Problem 16 - Calculus for Biology and Medicine

Question

A screening test for a disease shows a positive result in \(92 \%\) of all cases when the disease is actually present and in \(7 \%\) of all cases when it is not. Assume that the prevalence of the disease is 1 in 600 . If the test is administered to a randomly chosen individual, what is the probability that the result is positive?

Step-by-Step Solution

Answer

The probability of a positive result is 7.14%.

Step 1: Define Given Information

Let's define the information provided in the problem. We know the true positive rate (sensitivity) is 92%, the false positive rate is 7%, and the prevalence of the disease is 1/600. We are tasked with finding the overall probability of the test showing a positive result.

Step 2: Set Up Probabilities

Introduce symbols for the probabilities: let \( P(D) \) be the probability of having the disease, \( P(T^+) \) be the probability of a positive test, \( P(T^+|D) \) be the probability of a positive test given the disease is present, and \( P(T^+|\overline{D}) \) be the probability of a positive test given the disease is absent. Thus, \( P(D) = \frac{1}{600} \), \( P(\overline{D}) = 1 - \frac{1}{600} \), \( P(T^+|D) = 0.92 \), \( P(T^+|\overline{D}) = 0.07 \).

Step 3: Use Total Probability Theorem

To find \( P(T^+) \), apply the total probability theorem: \[ P(T^+) = P(T^+|D) \cdot P(D) + P(T^+|\overline{D}) \cdot P(\overline{D}) \] This equation allows us to calculate the probability of a positive test result by considering both cases of disease presence and absence.

Step 4: Calculate Probabilities

Calculate each component: \( P(T^+|D) \cdot P(D) = 0.92 \cdot \frac{1}{600} = \frac{0.92}{600} \) and \( P(T^+|\overline{D}) \cdot P(\overline{D}) = 0.07 \cdot \left(1 - \frac{1}{600}\right) = 0.07 \cdot \frac{599}{600} \).

Step 5: Solve for Total Probability of Positive Test

Combine the results from Step 4 into the formula from Step 3: \[ P(T^+) = \frac{0.92}{600} + \frac{0.07 \times 599}{600} \] Calculate this to find the final probability of a positive test result.

Step 6: Simplify and Compute Final Probability

Perform the computation: \[ \frac{0.92}{600} + \frac{41.93}{600} = \frac{0.92 + 41.93}{600} = \frac{42.85}{600} = 0.0714 \] Therefore, the probability that the result is positive is approximately 0.0714, or 7.14%.

Key Concepts

Screening Test

Screening tests are essential tools in medicine. They help in detecting diseases in individuals before symptoms appear, allowing for early intervention. A screening test's effectiveness is measured by its sensitivity and specificity.
Let's break these terms down:

• Sensitivity: The probability that the test will be positive when the disease is present. In our example, this is 92% which indicates that if someone does have the disease, the test indeed shows positive 92 out of 100 times.
• Specificity: The probability that the test will be negative when the disease is not present. This isn't given directly in the problem but can be deduced since we know the false positive rate is 7%. Thus, specificity would logically be 93%.

For a screening test, especially in widespread public health settings, these metrics are crucial. They influence how many people might get additional testing or treatment. The goal is to minimize false positives (indicating disease when there is none) and false negatives (missing a true case of disease). A good screening test has both high sensitivity and specificity to provide reliable results without causing unnecessary stress or harm.

Prevalence

Prevalence reflects how common a disease is in a population. It's the proportion of individuals who have the disease at a given time. Here, the prevalence is expressed as 1 in 600. This means that among 600 individuals, on average, one person has the disease.
Understanding prevalence helps to gauge the burden of a disease within a community. It influences how screening tests are interpreted and deployed. In populations where a disease is more common, positive test results are often due to true cases. However, in populations with low prevalence, as in our example, even reliable tests might frequently give false positives.
This inverse relationship between prevalence and accuracy of the positive test results is essential in clinical decision-making. High prevalence generally leads to higher predictive value of a positive test result, meaning a person with a positive result is indeed more likely to have the disease. Conversely, in low prevalence situations, many positive results might be false.

Bayes' Theorem

Bayes' Theorem provides a way to update our beliefs in light of new evidence. In the context of screening tests, it allows us to calculate the probability that someone who tests positive actually has the disease. This is called the positive predictive value (PPV).
The formula for Bayes' Theorem, especially when calculating predictive values, looks like:

• \[P(D|T^+) = \frac{P(T^+|D) \cdot P(D)}{P(T^+)}\]

Where:

• \( P(D|T^+) \) is the probability of having the disease given a positive test result.
• \( P(T^+|D) \) is the probability of a positive test when the disease is present.
• \( P(D) \) is the prevalence of the disease.
• \( P(T^+) \) is the overall probability of a positive test (calculated using the Total Probability Theorem).

Bayes' Theorem emphasizes that test results must be interpreted in context with how common the disease is. This insightful approach makes it possible to evaluate the true utility of tests beyond mere sensitivity and specificity.