Extra Problem 1.

  Recall that we defined D+ as the event a randomly selected person is positive for the disease of interest, and T+ the event the person tests positive. The events D- and T- are the corresponding complementary events of negative for disease and testing negative, respectively.

The table below gives the results on an experiment on 100,000 individuals to assess the accuracy of the Enzyme-linking ImmunoAssay (EIA) test for HIV antibodies:

True State

Test Results HIV negative HIV positive

EIA Positive 1,000 30

EIA Negative 98,969 1

Use these data to answer the following questions.

(a) Estimate the following quantities from the data:

1.

Sensitivity $\; = \;$ P(T+|D+)

2.

Specificity $\; = \;$ P(T-|D-)

3.

Positive Predictive Power $\; = \;$ P(D+|T+)

4.

Negative Predictive Power $\; = \;$ P(D-|T-)

5.

Prevalence $\; = \;$ P(D+) .

(b) Suppose that we apply the test in a population where surveys suggest the prevalence is about 10%. Assuming that the sensitivity and specificity do not change, what will be the positive and negative predictive powers?