MATHEMATICS FOR THE RELUCTANT

1. MATHEMATICS FOR THE RELUCTANT Deborah Hughes Hallett

2. For a General Education Course: • Use a context that is interesting to students • Ask students to make a decision; not only do calculations

3. For a General Education Course: • First provide evidence: data, numbers, graphs • Then summarize symbolically • Technology should be transparent; often useful to provide evidence

4. The Question: • Should All Pregnant Women Be Tested For HIV? The Mathematics: • Bayes’ Theorem and Rational Functions The Tools: • Spreadsheet and Simulation

5. BACKGROUND ON HIV AND TESTING

9. Reasons Widespread Testing Might Be Beneficial

10. Mother-to-child transmission of HIVEstimated number of children newly infected in the world 800 000 number of cases 700 000 other mother-to-child transmission of HIV 600 000 transmission of HIV through breast-feeding 500 000 400 000 300 000 200 000 100 000 0 1979 1982 1985 1988 1991 1994 1997 projected 98036-E-26 – 15 July 1998

11. The World Bank had estimated that Brazil would have some 1.2 million HIV/AIDS cases by 2000. Based on government statistics and projections, however, the number was about half that amount. Serra added, "The trouble is, the patent for AIDS drugs represents a pure monopoly condition in what's become a global epidemic. CNN August 14, 2001

12. Reasons Widespread Testing Might NOT Be Beneficial

13. "Two weeks ago, a 3-year-old child in Winston Salem, North Carolina, was struck by a car and rushed to a nearby hospital. Because the child's skull had been broken and there was a blood spill, the hospital performed an HIV test. As the traumatized mother was sitting at her child's bedside, a doctor came in and told her the child was HIV-positive. Both parents are negative. The doctor told the mother that she needed to launch an investigation into her entire family and circle of friends because this child had been sexually abused. There was no other way, the doctor said, that the child could be positive. A few days later, the mother demanded a second test. It came back negative. The hospital held a press conference where a remarkable admission was made. In her effort to clear the hospital of any wrongdoing, a hospital spokesperson announced that 'these HIV tests are not reliable; a lot of factors can skew the tests, like fever or pregnancy. Everybody knows that.'" Celia Farber, Impression Magazine, June 21, 1999 Reported by Christine Maggiore: Is the “AIDS test” Accurate? (http://healtoronto.com/testcm.html)

14. In regards to screening low-risk populations Xin M. Tu of Harvard School of Public Health estimated that 90 percent of positive tests are in fact false.(1) Although this may not be of much concern in certain situations, such as screening donated blood (where positive units are simply discarded), he notes, "Falsely labeling individuals applying for marriage licenses, pregnant women, health care workers and patients admitted to the hospital as carrying the virus is certainly irresponsible and can have an enormous psychological and social impact on the individuals.“ Christine Johnson: Mass HIV Testing: A Disaster in the Making (http://healtoronto.com/masstest.html)

15. Former Senator Lawton Chiles of Florida, at an AIDS conference in 1987, told of a tragic example from the early days of blood screening in Florida. Of 22 blood donors who were told they were HIV-positive by the ELISA test, seven committed suicide. Reported from a textbook (http://healtoronto.com/pospre.html)

16. “HIV tests are notoriously unreliable in Africa. A 1994 study published in the Journal of Infectious Diseases concluded that HIV tests were useless in central Africa, where the microbes responsible for tuberculosis, malaria and leprosy were so prevalent that they registered over 70% false positive." Sacramento Bee, October 30, 1994 Reported by Christine Maggiore: Is the “AIDS test” Accurate? (http://healtoronto.com/testcm.html)

17. TESTING • What is our HIV testing policy? • What should our HIV testing policy be?

18. TESTING: More Specifically • In 1996, the American Medical Association approved a recommendation for the mandatory testing of all pregnant women. Is this a good policy? • Should a home HIV test be developed?

19. Test Parameters • Sensitivity = Percent infected correctly identified • Specificity = Percent not infected correctly identified • Current HIV tests have sensitivity of 99.9% and specificity of 99.6%. From Center for Disease Control and Medical Univ S. Carolina www.cdc.gov/hiv/pubs/rt/rapidct.htmAccessed July 2001. www.musc.edu/dc/icrebm/sensitivity.html

20. Predictive Value as a Function of Incidence of InfectionSensitivity = a = 0.999, Specificity = b = 0.996, Incidence = x

21. Bayes’ Theorem

22. Other contexts involving the same mathematics: • Mammogram: 85%, 90%, 2.5% • Lie detector tests • DNA Testing • Airport screening • Search for missing persons

23. Questions you might have: • Couldn’t Bayes’ Theorem have been done sooner? • Is the data correct? • Why a spreadsheet and not a more powerful program? • Why not show probabilities instead of frequencies in the two-way table? (Then we don’t have to assume a population size.)

24. Article from Science, vol. 290, Dec 22, 2000: “Communicating Statistical Information” Describes research showing that expressing statistics as natural frequencies improves the statistical thinking of doctors and lawyers.