Example
Research data:

  1. 1% of al members of a population are infected with DIS
  2. a medicine company offers a test and claims that for this test
  • TPR (true positive rate) = 0.999
  • FPR (false positive rate) = 0.005

By definition:
TPR = P(TEST = POS | TRUTH = POS)
FPR = P(TEST = POS | TRUTH = NEG)

The question is, if a person is tested POS, what is the probability this person actually is infected (is POS).
In other words, what is P(TRUTH = POS | TEST = POS)?

Or: what percentage of the people who have tested POS are really POS?
This can be displayed in a tree-diagram (study this tree-diagram carefully, be sure you understand the ‘translation’ from the context to the tree-diagram).

Using this tree-diagram, it is possible to calculate other probabilities as well, e.g. what is the probability a random chosen person is POS and the test is POS?
\(P(TRUTH = POS ~ \cap ~ TEST = POS) = P(TRUTH = POS) \times P(TEST=POS \mid TRUTH = POS) =\) \(=0.01 \times 0.999 = 0.00999\)

Or: what is the probability that a random person is tested positive, i.e. what is P(TEST = POS)
\(P(TEST = POS) = P(TRUTH = POS ~ \cap ~ TEST = POS) + P(TRUTH = NEG ~ \cap ~ TEST = POS) =\) \(= 0.01 \times 0.999 + 0.99 \times 0.005 = 0.01494\)

Now it is also possible to calculate the probability that a random person who is tested POS is really POS.

\(P(TRUTH = POS \mid TEST = POS) = \frac{P(TRUTH = POS ~ \cap ~ TEST = POS)}{P(TEST = POS)} = \frac{0.00999}{0.01494} = 0.6687\)

As can be seen, this test is not really good in detecting a POS case, i.e. for a random selected and tested person, a POS test result doesn’t gives strong evidence that this person is infected.

How about detecting a NEG case?

\(P(TRUTH = NEG \mid TEST = NEG) = \frac{P(TRUTH = NEG ~ \cap ~ TEST = NEG)}{P(TEST = NEG)} = \frac{0.99 \times 0.995}{0.01 \times 0.001 + 0.99 \times 0.995} =\) 0.9999898 \(\approx\) 0.99999

Conclusion: this test is very useful to detect NEGATIVES; for POSITIVES an additional test is needed.
Such kind of a test can be very helpful as a home test, if tested NEG you can be quit sure that the truth is NEG as well; if tested POS, go to a test centre for an additional test.

The probabilities can also be calculated based on the expected numbers in each group in a population of 100,000 people (or another large number). Set op a two-way table based on the given probabilities and calculate the requested probabilities.

TRUTH TOTAL
POSITIVE NEGATIVE TOTAL
TEST POSITIVE
NEGATIVE
TOTAL 1,000 99,000 100,000

\(TOTAL(TRUTH = POS) = 0.01 * 100,000 = 1,000\)
\(TOTAL(TRUTH = NEG) = .....\)

\(TOTAL(TRUTH = POS ~ and ~ TEST = POS) =\)
\(=TOTAL(TRUTH = POS) \times P(TEST = POS \mid TRUTH = POS) = 1,000 \times 0.999 = 999\)

\(TOTAL(TRUTH = NEG ~ and ~ TEST = POS) =\)
\(=TOTAL(TRUTH = NEG) \times P(TEST = POS \mid TRUTH = NEG) = 99,000 \times 0.005 = 495\)

Enter these numbers into the two-way table.

TRUTH TOTAL
POSITIVE NEGATIVE TOTAL
TEST POSITIVE 999 495
NEGATIVE
TOTAL 1,000 99,000 100,000

Now the remaining numbers can be filled in quite easily.

TRUTH TOTAL
POSITIVE NEGATIVE TOTAL
TEST POSITIVE 999 495 1,494
NEGATIVE 1 98,505 98,506
TOTAL 1,000 99,000 100,000

As a final step calculate the required probabilities.

\(P(TRUTH = POS \mid TEST = POS) = \frac{999}{1,494} = 0.6687\)

\(P(TRUTH = NEG \mid TEST = NEG) = \frac{98,505}{98,506} = 0.99999\)