Count data regression models are used for special cases where the response variable takes count values or only non-negative values. Poisson regression models are commonly used to analyze count data. A frequent problem with the use of these models is that the observed variation is greater than expected and mixed Poisson models are alternative models that provide a means of explaining the extra-Poisson variation. Mixed Poisson regression models have extensive research and literature studies, and have been commonly used in fields such as epidemiology, medicine, genetics, economics, engineering, marketing, and in the physical and social sciences. However, in many cases, the analyst does not observe the entire distribution of counts. In such a case, the count data are truncated as the data are observed only over part of the range of the response variable. In this study, we formulate a class of regression models based on a Double Truncated Poisson regression model with random effects. Two different distributions for the random effects, Normal and Gamma, were studied through simulation. Misspecification of these distributions was addressed. Comparisons with the Left Truncated Mixed Poisson model and the regular Mixed Poisson model were presented. It was concluded that with Normal random effects, double and Left Truncated Mixed Poisson models provide a better fit to clustered double truncated count data compared to the regular mixed Poisson model. For Gamma random effects, the Double Truncated Mixed Poisson model provides a better fit to clustered double truncated count data. These models were used to analyze a Transitional Housing Facility data set
