Transcript Down`s Syndrome Example

Down’s Syndrome Example
Tobias, 2005
Econometrics 472
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 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 T=1 denote the event that the test
suggests a positive screen for Down's syndrome, and T=0
denote a negative screen.
Tobias, 2005
Econometrics 472
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(D=1) = 1/800 ≈ .0013
Pr(D=0) = 1 - Pr(D=1) = .9987
Tobias, 2005
Econometrics 472
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 (T=1). This tells us that:
Pr(T=1 | D=1) = .8 → Pr(T=0 | D=1) = .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(T=1 | D=0) = c → Pr(T=0 | D=0) = 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.
Tobias, 2005
Econometrics 472
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(D=1 | T=1) and Pr(D=0 | T=1)
which are the probabilities that the baby does or does not have
Down's Syndrome given that the screen was positive, and
Pr(D=1 | T = 0) and Pr(D = 0 | T=0 ),
the probabilities that the baby does and does not have Down's
Syndrome given that the test screen is negative.
Tobias, 2005
Econometrics 472
Working it Out
• Consider the quantity Pr(D=1 | T=1)
We reduce this quantity to a functions of known
quantities on the following page:
Tobias, 2005
Econometrics 472
Tobias, 2005
Econometrics 472
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(D=1|T=1) = .02.
How do we interpret this number? Does it make sense?
Tobias, 2005
Econometrics 472
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! (Recalculate this by making c much smaller. Interpret your results).
Tobias, 2005
Econometrics 472
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(D=1|T=0) and Pr(D=0|T=0)?
We can perform a similar calculation to evaluate
these quantities:
Tobias, 2005
Econometrics 472
Tobias, 2005
Econometrics 472
Interpreting a Negative Screen
• Evaluated at c = .05, we find
Pr(D=0|T=0) = .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 c=0 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.
Tobias, 2005
Econometrics 472