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

See Pagano- Chapter 6- section 1-4

credibile intervals
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
  • 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-)
slide5

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 tests1
Diagnostic Tests
  • Positive predictive value = P(D | T+)
  • Sensitivity = P(T+ | D)
  • Specificity = P(T- | Dc )
  • Prevalence = P(D)
example cotinine levels and smoking
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
slide34

Example: Cotinine levels andsmoking

Choose cutpoint using points closest to perfect.

Perfect

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