category: literaturenote citekey: laogeneralizednonlinearmodelsrear2014 title: Generalized nonlinear models for rear-end crash risk analysis authors: "Lao, Yunteng; Zhang, Guohui; Wang, Yinhai; Milton, John" year: 2014 date: 2014-00-00 2014 doi: 10.1016/j.aap.2013.09.004 publication: "Accident Analysis & Prevention" url: "https://doi.org/10.1016/j.aap.2013.09.004" zotero_key: IFACGKQ8 zotero_storage: P4XXINQS collections: transpordi_planeerimine folder: 001_artiklid/Road Safety Indicators firstAuthor: "Lao, Yunteng"

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Accident Analysis and Prevention

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

Generalized nonlinear models for rear-end crash risk analysis

Yunteng Laoa, Guohui Zhang b, Yinhai Wang^a,^∗, John Milton^c

^a Department of Civil and Environmental Engineering, University of Washington, Seattle, WA 98195, USA
^b Department of Civil Engineering, University of New Mexico, MSC01 1070, 1, Albuquerque, NM 87131, USA
^c Washington State Department of Transportation, 414 Olive Way, Suite 400, Seattle, WA 98101-1209, USA

a r t i c l e i n f o

Accepted 1 September 2013

Article history: Received 22 November 2012 Received in revised form 24 August 2013

Keywords: Generalized linear model (GLM) Crash data modeling Generalized nonlinear model (GNM) Traffic accidents Quasi-Poisson regression model

a b s t r a c t

A generalized nonlinear model (GNM)-based approach for modeling highway rear-end crash risk is formulated using Washington State traffic safety data. Previous studies majorly focused on causal factor identificationandcrashriskmodelingusing GeneralizedlinearModels (GLMs), suchas Poissonregression, Logistic regression, etc. However, their basic assumption of a generalized linear relationship between the dependent variable (for example, crash rate) and independent variables (for example, contribute factors to crashes) established via a link function can be often violated in reality. Consequently, the GLM-based modeling results could provide biased findings and conclusions. In this research, a GNM-based approach is developed to utilize a nonlinear regression function to better elaborate non-monotonic relationships between the independent and dependent variables using the rear end accident data collected from 10 highway routes from 2002 through 2006. The results show for example that truck percentage and grade have a parabolic impact: they increase crash risks initially, but decrease them after the certain thresholds. Such non-monotonic relationships cannot be captured by regular GLMs which further demonstrate the flexibility of GNM-based approaches in the nonlinear relationship among data and providing more reasonable explanations. The superior GNM-based model interpretations help better understand the parabolic impacts of some specific contributing factors for selecting and evaluating rear-end crash safety improvement plans.

1. Introduction

The number of highway incapacitating injuries and fatalities remains unacceptably high. Based on the traffic safety facts published by National Highway Traffic Safety Administration (NHTSA), in 2010, 32,885 people were killed and 2,239,000 people were injured in the estimated 5,419,000 police-reported motor vehicle traffic crashes (NHTSA, 2012). Rear-end crashes are among the more common accident types, accounting for almost one-third of all reported accidents in the U.S. (National Transportation Safety Board, 2011). In addition to traffic safety issues, when crashes occur, they temporarily reduce roadway capacity and cause congestion. The 2012 Urban Mobility Reportindicates the annual delay per person was 38 h and an average of \$818 per traveler resulted from congestion in the 498 surveyed-urban areas in 2011 (Schrank and Timothy, 2012). These data clearly illustrate the urgent needs of rear-end crash modeling and analysis for traffic safety improvements. Based on appropriate model specifications, we can identify significant factors to improve highway design and traffic operation regulation development, leading to a decrease in the frequency and severity of rear-end crashes.

