THE SITUATION: Suppose that the House of Delegates of the American Medical Association has decided to recommend that all physicians be tested for the Human Immunodeficiency Virus (HIV). This is the virus which causes AIDS. They want to include in their recommendation the format (or protocol) that testing should take.

If two tests are available and what is known is that

Unfortunately, choices about complex issues are rarely that clear-cut. A medical test usually has not one, but two percentages which describe its effectiveness. In fact, the tests above are more completely described by saying that

The first percentage in each description above gives the probability of the test detecting the virus when it is present. The second percentage gives the probability of the test returning a positive result when the virus is not really present, a false positive.

Initial reaction to this question might well be to simply say that there was a 5% error in effectiveness of the first test in detecting the disease when it is present and a 4% error due to false positive for a total 9% error for the first test. Similarly, one could add the 8% and the 2% for the second test to obtain an error value of 10%. Taking these numbers at face value, it appears that the first test is more effective.

To test this conclusion we decided to use a tree diagram to analyze the effect of the two tests on different populations of doctors. We chose a hypothetical population size of 100,000 doctors, and we chose two extreme hypothetical percentages of HIV infection of that population.

The total number of correct results from Test A, Case 1 is: ____________.

The total number of incorrect results from Test A, Case 1 is: ____________.

The probability of a correct result from Test A Case 1 is: (correct results / 100,000) = ____________.

The total number of correct results from Test B, Case 1 is: ____________.

The total number of incorrect results from Test B, Case 1 is: ____________.

The probability of a correct result from Test B Case 1 is: (correct results / 100,000) = ____________.
The scenario that 99% of doctors are HIV+ seems highly unlikely.

The total number of correct results from Test A, Case 2 is: ____________.

The total number of incorrect results from Test A, Case 2 is: ____________.

The probability of a correct result from Test A Case 2 is: (correct results / 100,000) = ____________.

The total number of correct results from Test B, Case 2 is: ____________.

The total number of incorrect results from Test B, Case 2 is: ____________.

The probability of a correct result from Test B Case 2 is: (correct results / 100,000) = ____________.

You probably have noticed that in Case 1, Test A is slightly more reliable while in Case 2, Test B is the more accurate test. Somewhere in between 99% HIV+ and 1% HIV+ is a population of doctors for which either test is equally effective. Complete the tree diagrams below, set the probability of correct results equal to one another, and solve for n to determine that percent.

For a population of doctors that is ____________% HIV+, Test A and Test B are equally correct.

In deciding on your recommendation to the AMA House of Delegates, you may want to include some ethical considerations as well as strictly mathematical ones. Is it more serious, in your opinion, for the test to fail to identify doctors who carry the HI virus or to identify as positive those doctors who do not carry the virus?

It is possible that you may want to recommend repeated testing. Doctors who test positive with one test, might be required to be retested. All doctors could be tested twice; in this case which test would be the best? Would both tests have to be positive or is one sufficient to designate a doctor as a carrier of the HI virus? Does it matter which test is given first and which is given as a follow-up?

You might want to make tree diagrams for several more scenarios to test your responses.

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