The negative Binomial-Lindley model with Time-Dependent Parameters: Accounting for temporal variations and excess zero observations in crash data.

Crash counts are non-negative integer events often analyzed using crash frequency models such as the negative binomial (NB) distribution. However, due to their random and infrequent nature, crash data usually exhibit unique characteristics, such as excess zero observations that the NB distribution cannot adequately model. The negative binomial-Lindley (NBL) and random parameters negative binomial-Lindley (RPNBL) models have been proposed to address this limitation. Despite addressing the issues of excess zero observations, these models may not fully account for unobserved heterogeneity resulting from temporal variations in crash data. In addition, many variables, such as traffic volume, speed, and weather, change with time. Therefore, the analyst often requires disaggregated data to account for their variations. For example, it is recommended to use monthly crash datasets to better account for temporally varying weather variables compared to yearly crash data. Using temporally disaggregated data not only adds the complexity of the temporal variations issue in data but also compounds the issue of excess zero observations. To address these issues, this paper introduces a new variant of the NBL model with coefficients and Lindley parameters that vary by time. The derivations and characteristics of the model are discussed. Then, the model is illustrated using a simulation study. Subsequently, the model is applied to two empirical crash datasets collected on rural principal and minor arterial roads in Texas. These datasets include several time-dependent variables such as monthly traffic volume, standard deviation of speed, and precipitation and exhibit unique characteristics such as excess zero observations. The results of several goodness-of-fit (GOF) measures indicate that using the NBL model with time-dependent parameters enhances the model fit compared to the NB, NBL, and the NB model with time-dependent parameters. Findings derived from crash data collected from both rural minor and principal arterial roads in Texas suggest that the variables denoting the median presence and wider shoulder width are associated with a potential decrease in crash occurrences. Moreover, higher variations in speed and wider road surfaces are linked to a potential increase in crash frequency. Similarly, a higher monthly average daily traffic (Monthly ADT) positively correlates with crash frequency. We also found that it is important to account for temporal variations using time-dependent parameters.