# 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!

• Not sure what the question is. Do you have trouble calculating the confidence interval or trouble with your definitions? BTW, is this homework? – cherub May 9 '18 at 12:21
• Hi cherub, this is not homework, it is a personal interest as the article in question investigates the long term effects of winter swimming. I'm teaching an Outdoor class, and I want to find out whether winter swimming might be harmful in the long run. I'm trying to find out whether the SMR of 1.09 has any statistical significance, with an observed number of deaths = 53 and an expected number of deaths of 48.68. This is out of a group of 894 people. The article doesn't specify how the expected deaths are calculated other than it is compared to the same age group of the general population. – Joakim May 9 '18 at 12:40

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

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.