# Calculating confidence intervals for "excess events"

I have a study that shows 100 deaths occurred in a cohort of high-risk people. Given mortality rates in the general population, you would only expect 70 deaths to occur. The standardised mortality ratio is 100/70 = 1.4 and I have 30 excess deaths.

I calculated the confidence interval around the ratio using an exact poisson method (in R):

poisson.test(100, 70)
# SMR = 1.43 (95% CI 1.16-1.74); p = 0.0006
# Similar to 1.43 +/- 1.96 * sqrt(100) / 70


Do you know if/how I can calculate a confidence interval around 30 excess deaths? One idea I had was to calculate the rate difference (my study has about 12000 person-years) using the same poisson method and then multiply by person years:

poisson.test(30, 12000)$conf.int * 12000 # 30 excess deaths (95% CI 20-43) # I then realised that this is the same as poisson.test(30, 1)  Is that correct? ## 1 Answer No, that’s not correct. (What would happen if you observed 70 deaths, or 69 deaths?) It’s the number of deaths that (we presume) has a Poisson distribution, not the excess number of deaths. So what you need is a confidence interval for the expected number of deaths in your high-risk population. You can than subtract the expected of number of deaths in the general population (which we’ll assume is estimated so precisely that we can treat it as a constant): > poisson.test(100)$conf.int - 70
[1] 11.4 51.6


This means that the confidence interval for the excess number of deaths is 11.4–51.6.

Note that if you observe few deaths, the lower, or even the upper, limit of the confidence interval may become negative. That’s OK. It just means that the there are negative ‘excess’ deaths (i.e., deficit deaths).