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Down

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Down s Syndrome Example The Variables Consider the case of prenatal testing for Down's syndrome. Let D=1 indicate that a baby has Down's syndrome and D=0 indicate ... – PowerPoint PPT presentation

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Title: Down


1
Downs Syndrome Example
2
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.

3
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

4
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.

5
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.

6
Working it Out
  • Consider the quantity Pr(D1 T1)
  • We reduce this quantity to a functions of known
    quantities on the following page

7
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8
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?

9
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).

10
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

11
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12
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.
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