# Unusual power analysis

#### stex3986

##### New Member
Hello,
does somebody have an idea how to perform the following sample size calculations?

I want to conduct a cross-sectional study in a special patient population to proof that screening for a disease does NOT make sense. So basically to exclude this disease in this special patient population.

So I want to include for example 500 to 1.000 patients and show that NOT a single patient has this disease.

any ideas?

#### hlsmith

##### Less is more. Stay pure. Stay poor.
Will you have a control group not screened? A comparator group helps to show a relative change in discovery. Though in your scenario you would look for an equivalency between discovery. Tell us a little more about the setting. Also, "NOT a singe patient" is a very strong hypothesis. I could live in Europe and say there are no black swans.

#### stex3986

##### New Member
Will you have a control group not screened? A comparator group helps to show a relative change in discovery. Though in your scenario you would look for an equivalency between discovery. Tell us a little more about the setting. Also, "NOT a singe patient" is a very strong hypothesis. I could live in Europe and say there are no black swans.
Thanks a lot for your input.

This disease (paroxysmal nocturnal hemoglobinuria) is very rare (prevalence of 1:100.00), but has a phenomenal treatment, so we want to detect it as early as possible to improve prognosis. But it does not make sense to screen for a disease with a prevalence of 1:100.000 in the general population.
Therefore, more defined patient populations have been suggested (here the prevalence might be 1:1000 or lower). My aim is to evaluate the role of screening for this disease in a patient cohort with unspecific signs of this disease. My hypothesis is that it does not make sense to screen for this disease at all, but I would like to show it by establishing a cohort of approximately 400-1000 patients and screen for this disease.

#### hlsmith

##### Less is more. Stay pure. Stay poor.
You could determine the positivity rate in the sample and slap confidence intervals on the estimate. If the interval was precise enough, you would make the conclusion to generalize it as an estimate of the true rate of population. That would be your metric for claiming the futility of the test. If you have a suspected positivity rate in your mind you can try to play around with sample sizes to see how many people you would need to screen in order to have reasonable confidence intervals.

Other things to consider would be the sensitivity, specificity, etc. of the test. If it has poor diagnostic/screening properties it would have lower utility as well. Traditionally if you were trying to examine the utility of a test you would take these things into consideration as well. A disease with low prevalence will typically also have poor gains in predictive probability.

#### katxt

##### Active Member
So I want to include for example 500 to 1.000 patients and show that NOT a single patient has this disease.
There is a "Rule of three" which says that if you want to show that the rate is less than 1 in n you need to see 3n patients without seeing the disease. So if you test 1500 patients without a positive, you can be 95% sure that the true rate is less than 1 in 500.
Any help?

#### hlsmith

##### Less is more. Stay pure. Stay poor.
@katxt - only slightly familiar with this concept. I will say what happens when their is at least one positive, so at the 501 trial? Would you do this confidence interval calculation based on 3/500, which seems off? I guess what I am getting at, is if @stex3986 has any positives how would they proceed?

#### stex3986

##### New Member
Thank you both for your thoughts. @hlsmith I made progress by thinking about your approach, but I am intrigued by the simplicity of @katxt approach. I think that is what I am looking for, and it seems there is some statistical work to back this up.
That's a great example for Occam's razor - I think I will use this approach.
If there are any positives in the population of 3n, screening might still be useful. If i have no positive, screening is likely not useful.

#### katxt

##### Active Member
Here is a site, but you have probably found your own - https://en.wikipedia.org/wiki/Rule_of_three_(statistics)
It is based on the Poisson approximation to the binomial distribution and the fact that P(0) = exp(-L) = exp(-3nx1/n)) = exp(-3) = 5% (near enough).
No doubt there will be other less attractive expressions if there are any positives but I don't know of them. You could probably use a binomial confidence interval calculator.

#### hlsmith

##### Less is more. Stay pure. Stay poor.
So, given the rule of three, if @stex3986 screens 1000 people and no positives are detected, then the inferred confidence interval for the population rate is 0-333 infections. How would you use that @stex3986?