Cumulative counts or counts for Poisson regression

Question

I have a set of data with measurements X1 and X2 across multiple time points, T1, T2 and T3. I would like to conduct a Poisson regression using X1 and X2 on the counts of a phenomenon. An example of how my dataset would like is as follows (pardon my inelegant code):

library(lme4)
set.seed(888)
df <- data.frame(SubID = sort(rep(1:10, 3)), Time = rep(1:3, 10), Count = sample(0:9, 30, replace = TRUE))
df <- data.frame(df, Cumulative.count = unlist(lapply(1:10, function(n) cumsum(df[cbind(c((1:10*3)-2), c(1:10*3))[n,1]:cbind(c((1:10*3)-2), c(1:10*3))[n,2],"Count"]))), x1 = rnorm(30), x2 = rnorm(30), x3 = rnorm(30))

As can be seen, I have 2 "counts" variable. One is cumulative, one is not. My question is should I be doing:

model.count <- glmer(Count ~ x1 + x2 + x3 + (Time|SubID), family = poisson(link = "log"), data = df)

OR:

model.cumulative.count <- glmer(Cumulative.count ~ x1 + x2 + x3 + (Time|SubID), family = poisson(link = "log"), data = df)

Answer 1

The approach generally taken is to regress the counts on features that were present during the intervals during which the counts accumulated. The length of the interval is used as an offset after applying log() to the values to match the default link for a Poisson model. The data situation described does not justify anything very fancy. A glm model would suffice. The first offset would be log(T1) ,,, assuming T0 was 0 ... and the second offset would be log(T2-T1).

Edit: with the better description/illustration of your data, I would say definitely to go with the first of your alternatives. You are not interested in the cumulative value per period but the incremental value per period. See comment below.