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Survival analysis, part 2: Kaplan-Meier method and the log-rank test
STATISTICS AND RESEARCH DESIGN
Survival analysis, part 2: Kaplan-Meier method and the log-rank test Despina Koletsia and Nikolaos Pandisb London, United Kingdom, Athens and Corfu, Greece, and Bern, Switzerland
W
hen analyzing survival data, time-to-event methods estimate the probability of not reaching the event, even though we are interested in reaching the event (ie, dental arch alignment). This occurs because these methods were developed for studies on death, where survival is the outcome of interest. The Kaplan-Meier method1 is often applied to estimate the probability of survival (not experiencing the outcome of interest—ie, dental arch alignment from our previous example2). Table I displays the probability of not reaching alignment at different time intervals (Kaplan-Meier estimate of the survivor function method).
“Beginning” time indicates the time interval, and “Total” denotes the number of participants at risk for the time interval. After each interval, the patient who reaches the event as well as those lost to follow-up are removed from the numbers at risk. Net failure and lost to follow up denote the number of patients who reached alignment and those who were lost to follow-up (they were censored), respectively. At the end of the study, censored patients are also those who did not reach alignment, but they are expected to align some time after the end of the trial. The inverse survival probability indicates the probability of not reaching the event at a particular interval.3
Table I. Probability of not reaching alignment at different time intervals Beginning time 28
Total 61
Net failures 1
Lost to follow up 0
Inverse survival probability (61-1)/61 5 0.9836
60 59
1 1
0 0
(60-1)/60 5 0.9833 (59-1)/59 5 0.9831
0.9836*0.9833 5 0.9672 0.9831*0.9672 5 0.9508
0.9672 0.9508
.. 95
.. 46
.. 1
.. 0
..
..
.. 0.7377
96 ..
45 ..
3 ..
0 ..
(45-3)/45 5 0.9333 ..
0.9333*0.7377 5 0.6885 ..
0.6885 ..
178 179
14 13
0 0
1 2
(13-0)/13 5 1
1*0.2976 5 0.2976
0.2976 0.2976
180 182
11 10
0 0
1 2
(11-0)/11 5 1 (10-0)/10 5 1
1*0.2976 5 0.2976 1*0.2976 5 0.2976
0.2976 0.2976
184 186
8 6
0 0
2 3
(8-0)/8 5 1 (6-0)/6 5 1
1*0.2976 5 0.2976 1*0.2976 5 0.2976
0.2976 0.2976
187 190
3 2
0 0
1 1
(3-0)/3 5 1 (2-0)/2 5 1
1*0.2976 5 0.2976 1*0.2976 5 0.2976
0.2976 0.2976
195
1
0
1
(1-0)/1 5 1
1*0.2976 5 0.2976
0.2976
32 55
Survivor function 0.9836
a London School of Hygiene and Tropical Medicine, University of London, London, United Kingdom; private practice, Athens, Greece. b Department of Orthodontics and Dentofacial Orthopedics, School of Dental Medicine/Medical Faculty, University of Bern, Bern, Switzerland; private practice, Corfu, Greece. Am J Orthod Dentofacial Orthop 2017;152:569-71 0889-5406/$36.00 Ó 2017 by the American Association of Orthodontists. All rights reserved. http://dx.doi.org/10.1016/j.ajodo.2017.07.008
569
Statistics and research design
570
Figure 1. Kaplan-Meier plot for overall survival.
Figure 2. Survival probabilities by type of wire. Red vertical lines indicate at what time point half of the events occurred.
When there are no events or there are only patients lost to follow-up, the probability of not reaching the event is 1, or the probability of reaching the event is 0. The estimated inverse survival probability of each interval is the probability that the persons at risk at the beginning of the period will not reach alignment (ie, fail) in that interval, and this is calculated as follows:
October 2017 Vol 152 Issue 4
number at risk number of events number at risk The survivor function denotes the cumulative probability of survival, and it decreases as the events accumulate with time. This is calculated by multiplying the inverse survival probability from the current interval
American Journal of Orthodontics and Dentofacial Orthopedics
Statistics and research design
571
Table II. The log-rank test Events observed (n)
Events expected (n)
A B
19 23
21.6 20.4
Total
42
42.0
Wire
P value 0.42
with the inverse survival probability of the previous interval as shown in Table I. This survival function is plotted with the KaplanMeier plot as shown in Figure 1. Figure 2 is a plot of survival probabilities by wire type. Assumptions for the Kaplan-Meier method
1.
