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 Europe
500 000500 000North Africa
& Middle East
210 000210 000sub-Saharan
Africa
22.5 million22.5 million
Eastern Europe & Central Asia
270 000270 000
South & South-East Asia
6.7 million6.7 million
Australia & New Zealand
12 00012 000
North America
890 000890 000Caribbean
330 000330 000
Latin America
1.4 million1.4 million
Total: 33.4 millionTotal: 33.4 million99000 -E -1 – 15 December 1998
East Asia & Pacific
560 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-feeding
other 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 would have some 1.2 million HIV/AIDS cases by 2000. Based on government statistics and by 2000. Based on government statistics and projections, however, the number was about projections, however, the number was about half that amount.half 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 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 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 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 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 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, 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 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 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 conference where a remarkable admission was made. In her effort to clear the hospital of any wrongdoing, a hospital spokesperson announced that 'these HIV 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. tests are not reliable; a lot of factors can skew the tests, like fever or pregnancy. Everybody knows that.'" 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 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 of much concern in certain situations, such as screening donated blood (where positive units are simply discarded), donated blood (where positive units are simply discarded), he notes, "Falsely labeling individuals applying for marriage he notes, "Falsely labeling individuals applying for marriage licenses, pregnant women, health care workers and patients licenses, pregnant women, health care workers and patients admitted to the hospital as carrying the virus is certainly admitted to the hospital as carrying the virus is certainly irresponsible and can have an enormous psychological and irresponsible and can have an enormous psychological and social impact on the individuals.“social impact on 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 Former Senator Lawton Chiles of Florida, at an AIDS conference in 1987, Florida, at an AIDS conference in 1987, told of a tragic example from the early told of a tragic example from the early days of blood screening in Florida. Of 22 days of blood screening in Florida. Of 22 blood donors who were told they were blood donors who were told they were HIV-positive by the ELISA test, seven HIV-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” Reported by Christine Maggiore: Is the “AIDS test” Accurate? 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 = Percent infected
correctly identifiedcorrectly identified• 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 TP
TPPred Pos
NIPNITPIPITP
IPITPTIP
NITPITP
ITP
⋅+⋅⋅=
∩+∩∩=
+=
++
++
++
+
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 statistics as natural frequencies improves the statistical thinking of doctors and the statistical thinking of doctors and lawyers.lawyers.