I have this idea in my head that is either bunk or has a name I don't know. (I'm not naive enough to think I'm breaking new ground here!)
Here's my scenario: I would like to know the proportion of a population that has disease x. Instead of taking a random sample of the population, I'm taking what I'm calling a priority sample, which is I start with the people who are most likely to have the disease, then after finding them I go to the next priority or the next most likely. After I test so many, I have a positivity proportion. The more I test, the further down the priority list I get, and the lower my positivity proportion. At some point I stop and then taking my positivity proportion and the number I've tested, I apply it to some distribution (skewed I would guess) and get an estimate of what the actual proportion of the population has the disease with some kind of confidence interval.
Already with this I could imagine I could guess some kind of upper bound after some 10-15 positive tests since the true proportion in the population is unlikely to be higher than my positivity proportion since I started with the highest priority.
Does this method exist? If so, can you steer me in the right direction to learn how to do it properly?
After mulling it over a bit, I realize there's one more detail I haven't included here, which is that I know the size of the total population and therefore can tell you the testing coverage.
In lieu of some fantastic mathematical proofs, I made a few simulations.
First I take a population of 1000 and divide them up into 100 groups of 10. Each of these have some proportion of the disease normally distributed around the population mean which I want to estimate. (I'm setting it at 0.2 here.)
I then order these groups into the highest proportion (priority) to lowest.
So if I were keeping track of my 10-person groups, when I reached the median group (50) I would have the population proportion. But the more common way is a cumulative collection. So if I graph the proportion cumulatively I get this.
This is surprisingly linear after about 20% coverage. And the true proportion seems to lie about 10% below the coverage at 20% for all but the extreme population proportions. So if this slope is consistent, then I could take the population coverage and extend it linearly to 100%.