Many diagnostic tests for detecting the presence of disease are based on biomarkers; broadly speaking, a biomarker is any biological indicator that can be measured, such as blood pressure or fasting blood glucose level.

Consider a diagnostic test for a hypothetical disease based on measuring the amount of a certain biomarker present in blood. High levels of the biomarker are often found in individuals with the disease, but a number of non-disease conditions can also cause high levels of the biomarker. Individuals without the disease have biomarker levels that are normally distributed with mean 1.6 ng/mL (nanograms per milliliter of blood), and standard deviation 0.50 ng/mL. Individuals with the disease have biomarker levels that are normally distributed with mean 5 ng/mL and standard deviation 1.2 ng/mL. Values of 2.5 ng/mL and higher constitute a positive test result.

a) Compute the accuracy of the test for those who have the disease and the accuracy of the test for those who do not have the disease.

b) In a population where 6% of individuals are thought to have the hypothetical disease, calculate the probability that an individual who tests positive has the disease.

c) In order to account for lab error, individuals who test positive are asked to return for another test at a later time. What is the probability that an individual who tests positive on two tests actually does not have the disease? If you make any assumptions in your calculation, be sure to state them and comment on whether they are reasonable.

d) Clinicians would like to increase the probability from part b) by changing the cutoff value that constitutes a positive test result. Suggest a reasonable new cutoff value x, such that values of x ng/mL and higher qualify as a positive test result. Explain the reasoning behind your answer.