Bayes\' Theorem deals with the calculation of posterior probabilities, which isn

Question

Bayes' Theorem deals with the calculation of posterior probabilities, which isn't always a natural thing to do. We're used to forward-chaining our probabilities (e.g., if we roll a 3 on a die, what's the probability the second roll will give us a total of 8?). Backward-chaining is less intuitive (e.g. if our total on the die was an 8, what's the probability that the first roll was a 3?). Since the rules of probability involve simple addition and multiplication, they work. fine in both directions. The thing that makes posterior probability more difficult is that we simply aren't used to thinking about things that way. Our reading in Chapter 2 provided an example of a diagnostic test for a rare disease. The resulting confidence in a positive test result is surprisingly low. Discuss why that is so. What is happening in the interaction of the various probabilities leads to this outcome? Post your response of 1-3 paragraphs (about 100-200 words) by the due date for this discussion assignment and then reply to at least two initial responses of your peers during the remainder of the week, particularly focusing on responses that might differ from your own. Also respond appropriately to anyone who posts questions against your own postings. Discuss the content! Keep responses focused on the substance of the issue, not simply on agreeing with a comment or encouraging each other.

Explanation / Answer

The false positive paradox is a statistical result where false positive tests are more probable than true positive tests, occurring when the overall population has a low incidence of a condition and the incidence rate is lower than the false positive rate. The probability of a positive test result is determined not only by the accuracy of the test but by the characteristics of the sampled population.[1] When the incidence, the proportion of those who have a given condition, is lower than the test's false positive rate, even tests that have a very low chance of giving a false positive in an individual case will give more false than true positives overall.[2]So, in a society with very few infected people—fewer proportionately than the test gives false positives—there will actually be more who test positive for a disease incorrectly and don't have it than those who test positive accurately and do. The paradox has surprised many.[3]

It is especially counter-intuitive when interpreting a positive result in a test on a low-incidence population after having dealt with positive results drawn from a high-incidence population.[2] If the false positive rate of the test is higher than the proportion of the new population with the condition, then a test administrator whose experience has been drawn from testing in a high-incidence population may conclude from experience that a positive test result usually indicates a positive subject, when in fact a false positive is far more likely to have occurred.

Not adjusting to the scarcity of the condition in the new population, and concluding that a positive test result probably indicates a positive subject, even though population incidence is below the false positive rate, is a "base rate fallacy".

High-incidence population[edit]

Imagine running an HIV test on population A of 1000 persons, in which 40% are infected. The test has a false positive rate of 5% (0.05) and no false negative rate. Theexpected outcome of the 1000 tests on population A would be:

Infected and test indicates disease (true positive)

1000 × 40/100 = 400 people would receive a true positive

Uninfected and test indicates disease (false positive)

1000 × 100 – 40/100 × 0.05 = 30 people would receive a false positive

The remaining 570 tests are correctly negative.

So, in population A, a person receiving a positive test could be over 93% confident (400/30 + 400) that it correctly indicates infection.

Low-incidence population[edit]

Now consider the same test applied to population B, in which only 2% is infected. The expected outcome of 1000 tests on population B would be:

Infected and test indicates disease (true positive)

1000 × 2/100 = 20 people would receive a true positive

Uninfected and test indicates disease (false positive)

1000 × 100 – 2/100 × 0.05 = 49 people would receive a false positive

The remaining 931 tests are correctly negative.

In population B, only 20 of the 69 total people with a positive test result are actually infected. So, the probability of actually being infected after one is told that one is infected is only 29% (20/20 + 49) for a test that otherwise appears to be "95% accurate".

A tester with experience of group A might find it a paradox that in group B, a result that had usually correctly indicated infection is now usually a false positive. The confusion of the posterior probability of infection with the prior probability of receiving a false positive is a natural error after receiving a life-threatening test result.

Number
of people Infected Uninfected Total Test
positive 400
(true positive) 30
(false positive) 430 Test
negative 0
(false negative) 570
(true negative) 570 Total 400 600 1000

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