Background InformationPart 1Clinical TrialsTotalsTotalsThe high percents for sensitivity and specificity appear to indicate that the ELISA is an excellent test. Let’s investigate this a bit further in Part 2.Part 2General PopulationTotalsTotalsPart 3Blood Donor PoolTotalsTotalsI.V. Drug UsersTotalsTotalsAccuracy and Apparent Accuracy in Medical TestingVersion 3.141Background InformationToday, testing for the presence or absence of a specific disease, medical condition, or illegal drug is common. The results of these tests are never as simple as they appear to be on many TV shows and movies. As patients become more and more critical consumers of medical information from their doctors, they must be aware of the quantitative and statistical reasoning that lurks behind the reported facts and figures. For example, if a medical test is reported as 99.9% accurate and you get a “positive” result, what is the chance that you have that medical condition?Medical researchers want to develop simpler, less expensive tests and screening tools for existing medical conditions. However, they do not want to sacrifice reliability when doing so. When a new test is developed, researchers need to compare its reliability with the existing “gold standard” test. In this activity, you can assume that the “gold standard” is perfect in detecting the medical condition under consideration.In this activity, you will explore the results of medical tests. The mathematics includes basic proportional reasoning, yet the real–world context is complex. Before beginning this activity, some definitions are needed.Definitionsfalse positive (FP): when a patient receives a positive test result for a disease but the patient does not have the diseasefalse negative (FN): when a patient receives a negative test result for a disease when the patient actually does have the diseasetrue positive (TP): when a patient receives a correct positive test result true negative (TN): when a patient receives a correct negative test result sensitivity: the probability that a test produces a positive test result when the patient does have the disease and this result is correctspecificity: the probability that a test produces a negative test result when the patient does not have the disease and this result is correctMath 101, Student Pages, Medical Tests, Page 1Math 101, Student Pages, Medical Tests, Page 2Part 1ELISA, or Enzyme-Linked ImmunoSorbent Assay, is a biochemical technique to detect the presence of certain antibodies. An ELISA is developed to diagnose HIV infections. This new procedure will be compared to the “gold standard” test, the Western Blot, which is a time–consuming test for HIV.Determining the ELISA’s sensitivity and specificity for 20,000 patients:a. Assume 10,000 patients that tested positive by Western Blot (the gold standard) were tested with the new ELISA and 9990 were found to be positive. These are true positives. Similarly, ELISA was used to test serum from 10,000 patients who were found by Western Blot not to be infected with HIV. Of these HIV-negative patients, ELISA returned 9990 negative results (true negatives) and 10 positives (false positives). A two-way table is a very succinct way oforganizing this information. Of the 20,000 patients tested with ELISA, fill in the number of patients in each of the four categories: true positive (TP), false positive (FP), false negative (FN), and true negative (TN)Clinical TrialsHIV–positive HIV–negativeTotalsELISA positive(TP) (FP)ELISA negative(FN) (TN)Totalsb. Fill in the cells for Totals.c. What percent of all the patients who tested positive were correctly diagnosed as HIV-positive? This is the sensitivity of ELISA.Math 101, Student Pages, Medical Tests, Page 3d. Refer to the definition box. Determine the specificity of ELISA.The high percents for sensitivity and specificity appear to indicate that the ELISA is an excellent test. Let’s investigate this a bit further in Part 2..Part 2Applying the ELISA to a different population:a. Let’s administer ELISA to a million people where 1% are believed to be infected with HIV. First, fill in cells labeled (p) and (n) by determining how many of the one million people are actually HIV-positive and HIV-negative. General PopulationHIV–positive HIV–negativeTotalsELISA positive(TP) (FP)ELISA negative(FN) (TN)Totals(p) (n)b. Now, use the sensitivity rates from Part 1 to fill in the cells labeled (TP) and (FN). c. Now, use the specificity rates from Part 1 to fill in the remaining cells.Investigating the ELISA’s false readings for these 1,000,000 patients:a. What percent of patients who tested positive were not HIV-positive? (This represents the percent of the population who are told they have HIV when in fact they do not.)Math 101, Student Pages, Medical Tests, Page 4b. Is your answer to a higher, lower, or about what you would have expected?c. What percent of patients who tested negative were HIV-positive? (This represents the percent of the population who are incorrectly told they are HIV-free.)d. Is your answer to d higher, lower, or about what you would have expected?Math 101, Student Pages, Medical Tests, Page 5Part 3Let’s investigate two other situations where the tests still have a sensitivity and specificity of 99.9%.Investigating the ELISA’s false readings for 1,000,000 patients from a blood donor pool:Case 1: Assume we administer the ELISA to one million patients who are part of a real blood donor pool. Patients in this population have already been screened for HIV risk factors before theyare even allowed to donate blood, so the prevalence of HIV in this population is closer to 0.1% (not the 1% used above). a. Create the appropriate two-way table. Remember: The sum of the row labeled Totals and the sum of the column labeled Totals should both be 1,000,000 (one million).Blood Donor PoolHIV–positive HIV–negativeTotalsELISA positive(TP) (FP)ELISA negative(FN) (TN)Totals(p) (n)b. What percent of the positive readings are false?c. What percent of the negative readings are false?d. What is the only factor that has changed in this case, leading to these different percents?Math 101, Student Pages, Medical Tests, Page 6Investigating the ELISA’s false readings for 1,000,000 patients from an “at-risk” pool:We will still assume that the test has a sensitivity and specificity of 99.9%.Case 2: The other case to examine is a drug–rehabilitation unit for I.V. drug users. In this case,