We studied disease testing in class, and in our stylized example, a patient had only two attributes: their true disease status \(D\) and their test result \(T\). In reality of course, patients have many more relevant attributes: genetics, prior medical history, lifestyle, charm and good looks, etc. So in an attempt to be slightly more realistic, we shall extend our little model to include a third attribute: whether or not you have symptoms \(S\).
Table 1 enumerates all of the possible states of the world together with their individual probabilities. These numbers come from a small COVID-19 study of undergraduates in Fall 2020. The test in question is an antigen test. Patients’ “true” disease status was determined by a PCR test that we assume is hella accurate.
| \(D\) | \(S\) | \(T\) | \(P(D\cap S\cap T)\) |
|---|---|---|---|
| - | - | - | 0.7650 |
| - | - | + | 0.0128 |
| - | + | - | 0.1685 |
| - | + | + | 0.0018 |
| + | - | - | 0.0091 |
| + | - | + | 0.0064 |
| + | + | - | 0.0073 |
| + | + | + | 0.0291 |
What is the overall disease prevalence?
What is the sensitivity of the test?
What is the specificity of the test?
Imagine you develop symptoms. So you visit Dr. Vinnie Boombatz to get tested, and the test comes back positive. Given everything we now know about you, what is the probability that you are truly infected?
What is the overall probability that the test is wrong?