
As 23 patients were alive at the start of the day in group 2,
the expected number of events at day 6 in group 2 was 23 ×
0.021739 = 0.5).
The total number
of expected events
in group
|
2 is sum of the
expected events calculated at different
time. The total number of expected
events in the other group (i.e.
E ) is calculated by subtracting the total number of expected
11.77 12.22
= 0.1263 +
0.1218 =
0.2481
Computations of all the
values in the above-mentioned formula will give test statistic value.
The test statistic and the significance can be drawn by comparing the calculated value
|
events in group 2 i.e. E
from the total of observed events in
with the critical
value (using chi-square table) for degree of
|
|
both the groups i.e. O
+
O .
freedom equal to one. The test statistic value is less than the
|
|
Death
(d/n)
|
Survival
(1- d/n)
|
time
(L)
|
||
|
9
|
1
|
23
|
0.043478
|
0.956522
|
0.9565
|
|
13
|
1
|
22
|
0.045455
|
0.954545
|
0.9130
|
|
27
|
1
|
21
|
0.047619
|
0.952381
|
0.8696
|
|
38
|
1
|
20
|
0.05
|
0.95
|
0.8261
|
|
49
|
2
|
18
|
0.111111
|
0.888889
|
0.7343
|
|
93
|
1
|
15
|
0.066667
|
0.933333
|
0.6853
|
|
126
|
1
|
12
|
0.083333
|
0.916667
|
0.6282
|
|
218
|
1
|
9
|
0.111111
|
0.888889
|
0.5584
|
|
301
|
1
|
5
|
0.2
|
0.8
|
0.4467
|
|
333
|
1
|
4
|
0.25
|
0.75
|
0.3351
|
|
|||||||
Table 3:
Log-rank statistic for patients mentioned in examples 1 and 2
|
Time of event
(t)
|
Total
no. of patients died in both group (D)
|
No.
of patients died in group 2 (O2)
|
Live
at the start of the day
(N)
|
Live
at the start of the day in group 2
(n2)
|
Probability
of death at the end of time (L)
|
Expected
probability of death in group 2 (E2)
|
Expected
probability of death in group
1 (E1)
|
|
6
|
1
|
0
|
46
|
23
|
0.021739
|
0.5
|
|
|
9
|
1
|
1
|
45
|
23
|
0.022222
|
0.511111
|
|
|
12
|
1
|
0
|
44
|
22
|
0.022727
|
0.5
|
|
|
13
|
1
|
1
|
43
|
22
|
0.023256
|
0.511628
|
|
|
21
|
1
|
0
|
42
|
21
|
0.02381
|
0.5
|
|
|
27
|
2
|
1
|
40
|
21
|
0.05
|
1.05
|
|
|
32
|
1
|
0
|
39
|
20
|
0.025641
|
0.512821
|
|
|
38
|
1
|
1
|
38
|
20
|
0.026316
|
0.526316
|
|
|
39
|
1
|
0
|
37
|
19
|
0.027027
|
0.513514
|
|
|
43
|
2
|
0
|
36
|
19
|
0.055556
|
1.055556
|
|
|
49
|
2
|
2
|
32
|
18
|
0.0625
|
1.125
|
|
|
89
|
1
|
0
|
31
|
16
|
0.032258
|
0.516129
|
|
|
93
|
1
|
1
|
29
|
15
|
0.034483
|
0.517241
|
|
|
126
|
1
|
1
|
25
|
12
|
0.04
|
0.48
|
|
|
218
|
1
|
1
|
19
|
9
|
0.052632
|
0.473684
|
|
|
261
|
1
|
0
|
17
|
8
|
0.058824
|
0.470588
|
|
|
263
|
1
|
0
|
15
|
7
|
0.066667
|
0.466667
|
|
|
270
|
1
|
0
|
14
|
7
|
0.071429
|
0.5
|
|
|
301
|
1
|
1
|
11
|
6
|
0.090909
|
0.545455
|
|
|
311
|
1
|
0
|
10
|
5
|
0.1
|
0.5
|
|
|
333
|
1
|
1
|
9
|
4
|
0.111111
|
0.444444
|
|
|
|
24
|
11
|
|
|
|
12.22015
|
11.77985
|
critical value (using chi-square
table) for degree of freedom equal to one. Hence, we can say that there is no
significant difference between the
two groups regarding the survival.
The log-rank
test is used to test whether the difference between
survival times between
two groups is statistically different or not,
but do not allow to test the effect of the other independent variables. Cox proportion hazard
model enables us to test the
effect of
other independent variables on survival
times of different groups of
patients, just like the multiple regression model. Hazard is nothing but the
dependent variable and can be defined as probability of dying at a given time assuming
that the patients
have survived up to that given time. Hazard ratio
is also an important term and defined
as the ratio of the risk
of hazard occurring at any given time in one group compared with another group at that very time i.e. if H1, H2, H3 … and
h1,
h2, h3 … are the hazards at a given
times T1, T2, T3… in group A and B, respectively, then hazard
ratio at times T1, T2, T3 are H1/h1, H2/h2, H3/h3 …, respectively. Both log-rank
test and Cox proportion hazard test assume that the hazard ratio is
constant over time i.e. in the above-mentioned scenario H1/ h1 = H2/h2
= H3/h3.
To conclude,
Kaplan-Meier method is a clever method
of statistical treatment of survival
times which not only makes proper allowances for those observations that are censored, but
also makes use of the information
from these subjects up to the time when they are censored. Such situations are common
in Ayurveda research when two interventions are used and outcome assessed as survival of
patients. So Kaplan-Meier method is a useful method
that may play a significant role in generating evidence-based information on survival time.
references
1. Armitage P, Berry G, Matthews JN.
Clinical trials. Statistical methods in medical research. 4th ed. Oxford
(UK): Blackwell
Science; 2002. p. 591.
2. Berwick
V, Cheek L,
Ball J. Statistics review 12: Survival analysis. Crit Care 2004;8:389-94.
3. Altman DG. Analysis of Survival times. In:Practical statistics for Medical research.
London (UK):
Chapman and Hall;
1992. p. 365-93.
4. Indrayan A, Surmukaddam SB. Measurement of Community
Health and Survival
analysis. In:
Chow SC,
editor. Medical Biostatistics. Vol. 7. New
York (US): Marcel Dekker; 2001. p.
232-42.
5. Marubini E, Valsecchi MG. Estimation of Survival
Probabilities.
Analysing survival data from clinical
trials and observational studies. Chichester (UK): John Wiley and Sons; 1995. p. 41-8.
Source
of Support: Nil, Conflict of Interest: None declared.
Optimized content for mobile and hand-held devices
HTML pages have been optimized for mobile and other hand-held devices (such as iPad, Kindle,
iPod) for faster browsing speed. Click on [Mobile Full text] from Table of Contents
page.
This is simple HTML
version for faster download on
mobiles (if viewed on desktop, it will be
automatically redirected to full HTML version)
E-Pub for hand-held devices
EPUB is an open e-book standard recommended by
The International Digital
Publishing Forum which is designed for reflowable content i.e. the text display can
be optimized for a
particular display device.
Click on [EPub] from Table of
Contents page.
There are various e-Pub readers
such as for Windows: Digital Editions, OS X: Calibre/Bookworm, iPhone/iPod Touch/iPad: Stanza,
and Linux: Calibre/Bookworm.
E-Book for desktop
One can also see the entire issue as printed here in a
‘flip
book’ version on
desktops. Links are available from Current
Issue as well as
Archives pages.
Click
on View as
eBook
No comments:
Post a Comment