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category: literaturenote citekey: wanginvestigatingexposuremeasuresfunctional2020 title: Investigating exposure measures and functional forms in urban and suburban intersection safety performance functions using generalized negative binomial - P model authors: "Wang, Kai; Zhao, Shanshan; Jackson, Eric" year: 2020 date: 2020-00-00 2020 doi: 10.1016/j.aap.2020.105838 publication: "Accident Analysis & Prevention" url: "https://doi.org/10.1016/j.aap.2020.105838" zotero_key: 58CFNRZT zotero_storage: SRN25FH2 collections: transpordi_planeerimine folder: 001_artiklid firstAuthor: "Wang, Kai"

status: converted

Contents lists available at ScienceDirect

Accident Analysis and Prevention

journal homepage: www.elsevier.com/locate/aap

Investigating exposure measures and functional forms in urban and suburban intersection safety performance functions using generalized negative binomial - P model

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: Exposure measures Generalized negative binomial - P model Safety performance functions Data heterogeneity Crash type

ABSTRACT

Selecting an appropriate exposure measure and functional form for Safety Performance Functions (SPFs) is critical in precisely predicting crash counts by different crash types for intersections. This study proposes a new approach, namely Generalized Negative Binomial-P (GNB-P) model, to model the complex relationship between crashes and different exposure measures by crash type for intersections, which helps not only identify the most reliable exposure measure for intersection SPFs, but also explore the most appropriate functional form of the NB models. To this end, three types of SPF functional forms, namely Power function, Hoerl function 1 and Hoerl function 2 with different exposure measures including major road AADT, minor road AADT and total AADT were estimated by crash type for stop-controlled and two types of signalized intersections. The over-dispersion of the SPF models was estimated using the exposure measures to account for crash data variation across different intersections.

The SPF estimation results highlighted that the mean-variance structure of NB models is not consistent and varies by crash data. The over-dispersion of SPFs by crash type is not constant and varies across different intersections. The minor road AADT is shown to be positively correlated with the over-dispersion of SPFs in estimating crash counts for Same-Direction Crashes (SDC), Intersecting-Direction Crashes (IDC) and Single-Vehicle Crashes (SVC). Estimating the over-dispersion using exposure measures results in more reliable SPF results. Furthermore, it is found that the Power function with major road and minor road AADT as the exposure measure performs the best in estimating SPFs for Opposite-Direction Crashes (ODC). The Hoerl function 2 with total AADT and the proportion of minor road AADT over the total as the exposure measure performs the best in estimating SVC SPFs for intersections. The Hoerl function 1 with major road and minor road AADT as the exposure measure is more accurate in estimating SPFs for both SDC and IDC.

1. Introduction and motivation

Due to the large number of conflict points generated between turning vehicles and vehicles going straight, drivers are more prone to make mistakes at intersections which leads to severe crash outcomes (Wang and Qin, 2015) and hence reducing traffic crashes at intersections has continuously been a top priority of the transportation agencies and engineers in the past few decades. According to the report published by National Highway Traffic Safety Administration (National Highway Traffic Safety Administration (NHTSA, 2010), intersection and intersection-related crashes contribute to about 40 % of total crashes per year, and lead to one of the largest economic and societal losses. The effort to precisely identify hazardous intersections and minimize the negative impacts of intersection crashes continue to be one of the most important topics in the traffic safety field.

Despite different methodological options, a crash prediction model (CPM) or a safety performance function (SPF) is still the primary approach used to predict the number of long-term expected crashes and identify intersections with higher expected crashes, which are candidates for safety improvement. The Highway Safety Manual (HSM, 2010) serves as a benchmark for intersection SPFs of multiple facilities, including rural two-lane highways, rural multi-lane highways, urban and suburban arterials and freeway ramp terminals. The HSM recommends a calibration procedure for individual agencies in using the

E-mail addresses: kai.wang@uconn.edu (K. Wang), shanshan.h.zhao@uconn.edu (S. Zhao), eric.d.jackson@uconn.edu (E. Jackson).

* Corresponding author.

intersection SPFs. However, the calibration might still violate the actual relationship between crashes and independent variables in the specific agency's data (Wang et al., 2019a). To achieve better crash predictions at intersections, States have estimated SPFs using their own data including, but are not limited to Colorado, Connecticut, Florida, Georgia, Illinois, Kansas, North Carolina, Oregon, Utah, Virginia and Michigan (Gates et al., 2018).

In terms of SPF estimation for intersections, traffic volumes or Annual Average Daily Traffic (AADT) is one of the key factors which reflects the exposure to the risk at intersections. Although research has demonstrated that the relationship between crashes and AADT is nonlinear (Qin et al., 2004; Jonsson et al., 2007; Ivan, 2004), their underlying relationship is more complex than a simple non-linear function and could vary by different crash types (Qin et al., 2004; Ivan, 2004). Therefore, identifying the appropriate measure of exposure is an important step forward for estimating reliable intersection SPFs. In addition, the Negative Binomial (NB) or Poisson-Gamma model has been widely employed to estimate the SPFs by addressing the over-dispersion issue of crash data (Lord and Mannering, 2010; Washington et al., 2011). However, most of the previous research focuses on investigating the relationship between crashes and exposure measures using the traditional NB model, which defines the relationship between the crash variance and mean as a quadric form (Wang et al., 2019b). The restricted mean-variance structure in traditional NB model may lead to biased SPF estimation results, which eventually fails to identify the true relationship between crash counts and exposure measures (Park, 2010; Wang et al., 2019a; Greene, 2008).

Consequently, this study proposes a new approach, namely Generalized Negative Binomial-P (GNB-P) model, to model the complex relationship between crashes and different exposure measures by crash type for intersections, which helps not only identify the most reliable exposure measure for intersection SPFs, but also explore the most appropriate functional form of the NB models.

