Kaplan Meier is derived from the names of two statisticians; Edward L. Kaplan and Paul Meier, in 1958 when they made a collaborative effort and published a paper on how to deal with time to event data.⁵ Therefore, they introduced the Kaplan-Meier estimator which serves as a tool for measuring the frequency or the number of patients surviving medical treatment. Later on, the Kaplan-Meier curves and estimates of survival data have become a better way of analyzing data in cohort study. Kaplan-Meier (KM) is non-parametric estimates of survival function that is commonly used to describe survivorship of a study population and to compare two study populations. KM estimate is one of the best statistical methods used to measure the survival probability of patients living for a certain period of time after treatment. It is an intuitive graphical presentation approach. In clinical trials or community trials, the intervention effect is assessed by measuring the number of participants saved or survived after that intervention over a period of time. KM estimate is the simplest procedure of determining the survival over time in spite of all the difficulties associated with subjects or situations. Curves are used in Kaplan Meier estimate to determine the events, censoring and the survival probability.

Kaplan-Meier survival curve is used in epidemiology to analyze time to event data and to compare two groups of subjects. The survival curve is used to determine a fraction of patients surviving a specified event, like death during a given period of time. This can be calculated for two groups of patients or subjects and also their statistical difference in the survivals. Below is an example of Kaplan-Meier survival curve:

The tick marks on the curve indicate censoring and the curve moves down when the event of interest occurs.

Product Limit estimate (PLI) is another name of Kaplan Meier estimate. The product-limit formula estimates the fraction of organisms or physical devices surviving beyond any age t, even when some of the items are not observed to die or fail, and the sample is rather small.⁶ It involves computing the probabilities of occurrence of event at a certain point of time. These successive probabilities will be multiplied by any earlier computed probabilities to determine the final estimate. For example, the probability of a sub-fertile woman surviving the pregnancy three months after laparoscopy and hydrotubation can be considered to be the probability of surviving the first month multiplied by the probabilities surviving the second and third months respectively given that the woman survived the first two months. The third probability is known as a conditional probability.

In survival analysis, intervals are defined by failures. For example, the probability of surviving intervals A and B is equal to the probability of surviving interval A multiplied by the probability of surviving interval B. thus, the PLI be:

$\frac{P\left(SurvivingintervalA\right)}{NumberofsubjectsatriskuptofailureA}{\rm X}\frac{P\left(SurvivingintervalB\right)}{NumberofsubjectsatriskuptofailureB}$

For each specified interval of time, survival probability is calculated as the number of participants surviving divided by the number of persons at risk. Participants who have dropped out, died, or move out are not counted as “at risk” that is, those who are lost (censored) will not be included in the denominator.

There are three assumptions used in this analysis.⁷ Firstly, it is assumed that at any time participants who are dropped out or censored have the same survival prospects as those who continue to be followed. Secondly, it is assumed that the survival probabilities are the same for participants recruited early and late in the study. Thirdly, it is assumed that the event occurs at the time specified.

The limitation of Kaplan Meier estimate is that it cannot be used for multivariate analysis as it only studies the effect of one factor at the time.

The log-rank test

Log-rank test is used to compare two or more groups by testing the null hypothesis. The null hypothesis states that the populations do not differ in the probability of an event at any time point. Thus, log-rank test is the most commonly-used statistical test to compare the survival functions of two or more groups. These groups can be treatment and control groups or different treatment groups in a clinical trial. The log rank test can be generated in form of table from the statistical softwares such as SPSS, SAS, Stata and R packages. The null hypothesis will be rejected when the p value is less than α value (α can be 0.05, etc.) or fail to be rejected when the p value is large. The log-rank test cannot provide an estimate of the size of the difference between a related confidence interval and groups as it is purely a significance test.

Benchmark problem

The tables below are the tables of fictive data generated from the SPSS software. (Table 1) contains the data of treatment group only while table 2 contains the data for both the two groups. The first group in the second table is the treatment group while the second group is the control group. Each group comprises ten participants who have been followed for the period of 24 months. The participants in the treatment and control groups were given Drug A and placebo respectively and they were given alphabetical names like A, B, C…, T. The data will be used to determine the Kaplan-Meier estimates (the product limit estimate) of the both the control and the treatment groups.

The product limit estimate is:

$\frac{P\left(SurvivingintervalA\right)}{NumberofsubjectsatriskuptofailureA}{\rm X}\frac{P\left(SurvivingintervalB\right)}{NumberofsubjectsatriskuptofailureB}$

From the curve above, the number of events (deaths) in the treatment group (those given drug A) is 6 while that of the control group (those given placebo) is 7. The number of censored for treatment and control groups are 4 and 3 respectively. The curve takes a step down when a participant dies and the tick marks on the curve indicate censoring, that is when they lost to follow-up or dropped out of the study.

In the treatment group, Subject D died at 2 months. The estimated survival probability [P(T>t)] will be: 9/10 = 0.9. Subject E died at 4 months, the estimated survival probability or fraction surviving this death is 8/9, and thus the product limit estimate (PLI) is: 0.9 × 8/9 = 0.8. Subject A also died at 6 months, therefore the PLI is: 0.8 × 7/8 = 0.7. Subjects B, Q and H were censored at 7, 8 and 14 months respectively. Subject F died at 19 months, the estimate will be: 0.7 × ¾ = 0.525. Subject L died at 20 months, the PLI will be 0.525 × 2/3 = 0.35. The next subject in the group, which is subject K, was censored at 22 months while subject N, the last subject in the group died at 24 months and that is the last month of the study. The product limit estimate will be 0.35 × 0 = 0.00.

In the control group, subject C died at the first month, the fraction surviving this death will be 9/10 = 0.90 while subject I was censored at the third month. Subject J died at 5 months, the estimated survival probability is 7/8 and thus, the product limit estimate will be 0.9 × 7/8 = 0.788. Subject P also died at 9 month, the estimated survival probability or fraction surviving this death is 6/7 = 0.8571, therefore the PLI will be 0.788 × 0.8571 = 0. 675. The next subject in the group, subject M died at 10 months, the fraction surviving this death is 5/6 = 0.8333 and the PLI will be 0.675 × 0.8333 = 0.562. Subject O was censored at 11 months. Subject G died at 12 months, the product limit estimate will be 3/4 × 0.562 = 0.422. Subject T was censored at 15 months. The next subject, which is R died at 17 months, the product limit estimate will be ½ × 0.422 = 0.211. S is the subject that died last in the group, the subject died at 18 months, therefore the product limit estimate will be 0 × 0.211 = 0.00.

Note: censored are assumed to be the participants who lost to followed-up or dropped out during the 24 month study.

It is seen from the curve

The curves for two different groups of participants can be compared. For example, compare the survival pattern for participants on a treatment with a control. We can identify the gaps in these curves in a vertical or horizontal direction. A vertical gap signifies that at a specific period of time, one group had a greater probability of participants surviving while a horizontal gap signifies that it took longer for one group to experience a certain fraction of deaths.

Now the two groups in figure 3 will be compared in terms of their survival curves. The null hypothesis is that “there is no difference between the groups’ survival curves”. The table below generated from the SPSS software will be used to test the hypothesis.

Table 2 indicates that all the three p-values are greater than 0.05, and this means that the null hypothesis is failed to be rejected. Therefore, statistically, the survival curves of the treatment and control groups do not differ. Survival curves here mean the population or the true survival curves. The Low Rank in the table place more emphasis on the events happening later in time, Generalized Wilcoxon place more emphasis on the events happening earlier in time while Taron-ware in between the two.