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8. Prostate cancer is the most common form of cancer found in men. The probability of developing prostate cancer over a lifetime is 16%. Many physicians routinely perform a PSA test, particularly for men over age 50. PSA is a protein produced only by the prostate gland and thus is fairly easy to detect Normally, men have PSA levels between 0 and 4 mg/ml. Readings above 4 may be considered high and potentially indicative of cancer. Studies have shown that the PSA test is not very accurate. In fact the probability of having an elevated PSA level given that the man does not have cancer (false positive) is 0.135. If the man does have cancer, the probability of a normal PSA level (false negative) is almost 0.300. If a physician concludes that the PSA is high, a biopsy is performed. Besides the concerns and health needs of the men, there are also financial costs. The cost of blood test is approximately $50. The cost of the biopsy is approximately $1,000. A false positive PSA will need an unnecessary biopsy. Because the PSA test is so inaccurate, some private and public medical plans do not pay for it. Suppose you are a manager in a medical insurance company and must decide on guidelines for whom should be routinely screened for prostate cancer. An analysis of prostate cancer incidence and age produces the following table of probabilities. (The probability for men under 40 developing prostate cancer is less than 0.0001, small enough to treat it as 0.) Probability of Developing Size of Population of Prostate Cancer men per age category Age 40-49 50-59 60-69 70 and older 0.010 0.022 0.046 0.079 1000000 1000000 1000000 1000000 Assume that a man in each of the age categories undergoes a PSA test with a positive result. a) Calculate the probability that each man actually has a prostate cancer and the probability that he does not. Enter your answers in the table below. Probabilities Given a Positive PSA Test Has Prostate Cancer Does not Have Prostate Cancer Age 40-49 50-59 60-69 70 and older

Answer:


Probability of false positive PSA test = 0.135

No cancer Age 40-49 = 0.135x(1-0.01) = 0.133

50-59 = 0.135x(1-0.022) = 0.132

60-69 = 0.135x(1-0.046) =0.128

70+ = 0.135x(1-0.079) = 0.124

 

Hence probability of having cancer=

40-49 = 0.867

50-59 = 0.868

60-69 = 0.872

70+ = 0.876

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