2. Literature review

Recent research has well demonstrated that the relationship between crash counts and exposure is non-linear (Hauer, 2004; Qin et al., 2004; Ivan, 2004; Jonsson et al., 2007). Moreover, studies have verified that the relationship between crashes and exposure varies vastly by crash type and crash severity. Qin et al. (2004) estimated the SPFs for rural two-lane highways by different types of crashes, including single-vehicle crashes, multi-vehicle same direction crashes, multi-vehicle opposite direction crashes and multi-vehicle intersecting-direction crashes. The results show that the relationship between crash counts and AADT is non-linear and varies by crash type. Ivan (2004) proposed a new approach for modeling the relationship between crashes and traffic volumes that can both estimate a crash risk per unit exposure and account for the nonlinear effect of traffic volumes on crash occurrences. The SPFs were estimated for single-vehicle crashes, same-direction crashes, opposite-direction crashes and intersecting-direction crashes. The study concluded that the relationship between traffic volumes and crash counts is complex and it relates to quantities such as the distribution of traffic through the day and the types of crashes experienced. Jonsson et al. (2007) estimated crash prediction models for intersections on rural multilane highways and compared the model differences by collision type. Their study illustrated that the differences in crash prediction models by collision type highly depend on traffic flow, crash severity and variables that best explain between-site variations. Zhang (2008) advanced the state of crash prediction modeling by redefining crash exposure to consider the practical relationship between traffic volume and crashes. The study identified different exposure measures for different crash types based on contributing factors which can result in more accurate and explainable prediction models. Qin et al. (2006) investigated the relationship between crash occurrence and hourly traffic volumes and found that at any time of day, the crash occurrence is

not proportional to the hourly traffic volume for all crash types. The recently accomplished National Cooperative Highway Research Program (NCHRP) 17-62 project (2017) confirmed that the relationship between AADT and crash counts varies among different crash types and crash severities, and SPFs should be estimated respectively by crash type and severity.

Given the non-linear relationship between crashes and exposure, identifying the appropriate measure of exposure and functional form is an important step forward for estimating reliable intersection SPFs because their underlying relationship is more complex than a simple non-linear function (Ivan, 2004; Srinivasan and Bauer, 2013). The most widely used functional form for estimating intersection SPFs is the so-called Power function, in which the relationship between crashes and exposure is estimated as a function of power (Srinivasan and Bauer, 2013). Hauer (2004) argued that the simple Power function may not be always appropriate, because the crash probability given the AADT might vary among different crash types, especially for single-vehicle crashes where the single-vehicle crash rate is not always shown to increase with increasing AADT (Qin et al., 2006). Hence a functional form that allows a point of inflection may be more appropriate. Kononov et al. (2011) then used the artificial neural network (ANN) to explore the underlying relationship between crashes and AADT and proposed that the sigmoidal or logistic function might be more appropriate to estimate SPFs. Another similar option providing more flexibility in the functional form of SPFs is the so-called Hoerl function (Hauer, 2015; Kononov et al., 2011). The Hoerl function allows the relationship between crash counts and AADT to have a convex/concave shape with inflection points. Kononov et al. (2015) introduced the power function and Hoerl function and used the Hoerl function to estimate SPFs for intersections.

In terms of the methodological frameworks, the NB or Poisson-Gamma model has been the most popular approach in estimating intersection SPFs in the past few decades. The NB model can address the over-dispersion issue of crash data (Lord and Mannering, 2010; Washington et al., 2011). The NB model has also been extended to more sophisticated forms as an effort to account for different crash prediction phenomena including, but are not limited to zero-inflated NB model, random parameter NB model, multivariate NB model and spatial-temporal NB model. Nevertheless, one of the drawbacks of these NB methodologies is that they define the relationship between the crash variance and mean as a quadric form, which is known as the NB-2 model (Wang et al., 2019b). Another restricted NB modeling framework is the so-called NB-1 model, which defines a linear relationship between the variance and mean. Multiple studies have been conducted to compare the NB-1 and NB-2 models in SPF estimation (Chang and Xiang, 2003; Mehta and Lou, 2013; Wang et al., 2019a; Lord and Park, 2015). However, the restricted mean-variance structure in both NB-1 and NB-2 models may lead to biased SPF estimation results, which eventually fails to identify the true relationship between crash counts and exposure (Park, 2010; Wang et al., 2019b; Greene, 2008). To address this issue, Greene (2008) extended the two restricted NB models and introduced an encompassing model formula entitled as the NB-P model, which parametrically nests both the NB-1 and NB-2 models while relaxing the mean-variance structure in NB models by allowing the structure to be estimated based on the data. Greene (2008) and Ismail and Zamani (2013) have compared the NB-P model with the NB-1 and NB-2 models in non-transportation research, and both studies confirmed that the NB-P model outperforms the NB-1 and NB-2 models in prediction performance. Our previous paper applied the NB-P model in estimating SPFs by crash type for the first time in crash prediction field (Wang et al., 2019a). The study concluded that the SPFs vary by crash type, and the NB-P model outperforms both NB-1 and NB-2 models in crash prediction accuracy.

Considering the lack of existing research on comparing different SPF functional forms and investigating the effects of model formula on the selection of appropriate exposure measures to estimate reliable intersection SPFs, this study proposes a new approach, namely Generalized

Negative Binomial-P (GNB-P) model to investigate the complex relationship between crashes and different exposure measures by crash type at intersections.

3. Objectives and contributions

The objectives and contributions of this study are as follows:

  • 1 Compare different functional forms for estimating intersection SPFs by crash type. Previous research has verified that the functional forms for intersection SPFs should vary by crash type. This study compares the Power function with two Hoerl functions to identify the most reliable functional form for each crash type at intersections.
  • 2 Compare different exposure measures for estimating intersection SPFs by crash type.

Most of the previous research used the AADT on both major and minor approaches as the exposure measure for intersection SPFs. This study extends the exposure measure by accommodating major and minor AADT with the total AADT, to identify the most reliable exposure measure for each crash type at intersections.

3 Implement the GNB-P model for estimating intersection SPFs, and identify the most reliable function form and exposure measure for estimating crash type counts at intersections.

Two advantages of SPF estimation are covered by the GNB-P model in this study. First, it relaxes the mean-variance structure in traditional NB models which can estimate more unbiased model results. Second, research has also noticed that the over-dispersion parameter in NB regression models may not always be fixed, due to the unobserved heterogeneity in the crash data (Mannering et al., 2016; Miaou and Lord, 2003; Lord and Park, 2008). The over-dispersion parameter for intersection SPFs depends greatly on how the mean-variance structure is modeled, and the over-dispersion is a function of covariates if the crash prediction model suffers from the omitted variables (Mitra and Washington, 2007). Therefore, other than the mean-variance structure, this study also relaxes the over-dispersion in the NB-P model by allowing it to vary by exposure and intersection.

To the best knowledge of the authors, this is the first study in traffic safety field to investigate the selection of NB modeling structures and exposure measures by crash type for intersection SPFs. The rest of paper is organized as follows. Section 3 presents the modeling frameworks and the consolidated exposure measures and SPF functions. Section 4 described the data for model estimation and Section 5 presents the SPF estimation results. The SPF comparisons are conducted in Section 6, and conclusions are provided in the final Section.

