Question

USE THREE DECIMALS FOR ALL OF THE ANSWERS.

Health experts’ estimate for the sensitivity of coronavirus tests, as they are actually used, is 0.7. They also think the specificity is very high. Suppose specificity is 0.99 and that the health experts’ estimated sensitivity is correct (0.7).

a. In a population where 20% of the population is infected with the coronavirus, what is the probability that a person who tests positive actually is infected?  

b. Continued. What is the probability that a person who tests negative actually is not infected?

In the US, testing initially was very selective. In other words, as of early April 2020, only patients (i) with symptoms (ii) who contacted the health care system were being tested. For the most part, tests were not obtainable on demand, and there was very limited testing of asymptomatic people, even if they had been in contact with someone who had tested positive.

When testing is selective, then for interpreting results of testing, what matters is not the fraction of the entire population who are infected, but rather the fraction of the tested population who are infected.

c. If the prevalence of infection in the tested population is 0.8 (in other words, if 80% of people tested have the infection), what is the probability that a person who tests positive actually is infected?

d. Continued. What is the probability that a person who tests negative actually is not infected?

What can you learn from comparing your answers to parts a and b with your answers to parts c and d? The article also makes this point:

“Dr. Smalley said a negative result is more likely to be accurate in places like Louisville where the prevalence is low, but could be virtually useless in New York, where it is high.”

In other words, Louisville’s situation is similar to parts a and b, and New York’s situation is similar to parts c and d (qualitatively).

Now let’s see what is implied by the study of Wuhan patients that the WSJ article describes. Here’s another quote:

“A February study of about 1,000 patients in Wuhan, China, who were hospitalized with suspected coronavirus there, where the pandemic began, found that about 60% tested positive using lab tests similar to those available in the U.S. But, almost 90% showed tell-tale signs of the virus in CT scans of their chests, the article, published in the journal Radiology, found, suggesting many patients in the group were testing negative despite active coronavirus infections.”

Here, the population is “patients hospitalized for something that seems like coronavirus”. In that population, 90% of people were infected (if we take the CT scans as definitive). But 60% of this population tested positive. Note that 60% is not the sensitivity of the test, because this includes people who were not infected and who tested positive.

e. Let’s “back out” the sensitivity of the test, instead of assuming a value for sensitivity. Using 0.9 as the prevalence of infection with Covid-19 in the tested population; using 0.99 as the specificity of the test (same as in the previous parts of this problem); and using 0.6 as the fraction of the population who tested positive, calculate the implied sensitivity of the test. HINT: Use the law of total probability.    

f. Continued. In this situation (in the situation of the Wuhan study), what is the probability that a patient who tested negative actually was not infected?

g. Suppose the goal was for the probability to be at least 0.75 that a patient who tested negative actually was not infected, in the conditions of the Wuhan study. What is the minimum value of sensitivity that would allow this goal to be achieved?

Answer 1