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In a sample of 2000 observations, no positive cases were found, but I want to still be able to provide an upper bound for the prevalence.

The general rule that people seem to use is to simply take 3/n, but this is the just same as the upper bound at 95% for 1 case found in the sample.

3/2000=0.15% and the upper bound at 95% for a prevalence of 1 in 2000 is 0.15%.

Surely if 0 cases are observed the upper bound should be lower than if 1 case was found? Are there any other methods I could use to keep in line with the upper bounds at a certain confidence level for other frequency results?

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    $\begingroup$ How do you arrive at your statement that the upper limit for 1 out of 2000 is 0.15%? Are you using a two-sided interval there but a one-sided interval for 0 out of 2000? $\endgroup$
    – mdewey
    Commented Sep 6 at 12:35
  • $\begingroup$ @mdewey two-sided for the upper limit of 1. I guess the upper bound for the 0 case is one sided, but this was just an approximation that I saw recommended in various places using the simple calculation mentioned. Ultimately, I'm not sure how I should create an upper bound for prevalence for 0 cases in 2000 that would be comparable to other results sets I have that have two-sided intervals for other higher prevalence $\endgroup$
    – philtttx3x
    Commented Sep 6 at 13:48
  • $\begingroup$ It does not make sense, at least to me, to calculate a two-sided CI when you know that values below zero are impossible. $\endgroup$
    – mdewey
    Commented Sep 6 at 14:54
  • $\begingroup$ How are you doing your calculations? If they're correct, they should be extremely close to the Poisson confidence limits illustrated at stats.stackexchange.com/a/488834/919. I obtain prevalences (as 95% UCLs) of $2.996/2000$ and $4.744/2000$ for observing zero and one positive cases, respectively. $\endgroup$
    – whuber ♦
    Commented Sep 6 at 15:14
  • $\begingroup$ @mdewey the lower bound would only be below zero for some of the results though and for these I would still like to have a two-sided CI. Are you suggesting I should use a one sided interval when the lower bound is below zero? But then it seems strange if I have any comparison between the two results - e.g. if I use one-sided for the 0 case, but my other test has 4 cases identified, then it feels like I should have a two-sided CI for it, but then the upper bound can't really be compared between the two tests $\endgroup$
    – philtttx3x
    Commented Sep 6 at 15:36

2 Answers 2

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You do not need any "rule of thumb" such as $\frac 3 n$, when you can compute it directly, using binomial confidence intervals.
You also should not use double sided CI's, as you want a "worse case" prevalence, so are only interested in the upper bound.
If you compute a single sided CI of the proportion, using 0 "failures" out of 2000, you get an upper prevalence of $2.9935$ per $2000$, or $0.15$%. (very close to @whuber's value, because for low proportions, Poisson is a very good approximation to the binomial). This is your upper prevalence bound, at 95% confidence (i.e. there is only a 5% chance that the prevalence would be greater). It "jumps" to $.23% for 99% confidence.
You can use this online calculator to play with the numbers.

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The general rule that people seem to use is to simply take $3/n$

(which in your case is 0.15%). This is sometimes called the "rule of three", and there are many questions here about this. (I do not close as duplicate, since this Q & answers also invoke the Poisson distribution).

and search the site.

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