Problem 18 - Calculus for Biology and Medicine

Question

A screening test for a disease shows a positive test result in \(95 \%\) of all cases when the disease is actually present and in \(20 \%\) of all cases when it is not. When the test was administered to a large number of people, \(21.5 \%\) of the results were positive. What is the prevalence of the disease?

Step-by-Step Solution

Answer

The prevalence of the disease is 2%.

Step 1: Understand the Problem

We need to determine the prevalence of a disease using given probabilities related to a screening test. We have information on the true positive rate (sensitivity), false positive rate, and overall positive test rate.

Step 2: Define the Variables

Let \( p \) represent the prevalence of the disease, which is the probability that a randomly selected individual has the disease. Also let: \( P(+|D) = 0.95 \) (probability of a positive test given that the disease is present), \( P(+|eg D) = 0.20 \) (probability of a positive test given that the disease is not present), and \( P(+) = 0.215 \) (the overall probability of a positive test).

Step 3: Apply Total Probability Theorem

Use the total probability theorem to express the overall positive test probability: \( P(+) = P(+|D)P(D) + P(+|eg D)P(eg D) \). This becomes: \[ 0.215 = 0.95p + 0.20(1-p) \].

Step 4: Solve the Equation for Prevalence

Simplify and solve the equation: \[ 0.215 = 0.95p + 0.20 - 0.20p \]. Combine terms: \[ 0.215 = 0.75p + 0.20 \]. Subtract 0.20 from both sides: \[ 0.015 = 0.75p \]. Finally, divide by 0.75: \[ p = \frac{0.015}{0.75} = 0.02 \].

Step 5: Interpret the Result

The prevalence \( p = 0.02 \) means that 2% of the population has the disease. Thus, the disease prevalence calculated from the information provided is 2%.

Key Concepts

Prevalence

Prevalence is an essential concept in epidemiology and statistics, referring to the proportion of a population that has a specific disease at a given time. In the context of Bayesian Probability, understanding prevalence helps in estimating the likelihood of a disease within a group. It is represented as a percentage or a decimal. For instance, if a disease's prevalence is 2%, it means that out of 100 people, approximately 2 individuals are expected to have the disease. This measure gives a snapshot of how widespread the disease is in the population.

• Prevalence provides insight into the health status of a population.
• It helps in resource allocation and planning public health strategies.
• Prevalence is distinct from incidence, which measures new cases over a period.

In our exercise, we found that only 2% of the population actually has the disease, despite many positive test results. This shows how prevalence helps in contextualizing test outcomes, important for informed health decisions.

Sensitivity (True Positive Rate)

Sensitivity, also known as the true positive rate, is crucial for evaluating a screening test's effectiveness. It quantifies how well the test correctly identifies those with the disease. A high sensitivity means most people who have the disease will get a positive test result.In mathematical terms, it is expressed as:\[P(+|D) = \text{The probability of a positive test given the disease is present.}\]

• A sensitivity of 95% means the test correctly identifies 95 out of 100 individuals with the disease.
• High sensitivity is important for ensuring that few cases are missed.
• It is a critical measure for diseases requiring early detection.

In our example, with a 95% sensitivity, the test is excellent at detecting the disease in those who have it. However, while sensitivity is vital, other factors like false positive rate also impact interpretation of test results.

False Positive Rate

The false positive rate reflects how often a test incorrectly indicates the presence of a disease in those who do not have it. This is crucial, as it affects the test's reliability.A false positive can lead to unnecessary stress, further testing, or even treatment. It's given by:\[P(+|eg D) = \text{Probability of a positive test when the disease is not present.}\]

• A false positive rate of 20% in our example signifies that 20 out of 100 non-diseased individuals are incorrectly labeled as having the disease.
• A high false positive rate increases the number of people mistakenly assumed to be sick.
• Balancing sensitivity with the false positive rate is key for effective screening tests.

In conclusion, the test's 20% false positive rate underscores the need for follow-up testing to confirm initial positive results, ensuring accurate diagnosis and treatment planning.