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Credible Intervals, Bayes Theorem + Diagnostic Tests

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- Credibile Intervals
- Posterior Distribution and Bayes Theorem

- Sensitivity
- Specificity
- Positive Predictive Value
- ROC curve

See Pagano- Chapter 6- section 1-4

- 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 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+)

- 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

- 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

Choose cutpoint using points closest to perfect.

Perfect