4. Statistical methodologies and SPF consolidations

To derive the GNB-P model, let's start with the Poisson and traditional NB models. Suppose the crash counts follow a Poisson distribution with the probability distribution as:

$$Prob[y_i|\mu_i] = \frac{exp(-\mu_i)\mu_i^{y_i}}{y_i!}$$ (1)

where Prob[ yiμi is the probability of y crashes occurring at intersection i, given *μi*expected number of crashes at intersection i. *μi*can then be estimated as:

$$E(y_i) = \mu_i = exp(\beta X_i)$$ (2)

where Xi is a vector of explanatory variables at intersection i and β is a vector of coefficients to be estimated. One limitation of the Poisson regression model is that the variance is restricted to be equal to the mean, i.e.

$$Var(y_i) = \mu_i \tag{3}$$

Previous research has proven that this constraint is not always true in crash prediction, as the variance of crash data is usually greater than the mean, which is known as the over-dispersion (Washington et al., 2011). To overcome this, the NB or Poisson-Gamma model is used by rewriting Eq. 2 as:

$$\mu_i = \exp(\beta X_i + \varepsilon_i) \tag{4}$$

where *εi*is an error term. If the exp(εi) is assumed to follow a gamma distribution with mean 1 and variance 1/σ, then the NB model is achieved which is known as the NB-2 model. The distribution of NB-2 model can be written as (Washington et al., 2011):

$$Prob[y{i}|\mu{i}] = \frac{\Gamma[(\sigma) + y{i}]}{\Gamma(\sigma)y{i}!} \left[ \frac{\sigma}{(\sigma) + \mu{i}} \right]^{\sigma} \left[ \frac{\mu{i}}{(\sigma) + \mu{i}} \right]^{y{i}}$$ (5)

where Γ is a gamma probability function. The mean and variance for the NB-2 model can be written as:

$$E(y_i) = \mu_i \text{ and } Var(y_i) = \mu_i + k\mu_i^2$$ (6)

Here k denotes the over-dispersion parameter in NB-2 model where k = 1/σ. The distribution of NB-2 model in Eq. 5 can be rewritten by replacing σ with k as:

$$Prob[y_i|\mu_i] = \frac{\Gamma\left[\left(\frac{1}{k}\right) + y_i\right]}{\Gamma\left(\frac{1}{k}\right)y_i!} \left[\frac{\frac{1}{k}}{\left(\frac{1}{k}\right) + \mu_i}\right]^{\frac{1}{k}} \left[\frac{\mu_i}{\left(\frac{1}{k}\right) + \mu_i}\right]^{y_i}$$ (7)

Following Greene (2008), in Eq. 7, let's replace 1 *k*with 1 kμi , then the NB-2 is converted to the so-called NB-1 model, with the distribution expressed as:

$$Prob[y{i}|\mu{i}] = \frac{\Gamma\left[\left(\frac{1}{k}\mu{i}\right) + y{i}\right]}{\Gamma\left(\frac{1}{k}\mu{i}\right)y{i}!} \left[\frac{\frac{1}{k}\mu{i}}{\left(\frac{1}{k}\mu{i}\right) + \mu{i}}\right]^{\frac{1}{k}\mu{i}} \left[\frac{\mu{i}}{\left(\frac{1}{k}\mu{i}\right) + \mu{i}}\right]^{y{i}}$$ (8)

The mean and variance for the NB-1 model can be written as:

$$E(y_i) = \mu_i \text{ and } Var(y_i) = \mu_i + k\mu_i$$ (9)

Comparing the mean-variance structure of NB-2 and NB-1 models in Eq. 6 and Eq. 9, the main difference between these two NB models is that the NB-2 model defines a quadric relationship to the variance and mean, while the NB-1 model define their relationship as linear. Nevertheless, both NB-1 and NB-2 restrict the mean-variance structure for all data used in SPF estimation. However, the restricted mean-variance structure in both NB-1 and NB-2 models may lead to biased SPF estimation results as the crash distribution might be different across different crash types (Park, 2010; Wang et al., 2019b; Greene, 2008). Greene (2008) introduced a NB-P regression model which nests both NB-1 and NB-2 models and allows the parameterization of the mean-variance function of the NB models. Similarly, in Eq. 7, let's replace 1 *k*by 1 kμi 2− P, then the NB-2 is converted to the new NB-P model, with the distribution expressed as:

$$Prob[y{i}|\mu{i}] = \frac{\Gamma\left[\left(\frac{1}{k^{\mu}}u{i}^{2-P}\right) + y{i}\right]}{\Gamma\left(\frac{1}{k^{\mu}}u{i}^{2-P}\right)y{i}!} \left[\frac{\frac{1}{k^{\mu}}u{i}^{2-P}}{\left(\frac{1}{k^{\mu}}u{i}^{2-P}\right) + \mu{i}^{2-P}}\right]^{\frac{1}{2}\mu{i}^{2-P}} \left[\frac{\mu{i}^{2-P}}{\left(\frac{1}{k^{\mu}}u{i}^{2-P}\right) + \mu{i}^{2-P}}\right]^{y{i}}$$ (10)

The mean and variance for the NB-P model can be written as:

$$E(y_i) = \mu_i \text{ and } Var(y_i) = \mu_i + k\mu_i^P$$ (11)

where P is a parameter to be estimated for the mean-variance function. The NB-2 and NB-1 are actually two special cases of NB-P model, where P is equal to 2 and 1 respectively. Given the probability function in Eq.

$$k = exp(\gamma1 \times \frac{AADT{major}}{10,000} + \gamma2 \times \frac{AADT{minor}}{10,000} + \gamma_0)$$ (15)

The third SPF structure included in this study is also a Hoerl function, which is denoted as Hoerl function 2. Instead of using major and minor road AADT as the exposure measure, the Hoerl function 2 used the total AADT (a.k.a. Total Entering Volume) and the proportion of minor road AADT over the total AADT as the exposure measure. This is because the total AADT might be more appropriate to estimate SPFs for some specific crash types such as single-vehicle crashes, since only one vehicle is involved in a single-vehicle crash. The formula for the Hoerl function 2 can be written as:v

Hoerl Function 2: $$Y = AADT_{total}^{\beta1} * \left(\frac{AADT{minor}}{AADT_{total}}\right)^{\beta_2} * exp(\beta_0 + \beta3 \frac{AADT{total}}{10,000} + \beta4 \frac{AADT{minor}}{AADT_{total}})$$ (16)

10, the log likelihood function of the NB-P model can be further derived, and the model coefficients can be estimated by the maximum likelihood estimation (MLE) approach (Lord and Park, 2015; Greene, 2008; Ismail and Zamani, 2013).

