# Compare incidence rate

During the follow-up time, event A will happen multiple times, and there are 3 types of A, I call it A1, A2 and A3. I wonder if there is a statistical method that I can calculate the incidence rates of A1, A2 and A3 separately and compare the incidence rate to each other?

Should I only consider A1 as the event of interest, and consider A2 and A3 as no event, then the incidence rate be N of A1 events/total Person-years during follow-up? And for A2, incidence rate is N of A2/total person-year during follow-up? Would it be problematic that the denominators are the same? How can I compare them?

Thank you so much for your input, Bernhard! I have provided more details as below:

Yes, one person can have more than one event A during follow-up.

Can they have different types of A at once or consequently?

They can only have one type of A at each time but could have different types of A during follow-up.

Can a patient have more then one A per year?

In our data, there is no such case, but theoretically patient can have multiple A in one year.

Maybe we can model this as a binomial experiment where each person-year comes with the same probability of each A? Then a proportions test would be the obvious choice. Are the person-years censored (like: Wie have some individuals with 20 person-years observed an some with only 2 person-years observed)? That would draw things towards survival models

*Yes, person-years could be censored, and some patients have quite short follow-up time.

• Please give more information on A. Can one person have mot then one event A? Can they have different types of A at once or consequently? Can a patient have more then one A per year? Maybe we can model this as a binomial experiment where each person-year comes with the same probability of each A? Then a proportions test would be the obvious choice. Are the person-years censored (like: Wie have some individuals with 20 person-years observed an some with only 2 person-years observed)? That would draw things towards survival models$\dots$ Please specify. – Bernhard Jan 23 '18 at 10:13

Let's assume, the incidence rate of A1, A2 and A3 are constant values (not increading or decreasing in time), then this could probably be modelled as $Poisson$-distributed. (Though you would need to check for overdispersion and then consider negative-binomial distribution or sth. similar. Let's stick to Poisson for the time being, just to show the approach).

A Poisson distribution is defined by the incidence rate $\lambda$ (the incidence per interval): https://en.wikipedia.org/wiki/Poisson_distribution

You should now computer plausible values for the three lamdas $\lambda_{A_1}$, $\lambda_{A_2}$ and $\lambda_{A_3}$ to check, how much they overlap.

One out of endless possibilities to do that practically in R would be an Baysian regression using MCMCpack to illustrate:

library(MCMCpack)
counts <- c(18,17,15,5,10,13,27,29,31) # i.e., counts per interval
treatment <- gl(3,3) # three different treatments, three of each type
posterior <- MCMCpoisson(counts ~ treatment-1)
plot(posterior)
summary(posterior)


If you are not into Baysian analysis, you could certainly find a function for a 95% confidence interval for $\lambda$ in your favourite statistcs program. In Rsomething similar to:

summary(glm(counts ~ treatment -1, family = "poisson"))

• Thanks a lot, Bernhard! I wonder what I can conclude if the 95% CI do overlap. There may still be a significant difference even the confidence intervals slightly overlap, right? Is there a statistical test or how I can calculate the 95% CI of rate ratios to see if there is a significant difference? – AprilCS Jan 25 '18 at 1:21
• As you speak of confidence intervals, I take it, that you are more interested in a Frequentist analysis. As you might have seen in my example, the summary of a glm object gives you $p$ values for each coefficient. In this simple case you could compute three models counts~treatment(no -1!) with just two different treatments/A-types each time and thus compute pairwise significance. That is certainly not the most elegant but probably simplest approach to $p$ values. – Bernhard Jan 25 '18 at 6:54
• Thank you for your reply, Bernhard! I am not very familiar with R codes, but I suppose "treatment" in the codes was considered as an independent variable. However, in my data, there is not a treatment variable that could divide all individuals into 3 groups. I only know the type of event, but for those without an event, we could not group them (such as cause-specific death, if we ignored the recurrent nature). So I think the model you provided is not suitable for my purpose? – AprilCS Jan 29 '18 at 20:42
• If you are not using R codes, the reasoning still is, that you could/should try to model this as Poisson or negative-binomially distributed and make inference on the incidence parameters in such a model. No matter what statistics software you use, there should be functions for generalized linear models - probably the place to look for Poisson- or NB-regression. When making my example code I started with some ready made example that I adjusted. There, they wanted to compare the incidences of some counts depending on a treatment. You will want to examine the incidences of three types of A. – Bernhard Jan 30 '18 at 7:54