The prevalence of a disease among a certain population is .40. That is, there is a 40 percent chance that a person randomly selected from the population will have the disease. An imperfect test that costs $250 is available to help identify those who have the disease before actual symptoms appear. Those who have the disease have a 90 percent chance of a positive test result; those who do not have the disease have a 5 percent chance of a positive test. Treatment of the disease before the appearance of symptoms costs $2,000 and inflicts additional costs of $200 on those who do not actually have the disease. Treatment of the disease after symptoms have appeared costs $10,000.
The government is considering the following possible strategies with respect to the disease:
S1. Do not test and do not treat early.
S2. Do not test and treat early.
S3. Test and treat early if positive and do not treat early if negative.
Find the treatment/testing strategy that has the lowest expected costs for a member of the population.
In doing this exercise, the following notation may be helpful: Let D indicate presence of the disease, ND absence of the disease, T a positive test result, and NT a negative test result. Thus, we have the following information:
P(D) = .40, which implies P(ND) = .60
P(T|D) = .90, which implies P(NT|D) = .10
P(T|ND) = .05, which implies P(NT|ND) = .95
This information allows calculation of some other useful probabilities:
P(T) = P(T|D)P(D)+P(T|ND)P(ND) = .39 and P(NT) = .61
P(D|T) = P(T|D)P(D)/P(T) = .92 and P(ND|T) = .08
P(D|NT) = P(NT|D)P(D)/P(NT) = .07 and P(ND|NT) = .93