At this point we consolidate the exposure measures and intersection SPF formulas. Since the objectives of this study are to investigate the interactions between different exposure measures and SPF structures for intersection crashes, only the AADTs were considered as the independent variables in SPF estimation. As mentioned earlier, we also relaxed the over-dispersion in the NB-P model in this study by allowing it to vary by exposure and intersection to account for unobserved heterogeneity.

The first SPF structure included in this study is the traditional Power function, which can be written as:

Power Function: $$Y = AADT_{major}^{\beta1} *AAADT{minor}^{\beta_2} *exp(\beta_0)$$ (12)

The exposure measure used by the Power function is the major and minor approach AADT. The over-dispersion of the Power function was estimated using the major and minor approach AADT as well, with the formula as (Geedipally and Lord, 2008):

$$k = exp(\gamma1 \times \frac{AADT{major}}{10,000} + \gamma2 \times \frac{AADT{minor}}{10,000} + \gamma_0)$$ (13)

Noting that both major and minor road AADT were normalized by 10,000 in the over-dispersion function. This is because the original value of AADT is very large, which creates challenges for operating the exponential function.

The second SPF structure included in this study is the Hoerl function. To differentiate with the next one, we referred this as Hoerl function 1. The formula can be written as:

The over-dispersion of the Hoerl function 2 was estimated using the same exposure measure as the SPF, i.e.

$$k = exp(\gamma1 \times \frac{AADT{total}}{10,000} + \gamma2 \times \frac{AADT{minor}}{AADT_{total}} + \gamma_0)$$ (17)

5. Data preparation and crash type categorization

SPFs for 4-leg intersections on urban and suburban arterials at the State of Connecticut were estimated in this study. The most recent fiveyear (2015–2019) crash data were collected from the Connecticut Crash Data Repository (CTCDR, 2020). SPFs were estimated by crash type to account for crash distribution variations. Following the previous research, crash types were categorized and aggregated into four categories, based on the original travel direction of involved vehicles in a crash (Wang et al., 2017, 2018, 2019a, 2019b; Qin et al., 2004, 2006; Ivan, 2004; Zhang, 2008). The crash type categorization and aggregation is summarized in Fig. 1.

Intersections were categorized into all-way stop-controlled (e.g. intersections with stop control on all roads) and signalized intersections. 1095 stop-controlled intersections and 1552 signalized intersections were included for SPF estimation in this study. Signalized intersections were further separated into 830 two-lane signalized intersections and 722 multi-lane signalized intersections. Furthermore, we randomly selected 80 % of data to estimate SPFs, and used the remaining 20 % of data to validate and compare SPFs for both intersection types. The summary statistics of major road AADT, minor road AADT, total AADT

$$\textit{Hoerl Function } 1: Y = AADT_{\textit{major}}^{\beta1} * AADT{\textit{minor}}^{\beta_2} * exp(\beta_0 + \beta3 \frac{AADT{\textit{major}}}{10,000} + \beta4 \frac{AADT{\textit{minor}}}{10,000}) \tag{14}$$

and crash type counts by intersection type are described in Table 1.

The exposure measure used in the Hoerl function 1 is still the major and minor road AADT. Comparing to the Power function, the Hoerl 6. SPF estimation results

Table 2 through Table 4 present the SPF estimation results for urban and suburban stop-controlled and two types of signalized intersections. SPFs were estimated using three types of functional forms, i.e. Power function, Hoerl function 1 and Hoerl function 2, for four crash types, i.e. SDC, ODC, IDC and SVC respectively, and for total crashes (TOT) as a

measure to have a convex/concave shape with inflection points (Hauer, 2015). Similar to the Power function, major and minor road AADT were normalized by 10,000 in the exponential function. The over-dispersion is the same as the Power function, i.e.

function allows the relationship between crash counts and exposure

Fig. 1. Crash Type Categorization and Aggregation.

Table 1 Summary Statistics of Connecticut Intersection Data (2019).

Intersection Variable Min. Max. Mean Sum
ODC 0 5 0.3 329
SDC 0 112 3.5 3,833
Stop-Controlled IDC 0 38 2.8 3,066
Intersections SVC 0 16 0.8 876
(1095) AADTmajor 550 34,400 9,442 –
AADTminor 20 20,300 2,633 –
AADTtotal 1,950 44,900 12,075 –
ODC 0 8 0.8 700
SDC 0 81 11.4 9,484
Two-Lane Signalized IDC 0 48 5.6 4,608
Intersections SVC 0 19 1.5 1,215
(830) AADTmajor 2,300 44,400 12,644 –
AADTminor 200 43,300 6,155 –
AADTtotal 3,500 54,600 18,799 –
ODC 0 11 1.2 846
SDC 0 138 23.4 16,895
IDC 0 71 9.8 7,062
Multi-Lane Signalized SVC 0 17 1.8 1,292
Intersections (722) AADTmajor 3,500 68,200 19,960 –
AADTminor 400 43,300 8,810 –
AADTtotal 5,900 96,700 28,769 –

reference. Coefficients with "—" are not statistically significant at the 10 % level of significance. All the P parameters in GNB-P models were reported for interpretation purpose. Here we briefly introduce how to compare different NB models using statistical tests. As mentioned earlier, The NB-2 and NB-1 are actually two special cases of NB-P model, where P is equal to 2 and 1 respectively. Therefore, the likelihood ratio test can be used in comparing the NB-1 or the NB-2 with the NB-P model, since the NB-P model parametrically nests both the NB-1 and NB-2 models (Greene, 2008; Wang et al., 2019a). To illustrate the testing procedure, the likelihood ratio between either the NB-1 or the NB-2 models and the NB-P model can be calculated as 2 × [LLNB− P − LLNB− 1 or NB− 2 ] and compared to the χ2 1. We didn't provide the detailed discussion of comparing NB-1 and NB-2 models with the NB-P model here because this is not within the scope of this study. If readers are

interested in comparisons among NB-1, NB-2 and NB-P models, please refer to our paper published previously (Wang et al., 2019b). In addition, the likelihood ratio test can also be used to test the Power function with the Hoerl function 1 since these two SPFs are also nested. However, the likelihood ratio test cannot be used to test any pairs of SPFs involving the Hoerl function 2 because they have different model structures.

