I have indicated that extended negative binomial models are generally developed to solve either a distributional or variance problem arising in the base NB-2 model. Changes to the negative binomial variance function were considered in the last chapter. In this chapter, we address the difficulties that arise when there are either no possible zeros in the data, or when there are an excessive number.

Zero-truncated negative binomial

Often we are asked to model count data that structurally exclude zero counts. Hospital length of stay data are an excellent example of count data that cannot have a zero count. When a patient first enters the hospital, the count begins. Upon registration the length of stay is given as 1. There can be no 0 days – unless we are describing patients who do not enter the hospital, and this is a different model where there may be two generating processes. This type of model will be discussed later.

The Poisson and negative binomial distributions both include zeros. When data structurally exclude zero counts, then the underlying probability distribution must preclude this outcome to properly model the data. This is not to say that Poisson and negative binomial models are not commonly used to model such data, the point is that they should not. The Poisson and negative binomial probability functions, and their respective log-likelihood functions, need to be amended to exclude zeros, and at the same time provide for all probabilities in the distribution to sum to one.