| Condition: Antibodies |
Condition positive (CP) | Condition negative (CN) | |
|---|---|---|---|
| Test outcome positive (OP) |
Sensitivity
True positive
|
False positive
|
All OP
|
| Test outcome negative (ON) |
False negative
|
Specificity
True negative
|
All ON
|
|
Base Rate
|
All CP
|
All CN
|
All Tested
100.00
|
P(H|E) = =
= = = = ≃ %;
Examples of Baysian inference
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By Bayes' Theorem, Bayesian Inference
Suppose, you do a COVID-19 blood test at home. Test kits are now available. Vendors guarantee around 99% specificity and 97-99% sensitivity. What does this mean? A vast majority of subjects, including academics would answer: “if the test finds antibodies, I am 99% sure it's true” and accordingly “if I have no antibodies, in 97% of the cases the test will come out negative”.
But it is wrong! Imagine, you were falsely tested positive (meaning, you have antibodies and are - probably - safe) and you go to work and don't wear a mask! This means, the test would cause you to run a big risk.
Let's do the math: suppose, the overall probabilty, the base rate, of having antibodies is 0.003 (US: 1.68 MM infected out of 330 MM are 0.005, Germany: 0.18MM / 83MM = .002 - data by end of May, 2020).
Suppose 10000 people are doing the test you have done. According to the base rate (.01), 100 of them will have antibodies. 99 of these get a positive test outcome (as the specificity is .99). The other 9900 should yield a negative test outcome. However, since the test specificity is not 100%, but only 98%, from these 9900 will 9000*0.02 (.02 = 1 - .98) yield a falsely positive outcome.
The result will come as a suprise! You gonna rub your eyes in disbelief! This app will help you to understand, what's going on here - before you learn the math. Toggle terms and definitions
Terms and definitions
Today's example: antibody COVID-19 tests. We are going to use three variables:
- How certain is it that someone having the "condition" (that is, having antibodies) will test positive (for an ideal test this would be 100% certain). This probability is denoted as sensitivity.
- How certain is it that someone not having the condition will test negative, as it should be (that is, she is clean, hasn't ever had COVID-19). This probablity is denoted as specificity.
- What is the magnitude of the outbreak? It is the fraction of all subjects and hence the fraction of "all", for instance, all inhabitants of a country. (The subjects should form a representative sample of the population.) This probabilty is denoted as base rate - or, if it is a disease, as prevalence
Then, we are dealing with two "binary" variables, meaning they can assume but two values: true or false.
- The condition, we call it H for Hypothesis. If the condition is given, the hypothesis is true.
- The test result, we call it E for Evidence. If the test outcome is positive, there is evidence for the hypotheses, the evidence is true
Finally, the quantity we are interested in is called P(H|E), meaning the probabilty of H given E (or, for H being true given that the there is evidence, namely a positve antibody test).
All this definitions and symbols may feel a little overwhelming. But the good new is: that's already it!