Problem 37 - Calculus for Biology and Medicine

Question

A screening test for a disease shows a positive result in \(95 \%\) of all cases when the disease is actually present and in \(10 \%\) of all cases when it is not. If the prevalence of the disease is 1 in 50 and an individual tests positive, what is the probability that the individual actually has the disease?

Step-by-Step Solution

Answer

The probability that the individual actually has the disease is approximately 16.24%.

Step 1: Understanding the Problem

We have a test with two scenarios: it correctly shows positive when the disease is present with a probability of 95%, and it shows a false positive (disease not present but test positive) with a probability of 10%. The prevalence of the disease is 1 in 50, meaning the probability of having the disease is 0.02. We need to find the probability that someone who tests positive actually has the disease.

Step 2: Define Probabilities

Denote the following events: - Let D be the event that the individual has the disease.- Let T+ be the event that the test result is positive.We are given:- Probability of having the disease: \(P(D) = \frac{1}{50} = 0.02\)- Probability of testing positive if the disease is present: \(P(T+|D) = 0.95\)- Probability of testing positive if the disease is NOT present: \(P(T+|eg D) = 0.10\)We need to find \(P(D|T+)\).

Step 3: Apply Bayes' Theorem

Bayes' Theorem allows us to calculate the probability of D given T+. The theorem is: \[P(D|T+) = \frac{P(T+|D) \cdot P(D)}{P(T+)}\]We have everything except \(P(T+)\), the overall probability that a test is positive.

Step 4: Calculate Total Probability of Positive Test

The total probability of a positive test, \(P(T+)\), can be found using the law of total probability: \[P(T+) = P(T+|D) \cdot P(D) + P(T+|eg D) \cdot P(eg D)\]Where:\(P(eg D) = 1 - P(D) = 0.98\).Substitute the values:\[P(T+) = 0.95 \times 0.02 + 0.10 \times 0.98\]\[P(T+) = 0.019 + 0.098 = 0.117\]

Step 5: Calculate Final Probability

Substitute the values into Bayes' Theorem:\[P(D|T+) = \frac{0.95 \times 0.02}{0.117}\]\[P(D|T+) = \frac{0.019}{0.117} \approx 0.1624\]Thus, the probability that an individual with a positive test actually has the disease is approximately 16.24%.

Key Concepts

Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has already occurred. It's a way to fine-tune the overall probability of an outcome based on new information. Symbolically, conditional probability is denoted as \( P(A|B) \), which represents the probability of event A occurring given that event B has occurred.
In the context of our problem, we used conditional probability when determining \( P(T+|D) \) and \( P(T+|eg D) \). These denote the probabilities of testing positive for the disease both when the disease is present and when it is not, respectively. Understanding conditional probability helps us apply concepts like Bayes' Theorem, which allows us to update initial predictions with new evidence.

False Positive Rate

A false positive rate is a measure of test accuracy that indicates the probability of a test erroneously identifying a condition when it is not actually present. Essentially, it tells us how often a test might mistakenly suggest a positive result.
The false positive rate in our exercise is seen in the probability \( P(T+|eg D) = 0.10 \), meaning that there is a 10% chance the test shows a positive result even when the disease isn't present. Reducing false positives is crucial in disease screening to avoid unnecessary stress and further tests for patients who are actually healthy.

Prevalence

In medical statistics, prevalence refers to the proportion of a population that has a particular disease at a given time. It is a fundamental metric that indicates how widespread a disease is in a specific population.
In this exercise, the disease's prevalence is given as 1 in 50, or 2%. This means that in any given group of 50 individuals, on average, 1 will have the disease. Prevalence data is essential in calculating the overall probability of testing positive using the total probability method, which is essential for using Bayes' Theorem effectively.

Disease Screening Tests

Disease screening tests are medical tests used to identify the presence of a disease in individuals who do not yet show symptoms. These tests are vital for early detection and treatment, potentially improving health outcomes.
Effective screening tests need to have a good balance between sensitivity (correctly identifying those with the disease) and specificity (correctly identifying those without the disease). In our example, the test has high sensitivity at 95% but a notable false positive rate of 10%. This highlights the importance of using statistical tools like Bayes' Theorem to interpret test results accurately. Utilizing such statistics helps healthcare providers offer more accurate diagnoses and limit false reassurances or unnecessary interventions.