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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
• 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
• 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 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.

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.