Table 2 shows the SPF estimation results for stop-controlled intersections. As we can see, the P parameters in GNB-P models are statistically significant in all SPF functional forms in items of SDC, IDC and SVC. It is not significant in ODC SPFs which might be due to the low sample size. This verifies one of the previous assumptions that the meanvariance structure in NB models is not consistent and thus should be estimated by the data. In terms of the SPFs estimated for ODC, the minor road AADT seems to be the most important exposure variable than the others because it is statistically significant in all three SPF models. The estimation of over-dispersion for ODC SPFs yields a constant value across all intersections. For SDC, major AADT is significant at the exponential term in Hoerl function 1, while the percentage of minor AADT over the total is significant at the exponential term in Hoerl function 2. The over-dispersion estimation for SDC yields different results from the ODC. One interesting finding is that the over-dispersion of SDC SPFs varies by intersection, and it highly depends on the minor road AADT. More specifically, greater minor road AADT leads to a larger over-dispersion in GNB-P models for SDC SPFs. The SPFs show complex relationships between different exposure measures and IDC counts. Minor AADT is shown to be significant at the exponential term in Hoerl function 1, and both minor road AADT and total AADT are significant at the exponential term in Hoerl function 2. The results highlight that the over-dispersion for IDC SPFs varies by intersection and can be estimated by both major and minor road AADT. One interesting finding in IDC SPFs is that minor road AADT increases the over-dispersion while major road AADT decreases the over-dispersion in GNB-P models. With regard to the SVC, one important conclusion here is that the major road AADT seems to be not correlated with the SVC counts, while the total AADT and the percentage of minor road AADT over the total are more important in estimating SVC SPFs. This verifies another assumption in this study that the total AADT might be more appropriate to estimate SPFs for some specific crash types compared to the major and minor road

Table 2 SPF Estimation Results for Stop-Controlled Intersections.

Power Function: Y = AADTmajorβ1 ∗AADTminorβ2 ∗exp⁡(β0)
(		 )
Overdispersion: k = exp γ1 × AADTmajor
+ γ2 × AADTminor
10, 000
10, 000		 + γ0
Parameters TOT ODC SDC IDC SVC
β1 0.67 (<0.01) 0.29 (0.04) 1.25 (<0.01) 0.34 (<0.01) —
β2 0.60 (<0.01) 0.55 (<0.01) 0.62 (<0.01) 0.60 (<0.01) 0.44 (<0.01)
β0 − 8.79 (<0.01) − 8.21 (<0.01) − 15.10 (<0.01) − 6.69 (<0.01) − 4.57 (<0.01)
γ1 — — — − 0.53 (<0.01) —
γ2 1.33 (<0.01) — 1.03 (0.04) 1.53 (<0.01) —
γ0 — — − 0.81 (<0.01) 0.73 (<0.01) —
P 1.47 (<0.01) 1.47 (0.46) 1.90 (<0.01) 1.95 (<0.01) 1.68 (<0.01)
¡2LL 4,820.70 1,083.20 3,514.00 3,594.90 2,063.50
Hoerl Function 1: Y = AADTmajorβ1 ∗AADTminorβ2 ∗exp⁡(β0 + β3 AADTmajor
AADTminor
10, 000 + β4
10, 000 )
Overdispersion: k = exp(γ1 × AADTmajor
+ γ2 × AADTminor
10, 000		 + γ0)
10, 000
Parameters TOT ODC SDC IDC SVC
β1 0.72 (<0.01) — 1.65 (<0.01) 0.51 (0.01) —
β2 0.62 (<0.01) 0.59 (<0.01) 0.54 (<0.01) 0.81 (<0.01) 0.49 (<0.01)
β3 — — − 0.41 (0.04) — —
β4 — — — − 0.86 (0.01) —
β0 − 9.00 (<0.01) − 10.37 (<0.01) − 17.85 (<0.01) − 9.39 (<0.01) − 6.59 (<0.01)
γ1 — — — − 0.64 (<0.01) —
γ2 1.28 (<0.01) — 1.08 (0.05) 0.88 (0.03) —
γ0 — — − 0.85 (<0.01) 0.68 (<0.01) —
P 1.45 (<0.01) 1.50 (0.38) 1.94 (<0.01) 1.30 (<0.01) 1.76 (0.07)
¡2LL 4,820.40 1,082.50 3,507.00 3,588.50 2,062.10
Hoerl Function 2: Y = AADTtotalβ1 ∗ (AADTminor
)β2
AADTtotal		 AADTtotal
AADTminor
∗exp⁡(β0 + β3
10, 000 + β4
AADTtotal		 )
Overdispersion: k = exp(γ1 × AADTtotal
10, 000 + γ2 × AADTminor		 + γ0)
AADTtotal
Parameters TOT ODC SDC IDC SVC
β1 1.30 (<0.01) 1.07 (0.02) 2.06 (<0.01) 1.39 (<0.01) 0.84 (<0.01)
β2 0.66 (<0.01) 0.68 (0.01) 0.69 (<0.01) 0.76 (<0.01) 0.45 (<0.01)
β3 — — — − 0.35 (0.05) —
β4 − 1.24 (0.01) — − 2.12 (<0.01) − 1.40 (0.06) —
β0 − 8.83 (<0.01) − 9.69 (0.02) − 16.48 (<0.01) − 9.94 (<0.01) − 7.00 (<0.01)
γ1 0.40 (0.05) — 0.51 (0.09) — —
γ2 2.30 (<0.01) — 1.34 (0.08) 1.83 (0.01) 5.00 (0.01)
γ0 − 0.74 (<0.01) — − 1.20 (<0.01) — − 2.45 (0.02)
P 1.54 (<0.01) 1.40 (0.37) 1.33 (<0.01) 1.24 (<0.01) 1.87 (<0.01)
¡2LL 4,822.50 1,081.70 3,513.80 3,599.30 2,058.70

Notes: "—" represents the coefficient is not statistically significant at 10 % significance level.

AADT. The estimation of over-dispersion for SVC SPFs yields a constant value across different intersections.

Table 3 shows the SPF estimation results for two-lane signalized intersections. The results are fairly consistent with stop-controlled intersections. The P parameters in GNB-P models are statistically significant in all SPF functional forms and crash types for two-lane signalized intersections, except for the ODC SPF in power function and both ODC and SVC in Hoerl function 1. Minor road AADT is the most significant exposure variable in all three SPF functions. For two-lane signalized intersections, none of the factors are correlated with the over-dispersion for ODC SPFs in all three functions. In terms of the SDC, both major and minor road AADT are found to be significant at the exponential term in Hoerl function 1. The over-dispersion of both Power function and Hoerl function 1 is significantly correlated with the minor road AADT, while the over-dispersion of the Hoerl function 2 is highly correlated with both the total AADT and the percentage of minor road traffic volume over the total. The three SPF functions show similar results for IDC, where the relationship between the major road AADT, minor road AADT or total AADT and the IDC counts is a power function. The over-dispersion is positively correlated with the minor road AADT in power function for IDC. The major road AADT is not significantly correlated with the SVC counts in Hoerl function 1. The minor road AADT is positively correlated with the over-dispersion of all three SPFs in estimating crash counts for SVC.