Over the past decades, significant research efforts have been made toward roadway traffic accident reduction. A number of studies have been performed to understand the relationships between crashes and potential causal factors including roadway geometric, environmental, traffic, and human factors (Lao et al., 2011a). For instance, various statistical modeling techniques have been developed to analyze collision characteristics under certain circumstances (Lord and Mannering, 2010). The majority of these models are generalized linear models (GLMs), such as Poisson regression (e.g. Jovanis and Chang, 1986; Miaou and Lum, 1993; Miaou, 1994), Gamma regression model (Winkelmann and Zimmermann, 1995; Oh et al., 2006), or GLM oriented models, such as negative binomial (NB) regression (or Poisson-gamma regression) (Miaou, 1994; Maher and Summersgill, 1996; Milton and Mannering, 1998; Chin and Quddus, 2003; Wang et al., 2003; Wang and Nihan, 2004; Donnell and Mason, 2006; Kim et al., 2007; Malyshkina and Mannering, 2010; Daniels et al., 2010; Geedipally et al., 2012), Random-parameters

^∗ Corresponding author. Tel.: +1 206 616 2696; fax: +1 206 543 5965. E-mail addresses: laoy@u.washington.edu (Y. Lao), guohui@unm.edu (G. Zhang), yinhai@u.washington.edu (Y. Wang), MiltonJ@wsdot.wa.gov (J. Milton).

models (Anastasopoulos and Mannering, 2009, 2011; El-Basyouny and Sayed, 2009, 2011; Anastasopoulos et al., 2012), Bayesian approaches (Deublein et al., 2013) and Bivariate/multivariate models (Park and Lord, 2007; Lao et al., 2011b). Shively et al. (2010) implemented a Bayesian semi-parametric model to analyze the relationships between crash frequencies and roadway characteristics. El-Basyouny and Sayed (2010) discussed the effect of various causal factors on the accident prediction based on a particular link function.

Meanwhile, various sensing technologies and control strategies have been utilized to enhance traffic regulation enforcement for safety improvements. Significant research efforts have been undertaken to utilize statistical modeling techniques for analyzing the characteristics of rear-end crashes. Wang (1998) modeled the expected values of rear-end crash risk at four legged signalized intersection based on rear-end crash probabilities. Abdel-Aty and Abdelwahad (2004) estimated the probability of a car-truck rear-end crash by using a nested logit model. Wang and Abdel-Aty (2006) proposed generalized estimation equations (GEM) to model rear-end crash frequencies at signalized intersections. Kim et al. (2007) estimated rear-end crash risk using a modified NB regression. Harb et al.(2008) proposed a conditional logistic regression model to estimate rear-end crash risk in work zone. Oh and Kim (2010) developed a method for estimating rear-end crash potential using individual vehicle trajectory data. Meng and Weng (2011) developed rear-end crash risk models to evaluate the rearend crash risk in work zone activity area based on the available work zone traffic data. Most of these previous studies can be classified as GLM-based approaches, and have provided valuable insights to investigate and examine rear-end crash occurrence. In general, a GLM-based approach utilize a linear regression to aggregate a series of independent variables, such as roadway curvature, shoulder width, traffic speed limit, etc. and establish a mapping relationship between independent variables and dependent variable (which is typically the expected value of crash rates) through a specific link function. However, such a GLM-based approach is constrained by its linear model specifications and may lead to biased model estimation and interpretation when the independent variable data demonstrates strong nonlinear features. For example, our studies show the parabolic impacts of truck percentage on rear-end crash risks: Higher truck percentages increase the likelihood of crash occurrence before a certain threshold is reached, and then continuously increased truck percentages have negative impacts on crash occurrence. GLM-based approaches are not designed to model such a relationship and further developments are needed. The elasticity analyses of specific factors and locations are important since transportation agencies heavily rely on estimated elasticity values to quantify marginal costs of various countermeasures for potential traffic safety improvements. Therefore, it is important to develop a more flexible modeling approach to enable nonlinear model specifications to better characterize and analyze rear-end crash occurrence and its associated causal factors.

