Thinking and Deciding

THE PSYCHOLOGY OF HYPOTHESIS TESTING 179
Table 7.1: ET scan results and incidence of three diseases in 100 patients
Test result Globoma Popitis Flapemia
Positive ET scan 40 10 0
Negative ET scan 40 0 10
All patients 80 10 10
Expected utility applies to a choice with probabilistic (uncertain) outcomes. It is
the average utility we would obtain if we made the same choice repeatedly and the
outcomes occurred with their respective probabilities. It is therefore the utility we
can expect on the average. In these problems, there are two choices: test or no test.
(There are also essentially two outcomes, correct treatment or incorrect treatment.)
To calculate expected utility, we consider all of the outcomes of a given choice, such
as not testing. We multiply the probability of each outcome (given the choice) by its
utility, and we add up these numbers across the outcomes. For example, in problem
1, for no test the probability of a correct treatment is .8 (assuming we treat globoma),
and the probability of an incorrect treatment is .2. The utility of a correct treatment
is (we have assumed) 1, and the utility of an incorrect treatment is 0. The expected
utility of this choice (no test) is thus .8 · 1+.2 · 0, or .8. Because we have defined
the utility of a good outcome as 1 and the utility of a bad outcome as 0, the expected
utility is equal to the probability of a good outcome.
Note that the utility of the other choice, testing, is the same. The probability of
correct treatment is still .8, with or without doing the test. This is because doing the
test cannot change the overall probability that the patient has globoma. Table 7.1
may make this clear. The top two rows show ET scan test results for 100 patients,
indicating how many patients with each disease had each test result. The bottom row,
the sum of the other two rows, shows the overall prevalence of the three diseases. If
no test is done, the figures in the row for “all patients” allow us to calculate the
probability of the three diseases: .8 for globoma and .1 for each of the others. If the
test is done and is positive, the top row indicates that the probability of globoma is
still .8. The same is found for a negative result. Therefore the test outcome cannot
affect our judgment of the probability of globoma, the most likely disease, and we
should treat globoma with or without knowledge of the test result.
Table 7.2 gives similar information for problem 2. In this case, the test can affect
the probability of the most likely disease, but it again cannot affect our decision about
treatment. If we do not do the test, we will have an 80% chance of giving the correct
treatment, so the expected utility of this choice would be .8 · 1+.2 · 0, or .8. If we do
the test and it is positive, the expected utility would be higher. Likewise, if the test
is negative, the expected utility would be lower: but we must decide whether or not
to do the test before we know what the outcome will be. Therefore, we must think