Table 4 presents the SPF estimation for multi-lane signalized intersections, which yields similar conclusions with the two-lane signalized intersections. In terms of the ODC SPFs, major AADT is not significant in all three SPF functions. Minor AADT is significant at the exponential term in Hoerl function 1. The P parameters in GNB-P models are statistically significant in both power function and Hoerl function 1. For SDC SPFs, minor AADT is shown to be significant at the exponential term in Hoerl function 1. Total AADT and the proportion of minor road AADT over the total are significant at the exponential term in Hoerl function 2. Minor AADT is positively correlated with the over-dispersion in power and Hoerl 1 functions, and both total AADT and the proportion of minor road AADT over the total are positively correlated with the over-dispersion in Hoerl function 2. The P parameters are statistically significant in all three SPF functions. With regard to IDC, only the total AADT is significant at the exponential term in Hoerl function 1, and minor AADT is positively correlated with the over-dispersion in all three

Table 3 SPF Estimation Results for Two-Lane Signalized Intersections.

Power Function: Y = AADTmajorβ1 ∗AADTminorβ2 ∗exp⁡(β0)
Overdispersion: k = exp(γ1 × AADTmajor + γ2 × AADTminor
10, 000
10, 000		 + γ0)
Parameters TOT ODC SDC IDC SVC
β1 0.82 (<0.01) 0.41 (0.01) 1.20 (<0.01) 0.33 (<0.01) 0.26 (0.05)
β2 0.42 (<0.01) 0.44 (<0.01) 0.42(<0.01) 0.49 (<0.01) 0.37 (<0.01)
β0 − 8.43 (<0.01) − 7.85 (<0.01) − 12.53(<0.01) − 5.54 (<0.01) − 5.24 (<0.01)
γ1 — — — — —
γ2 0.27 (<0.01) — 0.21 (<0.01) 0.30 (<0.01) 0.74 (<0.01)
γ0 − 1.58 (0.06) — − 1.68 (0.03) — —
P 1.72 (<0.01) 1.75 (0.45) 1.62 (<0.01) 1.36 (<0.01) 1.26 (<0.01)
¡2LL 4,849.90 1,639.00 4,220.70 3,597.00 2,154.90
Hoerl Function 1: Y = AADTmajorβ1 ∗AADTminorβ2 ∗exp⁡(β0 + β3 AADTmajor
AADTminor
10, 000 + β4
10, 000 )
Overdispersion: k = exp(γ1 × AADTmajor
+ γ2 × AADTminor		 + γ0)
Parameters 10, 000
TOT		 10, 000
ODC		 SDC IDC SVC
β1 0.61 (<0.01) — 1.21 (<0.01) 0.21 (<0.01) —
β2 0.30 (<0.01) 0.57 (<0.01) 0.27 (<0.01) 0.47 (<0.01) 0.42 (0.02)
β3 0.15 (<0.01) — − 0.02 (0.03) — —
β4 0.25 (<0.01) − 0.17 (0.09) 0.29 (<0.09) — —
β0 − 5.72 (<0.01) — − 11.46 (<0.01) − 4.49 (<0.01) − 10.76 (0.01)
γ1 — — — — —
γ2 — — 0.28 (0.08) — 0.16 (0.04)
γ0 — — − 1.56 (0.01) — —
P 1.76 (<0.01) 1.78 (0.57) 1.77 (<0.01) 1.80 (0.01) 1.77 (0.27)
¡2LL 4,846.10 1,637.30 4,217.30 3,597.50 2,154.50
Hoerl Function 2: Y = AADTtotalβ1 ∗ (AADTminor
)β2
AADTtotal		 AADTtotal
AADTminor
∗exp⁡(β0 + β3
10, 000 + β4
AADTtotal		 )
Overdispersion: k = exp⁡(γ1 × AADTtotal
10, 000 + γ2 × AADTminor		 + γ0)
Parameters TOT AADTtotal
ODC		 SDC IDC SVC
β1 1.01 (<0.01) — 1.76 (<0.01) 0.72 (0.08) 0.63 (0.04)
β2 0.31 (0.03) 1.22 0.02) 0.22 (0.01) 0.50 (0.08) 0.83 (0.01)
β3 — 0.55 (0.02) — — —
β4 — − 3.26 (0.07) — — − 2.04 (0.09)
β0 − 6.57 (0.01) — − 14.16 (<0.01) — —
γ1 0.08 (<0.01) — 0.03 (0.07) — 0.21 (0.07)
γ2 1.85 (<0.01) — 1.48 (0.02) — 2.74 (0.09)
γ0 — — − 2.07 (<0.01) — − 1.29 (0.07)
P 1.18 (<0.01) 1.03 (0.01) 1.81 (<0.01) 1.02 (0.01) 1.77 (<0.01)
¡2LL 4,851.60 1,633.00 4,216.60 3,601.30 2,152.90

Notes: "—" represents the coefficient is not statistically significant at 10 % significance level.

SPF functions. The P parameters are statistically significant in all SPFs. The most significant SPF for SVC is the Hoerl function 2 with both total AADT and the proportion of minor road AADT over the total positively correlated with the over-dispersion. The P parameters are statistically significant in power function and Hoerl function 2.

7. SPF comparison and summary

After estimating SPFs using the 80 % of data, we used the remaining 20 % of data to evaluate the SPF prediction performance. Two criteria were used in this study, i.e. the Mean Prediction Bias (MPB) and the Mean Absolute Deviation (MAD), which can be expressed as:

$$MPB = \frac{\sum\limits_{i=1}^{n} (\widehat{Y}_i - Y_i)}{n} \tag{18}$$

$$MAD = \frac{\sum\limits{i=1}^{n} \left| \widehat{Y}{i} - Y_{i} \right|}{n}$$ (19)

where *Y*̂*i*is the predicted crashes for intersection i; Yi is the observed crashes for intersection i; and n is the total number of intersections. A lower value for both MPB and MAD represents a better SPF (Wang et al., 2017).

