Credible Intervals, Bayes Theorem Diagnostic Tests

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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
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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 x4 out of n10 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
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Diagnostic Tests

• Diagnostic tests are routinely used to detect
  disease
• Events related to individuals health status
• Individual has disease (D)
• Individual is disease free (Dc)
• Outcomes of a diagnostic test
• Positive test result (T)
• Negative test result (T-)

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Diagnostic Tests
Diagnostic tests D have disease Dc do not
have disease Tpositive screening result
Find the probability that an individual who
tests positive actually has disease
Find P(D T)
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Diagnostic Tests

• Positive predictive value P(D T)
• Sensitivity P(T D)
• Specificity P(T- Dc )
• Prevalence P(D)

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Example X-ray screening for tuberculosis
X-ray Tuberculosis Tuberculosis
X-ray Yes
Positive x22
Negative 8
Total n30
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Example X-ray screening for tuberculosis
X-ray Tuberculosis Tuberculosis
X-ray Yes No
Positive 51
Negative x1739
Total n1790
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Example X-ray screening for tuberculosis
X-ray Tuberculosis Tuberculosis Total
X-ray Yes No Total
Positive 22 51 73
Negative 8 1739 1747
Total 30 1790 1820
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Example Using Bayes Theorem for PPV
X-ray Tuberculosis Tuberculosis Total
X-ray Yes(D1) No(D2) Total
Positive (T) 22 51 73
Negative (T-) 8 1739 1747
Total 30 1790 1820
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Example Cotinine levels and smoking

• Outcome of interest Smoking status
• Problem People may not report honestly
• Cotinine level may provide objective asessment
  of smoking
• Cotinine levels dont work perfectly
• Diagnostic test Concentration of cotinine
• i.e. If Cotinine level gt c ? Smoker
• If Cotinine level lt c ? NonSmoker

