I was asked to determine if the number of HAI infections in 2011 in statistically significant from the number of HAI infection in 2012. I performed a paired T-test using the actually numbers, but am now questioning myself as to whether I should be using the standardized incident rate instead. This is for a 6 hospital network so each hospital is reporting a number that I am adding up for a total yearly number.
You really have count data, so something like Poisson (or negative binomial) regression is appropriate. Also, you say you have actual counts and rates per 10000 patient days. That way you can calculate the actual number of patient days, which measures exposure, and can use Poisson rate regression. This is discussed multiple times here at Cross Validated, for instance here and here, more theoretical discussion here.
R code would look like
mod <- glm(Ninfections ~ offset(log(exposure))+Ihospital+<other predictors>, family=poisson(link="log"), data=<your data frame>, ...)