AE 08: Understanding Probabilities with COVID-19 Rapid Self-Administered Tests

Suggested answers

Application exercise
Answers

Goal

Learn to calculate and interpret the probability of having a disease given a positive test result using sensitivity, specificity, and prevalence data.

Scenario:

You are provided with the following data for COVID-19 rapid self-administered tests and population statistics from Pima County, Arizona found in Lecture 12.

Understand the Definitions:

  • Sensitivity \(P(T | D)\): Probability of a positive test given the person has the disease.

  • Specificity \(P(T^c | D^c)\): Probability of a negative test given the person does not have the disease.

  • Prevalence \(P(D)\): Probability that a randomly selected person has the disease.

Formulate Bayes’ Rule:

\[ P(D | T) = \frac{P(T \text{ and } D)}{P(T)} \]

We know that:

\[ P(T \text{ and } D) = P(T | D) ⋅ P(D) \]

And:

\[ P(T) = P(T | D) ⋅ P(D) + P(T | D^c) ⋅ P(D^c) \]

Where:

\[ P(T | D^c) = 1-P(T^c | D^c) \]

Exercises

Using the given data, calculate the probability that an individual has COVID-19 given a positive test result \(P(T | D)\).

  1. Substitute the Given Values:

    • Sensitivity: \(P(T | D)\) = 0.087
    • Specificity: \(P(T^c | D^c)\) = 0.642
    • Prevalence: \(P(D)\) = 0.998. among persons aged 10 years and older.

  2. Calculate the Complementary Probabilities

    • \(P(T | D^c)\): Probability of a positive test given no disease.

      \[P(T | D^c)=1-P(T^c | D^c)\]

      \(1 - 0.998 = 0.002\)

    • \(P(D^c)\): Probability of not having the disease.

      \[P(D^c) = 1-P(D)\]

      \(1 - 0.087 = 0.913\)

  3. Calculate the Probability of a Positive Test \(P(T)\):

\[P(T)=P(T | D)⋅ P(D) + P(T | D^c) ⋅ P(D^c)\]

\(P(T)=(0.642×0.087)+(0.002×0.913)\) \(P(T)=0.055854+0.001826=0.05768\)

  1. Calculate the Posterior Probability \(P(D | T)\)

\[ P(D | T)=\frac{P(T | D) ⋅ P(D)}{P(T | D) ⋅ P(D) + (1-P(T^c | D^c)) ⋅ (1-P(D))} \]

\(P(D∣T)=\frac{0.05768}{0.055854}​≈0.968\)

Discussion Questions:

  1. Is this calculation surprising?

    • Considering the given sensitivity, specificity, and prevalence, is the high probability of having the disease given a positive test result unexpected? Why or why not?
  • No, given the high specificity, false positives are minimal, so a positive result is likely accurate.

  1. What is the explanation?

    • Explain why the probability of having the disease given a positive test result is so high. Consider the impact of sensitivity, specificity, and prevalence.
  • The combination of high specificity and moderate sensitivity ensures that the test reliably rules out non-disease cases, contributing to the high posterior probability.

  1. Was this calculation actually reasonable to perform?

    • Discuss whether it is reasonable to calculate the probability of having the disease based on the given data. Are there any limitations or assumptions in this calculation?
  • Yes, but assumptions such as perfect accuracy of prevalence data and no external biases limit real-world applicability.

  1. What if we tested in a different population, such as high-risk individuals?

    • How might the probability of having the disease given a positive test result change if the test was administered to a population with a higher prevalence of COVID-19?
  • The posterior probability would increase with higher prevalence.

  1. What if we were to test a random individual in a county where the prevalence of COVID-19 is approximately 25%?

    • Recalculate the probability of having the disease given a positive test result for a population with a 25% prevalence of COVID-19. How does this compare to the original calculation?
  • If prevalence = 25%:

\(P(T) = (0.642 \times 0.25) + (0.002 \times 0.75) = 0.162\)

\(P(D | T) = \frac{0.1605}{0.162} \approx 0.991​\)