Down

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Downs Syndrome Example
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The Variables

• Consider the case of prenatal testing for Down's
  syndrome. Let
• D1 indicate that a baby has Down's syndrome and
  D0
• indicate that a baby does not have the disease.
• In addition, suppose there is a test available
  (denoted T)
• that can help to determine if the baby has
  Down's Syndrome
• prior to birth. Let T1 denote the event that
  the test
• suggests a positive screen for Down's syndrome,
  and T0
• denote a negative screen.

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Information About the Disease

• We know the following statistics (these numbers
  vary according to different reports, but we will
  use them in this example)
• Pr(D1) 1/800 .0013
• Pr(D0) 1 - Pr(D1) .9987

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Information About the Test

• In addition, we have some information regarding
  the accuracy of the test itself. In particular,
  we know that if the baby has the disease, then 80
  percent of the time, the test will screen
  positive (T1). This tells us that
• Pr(T1 D1) .8 ? Pr(T0 D1) .2
• We will define the false positive rate as the
  probability that the test screens positive given
  that the baby does not have the disease. We will
  denote this probability abstractly as c
• Pr(T1 D0) c ? Pr(T0 D0) 1-c.
• There are a variety of screening tests, with
  varying degrees of accuracy. One common test
  (CVS) reports a false positive rate of c.05. We
  will compute probabilities of interest for a
  variety of values of c.

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What Parents Care About

• What parents care about is the probability of
  disease (or no disease) given the result of the
  screening test. In particular, I would be most
  interested in the following probabilities
• Pr(D1 T1) and Pr(D0 T1)
• which are the probabilities that the baby does
  or does not have Down's Syndrome given that the
  screen was positive, and
• Pr(D1 T 0) and Pr(D 0 T0 ),
• the probabilities that the baby does and does
  not have Down's Syndrome given that the test
  screen is negative.

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Working it Out

• Consider the quantity Pr(D1 T1)
• We reduce this quantity to a functions of known
  quantities on the following page

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Working it out, continued

• The above can be calculated for a variety of
  values of the false positive rate c. In
  particular, at c .05, we evaluate
• Pr(D1T1) .02.
• How do we interpret this number? Does it make
  sense?

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Interpreting the Results

• The results suggest that the probability of
  having Down's syndrome, given a positive test
  result is only about 2 percent!
• The relatively high false positive rate makes it
  difficult for us to actually conclude that the
  baby has the disease. However, the revised
  probability (after finding out a positive screen)
  of 2 percent is 16 times greater than the
  unconditional probability of having the disease
  (1/800).
• So, children with a positive screen are indeed
  at greater risk, but are still relatively
  unlikely to actually have the disease!
  (Re-calculate this by making c much smaller.
  Interpret your results).

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Interpreting a Negative Screen

• What about the other quantity of interest, the
  probabilities of having or not having the disease
  if the test screen comes back negative, i.e.
• Pr(D1T0) and Pr(D0T0)?
• We can perform a similar calculation to evaluate
  these quantities

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Interpreting a Negative Screen

• Evaluated at c .05, we find
• 
  Pr(D0T0) .9997.
• Thus, if the test comes back negative, there
  is still a very small chance that the baby has
  Down's Syndrome.
• Also note that this number is larger than .9987,
  which is the overall probability of not having
  Down's Syndrome in the population.
• Finally, note that even if c0 so that there is
  no possibility of a false positive, you still can
  not be certain that your baby will not have the
  disease. This is because 20 percent of the time,
  the test will fail to identify a baby with Down's
  Syndrome when the baby does, in fact, have the
  disease.