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

Suggested 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:

$$P(T\text{ and }D)=P(T|D)\cdot P(D)$$

And:

$$P(T)=P(T|D)\cdot P(D)+P(T|D^c)\cdot P(D^c)$$

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 $P(T)$:

$$P(T)=P(T|D)\cdot P(D)+P(T|D^c)\cdot P(D^c)$$

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

4. Calculate the Posterior Probability $P(D|T)$

$$P(D|T)=\frac{P(T|D)\cdot P(D)}{P(T|D)\cdot P(D)+(1-P(T^c|D^c))\cdot (1-P(D))}$$

$P(D\mid T)=\frac{0.05768}{0.055854}\approx 0.968$

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.

2. 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.

3. 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.

4. 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.

5. 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%:

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

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