Credible intervals bayes theorem diagnostic tests
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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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Credible Intervals, Bayes Theorem + Diagnostic Tests

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Credible intervals bayes theorem diagnostic tests

Credible Intervals, Bayes Theorem +Diagnostic Tests


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-)


Credible intervals bayes theorem diagnostic tests

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)


Credible intervals bayes theorem diagnostic tests

Example: X-ray screening for tuberculosis


Credible intervals bayes theorem diagnostic tests

Example: X-ray screening for tuberculosis


Credible intervals bayes theorem diagnostic tests

Example: X-ray screening for tuberculosis


Credible intervals bayes theorem diagnostic tests

Example: Using Bayes Theorem for PPV


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


Credible intervals bayes theorem diagnostic tests

Example: Cotinine levels andsmoking


Credible intervals bayes theorem diagnostic tests

Example: Cotinine levels andsmoking

Choose cutpoint using points closest to perfect.

Perfect


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