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Comparison of confidence and prediction intervals for different mixed-Poisson regression models

Abstract A major focus for transportation safety analysts is the development of crash prediction models, a task for which an extremely wide selection of model types is available. Perhaps the most common crash prediction model is the negative binomial (NB) regression model. The NB model gained popularity due to its relative ease of implementation and its ability to handle overdispersion in crash data. Recently, many new models including the Poisson-Inverse-Gaussian, Sichel, Poisson-Lognormal, and Poisson-Weibull models have been introduced as they can also accommodate overdispersion and could potentially replace the NB model, because many have been found to perform better. All five of the aforementioned models, including the NB model, can be classified as mixed-Poisson models. A mixed-Poisson model arises when an error term, following a chosen mixture distribution, enters the functional form for the Poisson parameter. For the NB model, the mixture distribution is selected as gamma, hence the alternate model name of Poisson-Gamma model. In this paper, confidence intervals (CIs) for the Poisson mean ( ) as well as prediction intervals (PIs) for the Poisson parameter ( alternately referred to as the safety), and the predicted number of crashes at a new site ( ) are derived for each of the aforementioned types of mixed-Poisson models. After the derivations, the theory is put into practice when CIs and PIs are estimated for mixed-Poisson models developed from an animal-vehicle collision data set. Ultimately, this study provides safety analysts with tools to express levels of uncertainty associated with estimates from safety-modeling efforts instead of simply providing point estimates.