Statistics of a medical test

Problem

0.074% of newborn children have a specific metabolic disorder. If this disorder is identified at an early stage, a future disease can be prevented by means of an appropriate treatment. For an early diagnosis, one can begin with a simple test. If the test result indicates a metabolic disorder, we call it positive.

If a newborn child has a metabolic disorder, the test result is positive with a probability of 99.5%. If a newborn child does not have a metabolic disorder, the probability that the test result is erroneously positive is 0.78%.

The test is conducted with a randomly selected newborn child. One considers the following results:

\(S\): „There is a metabolic disorder.“

\(T\): „The test is positive.“

1. Describe the event \(\overline{S\cup T}\) in the present context.

2. Calculate the probabilities \(P(T)\) and \(P_T (S)\). Interpret the result for \(P_T(S)\) in the present context.

   (for checking purposes: \(P(T)\approx 0{.}85\%, P_T(S)<0{.}1\))

3. During a screening, a huge number of randomly selected newborn children is tested. Determine the number of children per million tested newborn children expected on average to have a metabolic disorder while the test shows a negative result.

Solution of part a

First, we simplify the statement:

\[\overline{S\cup T} = \overline{S} \cap \overline{T}.\]

This formula thus describes the event that there is no metabolic disorder and the test result is negative.

Solution of part b

\(P(T)\) describes the probability for a positive test result. It results from the probability of a positive test for a healthy newborn child as well as the probability of a positive test for an ill newborn child.

\[P(T) = (1-0{.}00074)\cdot 0{.}0078 + 0{.}00074 \cdot 0{.}995 \approx 0{.}00853\,.\]

\(P_T(S)\) is computed as follows:

\[P_T(S) = \frac{P(S \cap T)}{P(T)} = \frac{0{.}00074\cdot 0{.}995}{0{.}00853} \approx 0{.}0863\,.\]

This means that for a positive test, only in 8.63% of all cases a metabolic disorder is found.

Solution of part c

The probability that a randomly selected child has a metabolic disorder and is tested positively is:

\[P(S\cap\overline{T}) = 0{.}00074\cdot (1-0{.}995) \approx 3{.}7 \cdot 10^{-6}\]

Thus, for one million tested children this event occurs for about four children.

With Sage, we can simulate the test and determine the number of all occuring events.