Credible Intervals, Bayes Theorem + Diagnostic Tests

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# Credible Intervals, Bayes Theorem + Diagnostic Tests - PowerPoint PPT Presentation

Credible Intervals, Bayes Theorem + Diagnostic Tests. Outline. Credibile Intervals Posterior Distribution and Bayes Theorem Sensitivity Specificity Positive Predictive Value ROC curve. See Pagano- Chapter 6- section 1-4. Credibile Intervals. For Prob(Smoking)=p in a Population:

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Presentation Transcript
Outline
• Credibile Intervals
• Posterior Distribution and Bayes Theorem
• Sensitivity
• Specificity
• Positive Predictive Value
• ROC curve

See Pagano- Chapter 6- section 1-4

Credibile Intervals
• For Prob(Smoking)=p in a Population:
• p could be 0.05, 0.10, … 0.90, 0.95,1
• Prob of p: (prior probability)
• Data: x=4 out of n=10 people smoke
• Get Posterior Distribution using Bayes Theorem
• Credible Interval: 95% Credible Interval: 2.5th and 97.5th percentile of posterior distribution
• Example: Suppose the prior probability is the same for all p (uniform prior)

Posterior Distribution

Credible

Interval

Diagnostic Tests
• Diagnostic tests are routinely used to detect disease
• Events related to individual’s health status:
• Individual has disease (D)
• Individual is disease free (Dc)
• Outcomes of a diagnostic test:
• Positive test result (T+)
• Negative test result (T-)

Diagnostic Tests

Diagnostic tests

D = “have disease”

Dc =“do not have disease”

T+=“positive screening result”

Find the probability that an individual

who tests positive actually has disease

Find P(D |T+)

Diagnostic Tests
• Positive predictive value = P(D | T+)
• Sensitivity = P(T+ | D)
• Specificity = P(T- | Dc )
• Prevalence = P(D)
Example: Cotinine levels andsmoking
• Outcome of interest – Smoking status
• Problem: People may not report honestly
• Cotinine level may provide ‘objective’ asessment of smoking
• Cotinine levels don’t work perfectly
• Diagnostic test – Concentration of cotinine
• i.e. If Cotinine level > c  Smoker
• If Cotinine level <= c  NonSmoker

Example: Cotinine levels andsmoking

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