A diagnostic test for a certain disease is based on counting the number of antig

Question

A diagnostic test for a certain disease is based on counting the number of antigen cells in a certain amount of blood collected from the patient. It is believed that for a healthy patient, the number of antigen cells has a Poisson distribution withmean2. For a patient in the disease group, the number of antigen cells has aPoissondistribution with mean 4. For a randomly chosen patient, the test is declared positive if the number of antigen cells in the blood sample is greater or equal to 3. The test is declared negative otherwise.

a. Compute the sensitivity of this test? (Please clearly define your variables and distributions)

b. Compute the specificity of this test? Would you feel comfortable determining the disease status of an individual solely based on the result of this diagnostic test? Why?

c. A new technology was later introduced for the diagnostic test. For the disease free group, the result of a negative diagnostic test has a normal distribution withmean 10. In the disease group, the diagnostic test results also have a normal distribution with mean 20 and standard deviation 2. Suppose c is the cutoff value for the new test. What should c be so that the test correctly identifies 97.5% of the individual in the disease group. (i.e., find c so that the sensitivity of the test is .975)

d. Unfortunately, we don’t know the value of the standard deviation of the distribution of the negative test results. However, we know that the specificity of the test is .931. Given that information, compute the standard deviation of the distribution of the negative test results.

Explanation / Answer

X no of antigen cells for healthy is POisson 2

Y no of antigen cells for diseased is POisson  4

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Test positive if cells are >=3 is 0.3233 for a healthy individual

Test positive if cells are >=3 is 0.7619 for a diseased individual

This can be tabulated as

Sensitivity = True positive/(True positive+False negative)=0.7619/1 =0.7619

Specificity= True negative/(True negative+false positive)

= 0.6767

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