Statistics; conditional probability

Question

Tests for tuberculosis like all other diagnostic tests are not perfect. QFT-G is one of such tests for tuberculosis. Suppose that for the population of adults that is taking the test, 5% have tuberculosis. The test correctly identifies 74.6% of the time adults with a tuberculosis and correctly identifies those without tuberculosis 76.53% of the time. Suppose that POS stands for the test gives a positive result and S means that the adult really has tuberculosis. What is the probability of an adult getting a NEG result and truly having tuberculosis?

Answer 1

Hi Kai,

This is a great and very important question!

Firstly, the terms to consider:

1. Sensitivity: 74.6%
2. Specificity: 76.53%
3. Prevalence: 5%

Assume 100,000 adults are tested:

1. 5,000 have tuberculosis (5% of 100,000)
2. 95,000 don't have tuberculosis (95% of 100,000)

Now apply the sensitivity and specificity:

1. Correctly identified with tuberculosis: 0.746 * 5,000 = 3,730
2. Correctly identified without tuberculosis: 0.7653 * 95,000 = 72,704

The test will miss:

1. With tuberculosis: 5,000 - 3,730 = 1,270
2. Without tuberculosis: 95,000 - 72,704 = 22,296

You want to know the probability of getting a NEG result but truly having tuberculosis. This is the 1,270 individuals with tuberculosis who get a NEG result.

The total number of people who get a NEG result is 1,270 (missed cases) + 72,704 (correctly identified without tuberculosis) = 73,974.

The probability is then:

1. P = 1,270 / 73,974
2. P = 0.0172 or 1.72%

Thank you,

Benjamin M.

p.s. If this answer was helpful for you. I would greatly appreciate your feedback as I am new to this platform. :)