How to use the Calculator

1. To use this program for first time, work through the following example.

Suppose that 5% of people of your age and heredity have cancer.
Suppose that a blood test has been developed that correctly gives a positive test result in 80% of people with cancer, and gives a false positive in 20% of the cases of people without cancer. Suppose you take the test, and it is positive. What is the probability that you actually have cancer, given the positive test result?

2. First, you must identify the Hypothesis, H, the Datum, D, and the probabilities of the Hypothesis prior to the test, and the hit rate and false alarm rates of the test.

• H = the hypothesis; in this case H is the hypothesis that you have cancer, and H' is the hypothesis that you do not.
  
• D = the datum; in this case D is the positive test result.
  
• P(H) is the prior probability that you have cancer, which was given in the problem as .05.
  
• P(D|H) is the probability of a positive test result GIVEN that you have cancer. This is also called the HIT RATE, and was given in the problem as .80.
• P(D|H') is the probability of a positive test result GIVEN that you do not have cancer. This is also called the FALSE ALARM rate, and was given as .20.

3. Press the Clear Button. Note that probability is selected at the top of the second column.
In the first row, enter .05 for P(H).
In the second row, enter .80 for P(D|H).
In the third row, enter .20 for P(D|H').

4. Do not enter anything in the column for odds. When probability is selected, the odds are calculated for you. You should also not enter anything for the answer, P(H|D).

5. Press the compute button, and the answer will be computed in both probability and odds. In this example, the posterior probability given a positive test result is .174. In other words, the odds are almost 5:1 that you do NOT have cancer. You may decide not to undergo chemotherapy unless another test is positive, especially if the chemotherapy is dangerous, painful, and expensive.

• P(H|D) is the probability that you have cancer, given that the test was positive. This is also called the posterior probability, and it is what you computed. In this case it was .174.
• Bayes Theorem states P(H|D) = [P(D|H)P(H)]/[P(D|H)P(H)+P(D|H')(1-P(H))].
• W₀ is the prior odds of cancer, P(H)/(1-P(H)).
• LR is the likelihood ratio (or evidence ratio, sometimes called diagnosticity), P(D|H)/(P(D|H').
• W₁ is the posterior odds of cancer, given a positive test result, P(H|D)/(1-P(H|D).
• In odds form, Bayes Theorem can be written: W₁ = W₀*LR.

6. To do the same problem in terms of odds, click the Clear button.
Then click the radio button for ODDS.
Next, enter the prior odds [PH/(1-PH), in this case, .0526].
Next, enter the Liklihood ratio of the data given the hypotheses [P(D|H)/(P(D|H'), in this case, 4].
Press the compute button, and the solution is given in odds and probability.
Given only LR, the likelihood ratio (or diagnosticity of the test) does not uniquely identify P(H|D) and P(H|D').