C-Index from scratch in Python

Dimitrij Tschodu
5 min readJan 12, 2022

David Deutsch has this tweet:

If you can’t program it, you haven’t understood it.

If you do some work in medicine, statistics, or even biophysics, you are surely familiar with the C-Index, also called the C-statistics, or Harrell’s index . Frank Harrell introduced it to measure the ability of a model to discriminate patients with different prognosis. This means:

Consider two patients. Patient A lived longer than patient B.

If the predicted survival time for the patient A is longer than the predicted survival time for the patient B, the predictions for this pair A-B are concordant with the outcomes.

The definition of the C-Index is:

Here you can find a wonderful explanation of the C-Index and its interpretation, I highly recommend it.

If you go to the PySurvival website, you will find the following and more refined definition:


So, lets program it to understand it!

(Here is how you can type Greek letters on a Linux system.)

First, this expression:

δT = lambda Ti, Tj: 1. if Tj < Ti else 0.

The function δT takes two arguments: Ti and Tj. These are the survival times.

Second, this expression:

δη = lambda ηi, ηj: 1. if ηj > ηi else 0.

The function δη takes two arguments: ηi and ηj. These are the risk scores. The patient with a higher risk score should have a shorter predicted survival time.

Third, what is δj?

Mathematically, it can be either 1 or 0. So, lets assume for now it is always 1:

δj = [1, 1, 1, 1, …]

Assuming, the number of patients is n:

δj = np.array([1. for i in range(n)])

Lastly, we implement the sum over i and j:

numerator = 0
denominator = 0

for i in range(n):
for j in range(n):
numerator +=
denominator +=

The expression x+= 1 is identical to the expression x = x+1.

Putting it all together

This is now easy. Bur first, we have to define the actual survival times, predicted risk scores, and the vector…

Dimitrij Tschodu

PhD student in biophysics at Leipzig university | Physicist | ML & AI researcher | author | former blogger of PhysiPhi.com