Applet Exercise Refer to Exercises 2.125 and 2.126.

QUESTION:

Problem 127E

Applet Exercise Refer to Exercises 2.125 and 2.126. Suppose now that the disease is not particularly rare and occurs with probability .4 .

a If, as in Exercise 2.125, a test has sensitivity = specificity = .90, what is the positive predictive value of the test?

b Why is the value of the positive predictive value of the test so much higher that the value obtained in Exercise 2.125? [Hint: Compare the size of the numerator and the denominator used in the fraction that yields the value of the positive predictive value.]

c If the specificity of the test remains .90, can the sensitivity of the test be adjusted to obtain a positive predictive value above .87?

d If the sensitivity remains at .90, can the specificity be adjusted to obtain a positive predictive value above .95? How?

e The developers of a diagnostic test want the test to have a high positive predictive value.

Based on your calculations in previous parts of this problem and in Exercise 2.126, is the

value of the specificity more or less critical when developing a test for a rarer disease?

Reference

Applet Exercise Refer to Exercise 2.125. The probability that the test detects the disease given that the patient has the disease is called the sensitivity of the test. The specificity of the test is the probability that the test indicates no disease given that the patient is disease free. The positive predictive value of the test is the probability that the patient has the disease given that the test indicates that the disease is present. In Exercise 2.125, the disease in question was relatively rare, occurring with probability .01, and the test described has sensitivity = specificity = .90 and positive predictive value = .0833.

a In an effort to increase the positive predictive value of the test, the sensitivity was increased to .95 and the specificity remained at .90, what is the positive predictive value of the “improved” test?

b Still not satisfied with the positive predictive value of the procedure, the sensitivity of the test is increased to .999. What is the positive predictive value of the (now twice) modified test if the specificity stays at .90?

c Look carefully at the various numbers that were used to compute the positive predictive value of the tests. Why are all of the positive predictive values so small? [Hint: Compare the size of the numerator and the denominator used in the fraction that yields the value of the positive predictive value. Why is the denominator so (relatively) large?]

d The proportion of individuals with the disease is not subject to our control. If the sensitivity of the test is .90, is it possible that the positive predictive value of the test can be increased to a value above .5? How? [Hint: Consider improving the specificity of the test.]

e Based on the results of your calculations in the previous parts, if the disease in question is relatively rare, how can the positive predictive value of a diagnostic test be significantly increased?

Reference

A diagnostic test for a disease is such that it (correctly) detects the disease in 90% of the individuals who actually have the disease. Also, if a person does not have the disease, the test will report that he or she does not have it with probability .9. Only 1% of the population has the disease in question. If a person is chosen at random from the population and the diagnostic test indicates that she has the disease, what is the conditional probability that she does, in fact, have the disease? Are you surprised by the answer? Would you call this diagnostic test reliable?