2.
3.
Censoring is unrelated to the outcome. The KaplanMeier method assumes that the probability of censoring is not related to the outcome of interest. This might or might not be valid. The survival probabilities are the same for participants recruited early and late in the study. Circumstances that can affect survival (better or different treatments) are not assumed to change the baseline risk of survival in the patients as a group. The events occurred at the times specified. However, sometimes we do not know the exact date of an event, but only its status at each visit.
We can analyze the data by comparing the proportion of those who have survived at any given point during follow-up. This method is not appropriate, since it does not provide a comparison of the total survival experience of the 2 groups, because the difference in survival might vary with time. A more appropriate way to analyze these types of data is to use an approach similar to the chi-square test, called the log-rank test.2 Table II shows the number of observed and expected events per wire type and a statistical test that indicates that there is no difference between the 2 wires in terms of reaching alignment (P 5 0.42). These results derive from the log-rank test. Although the log-rank test appears to be similar to the chi-square test, it is calculated differently, because it takes into account the total follow-up period as well. The log-rank test is considered a nonparametric test and makes no assumptions about the shape of the survival curve (distribution of survival times), and the null hypothesis states that there is no difference between the populations in the probability of an event (alignment) at any time point. REFERENCES 1. Kaplan EL, Meier P. Non-parametric estimation from incomplete observations. J Am Stat Assoc 1958;53:57-81. 2. Koletsi D, Pandis N. Survival analysis, part 1: introduction. Am J Orthod Dentofacial Orthop 2017;152:428-30. 3. Kirkwood BR, Sterne JAC. Essential medical statistics. 2nd ed. Oxford, United Kingdom: Blackwell Science; 2003.
American Journal of Orthodontics and Dentofacial Orthopedics
October 2017 Vol 152 Issue 4
Survival analysis, part 2: Kaplan-Meier method and the log-rank test Despina Koletsia and Nikolaos Pandisb London, United Kingdom, Athens and Corfu, Greece, and Bern, Switzerland
W
hen analyzing survival data, time-to-event methods estimate the probability of not reaching the event, even though we are interested in reaching the event (ie, dental arch alignment). This occurs because these methods were developed for studies on death, where survival is the outcome of interest. The Kaplan-Meier method1 is often applied to estimate the probability of survival (not experiencing the outcome of interest—ie, dental arch alignment from our previous example2). Table I displays the probability of not reaching alignment at different time intervals (Kaplan-Meier estimate of the survivor function method).
“Beginning” time indicates the time interval, and “Total” denotes the number of participants at risk for the time interval. After each interval, the patient who reaches the event as well as those lost to follow-up are removed from the numbers at risk. Net failure and lost to follow up denote the number of patients who reached alignment and those who were lost to follow-up (they were censored), respectively. At the end of the study, censored patients are also those who did not reach alignment, but they are expected to align some time after the end of the trial. The inverse survival probability indicates the probability of not reaching the event at a particular interval.3
Table I. Probability of not reaching alignment at different time intervals Beginning time 28
Total 61
Net failures 1
Lost to follow up 0
Inverse survival probability (61-1)/61 5 0.9836
60 59
1 1
0 0
(60-1)/60 5 0.9833 (59-1)/59 5 0.9831
0.9836*0.9833 5 0.9672 0.9831*0.9672 5 0.9508
0.9672 0.9508
.. 95
.. 46
.. 1
.. 0
..
..
.. 0.7377
96 ..
45 ..
3 ..
0 ..
(45-3)/45 5 0.9333 ..
0.9333*0.7377 5 0.6885 ..
0.6885 ..
