Ch 3, Probability

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Ch 3, Probability

• Basic defns
• Sample space
• Events
• Mutually exclusive
• P(A or B)P(A)P(B)

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Ch 3, Probability

• A B are independent if P(A and B)P(A)P(B)
• P(A or B)P(A)P(B)-P(A and B)

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Ch 3, Conditional Probability

• P(BA)P(A and B)/P(A)
• What fraction of A is also B?
• Handy to do as table

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Ch 3, Conditional Probability
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Ch 3, Conditional Probability

• Ex 3.15, p 50
• P(A)0.10
• P(B)0.17
• P(A and B) 0.08

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Ch 3, Conditional Probability
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Ch 3, Conditional Probability
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Ch 3, Conditional Probability

• If A is pos, what is the chance B is pos?
• P(A and B) 0.08
• P(BA) 0.08/0.10 80
• If B is pos, what is the chance A is pos?
• P(AB) 0.08/0.17
• Numerators are the same

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Ch 3, Conditional Probability

• Relative risk of B given A
• P(BA)/P(B notA)

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Ch 3, Bayes Rule

• Relate P(AB) to P(BA)
• Note that P(A and B) P(AB)P(B)

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Ch 3, Bayes Rule

• Ex 3.27, p 62
• A symptoms
• B1normal
• B2lung cancer
• B3sarcoidosis

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Ch 3, Bayes Rule
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Ch 3, Med Testing

• Suppose we have a test for a disease
• Results are Pos or Neg
• P(Pos Disease), P(Neg no Disease)
• Sensitivity, Specificity
• Also need P(Disease)
• Note book calls this Prevalence
• Others call it Incidence Rate
• Book uses Cumulative Incidence for something else

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Ch 3, Med Testing
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Ch 3, Med Testing

• Ex 3.26, p 61
• Disease hypertension
• P(Pos Disease) 0.84
• P(Pos not D) 0.23
• So P(Neg not D) 0.77
• P(Disease) 0.20

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Ch 3, Med Testing
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Ch 3, Med Testing

• See Excel
• P(D Pos) 0.477
• Positive Predictive Value (PPV)
• P(not D Neg) 0.951
• Neg Pred Value (NPV)

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Ch 3, Med Testing

• Screening test
• Test everyone
• If test is neg, then theyre OK
• If test is pos, they are sent to doctor for more
  tests
• Want high NPV

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Ch 3, Med Testing

• Diagnostic test
• Someone is sick, want to know what it is
• PPV is more important
• If test is Pos, want to be sure that this is what
  is actually wrong with patient

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Ch 3, Med Testing

• Results depend on Prevalence (Incidence Rate)
• Try diff values

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Ch 3, Multiple Diagnoses

• So far, we have assumed either Sick or not
• Could be more options
• Cond A, Cond B, neither (Cond C)
• Need to know P(posA), P(posB) and P(posC)

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Ch 3, Med Testing
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Ch 3, Multiple Diagnoses

• Suppose P(A)0.60, P(B)0.25, P(C)0.15
• P(posA)0.45, P(posB)0.7, P(posC)0.9

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Ch 3, Med Testing
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Ch 3, Multiple Diagnoses

• Given the test results, what are the
  probabilities of A, B, C?

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Ch 3, Multiple Diagnoses

• A neg test suggests Cond A
• A pos test suggest Cond A, but could be B
• Note the high prevelance of A
• A pos test reduces the chance of A and increases
  the chance of C

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Ch 3, Multiple Diagnoses

• See Table 3.12 on p 74
• Note that along with X1 and X2, there is a chance
  that the patient is lt1 yr old
• Do 3.85-3.89