Question

Lyme disease is the leading tick-borne disease in the United States and Europe. Diagnosis of the disease is difficult and is aided by a test that detects particular antibodies in the blood. The article "Laboratory Considerations in the Diagnosis and Management of Lyme Borreliosis" (American Journal of Clinical Pathology [1993]: \(168-174\) ) used the following notation: + represents a positive result on the blood test \- represents a negative result on the blood test \(L\) represents the event that the patient actually has Lyme disease \(L^{C}\) represents the event that the patient actually does not have Lyme disease The following probabilities were reported in the article:\(\begin{aligned} P(L) &=0.00207 \\ P\left(L^{C}\right) &=0.99793 \\ P(+\mid L) &=0.937 \\ P(-\mid L) &=0.063 \\ P\left(+\mid L^{C}\right) &=0.03 \\ P\left(-\mid L^{C}\right) &=0.97 \end{aligned}\) a. For each of the given probabilities, write a sentence giving an interpretation of the probability in the context of this problem. b. Use the given probabilities to construct a "hypothetical \(1000 "\) table with columns corresponding to whether or not a person has Lyme disease and rows corresponding to whether the blood test is positive or negative. c. Notice the form of the known conditional probabilities; for example, \(P(+\mid L)\) is the probability of a positive test given that a person selected at random from the population actually has Lyme disease. Of more interest is the probability that a person has Lyme disease, given that the test result is positive. Use the table constructed in Part (b) to calculate this probability.

Short answer

In the context of this problem, for every 1000 people, 2 people are likely to have Lyme disease. If a person tests positive for Lyme disease, there is only approximately a 6.25% chance that they actually have the disease.

Step-by-step solution

Interpretation of the probabilities

Here's what each probability means in the context of this problem: 1. \(P(L) = 0.00207\): Out of the total population, around \(0.207\)% of people have Lyme disease. 2. \(P(L^{C}) = 0.99793\): Out of the total population, approximately \(99.793\)% do not have Lyme disease. 3. \(P(+ | L) = 0.937\): Given a person has the Lyme disease, there is a \(93.7\)% probability that he/she will test positive in the blood test. 4. \(P(- | L) = 0.063\): Given a person has the Lyme disease, there is a \(6.3\)% probability that his/her test will be a false negative. 5. \(P(+ | L^{C}) = 0.03\): Among those who do not have Lyme disease, around \(3\)% will have a false positive result, which means the test result will say they have Lyme disease even though they don't. 6. \(P(- | L^{C}) = 0.97\): Among those who do not have Lyme disease, \(97\)% will have a correct negative result, which means the test result will correctly say they don't have Lyme disease.

Construct a hypothetical 1000-person table

To create the hypothetical table, multiply the probabilities by 1000. Here is the probability table: - The number of people with Lyme disease = \(P(L) \times 1000 = 2.07\), round it to 2.- The number of people without Lyme disease = \(P(L^{C}) \times 1000 = 997.93\), round it to 998 (to total 1000).- The number of people with Lyme disease who test positive = \(P(+ | L) \times 2 = 1.874\), round it to 2.- The number of people with Lyme who test negative = \(P(- | L) \times 2 = 0.126\), round it to 0. - The number of people without Lyme disease but test positive = \(P(+ | L^{C}) \times 998 = 29.94\), round it to 30. - The number of people without Lyme disease and test negative = \(P(- | L^{C}) \times 998 = 969.06\), round it to 968 (to keep total 998).

Calculate the likelihood of having Lyme disease given a positive test.

The objective is to find the probability that a person has Lyme disease, given the test result is positive, denoted as \(P(L | +)\). This can be calculated by dividing the number of true positives by the total number of positives. This is \(P(L | +) = \frac{\text{Number of people with disease who test positive}}{\text{Total number of people who test positive}} = \frac{2} {2+30} \approx 0.0625\).