# Comparing incidence rates

I want to compare to incidence rate’s between two groups (one without a disease and one with).

I was planning to calculate the incidence rate ratio (IRR), i.e. incidence rate group B/ incidence rate group A, and then test if this rate equals to 1, and finally calculate 95% CI intervals for the IRR.

I found a method for calculation the 95% CI in a book (Rosner’s Fundamentals of Biostatistics):

where $a_1$ and $a_2$ are the number of events.
But this approximation is only valid for large enough sample sizes and i think the numer of event I have is to small (maybe for the total comparison it’s okay.)

So I think I should use another method.

Im using R and the exactci package and found that I could maybe use poisson.test(). But this function has 3 methods for defining the two sided p-values: central, minlike and blaker.

So my questions are:

1. Is it correct that to compare two incidence rate ratios im using a test for comparing poisson rates?

2. When in use the poisson.test function in R from the exactci package what method is best?

The vignette for exactci says:

central: is 2 times the minimum of the one-sided p-values bounded above by 1. The name ‘central’ is motivated by the associated
inversion con dence intervals which are central intervals, i.e., they
guarantee that the true parameter has less than $\alpha/2$ probability of
being less (more) than the lower (upper) tail of the 100(1-$\alpha$)%
confidence interval. This is called the TST (twice the smaller tail
method) by Hirji (2006).

minlike: is the sum of probabilities of outcomes with likelihoods less than or equal to the observed likelihood. This is called the PB
(probability based) method by Hirji (2006).

blaker: combines the probability of the smaller observed tail with the smallest probability of the opposite tail that does not exceed
that observed tail probability. The name ‘blaker’ is motivated by
Blaker (2000) which comprehensively studies the associated method for
con dence intervals. This is called the CT (combined tail) method by
Hirji (2006).

My data is:

Group A:
Age group 1: 3 cases    in 10459 person yrs.   Incidence rate: 0.29
Age group 2: 7 cases    in 2279 person yrs.    Incidence rate: 3.07
Age group 3: 4 cases    in 1990 person yrs.    Incidence rate: 2.01
Age group 4: 9 cases    in 1618 person yrs.    Incidence rate: 5.56
Age group 5: 11 cases   in 1357 person yrs.    Incidence rate: 8.11
Age group 6: 11 cases   in 1090 person yrs.    Incidence rate: 10.09
Age group 7: 9 cases    in 819 person yrs.     Incidence rate: 10.99
Total:    54 cases in 19612 person yrs.      Incidence rate: 2.75

Group B:
Age group 1: 3 cases    in 3088 person yrs.   Incidence rate: 0.97
Age group 2: 1 cases    in 707 person yrs.    Incidence rate: 1.41
Age group 3: 2 cases    in 630 person yrs.    Incidence rate: 3.17
Age group 4: 6 cases    in 441 person yrs.    Incidence rate: 13.59
Age group 5: 10 cases   in 365 person yrs.    Incidence rate: 27.4
Age group 6: 6 cases   in 249 person yrs.    Incidence rate: 24.06
Age group 7: 0 cases    in 116 person yrs.     Incidence rate: 0
Total:    28 cases in 5597 person yrs.      Incidence rate: 5.0