Accurately distinguishing lung cancer from benign lung disease remains challeng- ing, even with the use of imaging scans; computed tomography (CT) scans are known to have high sensitivity but poor specificity for lung cancer diagnosis. Tumor markers, molecules produced by a tumor associated with a cancer or by the body in response to a cancer, may be useful for clinical diagnosis.

Consider two tumor markers for lung cancer, CYFRA 21-1 and CEA, which tend to be ele- vated in patients with lung cancer relative to those with benign lung disease. A study was conducted on patients with known lung cancer status to assess how these tumor markers could be used for clinical diagnosis. The study team observed that in patients with lung cancer, CYFRA 21-1 is normally distributed with mean 4.7 ng/mL and standard deviation 9.2 ng/mL while CEA is normally distributed with mean 5.9 ng/mL and standard deviation 19.8 ng/mL. In patients with benign lung disease, CYFRA 21-1 is normally distributed with mean 1.6 ng/mL and standard deviation 4.3 ng/mL while CEA is normally distributed with mean 2.2 ng/mL and standard deviation 5.3 ng/mL.

Use the data from this study to answer the following questions.

i. Compute the sensitivity and specificity of a diagnosis test based on classifying patients with CYFRA 21-1 level greater than 3.3 ng/mL as having lung cancer.

ii. Compute the sensitivity and specificity of a diagnosis test based on classifying patients with CEA level greater than 5.0 ng/mL as having lung cancer.

iii. Explain the reasoning behind why a diagnostic test with low sensitivity may not be recommended for use in the general population but appropriate for use in high-risk groups, such as patients presenting with several risk factors or symptoms strongly pre- dictive of lung cancer. Use language accessible to someone who has not taken a statis- tics course. Limit your answer to no more than six sentences.

iv. Suppose a high-risk patient is tested for elevated CYFRA 21-1 level and found to have CYFRA 21-1 level below the cutoff in part i. Explain whether it seems reasonable to rule out lung cancer for this patient based on this test result and the test features computed in part i. Limit your answer to no more than six sentences, referencing numerical results as necessary.

The study team is interested in whether a diagnostic test based on both CYFRA 21-1 and CEA is an improvement over tests based solely on one of the markers. Suppose that a patient is classified as having lung cancer if at least one of the markers is above the cutoffs used in parts i. and ii.; i.e., a patient tests positive for lung cancer if CYFRA 21-1 level is greater than 3.3 ng/mL, CEA level is greater than 5.0 ng/mL, or both are elevated.

v. Compute the sensitivity and specificity of this diagnostic test. State any assumptions necessary to make the calculation and comment on whether those assumptions seem reasonable.

vi. Does the diagnostic test based on both markers represent an improvement over the tests in parts i. and ii. for use in high-risk patients? Explain your answer, referencing numerical results to support your reasoning.