Considerable research efforts have been undertaken to study the nonlinear models to extract more complex relationship. For example, Lindsey et al. (2000) applied a generalized nonlinear model (GNM) to analyze pharmacokinetic data. Some previous studies (e.g. Wong et al., 2007; Abdel-Aty and Haleem, 2011) used the logarithm of AADT instead of AADT to deal with the nonlinear relationship between the crash risk and AADT. Moreover, recently, Turner and Firth (2012) developed an R package to help on the estimating the parameters in GNMs. Thus, in this study, a generalized nonlinear model (GNM)-based approach is proposed to model rear-end accident risks to address the GLM inherent linear predictor constraint.Anonlinear predictor can be established to aggregate significantindependent variables to quantify their impacts on crash risks through a specific link function, which provide more flexible and greater explanatory power than GLMs.

2. Methodology

In this section, the GLM-based modeling principles, formulated by Nelder and Wedderburn (1972) and McCullagh and Nelder (1989), is introduced, and then the extended GNM-based approach is presented for crashdata analysis.Anelasticity estimationmethod is detailed for the significantly causal factors based on the proposed GNM nonlinear predictor.

2.1. GLMs for crash data

In classical linear regression models, the expectation of crash frequency is formulated as an ordinary linear model. This model specification can be expressed as follows (McCullagh and Nelder, 1989)

$$E(y_i) = \mui = L \sum{j=1}^{n} x_{ij} \beta_j; \quad i = 1, ..., n$$ (1)

where yi is the crash frequency along Roadway Segment i, E(yi) or ⁱ is the expected crash frequency along Segment i during a certain time period; L is the segment length, in mile; xij is the jth explanatory variable for Segment ⁱ; ^ˇ^j is the corresponding coefficient for the jth explanatory variable; n is the total number of explanatory variables considered in the model. Compared to the simplest linear regression, more complicated models, such as Poisson, Gamma, Gaussian, Logit, Probit, Negative Binomial regressions, etc. have been used to enhance their capability of approximating and interpreting crash data. These models can be generalized by using a smooth and invertible linearizing link function to transform the expectation of the response variable, ⁱ, to its linear predictor:

$$g(\mui) = L \sum{j=1}^{n} x_{ij} \beta_j \tag{2}$$

where, g(.) is the link function, which is monotonic, differentiable to connectthe linear predictor ofthe explanatory variables with the expected crash frequency in various formats, such as identity, log, logit, etc. Its inverse function is expressed by g−1(.). In this research, the log function is used for rear-end crash analysis. The distribution of yi is a member of a scaled exponential family, and its generalized density function can be expressed as

$$f(y_i:\theta:\psi) = \exp\left{\frac{y_i\theta - b(\theta)}{a(\psi)} + c(y_i:\psi)\right}$$ (3)

where, and are parameters characterizing the density, and ^a( ), ^b(), and ^c(yi : ) are real-valued functions. is the scale (or dispersion) factor. Depending on the specific formats of ^a( ), ^b(), and ^c(yi : ), yi could follow Gaussian, binomial, Poisson, gamma, inverse-Gaussian distributions, etc. The GLMs could be extended to multivariate exponential families (such as the multinomial distribution) and to certain non-exponential families (such as the two-parameter Negative-binomial distribution). One should note that the issues of data over-dispersion for crash data analysis can be well addressed by the Quasi-Poisson GLMs. The Quasi-Poisson model utilizes the dispersion parameter, , in Eq. (3), to model over-dispersion data compared to the Poisson model. If ^> 1, the variance of yi increases more rapidly than its mean.

Fig. 1. Rear-end crash rate (crash frequency per mile) in 5 years (2002–2006) from 10 highways in Washington State, by grade.

