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Bayes Theorem
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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
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
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)
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-
Hemophilia
c 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
Resources
Bayes Theorem (Wikipedia)
https://en.wikipedia.org/wiki/Bayes'_theorem
Bayes Theorem (Khan Academy)
https://www.khanacademy.org/math/precalculus/prob-comb/dependent-events-precalc/v/bayes-theorem-visualized
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
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