Decision of Uncertainty One of the Important

¶ … Decision of Uncertainty

One of the important features of medicine is diagnostic testing. The companies that produce diagnostic tests are multi-billion dollar companies. Not only is the effectiveness of their tests important for their bottom line, but it is also important for the health and well-being of those who rely on the tests for their health. One such test important diagnostic test is the test for colorectal cancer known as the fecal occult blood test. When analyzing the effectiveness of such tests, we consider the sensitivity and the specificity of the test. In this case, the sensitivity of the test is the proportion of results that correctly identify people with colorectal cancer. In symbolic terms, P (testing positive | have colorectal cancer). The specificity is the proportion of tests that correctly identify those people who do not have colorectal cancer. Symbolically, it is P (testing negative | do not have colorectal cancer).

In reality, we are very interested in the probability having the disease given that you tested positive. That is, we are interested in the false alarm rate. This is important from a treatment perspective, but also from a business perspective. For example, a test that identified everyone who took it has having colorectal cancer would have perfect sensitivity, but it would have a very high false alarm rate. Similarly, a test that identifies everyone has being healthy (i.e., does not have colorectal cancer) would have perfect specificity, but it would fail to diagnose everyone who has the disease.

So suppose a patient goes to the doctor for colorectal cancer screening. Suppose further that they test positive for the disease. What is the probability that they have colorectal cancer? Bayes' theorem can help us to determine this result. First we need some information. Based on "Sensitivity, Specificity, and Predictive Value of Fecal Occult Blood Testing (Hem occult II) for Colorectal Neoplasia in Symptomatic Patients: A Prospective Study with Total Colonoscopy" by Yaron Niv, M.D. In The American Journal of Gastroenterology Volume 90 Issue 11, Pages 1974 -- 1977, "The sensitivity, specificity for colorectal cancer were 69.2%, 73.2%, respectively." Finally, we need the overall incidence of colorectal cancer. The incidence of colorectal cancer is 1 in 1834 or 0.0005.

So we can use Bayes Theorem to determine the probability that we're interested in. This is as follows. Let us define event A as having colorectal cancer. Let us define + as testing positive for colorectal cancer. Then the calculations can be written out as follows:

P (A | +) = P (+ | A)*P (A) / (P (+ | A)*P (A) + P (+ | not A)*P (not A))

= 0.692*.005 / (0.692*.005 + (1-0.732)*0.995)

= 0.0128

So we see that a positive test result indicates only a very small probability of actually having the disease. This result might lead us to ask what use the test is. A similar calculation answers the question. What is the probability that you don't have colorectal cancer, given that you test negative. The calculation is similar:

P (not A | not +) = P (not + | not A)*P (not A) / (P (not + | not A)*P (not A) + P (not + | A)*P (A))

=0.732*.995 / (0.732*.995 + (1-.692)*.005)

=0.9979