2.2. GNMs for crash data

As we have discussed earlier, in many scenarios the relationship between the expected crash frequency and its associated factors cannot be simply expressed by GLMs. GNMs are proposed as an extension of a GLM to satisfy such specific requirement by changing the linear predictor, to be nonlinear in the parameters, ˇj, in Eq. (2). Withoutloss of any generality, assuming a Poisson model is formulated for analyzing rear-end crash data from 2002 to 2006 from 10 highways (US2, SR8, US12, SR20, I90, US97, US101, US395, SR525 and SR970) in Washington State. Its non-linearizing link function is ^g(i) ⁼ log(i). Fig. ¹ illustrates the association between the logarithm of the expectation of crash rates (number of crashes per mile) and roadway grades. The solid marker curve shows crash rate change tendencies.

Previous GLM-based studies assume the monotonic relationship between the dependent variable, average crash rate and the independent variable, grade through ^a linearizing link function, log(i). The logarithm of the expectation of crash rates is supposed to increase or decrease consistently when the variable, grade, changes as a linear function. In Fig. 1, such a linear relationship is modeled by the red dashed line L(x). In order to quantify how well the different models fit the data and measure the model performance, the coefficient of determination, R2, is used as the standardized model performance assessment criteria. R2, for generalized nonlinear models and piecewise linear functions can be calculated by Eq. (4)

$$R^2 = 1 - \frac{SS{err}}{SS{tot}} \tag{4}$$

SStot where the SSerr and the SStot the sum of squares of residuals and the total sum of squares (proportional to the sample variance), respectively, which can be calculated as follows.

$$SS{tot} = \sum{i} (x_i - \bar{x})^2 \tag{5}$$

$$SS{err} = \sum{i} (x_i - \overline{\hat{x}_i})^2 \tag{6}$$

where, xi is the ⁱth data value, ^xˆi is the associated modeled value based on the generalized non-linear models or piecewise linear functions, and x¯ is the mean of the observed data. The ^R² for the fitted line L(x) is nearly equal to zero (R² = 0.006) showing its poor goodness-of-fit. Consequently, L(x) could not model the fluctuation patterns associated with the variable, grade, and misrepresent the true relationship between the expected crash rate and the geometric variable, grade, assuming all other causal factors are approximately equal across all grade levels.

Therefore, a GNM-based approach is needed to reasonably model the nonlinear relationship between the expected crash rate and its associated factors. Based on the visualized comparisons, a piecewise linear function U(x) could be suitable to extract data change patterns. Based on the relationship between the logarithm of expected crash rate, and grade, two peaks should be appropriately modeled when grade = −4.4 and grade = 4. When the grade is small or moderate from −3.5 to 3.5, its impacts on crash occurrence are relatively stable. Its impacts become more severe when the grade increase beyond a certain threshold,for example grade = −4.4 and grade = 4. When the absolute value of the grade continuously increases (grade > 5 or grade < −5.5) the crash rate tends to be lower. This may result from the facts that the drivers will drive much slower and pay more attention to handle considerable grade changes in these situations. Additionally, we do see thatin a similar grade range, higher crash rate were observed on downhill highway sections than ones on uphill sections. This finding is consistent with the previous accident studies (Ahmed et al., 2011; Yu et al., 2013). In order to describe these unique impacts of grade on crashes occurrence and a piecewise linear function is developed to fit the data as shown as the red dashed line in Fig. 1:

$$U_g(x) \begin{cases} -1 & -7 \le x < -5.5 \ -1 + \frac{2.5(x+5)}{5-4.4} & -5.5 \le x < -4.4 \ -0.6 + \frac{2.1(x+5)}{4.4-3.5} & -4.4 \le x < -3.5 \ 0.1 & -3.5 \le x < 3.5 \ -0.6 + \frac{1.2(x-3.5)}{4-3.5} & 3.5 \le x < 4 \ -1 + \frac{1.6(5-x)}{5-4} & 4 \le x < \le 4.5 \ -1 & 4.5 \le x < 7 \end{cases}$$ (7)

