# http://www.fpnotebook.com/

## Bayes Theorem

*Aka: Bayes Theorem, Bayesian Statistics*

- See Also
- Screening Test
- Contingency Grid or Cross Tab (includes Statistics Example)
- Fagan Nomogram
- Experimental Error (Experimental Bias)
- Lead-Time Bias
- Length Bias
- Selection Bias
- Likelihood Ratio (Positive Likelihood Ratio, Negative Likelihood Ratio)
- Number Needed to Screen (Number Needed to Treat, Absolute Risk Reduction, Relative Risk Reduction)
- Negative Predictive Value
- Positive Predictive Value
- Pre-Test Odds or Post-Test Odds
- Receiver Operating Characteristic
- Test Sensitivity (False Negative Rate)
- Test Specificity (False Positive Rate)
- U.S. Preventive Services Task Force Recommendations

- Definitions
- Bayes Theorem (calculation)
- P (Disease | Positive Test) = P(Positive test | Disease) * P(Disease) / P(Positive Test)
- Where
- P (A | B) = Probability of A given B
- P(Positive test | Disease) = Test Sensitivity

- Bayes Theorem (calculation)
- Evaluation: Example - Probability of Disease Based on a Test
- Positive Test
- Disease Y Present in 75
- Disease Y NOT Present in 25

- Negative Test
- Disease Y Present in 10
- Disease Y NOT Present in 190

- Probabilities
- P(Positive test I Disease) = Test Sensitivity = 75 / (75 + 10) = 0.88
- P(Disease) = Pretest Probability in cohort tested = (75+10)/(75+10+25+190) = 0.28
- P(Positive Test) = True positives and False positives = (75 + 25)/(75+10+25+190) = 0.33

- Conclusion
- P (Disease | Pos Test) = P(Pos test I Disease) * P(Disease) / P(Pos Test) = 0.88 * 0.28 / 0.33 = 0.75
- In this case a patient from the given cohort has a 75% probability of Disease Y given a Positive Test

- Positive Test
- Evaluation: Example - Probability of a disease based on a group of findings
- The probability of a disease given one or more findings can be calculated from:
- Prevalence of a Disease (and of its differential diagnosis) AND
- Probability of findings when the disease is present (and when other conditions on the differential diagnosis are present)

- Assumptions
- Conditional independence of findings
- For a given disease, different findings do not have a relationship with one another
- Example: For Acute Coronary Syndrome, Chest Pain and Shortness of Breath are not dependently related

- Mutual exclusivity of conditions
- For a given presentation with specific findings, only one disease is present to explain those findings
- Example: The patient with Chest Pain, Tachypnea and Shortness of Breath
- Does NOT have both a Myocardial Infarction AND a Pulmonary Embolism

- Calculation
- P(D|F) = Probability of Disease (D) given Findings (F) = P(D) * P(F | D) / P(DDx) * P(F | DDx)
- Where
- P(D) = Probability of Disease (D)
- P(F | D) = Probability of Findings (F) given Disease (D)
- P(DDx) = Sum of probabilities of a group of Diseases including the Disease (D) of interest (Differential Diagnosis)
- P(F | DDx) = Probability of Findings (F) given the group of diseases (DDx)

- Conditional independence of findings

- The probability of a disease given one or more findings can be calculated from:
- Evaluation: Example of Family Tree and Hemophilia
- Setup
- A healthy woman has a brother with Hemophilia (xY)
- Hemophilia is X-linked and as she is unaffected she is either Xx (Hemophilia carrier) or XX (normal)
- She has two healthy male children without Hemophilia (each XY)
- What is the probability that she is XX (no Hemophilia gene)

- Assumptions
- P(xX) = p(XX) = probability mother is either Hemophilia carrier (xX) or normal (XX) = 0.5
- P(cXY and cXY|mXX) = probability that both children are XY (normal) given mother is XX = 1
- P(cXY and cXY|mxX) = probablity that both children are XY (normal) given mother is xX (Hemophilia carrier) = 0.5 * 0.5 = 0.25

- Bayes Formula
- P(A|B) = P(B|A) * P(A) / (P(B|A)*P(A) + P(B|not A)*P(not A) )
- P(mXX| cXY and cXY) = Probability mother has 2 normal X copies given 2 non-Hemophiliac sons
- P(mXX| cXY and cXY) = P(cXY and cXY|mXX) * P(XX) / ( P(cXY and cXY|mXX) * P(XX) + P(cXY and cXY|mxX) * P(xX) )
- P(mXX| cXY and cXY) = (1* 0.5) / ( 1 * 0.5 + 0.25 * 0.5 ) = 0.5 / 0.625 = 0.8 or 4/5

- References
- (2015) Columbia Statistical Thinking for Data Science and Analytics, EDX, accessed online 2/4/2017

- Setup
- Resources
- Bayes Theorem (Wikipedia)
- Bayes Theorem (Khan Academy)

- References
- Desai (2014) Clinical Decision Making, AMIA’s CIBRC Online Course
- Hersh (2014) Knowledge Acquisition and Use for Clinical Decision Support, AMIA’s CIBRC Online Course