Question

There is a genetic condition that affects 2.3% of the population, C = {c, ~c}. Tests can detect the condition with 72.1% (true positive rate) but have a false positive rate of 20.3%, T={pos, neg}.

1. If we want to know P(C|T), write out Bayes Rule for this problem:

2. Using the values supplied in the description of the problem, specify all priors and likelihoods for the version of Bayes Rule you just wrote out:

3. Calculate P(C|T). Please explain each individual probability and the difference between the prior and the posterior and the reason for the difference.

4. Assume the true positive rate is 97.3% but the false positive rate is 37.2%. Recalculate P(C|T) and describe how the posterior distribution changes between Part 1 and Part 2 and why.

Answer 1

1.

By Bayes Rule,

P(C|T) = P(T | C) P(C) / P(T)

2.

P(C = c) = 2.3/100 = 0.023

P(C = ~c) = 1 - 0.023 = 0.977

P(T = pos | C = c) = 72.1 / 100 = 0.721

P(T = pos | C = ~c) = 20.3 / 100 = 0.203

By law of total probability,

P(T = pos) = P(C = c) P(T = pos | C = c) + P(C = ~c) P(T = pos | C = ~c)

= 0.023 * 0.721 + 0.977 * 0.203 = 0.214914

3.

P(C = c | T = pos) = P(T = pos | C = c) P(C = c) / P(T = pos) = 0.721 * 0.023 / 0.214914 = 0.0772

Given the test result is positive, probability of genetic condition is 0.0772.

Prior probability of genetic condition is 0.023 which increased to 0.0772 for positive test results because of high true positive rate.

P(C = ~c | T = pos) = P(T = pos | C = ~c) P(C = ~c) / P(T = pos) = 0.203 * 0.977 / 0.214914 = 0.9228

Given the test result is positive, probability of not having genetic condition is 0.9228.

Prior probability of no genetic condition is 0.977 which decreased to 0.9228 for positive test results because of low false positive rate.

4.

P(T = pos | C = c) = 97.3 / 100 = 0.973

P(T = pos | C = ~c) = 37.2 / 100 = 0.372

By law of total probability,

P(T = pos) = P(C = c) P(T = pos | C = c) + P(C = ~c) P(T = pos | C = ~c)

= 0.023 * 0.973 + 0.977 * 0.372 = 0.385823

P(C = c | T = pos) = P(T = pos | C = c) P(C = c) / P(T = pos) = 0.973 * 0.023 / 0.385823 = 0.058

Given the test result is positive, probability of genetic condition is 0.058.

P(C = ~c | T = pos) = P(T = pos | C = ~c) P(C = ~c) / P(T = pos) = 0.372 * 0.977 / 0.385823 = 0.942

Given the test result is positive, probability of not having genetic condition is 0.942.

Given the test result is positive, probability of genetic condition is reduced in Part 2 because of high probability of positive test.

Given the test result is positive, probability of no genetic condition is increased in Part 2 because of high false positive rate.