# Survival analysis, part 2: Kaplan-Meier method and the log-rank test

When 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 method is often applied to estimate the probability of survival (not experiencing the outcome of interest—ie, dental arch alignment from our previous example ).

Table I displays the probability of not reaching alignment at different time intervals (Kaplan-Meier estimate of the survivor function method).

Table I
Probability of not reaching alignment at different time intervals
Beginning time Total Net failures Lost to follow up Inverse survival probability Survivor function
28 61 1 0 (61-1)/61 = 0.9836 0.9836
32 60 1 0 (60-1)/60 = 0.9833 0.9836*0.9833 = 0.9672 0.9672
55 59 1 0 (59-1)/59 = 0.9831 0.9831*0.9672 = 0.9508 0.9508
…… …… …… …… …… …… ……
95 46 1 0 0.7377
96 45 3 0 (45-3)/45 = 0.9333 0.9333*0.7377 = 0.6885 0.6885
…… …… …… …… …… …… ……
178 14 0 1 0.2976
179 13 0 2 (13-0)/13 = 1 1*0.2976 = 0.2976 0.2976
180 11 0 1 (11-0)/11 = 1 1*0.2976 = 0.2976 0.2976
182 10 0 2 (10-0)/10 = 1 1*0.2976 = 0.2976 0.2976
184 8 0 2 (8-0)/8 = 1 1*0.2976 = 0.2976 0.2976
186 6 0 3 (6-0)/6 = 1 1*0.2976 = 0.2976 0.2976
187 3 0 1 (3-0)/3 = 1 1*0.2976 = 0.2976 0.2976
190 2 0 1 (2-0)/2 = 1 1*0.2976 = 0.2976 0.2976
195 1 0 1 (1-0)/1 = 1 1*0.2976 = 0.2976 0.2976

“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. 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:

n u m b e r a t r i s k − n u m b e r o f e v e n t s n u m b e r a t r i s k