Question

Suppose that a certain disease is present in \(10 \%\) of the population, and that there is a screening test designed to detect this disease if present. The test does not always work perfectly. Sometimes the test is negative when the disease is present, and sometimes it is positive when the disease is absent. The following table shows the proportion of times that the test produces various results. $$ \begin{array}{lcc} \hline & \text { Test Is Positive }(P) & \text { Test Is Negative }(N) \\ \hline \text { Disease } & .08 & .02 \\ \text { Present }(D) & & \\ \text { Disease } & .05 & .85 \\ \text { Absent }(D 9 & & \\ & & \\ \hline \end{array} $$ a. Find the following probabilities from the table: \(P(D), P\left(D^{c}\right), P\left(N \mid D^{c}\right), P(N \mid D)\) b. Use Bayes' Rule and the results of part a to find \(P(D \mid N)\) c. Use the definition of conditional probability to find \(P(D \mid N)\). (Your answer should be the same as the answer to part b.) d. Find the probability of a false positive, that the test is positive, given that the person is disease-free. e. Find the probability of a false negative, that the test is negative, given that the person has the disease. f. Are either of the probabilities in parts d or e large enough that you would be concerned about the reliability of this screening method? Explain.

Short answer

Answer: The probability of having the disease given a negative test result (P(D|N)) is approximately 0.0026.

Step-by-step solution

1. Probability of disease presence (P(D))

By looking at the table, the disease is present in \(10\%\) of the population, which is equivalent to \(0.10\). Therefore, \(P(D)=0.10\).

2. Probability of disease absence (P(D^c))

Since the disease is present in \(10\%\) of the population, the complementary event Disease Absent (or disease-free) would be present in the remaining percentage of the population. \(P(D^c)=1-P(D)=1-0.10=0.90\).

3. Probability of a negative test result given disease absence (P(N|D^c))

According to the table, the probability of a negative test result given that the disease is absent is \(0.85\). Thus, \(P(N|D^c)=0.85\).

4. Probability of a negative test result given disease presence (P(N|D))

From the table, the probability of a negative test result given that the disease is present is \(0.02\). Hence, \(P(N|D)=0.02\). b. Use Bayes' Rule and the results in part a to find \(P(D|N)\)

1. Apply Bayes' Rule

Bayes' Rule states that: $$P(D|N)=\frac{P(N|D)P(D)}{P(N|D)P(D)+P(N|D^c)P(D^c)}$$

2. Substitute known probabilities and compute

From the results in part a: $$P(D|N)=\frac{0.02 \cdot 0.10}{0.02 \cdot 0.10 + 0.85\cdot 0.90}=\frac{0.002}{0.002 + 0.765}= \frac{0.002}{0.767}\approx 0.0026$$ c. Using the definition of conditional probability to find \(P(D|N)\)

1. Definition of Conditional Probability

The definition of conditional probability states that: $$P(D|N)=\frac{P(D \cap N)}{P(N)}$$

2. Finding probability of intersection (D ∩ N)

From the given table, \(P(D \cap N)=0.02\).

3. Finding probability of Negative test result (P(N))

By summing the probabilities of negative test results in the given table, we have: $$P(N)= P(N|D)P(D)+P(N|D^c)P(D^c) = 0.02\cdot0.10 + 0.85\cdot0.90 = 0.002 + 0.765 = 0.767$$

4. Compute P(D|N) using the known probabilities

Plugging the known probabilities into the equation, we get: $$P(D|N)=\frac{0.02}{0.767}\approx 0.0026$$ As expected, this result is the same as the one obtained using Bayes' Rule in part b. d. Find the probability of a false positive, that the test is positive, given that the person is disease-free.

1. False Positive

We are asked to find \(P(P|D^c)\). Looking at the table, the probability is \(0.05\). e. Find the probability of a false negative, that the test is negative, given that the person has the disease.

1. False Negative

We are asked to find \(P(N|D)\). From the table and part a, the probability is \(0.02\). f. Are either of the probabilities in parts d or e large enough that you would be concerned about the reliability of this screening method? Explain.

1. Analyzing the probabilities

The probability of a false-positive is \(0.05\), meaning the test will incorrectly indicate the disease's presence in \(5\%\) of the disease-free individuals. The probability of a false-negative is \(0.02\), signifying that the test will fail to detect the disease in \(2\%\) of the individuals with the disease.

2. Conclusion

While the numbers may seem small, it's important to consider the context and the possible consequences of false results. Depending on the severity of the disease and the potential treatment plans or follow-up tests, a false-positive or false-negative outcome could have significant consequences. Ultimately, whether these probabilities are considered concerning enough to question the reliability of the screening method will depend on the specific disease being tested and the medical professionals' evaluation.

Key concepts

Conditional Probability

Conditional probability is a fundamental concept in probability theory that helps us understand how the probability of an event changes when we have additional information. For example, when we know that a specific condition or event has occurred. In the context of our example, it refers to assessing the probability of having a disease given that the test result is negative, or vice versa.

To calculate conditional probabilities, we use the formula:

\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]

Here, \(P(A|B)\) represents the probability of event A occurring given that B has occurred. \(P(A \cap B)\) is the probability of both events A and B occurring together, and \(P(B)\) is the probability of event B happening. This formula is vital in medical testing because it allows us to compute the likelihood of a disease presence based on test results.

False Positives

A false positive in medical testing refers to a test result that incorrectly indicates the presence of a disease when it is not actually present. In the chart provided in the exercise, this scenario occurs with a probability of 0.05, or 5%, which means that 5% of disease-free individuals will receive a positive test result.

False positives can have significant implications:

• Unnecessary worry or stress for patients.
• Potentially unnecessary testing and medical procedures.
• Increased healthcare costs due to further evaluations.

Understanding false positive rates is crucial for evaluating the reliability and efficiency of medical tests. It helps us balance the risks and benefits of administering the test widely.

False Negatives

False negatives occur when a test indicates that a disease is absent despite the individual actually having the disease. In our scenario, it is stated that this happens with a probability of 0.02, or 2%.

Consequences of false negatives can be severe, including:

• Delayed diagnosis or treatment, which can worsen outcomes.
• A false sense of security for the patient potentially missing critical medical interventions.
• Misleading data for health studies or disease monitoring efforts.

It is crucial to minimize false-negative rates because an inaccurate "all-clear" signal can lead to serious consequences for patient health, especially when dealing with aggressive or fast-progressing diseases.

Probability Theory

Probability theory forms the foundation for understanding and solving problems related to uncertainty, like those encountered in medical statistics. It allows us to work with the likelihoods of various outcomes and is essential in fields like epidemiology, risk management, and diagnostics.

Key concepts include:

• Probability: The measure of how likely an event is to occur.
• Complementary Events: The idea that the probability of an event happening plus the event not happening equals one.
• Joint Probability: The probability of two or more events occurring simultaneously. This is often needed in both false positive and false negative calculations.

In medical contexts, probability theory helps healthcare professionals evaluate the likelihood of conditions and choose appropriate courses of action based on statistical evidence. This ensures decisions are informed, especially in critical healthcare scenarios.