There are many times when certain data elements are lost, discarded, ignored, or are otherwise excluded from analysis. Truncated and censored models have been developed to deal with these types of data. Both models take two forms: truncation or censoring from below, and truncation or censoring from above. Count model forms take their basic logic from truncated and censored continuous response data, in particular, from Tobit (Amemiya, 1984) and censored normal regression (Goldberger, 1983) respectively.

The traditional parameterization used for truncated and censored count data can be called the econometric or cut-point parameterization. This is the form of model discussed in standard econometric texts and is the form found in current econometric software implementations. I distinguish this from what I term a survival parameterization, the form of which is derived from standard survival models. This parameterization only relates to censored Poisson and censored negative binomial models. I shall first address the more traditional econometric parameterization.

Censored and truncated models – econometric parameterization

Censored and truncated count models are related, with only a relatively minor algorithmic difference between the two. The essential difference relates to how response values beyond a user-defined cut-point are handled. Truncated models eliminate the values altogether; censored models revalue them to the value of the cut-point. In both cases the probability function and log-likelihood functions must be adjusted to account for the change in distribution of the response. We begin by considering truncation.