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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:

Credible Intervals, Bayes Theorem + Diagnostic Tests

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### 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: X-ray screening for tuberculosis

Example: X-ray screening for tuberculosis

Example: X-ray screening for tuberculosis

Example: Using Bayes Theorem for PPV

### 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

Example: Cotinine levels andsmoking

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