Comparison of Confidence and Prediction Intervals

1 2 A major focus for transportation safety analysts is the development of crash prediction models, a 3 task for which an extremely wide selection of model types are available. Perhaps the most common 4 crash prediction model is the negative binomial (NB) regression model. The NB model gained 5 popularity due to its relative ease of implementation and its ability to handle overdispersion in 6 crash data. Recently, many new models including the Poisson-inverse-Gaussian, Sichel, Poisson7 lognormal, and Poisson-Weibull models have been introduced as they can also accommodate 8 overdispersion and could potentially replace the NB model, since many have been found to 9 perform better. All five of the aforementioned models, including the NB model, can be classified 10 as mixed-Poisson models. A mixed-Poisson model arises when an error term, following a chosen 11 mixture distribution, enters the functional form for the Poisson parameter. For the NB model, the 12 mixture distribution is selected as gamma, hence the alternate model name of Poisson-gamma 13 model. In this paper, confidence intervals (CIs) for the Poisson mean (μ) and Poisson parameter 14 (m, alternately referred to as safety), as well as prediction intervals (PIs) for the predicted number 15 of crashes at a new site are derived for each of the aforementioned types of mixed-Poisson models. 16 After the derivations, the theory is put into practice when CIs and PIs are estimated for mixed17 Poisson models developed from a Texas crash dataset. Ultimately, this study provides safety 18 analysts with tools to express levels of uncertainty associated with estimates from safety-modeling 19 efforts instead of simply providing point estimates. 20