Aren't all tests sensitive to the prevalence of a disease in the population?

Question

I'm trying to understand the difference between the false-positive rates of two kinds of COVID-19 tests: PCR and antibody.

The former indicates if someone is currently sick. The latter indicates if someone was sick in the past.

Per https://theconversation.com/coronavirus-tests-are-pretty-accurate-but-far-from-perfect-136671:

1. The author states that antibody tests "aren't any better" than PCR tests.
2. Antibody tests have a high false-positive rate due to a low true-positive rate in the population.

The article seems to imply that PCR tests do not suffer from the same problem, but how could that be?

Shouldn't the false positive rate of PCR tests be high because a low true-positive rate in the population?

Meaning, if a small number of people have been sick with COVID-19 in the past then certainly the number of people who are currently sick with it must be even smaller. Doesn't that imply that the sensitivity + specificity of PCR tests must be worse than those of antibody tests?

Answer 1

The blog you cite speaks mostly about sensitivity and specificity, shortly going towards positive and negative predictive value, without mentioning these terms.

Shouldn't the false positive rate of PCR tests be high because a low true-positive rate in the population?

Sensitivity and Specificity are defined for a sample of true positives and true negatives, without regard to the prevalence in the population. If you want to take the prevalence in the population into account, you will use the terms and definitions of "positive predictive value" (if the test is positive what are the chances of the patient being positive) and the "negative predictive value" (if the test is negative, what are the chances of the patient being negative).

As the maker and seller of the test, you can only advertise sensitivity and specificity as you cannot know, on which population a doctor will employ the test you sold them. As a doctor performing the test, you are not really interested in sensitivity or specificity but in positive and negative predictive value, which depend on the prevalence of the condition.

The formula to compute predictive values from prevalence and sensitivity/specificity is called Bayes' rule.

suggested reading

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3153801/ (Westbury CF. Bayes' rule for clinicians: an introduction. Front Psychol. 2010;1:192. Published 2010 Nov 16. doi:10.3389/fpsyg.2010.00192) https://en.wikipedia.org/wiki/Positive_and_negative_predictive_values (Wikipedia entry to the two terms that should have been in that blog post) https://en.wikipedia.org/wiki/Sensitivity_and_specificity

Answer 2

I find working in terms of odds works better. First you have the prior odds (or unconditional odds) of having the disease. If we call $D$ the event "has disease", the odds are..

$$\frac{P(D|I)}{1-P(D|I)}$$

Once we observe the test we then can update the odds to

$$\frac{P(D|I)}{1-P(D|I)}\times\frac{P(+|DI)}{P(+|D^cI)}= \frac{P(D|I)}{1-P(D|I)}\times\frac{\theta_{sensitivity}}{1-\theta_{specificity}}$$

This means the ratio $\frac{\theta_{sensitivity}}{1-\theta_{specificity}}$ is basically a measure of how useful a positive test is. Ideally we want this to be huge.

A similar calculation for having the disease given a negative test shows that the ratio $\frac{1-\theta_{sensitivity}}{\theta_{specificity}}$ is used instead. So this shows the value of a negative test.

Now suppose we put some numbers. A quick Google of "covid 19 specificity" gives $\theta_{specificity}=0.98,\theta_{sensitivity}=0.9$ as one set. Might not be applicable to the tests you're talking about, but probably close enough. This gives a ratio of $45$ for a positive test and $\frac{1}{9.8}$ for a negative test. So we have negative test means divide the odds (by $9.8$) and positive test means multiply the odds (by $45$).

Now suppose we introduce $R$ which indicates "had disease and recovered". Also suppose we have equally good tests (same specificity and sensitivity). At the early stages of an outbreak $P(R|I)<P(D|I)$ is possible Ie less recovered than have the disease. This occurs when the time taken to recover takes longer than the time to infect someone else. Over time the recovered $R$ will increase and $D$ may or may increase or decrease over a short time depending on how quickly it spreads, but will eventually decrease.

At the start of the outbreak we have say $\frac{P(R|I)}{1-P(R|I)}=10^{-6}$ - Ie a million to 1 odds against you already have caught the disease. We also have $\frac{P(D|I)}{1-P(D|I)}=10^{-4}$ - Ie ten thousand to 1 that you have the disease. A positive test multiplies both odds by $45$ but your actual chance of having the disease is still quite low. This is more likely a pre-symptomatic result, rather than symptomatic one - because symptoms are essentially addition tests from one perspective. For people who show symptoms the starting odds (ie general prevalence) are likely higher.