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Cotinine Level (ng/ml) Smokers Non-smokers
0--13 78 3300
14--49 133 72
50--99 142 23
100--149 206 15
150--199 197 7
200--249 220 8
250--299 151 9
300 412 11
Total 1539 3445
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Cotinine Level (ng/ml) Smokers Non-smokers Sensit-ivity
0--13 78 3300 1461/1539
14--49 133 72
50--99 142 23
100--149 206 15
150--199 197 7
200--249 220 8
250--299 151 9
300 412 11
Total 1539 3445
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Cotinine Level (ng/ml) Smokers Non-smokers Sensit-ivity
0--13 78 3300 1461/1539
14--49 133 72 1328/1539
50--99 142 23
100--149 206 15
150--199 197 7
200--249 220 8
250--299 151 9
300 412 11
Total 1539 3445
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Cotinine Level (ng/ml) Smokers Non-smokers Sensit-ivity
0--13 78 3300 1461/1539
14--49 133 72 1328/1539
50--99 142 23 1186/1539
100--149 206 15 980/1539
150--199 197 7 783/1539
200--249 220 8 563/1539
250--299 151 9 412/1539
300 412 11
Total 1539 3445
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Cotinine Level (ng/ml) Smokers Non-smokers Sensit-ivity
0--13 78 3300 0.95
14--49 133 72 1328/1539
50--99 142 23 1186/1539
100--149 206 15 980/1539
150--199 197 7 783/1539
200--249 220 8 563/1539
250--299 151 9 412/1539
300 412 11
Total 1539 3445
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Cotinine Level (ng/ml) Smokers Non-smokers Sensit-ivity
0--13 78 3300 0.95
14--49 133 72 0.86
50--99 142 23 1186/1539
100--149 206 15 980/1539
150--199 197 7 783/1539
200--249 220 8 563/1539
250--299 151 9 412/1539
300 412 11
Total 1539 3445
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Cotinine Level (ng/ml) Smokers Non-smokers Sensit-ivity
0--13 78 3300 0.95
14--49 133 72 0.86
50--99 142 23 0.77
100--149 206 15 980/1539
150--199 197 7 783/1539
200--249 220 8 563/1539
250--299 151 9 412/1539
300 412 11
Total 1539 3445
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Cotinine Level (ng/ml) Smokers Non-smokers Sensit-ivity
0--13 78 3300 0.95
14--49 133 72 0.86
50--99 142 23 0.77
100--149 206 15 0.64
150--199 197 7 783/1539
200--249 220 8 563/1539
250--299 151 9 412/1539
300 412 11
Total 1539 3445
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Cotinine Level (ng/ml) Smokers Non-smokers Sensit-ivity
0--13 78 3300 0.95
14--49 133 72 0.86
50--99 142 23 0.77
100--149 206 15 0.64
150--199 197 7 0.51
200--249 220 8 563/1539
250--299 151 9 412/1539
300 412 11
Total 1539 3445
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Cotinine Level (ng/ml) Smokers Non-smokers Sensit-ivity
0--13 78 3300 0.95
14--49 133 72 0.86
50--99 142 23 0.77
100--149 206 15 0.64
150--199 197 7 0.51
200--249 220 8 0.37
250--299 151 9 412/1539
300 412 11
Total 1539 3445
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Cotinine Level (ng/ml) Smokers Non-smokers Sensit-ivity
0--13 78 3300 0.95
14--49 133 72 0.86
50--99 142 23 0.77
100--149 206 15 0.64
150--199 197 7 0.51
200--249 220 8 0.37
250--299 151 9 0.27
300 412 11
Total 1539 3445
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Cotinine Level (ng/ml) Smokers Non-smokers Sensit-ivity 1Specif-icity
0--13 78 3300 0.95 145/3445
14--49 133 72 0.86
50--99 142 23 0.77
100--149 206 15 0.64
150--199 197 7 0.51
200--249 220 8 0.37
250--299 151 9 0.27
300 412 11
Total 1539 3445
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Cotinine Level (ng/ml) Smokers Non-smokers Sensit-ivity 1Specif-icity
0--13 78 3300 0.95 145/3445
14--49 133 72 0.86 73/3445
50--99 142 23 0.77
100--149 206 15 0.64
150--199 197 7 0.51
200--249 220 8 0.37
250--299 151 9 0.27
300 412 11
Total 1539 3445
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Cotinine Level (ng/ml) Smokers Non-smokers Sensit-ivity 1Specif-icity
0--13 78 3300 0.95 145/3445
14--49 133 72 0.86 73/3445
50--99 142 23 0.77 50/3445
100--149 206 15 0.64 35/3445
150--199 197 7 0.51 28/3445
200--249 220 8 0.37 20/3445
250--299 151 9 0.27 11/3445
300 412 11
Total 1539 3445
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Cotinine Level (ng/ml) Smokers Non-smokers Sensit-ivity 1Specif-icity
0--13 78 3300 0.95 0.04
14--49 133 72 0.86 73/3445
50--99 142 23 0.77 50/3445
100--149 206 15 0.64 35/3445
150--199 197 7 0.51 28/3445
200--249 220 8 0.37 20/3445
250--299 151 9 0.27 11/3445
300 412 11
Total 1539 3445
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Cotinine Level (ng/ml) Smokers Non-smokers Sensit-ivity 1Specif-icity
0--13 78 3300 0.95 0.04
14--49 133 72 0.86 0.02
50--99 142 23 0.77 50/3445
100--149 206 15 0.64 35/3445
150--199 197 7 0.51 28/3445
200--249 220 8 0.37 20/3445
250--299 151 9 0.27 11/3445
300 412 11
Total 1539 3445
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Cotinine Level (ng/ml) Smokers Non-smokers Sensit-ivity 1Specif-icity
0--13 78 3300 0.95 0.04
14--49 133 72 0.86 0.02
50--99 142 23 0.77 0.01
100--149 206 15 0.64 35/3445
150--199 197 7 0.51 28/3445
200--249 220 8 0.37 20/3445
250--299 151 9 0.27 11/3445
300 412 11
Total 1539 3445
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Cotinine Level (ng/ml) Smokers Non-smokers Sensit-ivity 1Specif-icity
0--13 78 3300 0.95 0.04
14--49 133 72 0.86 0.02
50--99 142 23 0.77 0.01
100--149 206 15 0.64 0.01
150--199 197 7 0.51 28/3445
200--249 220 8 0.37 20/3445
250--299 151 9 0.27 11/3445
300 412 11
Total 1539 3445
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Cotinine Level (ng/ml) Smokers Non-smokers Sensit-ivity 1Specif-icity
0--13 78 3300 0.95 0.04
14--49 133 72 0.86 0.02
50--99 142 23 0.77 0.01
100--149 206 15 0.64 0.01
150--199 197 7 0.51 0.01
200--249 220 8 0.37 20/3445
250--299 151 9 0.27 11/3445
300 412 11
Total 1539 3445
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Cotinine Level (ng/ml) Smokers Non-smokers Sensit-ivity 1Specif-icity
0--13 78 3300 0.95 0.04
14--49 133 72 0.86 0.02
50--99 142 23 0.77 0.01
100--149 206 15 0.64 0.01
150--199 197 7 0.51 0.01
200--249 220 8 0.37 0.006
250--299 151 9 0.27 11/3445
300 412 11
Total 1539 3445
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Cotinine Level (ng/ml) Smokers Non-smokers Sensit-ivity 1Specif-icity
0--13 78 3300 0.95 0.04
14--49 133 72 0.86 0.02
50--99 142 23 0.77 0.01
100--149 206 15 0.64 0.01
150--199 197 7 0.51 0.01
200--249 220 8 0.37 0.006
250--299 151 9 0.27 0.003
300 412 11
Total 1539 3445
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Example Cotinine levels and smoking
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Example Cotinine levels and smoking
Choose cutpoint using points closest to perfect.
Perfect