In order to verify its goodness-of-fit ofthe piecewise linear function U(x), its R² is calculated and aggregated for multiple linear pieces. The results of R² = 0.68 indicate the piecewise linear function significantly outperforms the linear regression function and provides a better modeling performance for fitting the expected crash rate. The piecewise linear function is a special case of the developed GNM under certain circumstances. The GNM-based method uses a user-defined, customized function to extract the relationship between crash risks and its contributing factors with more general assumptions. For the other explanatory variables, the diverse nonlinear predictor, U(x), such as polynomial function, exponentialfunction, parabolic function, logarithmic function, etc. may be utilized to extract proper data features. In general, the model format of U(x) can be determined based on statistical analysis of the crash rate and the specific explanatory variable. Notice that the defined nonlinear function U(x) is an assumed relationship. This defined function can be revised based on more statistical analysis. The concept of utility function was initially formulated in economic disciplines. It has been widely used in many research areas in transportation, such as crash severity analysis, driver behavior studies (Koppelman, 1981), toll lane operations (Zhang et al., 2008), etc. Aggregating the nonlinear predictors for all the independent variables, and Eq. (2) can be rearranged as

$$E(y_i) = \mui = L \sum{j=1}^n Uj(x{ij}) w_j; \quad i = 1, ..., n,$$ (8)

where Uj(xij) is ^a nonlinear predictor for the ^jth explanatory variable; and wj is the corresponding weight for Uj. Consequently, the GNM link functions becomes

$$g(\mui) = \sum{j=1}^{p} Uj(x{ij}) w_j$$ (9)

If all the Uj(xij) in the model are linear regressions of xij, ^a GNM will degrade to a GLM. Therefore, in this research, GLMs are special cases of GNMs.

2.3. Elasticity calculation

In order to understand the influence and magnitude of causal factors for potential safety improvements along a particular segment, the marginal effects of significant independent variables need to be calculated. Elasticity, a measure of a percentage change ofthe dependent variable resulting from a 1% change of an independent variable (Washington et al., 2003), is suitable for proportional marginal effect analysis here. The elasticity value can be calculated as (Shankar et al., 1995; Abdel-Aty and Radwan, 2000; Washington et al., 2003):

$$E{x{ij}}^{\mu_i} = \frac{\partial \mui}{\partial x{ij}} \frac{x_{ij}}{\mui} = \frac{\partial g^{-1} \left( \sum{j=1}^p Uj(x{ij}) wj \right)}{\partial x{ij}} \frac{x{ij}}{g^{-1} \left( \sum{j=1}^p Uj(x{ij}) w_j \right)}$$ (10)

where ⁱ is the expected crash frequency for Roadway Segment i, and xij is the jth explanatory variable for Segment i. One can find in Eq. (10) that elasticity is determined by the value of the variable xij, the inverse link function, g−1(.), and the (non) linear predictor Uj(.). Assume that crash frequency follows the Poisson distribution, then the inverse link function g−1(.) = exp(.), then Eq. (10) can be exemplified as follows:

$$E{x{ij}}^{\mui} = \frac{\partial \exp\left(\sum{j=1}^{p} Uj(x{ij})wj\right)}{\partial x{ij}} \frac{x{ij}}{\exp\left(\sum{j=1}^{p} Uj(x{ij})w_j\right)}$$

$$= \frac{\partial \left(\sum_{j=1}^{p} Uj(x{ij})wj\right)}{\partial x{ij}} x_{ij} = wj x{ij} \frac{\partial Uj(x{ij})}{\partial x_{ij}}$$ (11)

Eq. (11) emphasizes the importance of the (non) linear predictor Uj(). It could introduce significant errors if inappropriate model specifications are used to estimate the values of elasticity without considering the nonlinear contribution function. For GLM, Uj(xij) = xij. Then Eq. (11) can be rewritten as:

$$E{x{ij}}^{\mu_i} = \frac{\partial \mui}{\partial x{ij}} \frac{x_{ij}}{\mu_i} = wj x{ij} \tag{12}$$

