# Choosing sample sizes when working with different sensitivity levels

I am currently ramping up a project to determine the rate of immunity in a population with SARS-CoV-2 using the new Roche Diagnostice test kits.

Now, we will be testing people over the course of 6 weeks and will have a max of 200 tests per week available due to reagent delivery restrictions, but we estimate to have 140 eligible test subjects per week. We are still in the setup-phase, but I am struggling to judge the necessary sample size.

Our final goal is to determine the [%] of people that have an active SARS-CoV-2 infection using PCR and the [%] that are currently immune using ELISA per week and monitor this development over the course of the 6 weeks. The problem here is that Roche gives different sensitivity levels of their testing kits depending on the time that has passed since infection was confirmed with PCR as shown below (specifity is given at 99.81 % (99.65 – 99.91 %) ):

Days post PCR confirmation  Sample number (N)   Sensitivity (95 % CI)
0 – 6 days                  116                 65.5 % (56.1 – 74.1 %)
7 – 13 days                 59                  88.1 % (77.1 – 95.1 %)
≥14 days                    29                  100 % (88.1 – 100 %)


For the people that come to us without symptoms but positive immunity status, we will either try to gauge when the infection happened using a questionnaire but for the ones that had an asymptomatic infection, we will most probably say that the infection was >7 days past the onset of symptoms, so we don't assume a sensitivity that is too high (as shown by Roche). Now, the problem I am having is actually two-fold:

a) how can I estimate an appropriate sample size if I can't know the distribution of patients that will fall in the 1. PCR (-) and ELISA (-), which means no infection or the 2. PCR (+) and ELISA (-) or PCR (-) and ELISA (+), which means infection has taken place and I need to deal with the different sensitivities given by Roche?

b) how can I determine the CIs in the analysis? Will I need to stratify my data?