For human subjects, to match efficacy and safety, controlled experiments are conducted which are called as clinical trials. In clinical or community trials, the effect of an intervention is assessed by measuring the amount of subjects survived or saved then intervention over a period of your time . Sometime it’s interesting to match the survival of subjects in two or more interventions. In situations where survival is that the issue then the variable of interest would be the length of your time that elapses before some event to occur. In many of the situations this length of your time is extremely long for instance in cancer therapy; in such case per unit duration of your time number of events like death are often assessed. In other situations, the duration for a way long until a cancer relapses or how long until an infection occurs are often assessed. Sometimes it can even be used for a selected outcome, like how long it takes for a few to conceive. The time ranging from an outlined point to the occurrence of a given event is named because the survival time and therefore the analysis of group data because the survival analysis.
These analyses are often complicated when subjects under study are uncooperative and refused to be remained within the study or when a number of the themes might not experience the event or death before the top of the study, although they might have experience or died, or we lose touch with them midway within the study. We label these situations as right-censored observations. For these subjects we’ve partial information. We all know that the event occurred (or will occur) sometime after the date of last follow-up. We don’t want to ignore these subjects, because they supply some information about survival. We’ll know that they survived beyond a particular point, but we don’t know the precise date of death.
Sometimes we’ve subjects that become a neighborhood of the study later, i.e. a big time has elapsed from the beginning . we’ve a shorter observation time for those subjects and these subjects may or might not experience the event therein short stipulated time. However, we cannot exclude those subjects since otherwise sample size of the study may become small. The Kaplan-Meier estimate is that the easiest method of computing the survival over time in spite of of these difficulties related to subjects or situations.
The Kaplan-Meier survival curve is defined because the probability of surviving during a given length of your time while considering time in many small intervals. There are three assumptions utilized in this analysis. Firstly, we assume that at any time patients who are censored have an equivalent survival prospects as those that still be followed. Secondly, we assume that the survival probabilities are an equivalent for subjects recruited early and late within 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 daily examination. All we all know is that the event happened between two examinations. Estimated survival are often more accurately calculated by completing follow-up of the individuals frequently at shorter time intervals; as short as accuracy of recording permits i.e. for at some point (maximum). The Kaplan-Meier estimate is additionally called as “product limit estimate”. It involves computing of probabilities of occurrence of event at a particular point of your time . We multiply these successive probabilities by any earlier computed probabilities to urge the ultimate estimate. The survival probability at any particular time is calculated by the formula given below:
For each interval , survival probability is calculated because the number of subjects surviving divided by the amount of patients in danger . Subjects who have died, dropped out, or move out aren’t counted as “at risk” i.e., subjects who are lost are considered “censored” and aren’t counted within the denominator. Total probability of survival till that point interval is calculated by multiplying all the possibilities of survival in the least time intervals preceding that point (by applying law of multiplication of probability to calculate cumulative probability). for instance , the probability of a patient surviving two days after a kidney transplant are often considered to be probability of surviving the at some point multiplied by the probability surviving the second day as long as patient survived the primary day. This second probability is named as a contingent probability . Although the probability calculated at any given interval isn’t very accurate due to the tiny number of events, the general probability of surviving to every point is more accurate. allow us to take a hypothetical data of a gaggle of patients receiving standard anti-retroviral therapy. the info shows the time of survival (in days) among the patients entered during a clinical test – (E.g. 1)- 6, 12, 21, 27, 32, 39, 43, 43, 46F*, 89, 115F*, 139F*, 181F*, 211F*, 217F*, 261, 263, 270, 295F*, 311, 335F*, 346F*, 365F* (* means these patients are still surviving after mentioned days within 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’s at different times. There also are a couple of subjects who are still surviving i.e. at the top of the trial. Even in these conditions we will calculate the Kaplan-Meier estimates as summarized in Table 1.
The time ‘t’ that the worth of ‘L’ i.e. total probability of survival at the top of a specific time is 0.50 is named 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. The survival curve is drawn as a step function: the proportion surviving remains unchanged between the events, albeit there are some intermediate censored observations. it’s incorrect to hitch the calculated points by sloping lines [Figure 1].
We can compare curves for 2 different groups of subjects. for instance , compare the survival pattern for subjects on a typical therapy with a more moderen therapy. we will search for gaps in these curves during a horizontal or vertical direction. A vertical gap means at a selected time point, one group had a greater fraction of subjects surviving. A horizontal gap means it took longer for one group to experience a particular fraction of deaths.
Let us take another hypothetical data for instance of a gaggle of patients receiving new Ayurvedic therapy for HIV infection. the info shows the time of survival (in days) among the patients entered during a clinical test (as in e.g. 1) 9, 13, 27, 38, 45F*, 49, 49, 79F*, 93, 118F*, 118F*, 126, 159F*, 211F*, 218, 229F*, 263F*, 298F*, 301, 333, 346F*, 353F*, 362F* (* means these patients are still surviving after mentioned days within the trial.)
The Kaplan-Meier estimate for the above example is summarized in Table 2.
The two survival curves are often compared statistically by testing the null hypothesis i.e. there’s no difference regarding survival among two interventions. This null hypothesis is statistically tested by another test referred to as log-rank test and Cox proportion hazard test. In log-rank test we calculate the expected number of events in each group i.e. E1 and E2 while O1 and O2 are the entire number of observed events in each group, respectively [Figure 2] . The test statistic is
The total number of expected events during a group (e.g. E2) is that the sum of expected number of events, at the time of every event in any of the group, taking both groups together. At the time of event in any group the expected number of events is that the product of risk of event at that point with the entire number of subjects alive at the beginning of the time of event therein very group (e.g. at day 6, 46 patients were alive at the beginning of the day and one died, therefore the risk of event was 1/46 = 0.021739. As 23 patients were alive at the beginning of the day in group 2, the expected number of events at day 6 in group 2 was 23 × 0.021739 = 0.5). the entire number of expected events in group 2 is sum of the expected events calculated at different time. the entire number of expected events within the other group (i.e. E1) is calculated by subtracting the entire number of expected events in group 2 i.e. E 2, from the entire of observed events in both the groups i.e. O1 + O2. Considering the above example the log-rank test can be applied as shown in Table 3.
Log-rank statistic for patients mentioned in examples 1 and 2
Computations of all the values within the above-mentioned formula will give test statistic value. The test statistic and therefore the significance are often drawn by comparing the calculated value with the critical value (using chi-square table) for degree of freedom adequate to one. The test statistic value is a smaller amount than the critical value (using chi-square table) for degree of freedom adequate to one. Hence, we will say that there’s no significant difference between the 2 groups regarding the survival.
The log-rank test is employed to check whether the difference between survival times between two groups is statistically different or not, but don’t allow to check the effect of the opposite independent variables. Cox proportion hazard model enables us to check the effect of other independent variables on survival times of various groups of patients, a bit like the multiple correlation model. Hazard is nothing but the variable and may be defined as probability of dying at a given time assuming that the patients have survived up thereto given time. Hazard ratio is additionally a crucial term and defined because the ratio of the danger 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 A and B, respectively, then hazard ratio sometimes 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. within the above-mentioned scenario H1/h1 = H2/h2 = H3/h3.
To conclude, Kaplan-Meier method may be 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 knowledge from these subjects up to the time once 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 may be a useful method which will play a big role in generating evidence-based information on survival time.