Group Fuchsia Presents:Bayes’ Rule

Neuropsychiatric DecisionMaking: The Role of

Disorder Prevalence in Diagnostic Testing

Heather Jacobson, Jessica Landin, Will Liu, Brian Wiggs

Thomas Bayes 1702-1761 English Mathematician Presbyterian Minister Deep interest in probability Bayes’ solution to “inverse

probability” was presented after his death, became “Bayes’ Theorem”

Bayes’ Rule Basics Purpose: To revise probabilities as new

information becomes available- i.e., the probability of our prior probability, given the result of our conditional probability• Example: the probability of condition in a patient

given the probability of this condition in the general population

The theorem states that the post-test likelihood of a condition is a function of the test’s accuracy and the pretest likelihood that the condition was present.

Bayes’

Related Vocab P(A|B)=(P(B|A) * P(A)) /P(B/A)P(A) + P(B/Not A)P(Not A)

Where the denominator is an “unconditional probability” Prior probability- known before current state

• P(A) Likelihood probability- depends on prior node

• P(B/A) Posterior Probability-revised probability given new info

• P(A/B) Sensitivity- probability that the test will have a positive

result (has a disease) when the person actually has the disease

Specificity-Probability that the test will be negative when the person actually lacks the disease

Mr. A Example Mr. A Is a 65 year-old male has

dementia. Dr. X administers the Short Test of

Mental Status (STMS) For persons in Mr. A’s age group,

scores<30 are considered diagnostic of dementia

STMS has a sensitivity of 95% and specificity of 88%

Mr. A Example Mr. A scores below 30. Dr. X should conclude that:

• A.) Mr. A. has a 95% chance of having dementia

• B.) Mr. A has an 88% chance of having dementia

• C.)Mr. A has a 92% chance of having dementia

• D.)She needs more information to specify the post-test likelihood of dementia

Mr. A Example Answer: D

To specify the post-test likelihood of having dementia, Dr. X needs to know what the likelihood of his having dementia before he was tested

Dr. X needs to use Bayes’ Rule!!

Mr. A Example Dr. X estimates that before testing Mr. A has a 75% chance of having dementia• Prior probability of dementia:

P(D+)=75%• Prior probability of no dementia:

P(D-)=25%

Mr. A Example• Bayes’ Rule:

• D+: has dementia• D-: does not have dementia• T+: Test results positive for dementia• T-: Test results negative for dementia

Bayes’ Example 1

Bayes’ Example 1

Counts Method 10,000 people in Mr. A’s age group 75% probability of having dementia

=• 7,500 people with dementia• 2,500 people without dementia

Counts Method Administer STMS

• Expect 300 people without dementia to test positive, given false positive rate of 12%

2,500*12%=300• Expect 7,125 with dementia to test

positive , given true positive rate is 95%7,500*95%=7,125

• 7,125+300=7425 positive tests• P(D+/T+)=7,125/7,425=0.96

Same result as Bayes’ Rule calculation!

Bayes’- Example 2

Bayes’- Example 2

Questions?