Question

A Screening test has a sensitivity of 0.89 and a false-positive rate of 0.01. The test is used in a population that has a disease prevalence of 0.002. Given a positive result, find the probability of having the disease.

• A. 14.14%
• B. 15.14%
• C. 13.14%
• D. 12.14%

Hint:

Positive Predictive Value (PPV) depends heavily on prevalence. Even highly sensitive tests may give low PPV when disease prevalence is rare.

Answer 1

The Correct Option is B

Step 1: Apply Bayes’ theorem.
We are asked to calculate the Positive Predictive Value (PPV), i.e., $P(\text{Disease}|\text{Test Positive})$.
Bayes’ theorem:
\[ P(D|+) = \frac{P(+|D) \cdot P(D)}{P(+|D) \cdot P(D) + P(+|\overline{D}) \cdot P(\overline{D})} \] Step 2: Substitute values.
- $P(+|D) =$ Sensitivity $= 0.89$
- $P(+|\overline{D}) =$ False Positive Rate $= 0.01$
- $P(D) =$ Prevalence $= 0.002$
- $P(\overline{D}) = 1 - 0.002 = 0.998$
Step 3: Calculate numerator and denominator.
Numerator = $0.89 \times 0.002 = 0.00178$
Denominator = $0.00178 + (0.01 \times 0.998) = 0.00178 + 0.00998 = 0.01176$
Step 4: Final probability.
\[ P(D|+) = \frac{0.00178}{0.01176} \approx 0.1514 = 15.14% \] Step 5: Conclusion.
The probability that a person truly has the disease given a positive test is 15.14%.