mathematics for the reluctant

MATHEMATICS FOR MATHEMATICS FOR THE RELUCTANTTHE RELUCTANT

Deborah Hughes HallettDeborah Hughes Hallett

For a General Education Course:For a General Education Course:

• Use a context that is Use a context that is interesting to studentsinteresting to students

• Ask students to make a Ask students to make a decision; not only do decision; not only do calculationscalculations

For a General Education Course:For a General Education Course:

• First provide evidence: data, First provide evidence: data, numbers, graphsnumbers, graphs

• Then summarize symbolicallyThen summarize symbolically• Technology should be Technology should be

transparent; often useful to transparent; often useful to provide evidenceprovide evidence

The Question:The Question:• Should All Pregnant Women Be Tested For Should All Pregnant Women Be Tested For

HIV?HIV?

The Mathematics:The Mathematics:• Bayes’ Theorem and Rational FunctionsBayes’ Theorem and Rational Functions

The Tools:The Tools:• Spreadsheet and SimulationSpreadsheet and Simulation

BACKGROUNDBACKGROUND ON HIV AND TESTING ON HIV AND TESTING

Adults and children estimated to be living Adults and children estimated to be living with HIV/AIDS as of end 1998with HIV/AIDS as of end 1998

Western Europe500 000500 000

North Africa& Middle East210 000210 000

sub-Saharan Africa

22.5 million22.5 million

Eastern Europe & Central Asia270 000270 000

South & South-East Asia

6.7 million6.7 millionAustralia & New Zealand

12 00012 000

North America890 000890 000

Caribbean330 000330 000

Latin America1.4 million1.4 million

Total: 33.4 millionTotal: 33.4 million99000 -E-1 – 15 December 1998

East Asia & Pacific560 000560 000

A global view of HIV infection30 million adults living with HIV/AIDS as of end 1997

Estimated percentageof adults (15–49) infected with HIV

8.00% – 32.00%2.00% – 8.00%0.50% – 2.00%0.13% – 0.50%0.03% – 0.13% 0% – 0.03% not available

98036 -E-11 – 15 July 1998

Proportional increase in country HIV prevalence rates between 1994 and 1997

Over 100% (27)10% to 100% (47)0.01% to 10% (18)No growth (44)No 1997 data (36)

98036-E-15 – 15 July 1998

Reasons Widespread Testing Reasons Widespread Testing Might Be BeneficialMight Be Beneficial

Mother-to-child transmission of HIVMother-to-child transmission of HIVEstimated number of children newly infected in the worldEstimated number of children newly infected in the world

0

100 000

200 000

300 000

400 000

500 000

600 000

700 000

800 000

1979 1982 1985 1988 1991 1994 1997 projected

transmission of HIV through breast-feedingtransmission of HIV through breast-feedingother mother-to-child transmission of HIV

number of cases

98036-E-26 – 15 July 1998

The World Bank had estimated that Brazil The World Bank had estimated that Brazil would have some 1.2 million HIV/AIDS cases by would have some 1.2 million HIV/AIDS cases by 2000. Based on government statistics and 2000. Based on government statistics and projections, however, the number was about half projections, however, the number was about half that amount.that amount.

Serra added, "The trouble is, the patent for Serra added, "The trouble is, the patent for AIDS drugs represents a pure monopoly AIDS drugs represents a pure monopoly condition in what's become a global epidemic.condition in what's become a global epidemic.CNN August 14, 2001CNN August 14, 2001

Reasons Widespread Testing Reasons Widespread Testing Might NOT Be BeneficialMight NOT Be Beneficial

"Two weeks ago, a 3-year-old child in Winston Salem, North Carolina, was struck "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 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 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 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 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 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 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 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 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 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 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.'" factors can skew the tests, like fever or pregnancy. Everybody knows that.'"

   Celia Farber, Impression Magazine, June 21, 1999 Celia Farber, Impression Magazine, June 21, 1999

Reported by Christine Maggiore: Is the “AIDS test” Accurate? Reported by Christine Maggiore: Is the “AIDS test” Accurate? ((http://healtoronto.com/testcm.html))

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

Christine Johnson: Mass HIV Testing: A Disaster in the Making Christine Johnson: Mass HIV Testing: A Disaster in the Making ((http://healtoronto.com/masstest.html))

Former Senator Lawton Chiles of Florida, Former Senator Lawton Chiles of Florida, at an AIDS conference in 1987, told of a at an AIDS conference in 1987, told of a tragic example from the early days of tragic example from the early days of blood screening in Florida. Of 22 blood blood screening in Florida. Of 22 blood donors who were told they were HIV-donors who were told they were HIV-positive by the ELISA test, seven positive by the ELISA test, seven committed suicide.committed suicide.

