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I thought of this as I'm reading about covid tests accuracy, and I am thinking how taking multiple tests influences the chance of correctly detecting illness/no illness. So, if I understand this concepts correctly, I want to see how specificity and sensitivity change (and I assume they increase) when taking two tests.

I'm going to denote testing positive with + and having the illness as I, not having as NI. I'm going to limit the question to seeing one positive test result vs seeing two.

So let's say P(I|+)=0.8 and I want to first calculate P(I|++). From Bayes theorem: equation - bayes theorem

Then since P(++)=P(+)*P(+) as events are independent, I have

bayes theorem for two tests

and now I'm not sure where to take it, and how Bayes theorem was even useful. It seems I am missing something. I think I also need the prevalence of the disease in the general population to compute this, that is, P(I).

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    $\begingroup$ Are you suggesting getting tested twice with the same test? (In this case whatever may have cause an error in the first test result may also cause an error in the second--like not enough time elapsed since infection to make antibodies.) Or are you suggesting getting tested with two kinds of test? // Big difference in situation, analysis, and numerical results. $\endgroup$ – BruceET Jun 27 '20 at 21:42
  • $\begingroup$ Also with two different tests, the same problems may occur (e.g. not enough time elapsed). You are always gonna get this. In the end, the analysis is gonna look ugly where you do not define a single type of infected but also have a nuisance parameter that varies among the different infected (and non-infected) which will describe the success probability of the test. A toy-problem/example could be made by using two beta distributions, but a good treatment requires decent information of the proper distributions (and their parameters) to use. $\endgroup$ – Sextus Empiricus Jun 29 '20 at 12:06
  • $\begingroup$ no, I was thinking same test twice, immediately after. I'm thinking intuitively if it shows positive both times, it's a higher chance you have the disease. is my intuition wrong? $\endgroup$ – vvv Jun 29 '20 at 20:23
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Given that you're assuming the two tests are the same and independent Bayes formula becomes

enter image description here

As you thought, you'll need an estimate for P(I), which I'm sure there are a number of estimates floating around for COVID.

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