Baye

1
Bayes Rule and Medical Screening Tests
2
Bayes Rule

• Bayes Rule is used in medicine and epidemiology
  to calculate the probability that an individual
  has a disease, given that they test positive on a
  screening test.
• Example Down syndrome is a variable combination
  of congenital malformations caused by trisomy 21.
  It is the most commonly recognized genetic cause
  of mental retardation, with an estimated
  prevalence of 9.2 cases per 10,000 live births in
  the United States. Because of the morbidity
  associated with Down syndrome, screening and
  diagnostic testing for this condition are offered
  as optional components of prenatal care.
• Many studies have been conducted looking at the
  effectiveness of screening methods used to
  identify likely Down syndrome cases.

3
Study of Triple Test Effectiveness

• The results of a study looking at the
  effectiveness of the triple-test are presented
  below
• How well does the triple-test perform?

4
The General Situation

• Patient presents with symptoms, and is suspected
  of having some disease. Patient either has the
  disease or does not have the disease.
• Physician performs a diagnostic test to assist in
  making a diagnosis.
• Test result is either positive (diseased) or
  negative (healthy).

5
The General Situation
Test Result
True Disease
Diseased
Healthy
(-)
Status
()
False
Diseased ()
Correct
Negative
False
Healthy (-)
Correct
Positive
6
Definitions

• False Positive Healthy person incorrectly
  receives a positive (diseased) test result.
• False Negative Diseased person incorrectly
  receives a negative (healthy) test result.

7
Goal

• Minimize chance (probability) of false positive
  and false negative test results.
• Or, equivalently, maximize probability of correct
  results.

8
Accuracy of Tests in Development

• Sensitivity probability that a person who truly
  has the disease correctly receives a positive
  test result.
• Specificity probability that a person who is
  truly healthy correctly receives a negative test
  result.

9
Triple-Test Performance
How does the triple-test perform using these
measures?
87/118 .7373
31/118 .2627
203/4072 .0499
3869/4072 .9501
10
Test Result ? What ?

• Now suppose you are have just been given the news
  the results of the triple test are positive for
  Down syndrome.
• What do you want to know now?
• You probably would like to know what the
  probability that your unborn child actually has
  Down syndrome.

11
Accuracy of Tests in Use

• Positive predictive value probability that a
  person who has a positive test result really has
  the disease.
• Negative predictive value probability that a
  person who has a negative test result really is
  healthy.
• To find these we use Bayes Rule to reverse the
  conditioning.

12
Case-Control Nature of Study

• Generally these studies are case-control in
  nature, meaning that the disease is not random!
• We specifically choose individuals with the
  disease to perform the screening test on,
  therefore we cannot talk about or calculate P(D)
  or P(D-) using our results.
• To find these we use Bayes Rule to reverse the
  conditioning.

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Bayes Rule

• This requires prior knowledge about the
  probability of having the disease, P(D), and
  hence the probability of not having the
  disease, P(D-).
• This probability is called the positive
  predictive value (PPV) of the triple-test.

14
Negative Predictive Value
This also requires prior knowledge of the
probabilities of having and not having the
disease.
15
Example PPV for Triple-Test

• Prior knowledge about Downs Syndrome suggests
  P(D) .00092
• or roughly 1 in 1,000 (i.e. P(D) .001).
• We also now from our earlier work that
• P(T D) .7373 P(T -D) .2627
• P(T -D -) .9501 P(T D -).0499

16
Comparing to Prior Probability

• In the absence of any test result the probability
  of having a child with Downs Syndrome is
  P(D) .00092
• Given a positive test result we have
• P(DT) .0134
• Comparing these two probabilities in the form of
  a ratio we find .0134/.00092 14.56.
• When we look at in practical terms however there
  is still only a 1.34 chance that the unborn
  child has Downs Syndrome.

17
Example NPV for Triple-Test

• Prior knowledge about Downs Syndrome suggests
  P(D-) .99908
• or roughly 999 in 1,000 (i.e. P(D-) .999).
• We also now from our earlier work that
• P(T D) .7373 P(T -D) .2627
• P(T -D -) .9501 P(T D -).0499

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Comparing to Prior Probability

• In the absence of any test result the probability
  of NOT having a child with Downs Syndrome is
  P(D-) .99908
• Given a negative test result we have
• P(D-T-) .99975
• Comparing these two probabilities in the form of
  a ratio we find .99975/.99908 1.00067.
• When we look at in practical terms the
  probability only increases by .00067 when we have
  a negative test result.

19
Questions to Think About

• What do you think of the Triple-Test for Downs
  Syndrome?
• Would you advocate having it administered to all
  women during their first trimester of pregnancy?
• Would you have it done if you were pregnant?