# Count regression with censored counts that aren't necessarily zero.

Say you have count outcomes $Y_1, ..., Y_n$ out of different population sizes (e.g. offsets in a usual count model) $p_1, ..., p_n$. I want to relate this to a covariate $X_1, ..., X_n$ using a usual log-linear model

$${\rm log}(\lambda_i) = \log(p_i) + \beta_0 + \beta_1 X_i$$

where $\lambda_i$ is the poisson rate of $Y_i$. The problem is a decent proportion of the $Y_i$ are censored, e.g. you don't get an exactly value if $Y_i < m$. So, when $Y_i$ is censored, all you know is $Y_i \in \{0, 1, ..., m-1 \}$. If I wrote my own, this could enter directly into the likelihood and fit that way but I don't really want to do that unless I have to.

Is there existing methods for doing this? I'm assuming there's a buzzword I'm missing here (is it "hurdle model"?? Seems like those are all only when the cut off is zero...) that's making it hard for me to find a canned way to do this. Any thoughts are appreciated.

• BTW, a lot of the censored counts probably are zero, so I could just do zero inflation if it really came down to it ... – within_person Dec 19 '17 at 21:29

Stata has a censored Poisson model (cpoisson) that would allow you to do this pretty easily. It can handle both bottom and top censoring with individual-specific thresholds. Details and references can be found here.

The log-likelihood is pretty straightforward, so it might be be pretty easy to hack together.

• Thanks. Also you're right that the likelihood for the censored data was super simple. I coded it in R and got it to work using optim() – within_person Dec 20 '17 at 19:04