Homework: Decision Analysis
                              
            (Introduction to Medical Informatics)
         (http://www.cpmc.columbia.edu/edu/textbook)
                              

1. Prostate-specific antigen (PSA) is a common test for
     prostate cancer. It tells you the concentration in the
     blood of a protein that is produced by the cancer;
     unfortunately, the same protein is produced by other
     conditions. There is concern in the medical community
     that the PSA is being overused and does not really give
     that much information. Two reports were reviewed to
     determine the sensitivity and specificity of the test:
     
     report      cutoff(ng/dl)    sensitivity    specificity
     
        A           2.5               91%            32%
                   15.0               50%            85%
     
        B           3.6               78%            92%
     
     The prevalence (prior probability) of prostate cancer
     in 60 to 75 year old men is 1.4%.
     
     a. find the posterior probability of having cancer
          given a positive test (ie, the patient's value is
          greater than the cutoff) for each of the three
          sets of values.
     
     b. draw an ROC graph with two "curves," where there are
          two points for curve A, and only one point for
          curve B
     
     c. the two curves are not consistent; which report (A
          or B) makes the PSA seem like a better test?
     
     d. what might account for the discrepancy?

2. Your patient has a positive PSA test. A prostate biopsy
     is done and shows prostate cancer. Should you operate?
     If you do not operate, then the patient will eventually
     die of cancer. If you do operate, then the patient may
     die at surgery. If the patient survives surgery, then
     he may be cured, or he may have incurable cancer and
     die eventually anyway. Whether he is cured or has
     incurable cancer, the operation may affect his ability
     to urinate: he may be able to urinate normally, or he
     may be incontinent.

     life expectancy
          cancer without operation = 7 years
          death at operation = 0 years
          operation is a cure = 15 years
          operation does not cure = 3 years
     
     the patient values living 10 years with incontinence to
          be equal to living 9 years with normal urination
          (eg, living 15 years with incontinence = living
          13.5 healthy years)
     
     probabilities
          P(death at surgery) = 0.05
          P(cure | patient survives surgery) = 0.7
          P(incontinence | patient survives surgery) = 0.4
     
     a. draw the decision tree for this problem
     b. what is expected value of no surgery and of surgery
     c. which is the preferred choice?
     d. what if the patient said 10 incontinent years = 
          7 normal years