hw1_soln - CME308: Assignment 1 Due: Thursday, 8 April,...

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Unformatted text preview: CME308: Assignment 1 Due: Thursday, 8 April, 2010 Due Date: This assignment is due on Thursday, 8 April, 2010, by 5pm in the box outside Durand 112. See the course website for the policy on incentives for L A T E X solutions. Problem 1 (10 pts): Suppose there are 60,000 people in the US population carrying the syphilis virus. We wish to evaluate the performance of a new diagnostic test for this virus. Lab tests have show the following probabilities for the diagnostic test: Positive Test Negative Test Virus present 0.995 0.005 Virus absent 0.015 0.985 1. For an ideal test, the above matrix would be the identity. To evaluate the test, compute P { virus | positive } and discuss the tests accuracy. (Assume the US population is 300,000,000.) (This probability will tell us how likely it is that person who tests positive under the proposed diagnostic test actually has the virus. Given the cost of treatment and additional testing for individuals who test positive, this probability is of central interest to health professionals.) 2. Assume that the test costs $25 per person and that each individual will be tested every 3 years over a 80 year period. Individuals who are detected, through the test, as having the disease are subjected to a drug treatment that costs $100 per individual. Applying this treatment within three years of initiation avoids the need for intensive treatment. Intensive treatment costs $1,000,000 over the lifetime of the individual. From a purely economical viewpoint, is this test accurate enough to use across the entire population? What assumptions have you made? Solution: 1. The proportion of infected people gives the probability any given person has syphillis. p = 60 , 000 300 , 000 , 000 = 0 . 0002 = P ( V + ) From Bayes rule, P ( V + | T + ) = P ( T + | V + ) P ( V + ) P ( T + | V + ) P ( V + ) + P ( T + | V- ) P ( V- ) = . 995 . 0002 . 995 . 0002 + 0 . 015 . 9998 = 0 . 0131 The test is poor since the percentage of true positives is 1.31%. Or to say it another way, 99% of the time, a person will be receiving intensive treatment when they shouldnt. This is not a good test. To improve this, a smaller value of P T + | V- would be needed. 2. There are a number of ways to consider this problem. We offer this as one particularly rich method. The first assumption to be made is that testing the whole population spread out over 3 years is equivalent to testing them all in one year (or even at the very same instant). This is for ease of computation. The second assumption is that if the disease is undetected after 3 years, it becomes very noticeable that a person has the disease and therefore automatically qualifies for intensive treatment....
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hw1_soln - CME308: Assignment 1 Due: Thursday, 8 April,...

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