  Reported from a textbook Reported from a textbook ((http://healtoronto.com/pospre.html))

““HIV tests are notoriously unreliable in Africa. HIV tests are notoriously unreliable in Africa. A 1994 study published in the Journal of A 1994 study published in the Journal of Infectious Diseases concluded that HIV tests Infectious Diseases concluded that HIV tests were useless in central Africa, where the were useless in central Africa, where the microbes responsible for tuberculosis, malaria microbes responsible for tuberculosis, malaria and leprosy were so prevalent that they and leprosy were so prevalent that they registered over 70% false positive." registered over 70% false positive."

Sacramento Bee, October 30, 1994Sacramento Bee, October 30, 1994Reported by Christine Maggiore: Is the “AIDS test” Accurate? Reported by Christine Maggiore: Is the “AIDS test” Accurate? ((http://healtoronto.com/testcm.html))

TESTINGTESTING

• What is our HIV testing What is our HIV testing policy?policy?

• What should our HIV testing What should our HIV testing policy be?policy be?

TESTING: More SpecificallyTESTING: More Specifically• In 1996, the American Medical In 1996, the American Medical

Association approved a Association approved a recommendation for the recommendation for the mandatory testing of all pregnant mandatory testing of all pregnant women. Is this a good policy?women. Is this a good policy?

• Should a home HIV test be Should a home HIV test be developeddeveloped??

Test ParametersTest Parameters• Sensitivity Sensitivity = Percent infected correctly = Percent infected correctly

identifiedidentified• Specificity Specificity = Percent not infected = Percent not infected

correctly identifiedcorrectly identified• Current HIV tests have sensitivity ofCurrent HIV tests have sensitivity of

99.9% 99.9% and specificity ofand specificity of 99.6%. 99.6%.From Center for Disease Control and Medical Univ S. Carolina From Center for Disease Control and Medical Univ S. Carolina

www.cdc.gov/hiv/pubs/rt/rapidct.htm Accessed July 2001. Accessed July 2001. www.musc.edu/dc/icrebm/sensitivity.htmlwww.musc.edu/dc/icrebm/sensitivity.html

Predictive Value as a Function Predictive Value as a Function of Incidence of Infectionof Incidence of Infection

Sensitivity = a = 0.999, Specificity = b = 0.996, Incidence = x

4.0995.00.999 Value Predictive Pos

)1(100)1( Value Predictive Pos

+=

−++−=

xx

xbabax

Bayes’ TheoremBayes’ Theorem

)()|()()|()()|()|(

)()()(

FP TPTPPred Pos

NIPNITPIPITPIPITPTIP

NITPITPITP

⋅+⋅⋅=

∩+∩∩=

+=

++

++

++

+

Other contexts involving the Other contexts involving the same mathematics:same mathematics:

• Mammogram: Mammogram: 85%, 90%, 2.5%85%, 90%, 2.5%• Lie detector testsLie detector tests• DNA TestingDNA Testing• Airport screening Airport screening • Search for missing personsSearch for missing persons

Questions you might have:Questions you might have:

• Couldn’t Bayes’ Theorem have been done Couldn’t Bayes’ Theorem have been done sooner?sooner?

• Is the data correct?Is the data correct?• Why a spreadsheet and not a more powerful Why a spreadsheet and not a more powerful

program?program?• Why not show probabilities instead of Why not show probabilities instead of

frequencies in the two-way table? (Then we frequencies in the two-way table? (Then we don’t have to assume a population size.)don’t have to assume a population size.)

Article from Article from ScienceScience, vol. 290, Dec 22, 2000:, vol. 290, Dec 22, 2000:

““Communicating Statistical Communicating Statistical Information”Information”

Describes research showing that expressing Describes research showing that expressing statistics as natural frequencies improves the statistics as natural frequencies improves the statistical thinking of doctors and lawyers.statistical thinking of doctors and lawyers.