Problem 15 - Calculus for Biology and Medicine

Question

A screening test for a disease shows a positive test result in \(90 \%\) of all cases when the disease is actually present and in \(15 \%\) of all cases when it is not. Assume that the prevalence of the disease is 1 in 100 . If the test is administered to a randomly chosen individual, what is the probability that the result is negative?

Step-by-Step Solution

Answer

The probability of a negative test result is approximately 84.25%.

Step 1: Understand the Problem

We are given the sensitivity (true positive rate) and the false positive rate for a test, alongside the prevalence of the disease. Our task is to find the probability of getting a negative test result.

Step 2: Define Variables and Probabilities

Let \( D \) be the event that a person has the disease and \( eg D \) the event they do not.- \( P(D) = 0.01 \)- \( P(\text{Positive} | D) = 0.9 \)- \( P(\text{Positive} | eg D) = 0.15 \).We need to find the probability of a negative result: \( P(\text{Negative}) \).

Step 3: Breakdown of Outcomes

We need to consider two possibilities for a negative test result: 1. The person does not have the disease and tests negative. 2. The person has the disease and tests negative.

Step 4: Calculate Probability of Testing Negative Given Disease is Absent

If the disease is absent, the probability of testing negative is one minus the false positive rate:\[ P(\text{Negative} | eg D) = 1 - P(\text{Positive} | eg D) = 1 - 0.15 = 0.85 \]

Step 5: Calculate Probability of Testing Negative Given Disease is Present

If the disease is present, the probability of testing negative (false negative rate) is one minus the sensitivity:\[ P(\text{Negative} | D) = 1 - P(\text{Positive} | D) = 1 - 0.9 = 0.1 \]

Step 6: Apply Total Probability Theorem

The total probability of a negative test result can be calculated using:\[P(\text{Negative}) = P(\text{Negative} | D) \cdot P(D) + P(\text{Negative} | eg D) \cdot P(eg D)\]Substitute the values:\[P(\text{Negative}) = (0.1 \cdot 0.01) + (0.85 \cdot 0.99)\]

Step 7: Perform Calculations

Calculate each term separately:- \( 0.1 \cdot 0.01 = 0.001 \)- \( 0.85 \cdot 0.99 = 0.8415 \)Now sum these values:\[P(\text{Negative}) = 0.001 + 0.8415 = 0.8425\]

Step 8: Conclusion

The probability that a randomly chosen individual tests negative is \(0.8425\), or approximately \(84.25\%\).

Key Concepts

Sensitivity and Specificity

When dealing with medical tests, sensitivity and specificity are key terms.
Sensitivity refers to the probability that a test will correctly return a positive result for an individual who has the disease. In other words, it measures the true positive rate.
For the exercise above, the sensitivity is given as 90%, meaning if someone has the disease, there's a 90% chance the test will show a positive result.

Specificity, on the other hand, measures the true negative rate. This is the probability that a test will return a negative result for someone who does not have the disease.
In the exercise, while specificity is not directly mentioned, we can relate it to the false positive rate.
The false positive rate is 15%, meaning 15% of the time, those without the disease get a positive result. Thus, the specificity is the complement of this rate, which is 85%.

High sensitivity and specificity are desirable as they indicate that a test is very good at detecting the disease and very accurate in ruling it out in healthy people. Medical tests aim to optimize these values to ensure reliability.

False Positive Rate

The false positive rate is a measure of how often a test incorrectly identifies the presence of a condition or disease when it is not actually present.
This rate is crucial since it impacts the reliability of the test in diagnosing those who are not sick.

In our example, the false positive rate stands at 15%.
This indicates that if 100 people who do not have the disease are tested, 15 of them could receive a positive result despite being healthy.

• A high false positive rate can lead to unnecessary stress for patients.
• It can lead to further unnecessary medical testing.
• Can also increase healthcare costs.

Minimizing the false positive rate is important to increase confidence in medical test results, reducing the chances of false alarms.

Total Probability Theorem

The total probability theorem helps in determining the likelihood of an event by considering all possible pathways to it.
This theorem is particularly useful in medical testing to find the probability of different outcomes.

For the exercise given, we used the total probability theorem to find the probability of a negative test result. The theorem considers:

• The chance of getting a negative result if the disease is truly present (false negative rate).
• The chance of getting a negative result if the disease is absent (true negative rate).

Both scenarios involve multiplying these specific probabilities by the overall likelihood of the disease being present or absent in the tested population.

Mathematically expressed as:

\[ P(\text{Negative}) = P(\text{Negative} | D) \cdot P(D) + P(\text{Negative} | eg D) \cdot P(eg D) \]

This calculation combines the individual probabilities to give a comprehensive view of how often a negative test result is expected in practice. It’s a powerful tool for understanding compound events.