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. 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.
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. For each specified interval of time, survival probability is calculated as the number of participants surviving divided by the number of persons at risk.
There are three assumptions used in this analysis. 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. 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 cannot provide an estimate of the size of the difference between a related confidence interval and groups as it is purely a significance test.
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. 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. Subject A also died at 6 months, therefore the PLI is: 0.
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. Subject L died at 20 months, the PLI will be 0. 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. Subject O was censored at 11 months. Subject T was censored at 15 months. Note: censored are assumed to be the participants who lost to followed-up or dropped out during the 24 month study. 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 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. 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.
We label these situations as right-censored observations. 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. 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 survival curve is defined as the probability of surviving in a given length of time while considering time in many small intervals. 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.
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:. For each time interval, survival probability is calculated as the number of subjects surviving divided by the number of patients at risk.
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. We know about the time of the event, i. There are also a few subjects who are still surviving i. Even in these conditions we can calculate the Kaplan-Meier estimates as summarized in Table 1.
The estimates obtained are invariably expressed in graphical form. It is incorrect to join the calculated points by sloping lines [ Figure 1 ]. Plots of Kaplan-Meier product limit estimates of survival of a group of patients as in e. 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. 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. The steepness of the curve is determined by the survival durations.
Looking at the censored objects, the one subject that censored in group female materially reduced the cumulative survival between the intervals. Whereas, the terminally censored subject in the male group did not change the survival probability and the interval was not terminated by an event. The table above shows what happens behind the production of the KM curve.
When the above table is cross-referenced with the KM curve, it is evident that intervals and the attendant probabilities are only constructed for events of interest and not for censored subjects. Because an event ends one interval and begins another interval, there should be more intervals than events. The table explains the way the curves end. In group male, the curve ends without creating another interval below. The cumulative probability of surviving this long is determined by the last horizontal, sixth interval and is 0.
In the other group, the curve drops to zero after the fifth interval to cause the sixth interval horizontal to be on the X-axis. Looking at the probabilities of survival, it could be a little confusing that there are two probabilities 1. Cumulative probability 2. Interval probability. The cumulative probability defines the probability at the beginning and throughout the interval. This is graphed along the Y-axis of the curve. The interval survival rate defines the probability of surviving past the interval.
Censoring affects survival rates. Censored observations that coincide with an event are usually considered to fall immediately after the event.
Censoring removes the subject from the denominator, i. For example, in Group 2, there were three surviving intervals four and available to be at risk in interval five. However, during interval four one was censored; therefore, only two were left to be at risk in interval five, i. Thus, we calculated the survival probabilities of each subject of two different groups. The null hypothesis is there is no difference, and the alternate hypothesis is the groups are significantly different.
Since the p-value is greater than 0. This brings us to the end of the blog on the Kaplan Meier Curve. We hope you enjoyed it. This data science course will help you learn comprehensive curriculum from a top-ranking global school and to build job-ready data science skills. The program offers a hands-on learning experience with top faculty and dedicated mentor support. Remember Me! Great Learning is an ed-tech company that offers impactful and industry-relevant programs in high-growth areas.
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