178 179
14 13
0 0
1 2
(13-0)/13 5 1
1*0.2976 5 0.2976
0.2976 0.2976
180 182
11 10
0 0
1 2
(11-0)/11 5 1 (10-0)/10 5 1
1*0.2976 5 0.2976 1*0.2976 5 0.2976
0.2976 0.2976
184 186
8 6
0 0
2 3
(8-0)/8 5 1 (6-0)/6 5 1
1*0.2976 5 0.2976 1*0.2976 5 0.2976
0.2976 0.2976
187 190
3 2
0 0
1 1
(3-0)/3 5 1 (2-0)/2 5 1
1*0.2976 5 0.2976 1*0.2976 5 0.2976
0.2976 0.2976
195
1
0
1
(1-0)/1 5 1
1*0.2976 5 0.2976
0.2976
32 55
Survivor function 0.9836
a London School of Hygiene and Tropical Medicine, University of London, London, United Kingdom; private practice, Athens, Greece. b Department of Orthodontics and Dentofacial Orthopedics, School of Dental Medicine/Medical Faculty, University of Bern, Bern, Switzerland; private practice, Corfu, Greece. Am J Orthod Dentofacial Orthop 2017;152:569-71 0889-5406/$36.00 Ó 2017 by the American Association of Orthodontists. All rights reserved. http://dx.doi.org/10.1016/j.ajodo.2017.07.008
569
Statistics and research design
570
Figure 1. Kaplan-Meier plot for overall survival.
Figure 2. Survival probabilities by type of wire. Red vertical lines indicate at what time point half of the events occurred.
When there are no events or there are only patients lost to follow-up, the probability of not reaching the event is 1, or the probability of reaching the event is 0. The estimated inverse survival probability of each interval is the probability that the persons at risk at the beginning of the period will not reach alignment (ie, fail) in that interval, and this is calculated as follows:
October 2017 Vol 152 Issue 4
number at risk number of events number at risk The survivor function denotes the cumulative probability of survival, and it decreases as the events accumulate with time. This is calculated by multiplying the inverse survival probability from the current interval
American Journal of Orthodontics and Dentofacial Orthopedics
Statistics and research design
571
Table II. The log-rank test Events observed (n)
Events expected (n)
A B
19 23
21.6 20.4
Total
42
42.0
Wire
P value 0.42
with the inverse survival probability of the previous interval as shown in Table I. This survival function is plotted with the KaplanMeier plot as shown in Figure 1. Figure 2 is a plot of survival probabilities by wire type. Assumptions for the Kaplan-Meier method
1.
2.
3.
Censoring is unrelated to the outcome. The KaplanMeier method assumes that the probability of censoring is not related to the outcome of interest. This might or might not be valid. The survival probabilities are the same for participants recruited early and late in the study. Circumstances that can affect survival (better or different treatments) are not assumed to change the baseline risk of survival in the patients as a group. The events occurred at the times specified. However, sometimes we do not know the exact date of an event, but only its status at each visit.
We can analyze the data by comparing the proportion of those who have survived at any given point during follow-up. This method is not appropriate, since it does not provide a comparison of the total survival experience of the 2 groups, because the difference in survival might vary with time. A more appropriate way to analyze these types of data is to use an approach similar to the chi-square test, called the log-rank test.2 Table II shows the number of observed and expected events per wire type and a statistical test that indicates that there is no difference between the 2 wires in terms of reaching alignment (P 5 0.42). These results derive from the log-rank test. Although the log-rank test appears to be similar to the chi-square test, it is calculated differently, because it takes into account the total follow-up period as well. The log-rank test is considered a nonparametric test and makes no assumptions about the shape of the survival curve (distribution of survival times), and the null hypothesis states that there is no difference between the populations in the probability of an event (alignment) at any time point. REFERENCES 1. Kaplan EL, Meier P. Non-parametric estimation from incomplete observations. J Am Stat Assoc 1958;53:57-81. 2. Koletsi D, Pandis N. Survival analysis, part 1: introduction. Am J Orthod Dentofacial Orthop 2017;152:428-30. 3. Kirkwood BR, Sterne JAC. Essential medical statistics. 2nd ed. Oxford, United Kingdom: Blackwell Science; 2003.
American Journal of Orthodontics and Dentofacial Orthopedics
October 2017 Vol 152 Issue 4