The Poisson inverse Gaussian and generalized Poisson distributions are widely used in modelling overdispersed count data which are commonly found in healthcare, insurance, engineering, econometric and ecology. The inverse trinomial distribution is a relatively new count distribution arising from a one-dimensional random walk model (Shimizu & Yanagimoto, 1991). The Poisson inverse Gaussian distribution is a popular count model that has been proposed as an alternative to the negative binomial distribution. The inverse trinomial and generalized Poisson models possess a common characteristic of having a cubic variance function, while the Poisson inverse Gaussian has a quadratic variance function. The nature of the variance function seems to be an important property in modelling overdispersed count data. Hence it is of interest to be able to select among the three models in practical applications. This paper considers discrimination of three models based on the likelihood ratio statistic and computes via Monte Carlo simulation the probability of correct selection.