# Poisson Standard Deviation

#### jamesmartinn

##### Member
Hi!

Just a sanity check as I have to do a few computations for a report manually.

Our indicator is defined in the following way: # re-admissions / #initial visits. I have a person level data set that has this information already summarized.

Here is my data set (mock)

Code:
ID	Visits	Readmission
1	3	2
2	0	0
3	1	0
4	2	2
5	0	0
The indicator of interest is then 4/6 = 2/3 or 0.67. The denominator is the number of initial visits, not patients, although we aggregate up by all patients.

How would I arrive at a standard deviation of this statistic? I'm assuming the counts are Poisson so would it just be the square root of 2/3? sqrt(2/3) = 0.816?

#### rogojel

##### TS Contributor
hi,
This would definitely not be Poisson distributed imo. The sums could be approximatively normal, so the number would be the ratio of two normally distributed variables.

regards

#### hlsmith

##### Omega Contributor
I think that is if the underlying data generating process is Poisson, though many datasets have over or under-dispersion.

Though, your outcome of interest seems to be a percentage contained within {0,1}, not a count per se.

#### jamesmartinn

##### Member
I think that is if the underlying data generating process is Poisson, though many datasets have over or under-dispersion.

Though, your outcome of interest seems to be a percentage contained within {0,1}, not a count per se.
Right. The outcome will always be less than or equal to 1 because the numerator is a subset of the denominator by definition. I'm still a bit unsure on how to treat this.

At one point I thought binomial, however many patients (in the full dataset) tend not to have just a single visit - they may have multiple, which seems like the assumption of independence of 'trials' could be violated. Then again, we're not analyzing the data at the patient level, just all the records from all patients at a given hospital, so i'm not sure whether this is even an issue.

#### rogojel

##### TS Contributor
Seems to be a good case for bootstrapping.

regards

#### hlsmith

##### Omega Contributor
I think it would drastically help if you write out your actual overall study question, not just this component of it. I believe many times people will have DVs of rates in models and use Poisson. Such an approach allows you to get risks out of the model instead of odds, since they are looking at incidents. Large enough samples and these are approximately normal, also given that the rates are not really close to the bounds, since there will be CI issues spilling over into non-feasible values.

What is your overall outcome, percentage readmission per patient or you allude to looking at facilities. Yes, treating them all as independent trials could be troublesome depending on your methods and approaches (e.g., a patient has 30 encounters, if not controlled for, this patients contributes 30 observation, so if he has polypharmacy, comorbidies, etc., it makes those covariates seem more predictive of the binary outcome of readmission yes/no).

#### toks

##### New Member
Thanks for the response and explanation