Let’s say the prevalence of a disease among those likely to take the test for that disease is 1%. So if you test a million people, you will find 10,000 people actually have the disease.
However, what does it mean if the provider of the test says that there is a 13% chance of a false positive on the test? If you test a million people and 10,000 actually have the disease, does it mean that you will get 11,494 positives of which 13% (1,494) are false, leaving 10,000 actual positives (if my math is correct)? (Note: I realize the percentage is an estimate, I am using specific numbers for simplicity.)
That seems the most intuitive to me, but I am suspicious of any commercial claim that includes percentages. Is there an industry standard for what this formula-of-words means?
(Not to be coy, this is about Cologuard. I got a positive result but have no externally-detectable actual symptoms, so I can’t have the follow-up colonoscopy until elective procedures are allowed. I would like to know what my odds are.)
I.e., the probability of a negative testing positive. (To be distinguished from the probability of a positive result being wrong-- which is what the op. was worried about, NOT the false positive rate.)
However, not to impugn your math, but if there are 10000 people who actually have the condition and 990000 true negatives, the number of false positives must be 0.
To put some numbers on it, if there are 1000000 people and 990000 are negative, a false positive rate of 0.13 means that 128700 of those 990000 test positive. However, that does not answer the OP’s question, since he did not give the rate of false negatives. So his odds (of being positive) could be anything between 0 and 7.2%
To put it another way, the false positive rate given for a test does not take prevalence into consideration.
Imagine if it did. Then it would involve not just the uncertainties of the procedure used to estimate how many true negatives test positive, but the uncertainty of estimating the real prevalence. And you wouldn’t be able to know what it meant without knowing both the prevalence used to estimate it, and what the current best estimate for the prevalence was.
The other way round the rate is easily applicable, even when what you want is to calculate the odds for a specific subgroup with an estimated prevalence different from the whole population.
Just an aside - there are serious concerns these days about the negative effect of false positives. The problem is that some people are so badly affected by being told that they have a serious medical condition that they become ill.
Of course, most medical tests also aren’t given to just everyone. Usually, a test is only given if there’s some reason to suspect a particular condition. So the prevalence you want to look at isn’t the prevalence among the general population, but the prevalence among the subset of the population who’s getting the test, and that’s probably much higher.
The statistic you’re most interested in here is known as the Positive Predictive Value (PPV) - it answers the question “If I get a positive result from this test, how likely is it that that I actually have the condition?”
PPV depends on:
The prevalence P of the disease in the population taking the test.
The test sensitivity SN - the fraction of people who actually have the disease, who test positive.
The test specificity SP - the fraction of people who do not have the disease, who test negative.
You can then calculate PPV as:
PPV = SN * P / [(SN * P) + (1 - SP) * (1 - P)]
Cologuard cites a study reporting their sensitivity as 92% overall and and specificity as 87% overall (https://www.cologuardtest.com/hcp/about/clinical-offer). Since they can cherry pick which studies they report numbers from, one might assume these numbers are on the high end of the actual expected performance.
Using your own assumption of a 1% prevalence (P = 0.01) and SN = 0.92, SP = 0.87 this gives a PPV of about 6.67%.
Meaning, a person receiving a positive result can be expected to actually have the disease roughly 6.67% of the time.
Welcome to the world of medical testing! In addition to test availability, this is one reason why, early in the pandemic, the medical system was unwilling to test every person with a fever and cough for coronavirus.
I got most of that, and thank you very much for the formula; but I don’t understand the last sentence. Is it because it would have resulted in too many false positives?
[soap box]I’m not a huge fan generally of medical testing, although of course properly used it helps diagnosis. I once had to endure a painful and unnecessary prostate biopsy based on the highly unreliable PSA test, which was a little elevated over what was normal for me. I call it unnecessary because neither my PCP nor the specialist told me about other factors that can elevate PSA, such as age and sexual activity. The specialist was surprised and perhaps a little disappointed when I was later able to reduce the PSA number by refraining from sexual activity for 72 hours before the blood draw.[/soap box]
More to the point, it’d be too many false positives compared to the true positives. When there were only a few people in the country (say, 1 in a million) who were actually infected, the test would find 13.0001% positive, instead of finding 13.0000% positive, impossible to tell the difference. Even once 1% of the population is actually infected, the test would show 14% instead of 13%, which would be very, very difficult to tell the difference, unless you were testing a great many people. But once 10% of the population is infected, showing 23% instead of 13% is definitely distinguishable (even though, in that case, getting a positive test result still means you’re more likely not to not have it than to have it).
Of course, the numbers are more amenable if you’re just looking at the population of people with fevers and coughs, instead of the general population. But even there, the tests wouldn’t have been much use at first.
I thought I might as well follow up here with the results, since I finally had my colonoscopy in December (I thought they were going to call me when they were doing optional screenings again, but they didn’t, so I finally called them).
I had two large polyps of the type that can possibly become cancerous (there are three types: completely benign, already cancerous, and these in the middle). I’m not sure if that explains the positive result on the Cologuard test, but clearly the colonoscopy was necessary, and I have the great privilege of going back in one year to do it all again. I think the doctor was more concerned than he let on, since he was all “Good news, no cancer, but come back in a year.”