# The confidence interval for the ratio of median survival times

If a survival curve goes down to less than 50% survival, Prism computes the median survival -- the time it takes to reach 50% survival. If more than 50% of the subjects are alive at the end of the study, then the median survival time is simply not defined.

If you are comparing two survival curves, Prism will also compute the ratio of the two survival times. This value is also called the *hazard ratio from median survival*. Along with that Prism reports the 95% confidence interval for the ratio of median survivals.

This calculation is based on an assumption that is not part of the rest of the survival comparison. The calculation of the 95%CI of ratio of median survivals assumes that the survival curve follows an exponential decay -- that the chance of dying in a small time interval is the same early in the study and late. In other words, it assumes that the survival of patients or animals in your study follows the same model as radioactive decay. If your survival data follow a very different pattern, then the 95% CI for the ratio of median survivals will not be helpful.

Due to a subtle bug in Prism, the 95% confidence interval of the ratio of median survivals has been computed incorrectly in all versions of Prism up to 5.04 and 5.0d.

Download this Excel file to do the correct calculations (from pages 67-68 of Machim; details below). All you have to do is enter the two median survial times (from the Curve comparison page of results) and the number of deaths or events in each group (from the Data summary results page).

The calculations are also not hard to do by hand. The calculations work on a log scale, followed by the antilog. Follow these steps:

- Compute the natural logarithm of the ratio of median survival time. For the sample data, the ratio is 0.7934 (from the Curve Comparison page of results), and ln(0.7934) = -0.2314. Be sure to use the natural logarithm (ln) rather than the common logarithm (base 10).
- Compute the standard error of the logarithm of the ratio of survival times. To do this first find the number of deaths in each group. This is shown at the top of the Data Summary page. For the sample data, number of deaths/events in the two groups are 11 and 5. Call these values O1 and O2. The SE of the logarithm of the ratio of survival times is the square root of the sum of 1/O1 + 1/O2. For the sample data sqrt(1/11 + 1/5) = 0.5396.
- Compute the margin of error by multiplying the SE by the appropriate value from the z distribution. For 95% confidence, this value is 1.96. For the sample data, the margin of error equals 1.96*0.5396 = 1.0576.
- Add and subtract the margin of error from the logarithm of the median survivals to create a confidence interval on a logarithm scale. For the sample data, this interval runs from -0.2314 - 1.0576 to -0.2314 + 1.0576, or from -1.289 to 0.8263.
- Convert back from a logarithmic scale to the original ratio scale. Since we used natural logarithms, convert back by taking e to that power, abbreviated exp(x). For the sample data the 95% confidence interval for the ratio of median survival runs from exp(-1.289) to exp(0.8263), or from 0.2755 to 2.2848. (This contrasts with the interval computed by Prism up to 5.04 and 5.0d which is 0.4460 to 1.141.)

Sample Prism file. Excel file.