Confidence interval of Standardized Mortality Ratio I'm trying to calculate the 95% confidence interval of a standardized mortality ratio, from this article: https://www.sciencedirect.com/science/article/pii/S0167527314018658#bb0020
The observed number of deaths are: 53 and the expected are 48.62, which gives a ratio of: 1.09. I have calculated the confidence using this example:
https://ibis.health.state.nm.us/resource/SMR_ISR.html
but I am not sure it is correct. I have done the following:
SMR = O/E, where SMR is the Standardized Mortality Rate, O is the observed deaths and E is the expected deaths.
CI = 1.96*sqrt(O)/E, where CI is the Confidence Interval. This gives us:
1.96*sqrt(53)/48.62 = 0.29.
Are these calculations correct, or have I forgotten some basic assumptions?
Any help would be much appreciated!
 A: I had the same problem myself. After a few hours of searching and reading, I managed to work out the approximate 95% confidence interval (CI) for the SMR by using the method proposed by     

Vandenbroucke JP. A shortcut method for calculating the 95 percent confidence interval of the standardized mortality ratio. (Letter). Am
  J Epidemiol 1982; 115:303-4.

I found the formulas of all possible CI calculations for the SMR here. It is the documentation of this and this online calculators
I tried out a few formulas in the documentation, specifically the ones that are called approximations
I haven’t tried the Exact Tests calculations - I think they need more programming + I couldn’t figure out how I am supposed to do the iterative process. Perhaps it is easy and but didn't have time to think about it :). If anyone finds a worked example I would be happy to try it out
I have tried this:   
 
Beware that the (√α) refers to the (sqrt) of the observed mortality and the λ to the predicted mortality. DO NOT confuse with the α of the Z score (i.e. the 1.96)
I created this function, that reads the observed and predicted mortality (actual counts) and returns a vector of 2 values. The 1st is the lower limit, and the 2nd the upper limit
  smr_conf =  function(observed, predicted){

  lower = ((sqrt(observed) - 1.96*0.5)^2)/ predicted
  upper = ((sqrt(observed) + 1.96*0.5)^2)/ predicted

  return(c(lower, upper))

}

It worked well    
You can also this r-publication of mine, with a full implementation of working out the SMR and the 95% CI
A: I don't have much experience with this medically related data. 
In principle what you're doing is right. You have a model for the rate of deaths (Poissonian from the description), and you calculate the variance from the expected value. The given prescription from the webpage basically does a normalization in terms of numbers. 
Your result for the "normal" number of deaths for your data sample is 48.62, referring to a ratio of 1. Since fractional people can't die, this has an uncertainty. 
Your sampled ratio is $1.09 \pm 0.29$, well including the ratio of one.
Without further analysis it's hard to argue, that winter swimming is more dangerous than breathing or any other activity.