Table 5 presents the SPF prediction comparison results based on the two criteria. Overall, the Power function outperforms the other two Hoerl functions in predicting ODC for stop-controlled and two types of signalized intersections. The Hoerl function 2 performs the best among the three SPF functional forms in predicting SVC, while the Hoerl function 1 generally performs the best in predicting both SDC and IDC.

Based on both the SPF estimation and prediction performance comparison results, the major findings of this study are:

1 In terms of the methodological framework for SPF estimation, the mean-variance structure of NB models is not consistent and should be estimated based on the data. Furthermore, the over-dispersion of SPFs by crash type is not constant and varies across different intersections. The minor road AADT is found to be positively correlated with the over-dispersion of SPFs in estimating crash counts for SDC,

Table 4 SPF Estimation Results for Multi-Lane Signalized Intersections.

Power Function: Y = AADTmajorβ1 ∗AADTminorβ2 ∗exp⁡(β0)
Overdispersion: k = exp(γ1 × AADTmajor + γ2 × AADTminor
10, 000
10, 000		 + γ0)
Parameters TOT ODC SDC IDC SVC
β1 0.56 (<0.01) — 0.58 (<0.01) 0.51 (<0.01) 0.53 (0.01)
β2 0.60 (<0.01) 0.48 (<0.01) 0.79 (<0.01) 0.31 (<0.01) —
β0 − 7.36 (<0.01) − 5.09 (<0.01) − 9.93(<0.01) − 5.41 (<0.01) − 4.91 (<0.01)
γ1 — — — — —
γ2 1.04 (<0.01) — 0.85 (<0.01) 0.93 (<0.01) 1.60 (<0.01)
γ0 1.69 (0.06) — — 1.52 (0.06) —
P 1.21 (<0.01) 1.30 (0.02) 1.83 (<0.01) 1.44 (<0.01) 1.01 (<0.01)
¡2LL 4,889.30 1,698.20 4,439.10 3,636.60 2,057.80
Hoerl Function 1: Y = AADTmajorβ1 ∗AADTminorβ2 ∗exp⁡(β0 + β3 AADTmajor
AADTminor
10, 000 + β4
10, 000 )
Overdispersion: k = exp(γ1 × AADTmajor
10, 000 + γ2 × AADTminor		 10, 000 + γ0)
Parameters TOT ODC SDC IDC SVC
β1 0.61 (<0.01) — 0.86 (<0.01) 0.52 (0.07) —
β2 0.40 (<0.01) 0.75 (<0.01) 0.40 (<0.01) 0.53 (<0.01) 0.13 (<0.01)
β3 — — — — —
β4 0.22 (0.09) − 0.33 (0.09) 0.24 (0.07) — —
β0 − 6.18 (0.01) — − 9.16 (<0.01) − 7.38 (0.01) —
γ1 — — — — —
γ2 1.37 (<0.01) — 1.06 (0.08) 0.82 (0.02) —
γ0 — — — — —
P 1.60 (<0.01) 1.79 (0.07) 1.03 (<0.01) 1.25 (<0.01) 1.87 (0.70)
¡2LL 4,886.30 1,696.40 4,435.70 3,636.00 2,054.00
Hoerl Function 2: Y = AADTtotalβ1 ∗ (AADTminor
)β2
AADTtotal		 AADTtotal
AADTminor
∗exp⁡(β0 + β3
10, 000 + β4
AADTtotal		 )
Overdispersion: k = exp⁡(γ1 × AADTtotal
10, 000 + γ2 × AADTminor		 + γ0)
Parameters TOT AADTtotal
ODC		 SDC IDC SVC
β1 1.29 (<0.01) — 1.50 (<0.01) 1.48 (<0.01) 0.55 (0.04)
β2 0.31 (0.02) 0.79 (0.05) 0.30 (0.04) 0.41 (0.03) 0.21 (0.01)
β3 — — — − 0.22 (0.09) —
β4 — — — — —
β0 − 9.18 (0.01) — − 11.76 (<0.01) − 11.72 (<0.01) − 5.06 (0.07)
γ1 0.67 (0.02) — 0.62 (0.04) — 1.61 (0.07)
γ2 3.48 (<0.01) — 3.02 (<0.01) 2.56 (0.02) 4.96 (0.05)
γ0 — — — — − 1.73 (0.01)
P 1.73 (<0.01) 1.58 (0.49) 1.38 (<0.01) 1.15 (<0.01) 1.38 (<0.01)
¡2LL 4,888.80 1,698.50 4,437.10 3,637.50 2,061.70

Notes: "—" represents the coefficient is not statistically significant at 10 % significance level.

IDC and SVC. Estimating the over-dispersion using exposure measures offers more reliable SPF results.

2 Exposure measure and SPF functional form play a dominant role in precisely predicting crash counts by different crash types for intersections. Specifically, the Power function with major road and minor road AADT as the exposure measure performs the best in estimating ODC SPFs. The Hoerl function 2 with total AADT and the percentage of minor road AADT over the total as the exposure measure performs the best in estimating SVC SPFs for intersections. The Hoerl function 1 with major road and minor road AADT as the exposure measure is more accurate in estimating SPFs for both SDC and IDC.

Consequently, we summarized the most reliable exposure measure and functional form for intersection SPFs by each crash type consolidated in this study in Table 6.

8. Conclusions

This study proposes a new approach, namely Generalized Negative Binomial-P (GNB-P) model, to model the complex relationship between crashes and different exposure measures by crash type for intersections, which aims to not only identify the most reliable exposure measure for intersection SPFs, but also explore the most appropriate functional form of the NB models. Three types of SPF functional forms, i.e. Power function, Hoerl function 1 and Hoerl function 2 with different exposure measures including major road AADT, minor road AADT and total AADT were estimated by crash type for stop-controlled and two types of signalized intersections respectively using the GNB-P models. The overdispersion of the SPF models was estimated using the exposure measures to account for crash data variation across different intersections.

The SPF estimation results demonstrated that the mean-variance structure of NB models is not consistent and should be estimated based on the data. The over-dispersion of SPFs by crash type is not constant and varies across different intersections. One important finding of this study is that the minor road AADT is positively correlated with

Table 5 SPF Prediction Performance Comparison.

