KM Curves , Median Survival Times and Ratio of the Median Survival Times

 1. An Example of Kaplan-Meier Curves

survival probabilities: we count the number of subjects surviving past the specified time being considered and divide this number by 21, the number of subjects at the start of follow-up.

There is alternative formula is called the Kaplan-Meier (KM) approach and can be illustrated using the group 2 data even though all values of q are zero.

KM formula =product limit formula

For a specified failure time t(f), the fraction may be generally expressed as the conditional probability

of surviving past time t(f), given availability (i.e., in the risk set) at time t(f).

2. Median Survival Times: 

The median survival time is the time point at which the probability of survival equals 50%. 

Some things to keep in mind:

•If the probability of survival exceeds 50% at the longest time point, then the median survival time cannot be computed. Prism reports that the median survival is “undefined”. The logrank comparison of curves compares entire curves, and does not compare median survival times. So the P value computed by the logrank test is still valid even if one or both median survival times are undefined

•If the survival curve is horizontal at 50% survival, then the median survival time is not clearly defined. In the survival curve below, the curve is horizontal at Y=50% between 9 and 17 months. Prism follows the suggestion of Machin and reports that the median survival is the average of those two values, or 13 months in this case.

3. the ratio of the median survival times

The ratio of median survival times, also known as the 

median ratio (MR), compares the median survival of two groups in a clinical or research study. It is often used as a more intuitive and understandable alternative to the hazard ratio (HR), which compares the instantaneous risk of an event.

This ratio, sometimes called the hazard ratio from median survival, helps assess a treatment's effectiveness, but its interpretation depends on assumptions about the survival curve, such as the curve following an exponential decay for accurate confidence interval calculations. 

3.1 Calculating the median survival ratio 

To calculate the median ratio, you first must determine the median survival time for each group being compared. In a study with two groups, a treatment group (T) and a reference group (R), the median ratio is calculated as:

$MR=\frac{\text{Median survival time of T}}{\text{Median survival time of R}}\text{[1.3.8]}$

 

For example, if the median survival is 8 months for the treatment group and 4 months for the reference group, the median ratio is 2. A clinician could then say that the treatment "will double your survival time". 

3.2 Comparing median ratio and hazard ratio 

The median ratio and hazard ratio can both be used to compare survival curves, but they offer different interpretations. 

• Median Ratio: Compares the horizontal distance between survival curves, representing the difference in time survived. It's often easier for patients to understand.
• Hazard Ratio: Compares the vertical distance between survival curves, representing the instantaneous risk of an event at any given time. If the survival curves are not proportional (i.e., they cross), the hazard ratio will change over time, making it a poor summary measure. In such cases, the median ratio may be more informative.