# Calculate incidence rate per person or in total

For my study I am doubting about the following:

1. First I want to present the incidence rate for patients who received an CT-scan.

I thought I would just count all the CT-scans that took place and divide this by the time all patients are at risk. So, e.g.:

Patient 1: 1 CT-scan in 365 days;
Patient 2: 3 CT-scans in 250 days;
Patient 3: 1 CT-scan in 10 days;
Total --> 5/625 = 0.008

2. But then I thought about what would happen if I calculated the incidence rate for each patient separately and add this (instead of just the total as showed above).

We would get something like this for the patients above:

1/365 + 3/250 + 1/10 = 0.002739 + 0.0120 + 0.1 = 0.114739


What I understand is that the first way gives more weight to patients who are a long time at risk for getting a CT-scan, while the second way gives more weight to patients who are a short time at risk for getting a CT-scan.

What I don't know is if it is even justified to use the second way of calculating? And where I can't get my head around is what it means for my results if I use the second way instead of the first way - what is the difference between the first answer (0.008) and the second answer (0.114739)?

These are two different ways of summarizing incidence. The first approach is called a geometric mean. You would get 0.008 from $$\exp(\log(1/365) + \log(3/250) + \log(1/10))$$. The geometric mean tends toward the median more than the arithmetic mean. You can also refer to the geometric mean in this case as a Poisson intensity. It's desirable that the subject with the highest incidence: 1 scan every 10 days, is heavily downweighted by the fact they're only observed for 10 days. That's because this very high incidence has very high variability.