Contents lists available at ScienceDirect Accident Analysis and Prevention journal homepage: www.elsevier.com/locate/aap Functional forms of the negative binomial models in safety performance functions for rural two-lane intersections Kai Wang ⁎ , Shanshan Zhao, Eric Jackson Connecticut Transportation Safety Research Center, Connecticut Transportation Institute, University of Connecticut, 270 Middle Turnpike, Unit 5202, Storrs, CT, 06269- 5202, USA ARTICLE INFO Keywords: Intersection crashes Rural two-lane Safety performance functions Negative binomial Data heterogeneity Over-dispersion parameterization ABSTRACT Safety Performance Functions (SPFs) play a prominent role in estimating intersection crashes, and identifying the sites with the highest potential for safety improvement. To maximize the crash prediction accuracy, this paper describes the application of different functional forms of the Negative Binomial (NB) models (i.e. NB-1, NB-2 and NB-P) in estimating safety performance functions by crash type for three types of rural two-lane intersections, including three-leg stop-controlled (3ST) intersections, four-leg stop-controlled (4ST) intersections and four-leg signalized (4SG) intersections. Crash types were aggregated into same-direction, opposite-direction, intersecting-direction and single-vehicle crashes. Major and minor road Annual Average Daily Traffic (AADT) were used as predictors in the SPF estimation. In addition, major and minor road AADT were also used as predictors in the estimation of the over-dispersion parameter of the NB models to account for the crash data heterogeneity. In the end, all NB models were compared based on both the model estimation goodness-of-fit and the prediction performance. The model goodness-of-fit indicates that the NB-P model outperforms the NB-1 and NB-2 models for most crash types and intersection types, by providing a flexible variance structure to the NB approaches. The para- meterization of the over-dispersion factor verifies that the over-dispersion parameter of the NB models highly depends on how the variance structure is defined in the model, and the over-dispersion parameter is shown to vary among different intersections for each crash type and can be estimated using both the major and minor road AADT at rural two-lane intersections. The NB-P model is found to more effectively capture the variation of over- dispersion among intersections in NB models, which benefits the accommodation of data heterogeneity in in- tersection SPF development. The prediction performance comparison illustrates that the NB-P model slightly improves the crash prediction accuracy compared with the other two models, especially for the 3ST and 4SG intersections. In conclusion, the NB-P model with parameterized over-dispersion factor is recommended to provide more unbiased parameter estimates when estimating SPFs by crash type for rural two-lane intersections. 1. Introduction and motivation In the United States, reducing traffic crashes at intersections has continuously been a high priority of the transportation agencies in the past few decades, due to the fact that intersection and intersection-re- lated crashes contribute to about 50% of total crashes per year, and lead to one of the largest economic and societal losses (National Highway Traffic Safety Administration (NHTSA, 2015). In order to improve traffic safety at intersections, there has been increasing interest in es- timating crash prediction models and identifying locations with the highest potential for safety improvement. The Highway Safety Manual (HSM) (2010) provides the Safety Performance Functions (SPFs) for intersection crash predictions of several highway facilities including rural two-lane highways, rural multi-lane highways, urban and suburban arterials and freeway ramp terminals. Of which, traffic volumes for both major and minor roads are used as predictors to estimate the crash counts. The SPFs in HSM were estimated using data collected from a limited number of States, in- cluding Washington, Minnesota, Texas and Ohio. Because crash re- lationships in these states are not necessarily representatives of those in the entire country, the HSM recommends a calibration procedure to adjust the predicted crash counts for individual jurisdictions in using the prediction from the intersection SPFs. To achieve a better crash prediction, instead of calibrating the HSM SPFs, a variety of states have collected sufficient intersection data, and estimated their own inter- section SPFs, including Colorado, Florida, Georgia, Illinois, Kansas, https://doi.org/10.1016/j.aap.2019.01.015 Received 18 October 2018; Received in revised form 20 December 2018; Accepted 11 January 2019 ⁎ Corresponding author. E-mail addresses: kai.wang@uconn.edu (K. Wang), shanshan.h.zhao@uconn.edu (S. Zhao), eric.d.jackson@uconn.edu (E. Jackson). Accident Analysis and Prevention 124 (2019) 193–201 0001-4575/ © 2019 Elsevier Ltd. All rights reserved. T