Bayes and Medical Testing

M. Drew LaMar
February 12, 2016

“It’s easy to think of medical tests as black and white. If the test is positive, you have the disease; if it’s negative, you don’t. Even good clinicians sometimes fall into that trap.”

- Harriet Hall

Introduction to Biostatistics, Spring 2016

Example 5.9: Detection of Down Syndrome

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Question: What is Pr[DS | +]?

Discuss: What probabilities are given to us here?

Example 5.9: Detection of Down Syndrome

       result
status       +      -    Sum
  DS        60     40    100
  no DS   4995  94905  99900
  Sum     5055  94945 100000

Example 5.9: Detection of Down Syndrome

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Discuss: Graphically, what is:

  Pr[DS]

  Pr[+ | DS]

  Pr[+ | no DS]

  Pr[DS | +]

Example 5.9: Detection of Down Syndrome

Example 5.9: Detection of Down Syndrome

Example 5.9: Detection of Down Syndrome (Visual Check)

 

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Example 5.9: Detection of Down Syndrome (Calculate)

 

Discuss: Pr[DS | +] = ???

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Example 5.9: Detection of Down Syndrome (Calculate)

 

Discuss: Pr[DS | +] = ???

 

 

\[ \begin{align*} \mathrm{Pr[DS \ | \ +]} & = \frac{\mathrm{Pr[DS \ and \ +]}}{\mathrm{Pr[+]}} \\ & = \frac{\mathrm{Pr[DS \ and \ +]}}{\mathrm{Pr[DS \ and \ +] + Pr[no DS \ and \ +]}} \\ & = \frac{0.0006}{0.0006 + 0.04995} = 0.0118694 \\ \end{align*} \]

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Example 5.9: Detection of Down Syndrome (Calculate)

 

Discuss: Pr[DS | +] = ???

 

 

\[ \begin{align*} \mathrm{Pr[DS \ | \ +]} & = \frac{\mathrm{Pr[DS \ and \ +]}}{\mathrm{Pr[+]}} \\ & = \frac{\mathrm{Pr[DS \ and \ +]}}{\mathrm{Pr[DS \ and \ +] + Pr[no DS \ and \ +]}} \\ & = \frac{\mathrm{Pr[DS \ and \ +]}}{\mathrm{Pr[DS]\times Pr[+ \ | \ DS] + Pr[no \ DS]\times Pr[+ \ | \ no \ DS]}} \\ \end{align*} \]

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Example 5.9: Detection of Down Syndrome (Calculate)

 

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