Your rapid test is positive. Does that mean you have covid?
Here's the math you need to figure it out. A thread. π§΅
Here's the math you need to figure it out. A thread. π§΅
Let's define some terms that you might have heard of.
A "true positive" means the test result was positive and the person tested does indeed have covid.
A "false positive" means the test was positive but the person doesn't have covid.
A "true positive" means the test result was positive and the person tested does indeed have covid.
A "false positive" means the test was positive but the person doesn't have covid.
A "true negative" means the test was negative and the person tested doesn't have covid.
A "false negative" means the test was negative but the person does have covid.
Got it? Good!
A "false negative" means the test was negative but the person does have covid.
Got it? Good!
The "sensitivity" of a test is the percent of cases that test positive out of the total number of infected people in the group being tested.
The "specificity" of a test is the proportion of uninfected people who were tested and who were negative according to the test.
The "specificity" of a test is the proportion of uninfected people who were tested and who were negative according to the test.
Finally, the "prevalence" is the proportion of people who are infected.
None of those numbers are really what we want to know though.
What we actually care about is called the "positive predictive value" of the test. It's the proportion of people who *test positive* who actually have the disease.
What we actually care about is called the "positive predictive value" of the test. It's the proportion of people who *test positive* who actually have the disease.
When you test a lot of people, positive results come from two sources: false positives and true positives.
The false positives depend on how many people don't have covid. For instance, if everyone was infected then it would be logically impossible to get a false positive.
The false positives depend on how many people don't have covid. For instance, if everyone was infected then it would be logically impossible to get a false positive.
Similarly, the number of true positives depends on how many people actually have covid. If nobody has covid then you can't logically have a true positive either.
It takes a bit of thinking to see this but the total number of true positives is just the sensitivity times the prevalence and the total number of false positives is just (1-specificity) times (1-prevalence), which is the number of uninfected people.
Putting everything together with the previous formula for positive predictive value, the probability of being positive given you have tested positive is given by the following formula:
This idea is a little counterintuitive because how well the test detects positives and negatives is a property of the test. But that's not really what we want to know.
Instead, we want to know the probability that you have covid after getting a positive test result.
Instead, we want to know the probability that you have covid after getting a positive test result.
That number is related to the number of people who actually have covid in the first place.
The more likely people around you are to have covid, the more confident you should be that you have covid if you get a positive test.
The more likely people around you are to have covid, the more confident you should be that you have covid if you get a positive test.
Let me repeat that idea again. A positive test result is a BETTER predictor when more people have covid. As covid surges due to omicron, positive rapid tests are actually getting *more* believable.
OK. Let's try using the PPV formula with some real numbers.
OK. Let's try using the PPV formula with some real numbers.
First, we need to know some actual test sensitivities and specificities. I got this list of test sensitivities and specificities from a very informative NYT article by @helloellenlee and @tracyvence which you can find here: nytimes.com/wirecutter/revβ¦
The Abbott BinaxNow test is the most common one where I live so I'm going to use it as an example. If you have a different test, feel free to plug in your own numbers for sensitivity and specificity into the formula. Using the BinaxNow numbers, I got...
Look at how the positive predictive value of the test goes up as covid becomes more common. It crosses 50% (more likely to be a true positive than not) at about 1.7% prevalence.
If you don't know prevalence, you can use test positivity for your area instead. In my state MA, it's around 5.6%. Here's a CDC link to test positivity rates by state and county for the US: covid.cdc.gov/covid-data-traβ¦
Test positivity is probably an overestimate of prevalence for the overall population but since you're concerned that you might have covid, you're probably in a similar risk category to the types of people that are being tested at your local healthcare provider.
Hope this thread gives you some tools to help you think about any positive tests that may come up. Happy Holidays and may all your tests be negative (or at the very least false positives)!
Disclaimer: This thread is not official advice. I'm not an MD. I do have a graduate degree in biostatistics but I'm in no way attempting to give you medical advice or contradict the advice of your local public health authority.
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