Looking at Bayes’ Theorem with a Tree Diagram An alternate way of “reversing the conditioning” on conditional probabilities

Pulmonary Tuberculosis Screening The prevalence of latent pulmonary tuberculosis (TB) infection in the population is about 6%. The Roche AMPLICOR is a test used to screen for TB. If the subject is infected with TB, the probability that the test will be positive is 2/3. (Sensitivity) If the subject is not infected with TB, the probability of a negative result is (Specificity) Source: California State University, Hayward, Statistics Department

Pulmonary Tuberculosis Screening 1.A patient gets a positive result. What is the probability that he does not have the disease*? (False Positive) 2.A patient gets a negative result. What is the probability that he has the disease*? (False Negative). * in any form, including mild cases.

Sorting out the data “ The prevalence of latent pulmonary tuberculosis (TB) infection is about 6%. ” P(TB) = P(no TB) = 0.94.

Sorting out the data “ If the subject has TB, the probability that the test will be positive is 2/3.“ P(+ | TB) = 2/3. P(- | TB) = 1/3.

Sorting out the data “ If the subject does not have TB, the probability of a negative result is 0.996” P(- | No TB) = P(+ | No TB) =

Pulmonary Tuberculosis Screening 1.A patient gets a positive result. What is the probability that he does not have the disease? (False Positive) P(No TB | +)? 2.A patient gets a negative result. What is the probability that he has the disease? (False negative). P(TB | -)?

We have / We want We Have: 1.P(TB) = P(no TB) = P(+ | TB) = 2/3. 4.P(- | TB) = 1/3. 5.P(- | No TB) = P(+ | No TB) = We Want: a)P(No TB | +)? b)P(TB | -)?

Build the tree TB? Yes No

Build the tree TB? Yes No

Build the tree TB? Yes No (Test) Results

Build the tree TB? Yes No Results

Build the tree TB? Yes No Results /3 1/

Build the tree TB? Yes No Results /3 1/ Yes+ - No+ -

Build the tree TB? Yes No Results /3 1/ Yes+ (2/3)*.06 Yes- No+ -

Build the tree TB? Yes No Results /3 1/ Yes+ (2/3)*.06 Yes- (1/3)*.06 No+ -

Build the tree TB? Yes No Results /3 1/ Yes+ (2/3)*.06 Yes- (1/3)*.06 No *0.94 No-

Build the tree TB? Yes No Results /3 1/ Yes+ (2/3)*.06 Yes- (1/3)*.06 No *0.94 No *0.94

Build the tree TB? Yes No Results /3 1/ Yes Yes No No

A patient gets a positive result. What is the probability that he does not have the disease? (False Positive) TB? Yes No Results /3 1/ Yes Yes No No P(No TB | +)?

A patient gets a positive result. What is the probability that he does not have the disease? (False Positive) TB? Yes No Results /3 1/ Yes Yes No No P(No TB | +)?

A patient gets a positive result. What is the probability that he does not have the disease? (False Positive) TB? Yes No Results /3 1/ Yes Yes No No =

A patient gets a positive result. What is the probability that he does not have the disease? (False Positive) TB? Yes No Results /3 1/ Yes Yes No No = Conclusion in context: If a patient gets a positive result, the probability that he does not have TB is (Therefore, the probability that such a patient does have TB is )

A patient gets a negative result. What is the probability that he has the disease? (False negative). TB? Yes No Results /3 1/ Yes Yes No No P(TB | - )?

A patient gets a negative result. What is the probability that he has the disease? (False negative). TB? Yes No Results /3 1/ Yes Yes No No P(TB | - )?

A patient gets a negative result. What is the probability that he has the disease? (False negative). Yes Yes No No TB? Yes No Results /3 1/ =

A patient gets a negative result. What is the probability that he has the disease? (False negative). Yes Yes No No TB? Yes No Results /3 1/ = Conclusion in context: If a patient gets a negative result, the probability that he has TB is (Therefore, the probability that such a patient does not have TB is )

Two ways they can make an error - Which is worse? First type: False positive –  Diagnosing someone with a serious illness that he does not have. (p = )  In the long run, out of 10,000 people who receive a positive diagnosis, about 859 patients will be misdiagnosed. Second type: False negative –  Failure to catch and diagnose a serious illness that he does have. (p = )  In the long run, out of 10,000 patients who receive a “clean bill of health”, about 209 are actually infected with TB.

A patient gets a positive result. What is the probability that he does not have the disease? (False Positive) We can use Bayes’ Theorem:

A patient gets a negative result. What is the probability that he has the disease? (False negative). By Bayes’ Theorem,