Stop-Controlled Intersections Two-Lane Signalized Intersections Multi-Lane Signalized Intersections
Crash
Type		 Performance
Matric		 Power
Function		 Hoerl
Function 1		 Hoerl
Function
2		 Power
Function		 Hoerl
Function 1		 Hoerl
Function 2		 Power
Function		 Hoerl
Function 1		 Hoerl
Function 2
ODC MPB ¡0.007 − 0.055 − 0.007 0.042 0.143 − 0.052 0.045 0.059 − 0.053
MAD
MPB		 0.363
0.025		 0.383
− 0.315		 0.384
− 0.462		 0.942
1.617		 1.013
0.896		 0.9783
1.540		 0.986
1.819		 1.135
1.531		 1.124
2.102
SDC MAD 2.557 2.447 2.460 9.917 8.543 8.265 8.719 8.153 9.261
MPB 0.339 0.158 0.199 0.371 0.207 0.236 0.506 0.379 0.382
IDC MAD 2.168 2.150 2.137 4.473 4.455 4.479 5.012 4.739 4.897
SVC MPB − 0.410 0.411 0.068 0.208 0.245 0.137 0.550 0.577 0.321
MAD 0.739 0.891 0.707 1.239 1.236 1.187 2.195 2.786 1.958

Table 6 Summary of Exposure Measures and Safety Performance Function Frameworks.

Stop-Controlled Intersections
Crash Type Exposure Measure Safety Performance Function Framework
ODC Major AADT Mean = Y = AADTmajor0.29 ∗ AADTminor0.55 ∗ exp⁡(− 8.21)
Minor AADT Variance = Y + k
Overdispersion = k = 1
SDC Major AADT
Minor AADT		 Mean = Y = AADTmajor1.65 ∗ AADTminor0.54 ∗ exp⁡(− 17.85 − 0.41 ∗ AADTmajor
10, 000 )
Variance = Y + k ∗ Y1.94
Overdispersion = k = exp⁡(1.08 ∗ AADTminor
10, 000 − 0.85)
IDC Major AADT
Minor AADT		 Mean = Y = AADTmajor0.51 ∗ AADTminor0.81 ∗ exp⁡(− 9.39 − 0.86 ∗ AADTminor
10, 000 )
Variance = Y + k ∗ Y1.30
Overdispersion = k = exp⁡(− 0.64 ∗ AADTmajor
10, 000 + 0.88 ∗ AADTminor
10, 000 + 0.68)
)0.45
SVC Minor AADT
Total AADT		 (AADTminor
Mean = Y = AADTtotal0.84 ∗
∗ exp⁡(− 7.00)
AADTtotal
Variance = Y + k ∗ Y1.87
Overdispersion = k = exp⁡(5.00 × AADTminor
− 2.45)
AADTtotal
Two-Lane Signalized Intersections
Crash Type Exposure Measure Safety Performance Function Framework
ODC Major AADT
Minor AADT		 Mean = Y = AADTmajor0.41 ∗ AADTminor0.44 ∗ exp⁡(− 7.85)
Variance = Y + k
Overdispersion = k = 1
SDC Major AADT Mean = Y = AADTmajor1.21 ∗ AADTminor0.27 ∗ exp⁡(− 11.46 − 0.02 ∗ AADTmajor
10, 000 + 0.29 ∗ AADTminor
Minor AADT 10, 000 )
Variance = Y + k ∗ Y1.77
Overdispersion = k = exp⁡(0.28 × AADTminor
10, 000 − 1.56)
IDC Major AADT Mean = Y = AADTmajor0.21 ∗ AADTminor0.47 ∗ exp⁡(− 4.49)
Minor AADT Variance = Y + k ∗ Y1.80
Overdispersion = k = 1
SVC Minor AADT (AADTminor
)0.83
Total AADT Mean = Y = AADTtotal0.63 ∗
∗ exp⁡(− 2.04)
AADTtotal
Variance = Y + k ∗ Y1.77
Overdispersion = k = exp⁡(0.21 ∗ AADTtotal
10, 000 + 2.74 ∗ AADTminor
− 1.29)
AADTtotal
Multi-Lane Signalized Intersections
Crash Type
ODC		 Exposure Measure
Major AADT		 Safety Performance Function Framework
Minor AADT Mean = Y = AADTminor0.48 ∗ exp⁡(− 5.09)
Variance = Y + kY1.30
Overdispersion = k = 1
SDC Major AADT
Minor AADT Mean = Y = AADTmajor0.86 ∗ AADTminor0.40 ∗ exp⁡(− 12.24 + 0.24 ∗ AADTminor
10, 000 )
Variance = Y + k ∗ Y1.03
Overdispersion = k = exp⁡(1.06 × AADTminor
10, 000 )
IDC Major AADT Mean = Y = AADTmajor0.52 ∗ AADTminor0.53 ∗ exp⁡(− 7.38)
Minor AADT Variance = Y + k ∗ Y1.25
Overdispersion = k = exp⁡(0.82 × AADTminor
10, 000 )
SVC Minor AADT (AADTminor
)0.21
Mean = Y = AADTtotal0.55 ∗
∗ exp⁡(− 5.06)
Total AADT AADTtotal
Variance = Y + k ∗ Y1.38
Overdispersion = k = exp⁡(1.61 ∗ AADTtotal
10, 000 + 4.96 ∗ AADTminor
− 1.73)
AADTtotal

the over-dispersion of SPFs in estimating crash counts for SDC, IDC and SVC for both stop-controlled and signalized intersections. Estimating the over-dispersion using exposure measures offers more reliable SPF results. Furthermore, selecting an appropriate exposure measure and SPF functional form plays an important role in precisely predicting crash counts by different crash types for intersections. This study highlighted that the Power function with major road and minor road AADT as the exposure measure performs the best in estimating ODC SPFs. The Hoerl function 2 with total AADT and the percentage of minor road AADT over the total as the exposure measure performs the best in estimating SVC SPFs for intersections. The Hoerl function 1 with major road and minor road AADT as the exposure measure is more accurate in estimating SPFs for both SDC and IDC.

We expect that this study will offer new insight into selecting the most appropriate exposure measure and modeling functional form for estimating reliable intersection SPFs. Considering the complexity of estimating the GNB-P model, this study focuses on the estimation of planning-level SPFs where only the traffic volume related factors were considered in the models. Future work can be considered to extend the modeling framework to project-level SPFs, where roadway geometric factors can be investigated by the GNB-P model.

CRediT authorship contribution statement

Kai Wang: Conceptualization, Methodology, Writing - review & editing. Shanshan Zhao: Conceptualization, Investigation, Writing review & editing. Eric Jackson: Conceptualization, Supervision, Writing - review & editing.

Declaration of Competing Interest

None.

Acknowledgments

The authors would like to thank the Connecticut Department of Transportation (CTDOT) for providing the intersection data, and the Connecticut Crash Data Repository (CTCDR) for providing the crash data.

References