The elasticity in Eqs. (11) and (12) apply when the explanatory variable xij is continuous. In case of a discrete variable, pseudoelasticity is estimated as an approximate elasticity of this variable (Washington et al., 2003):

$$E{k{ij}}^{\mu_i} = \frac{\exp(w_j) - 1}{\exp(w_j)} \tag{13}$$

3. Data description

To verify the effectiveness ofthe GNM-based approach,the rearendcrashdata from10highway routes (US 2, SR8, US 12, SR20,I-90, US 97, US 101, US 395, SR 525 and SR 970)inWashington State from 2002 to 2006 are used in this study. The rear-end crash data have been extracted from theWashington State accident file provided by Highway Safety Information System (HSIS), which is operated by the University of North Carolina Highway Safety Research Center

Fig. 2. Rear end crash rate (crash frequency per mile) in 5 years (2002–2006) from 10 highways in Washington State, by AADT.

and the LENDIS corporation under a contract with Federal Highway Administration (FHWA) (HSIS, 2012). The HSIS collision data of Washington were compiled from the State Trooper filed field reports and citizen reports. Roadlog data provided by HSIS provides geometric information for the roadway, such as terrain type, lane width and shoulder width. Additionally, traffic flow data, and accident-involved vehicle data are also collected from the Washington State Department of Transportation (WSDOT). To increase the model accuracy, only the segments with roadway length equal or bigger than 0.05 mile are included in the model. Table 1 lists all explanatory variables used in the modeling process. A total of 2317 crashes are recorded on 6630 highway segments during the five years. The minimum, maximum, mean, and standard deviation (SD) of each variable are calculated as shown in Table 1. One canfind that the number of rear-end accident per mile is over-dispersed as indicated by the variance (25) being higher than the mean (4.7).

4. Model estimation

One of the superiority of the proposal GNM-based approach is to provide the sufficient modeling flexibility to extract the complex relationship between dependent and independent variables. The model specifications and function structures can be defined and developed as needed. For example, for the variable, AADT, the logarithmic function should be used based on the statistical analysis, observation relationship between AADT and log (crash rate), and visual illustration showed in Fig. 2. Then the parameters for the logarithmic function are estimated during the model calibration and validation processes. For the other specific explanatory variables, the diverse functions, such as polynomial function, exponential function, parabolic function, logarithmic function, etc. may be utilized to extract proper data features. Thus, the model estimation consists of two interactive steps: (1) model specification estimation of nonlinear predictors, and (2) parameter calibration and estimation, which are detailed as follows.

4.1. Estimation of nonlinear predictors

It is necessary to determine the appropriate predictors Uj in Eqs. (8) and (9) before the corresponding coefficients wj are calibrated. To better illustrate the nonlinear contribution function estimation process, an example is detailed to formulate the contribution function for the variable, AADT, as follows. Assume the number of rear-end crashes follows the Poissonmodel, and all other dependent variables are approximately consistent across different AADT levels when the sample data is large enough. To develop an appropriate format of the predictor Uj, the visualized comparisons

Table 1 Description of explanatory variables in the models.

Variable	Min	Max	Mean	SD
Y, Number of rear end accidenta	0	114	0.18	1.22
z1, Annual average daily traffic (in thousands)	0.31	148.8	13.24	18.70
z2, Restrictive access control (yes: 1; no: 0)	0	1	0.20	–
z3, Speed limit (mph)	25	70	55.1	11.16
z4, Truck percentage	0	60.37	13.74	9.06
z5, Median width (>6 ft: 1; others: 0)	0	1	0.31	–
z6, Total number of lanes (in both directions)	2	9	2.85	1.25
z7, Roadway length (mile)	0.05	6.99	0.32	0.46
z8, Terrain type (rolling: 1; otherwise: 0)	0	1	0.73	–
z9, Terrain type (mountainous: 1; otherwise: 0)	0	1	0.10	–
z10, Lane width (ft)	10	38	12.5	2.7
z11, Left shoulder width (ft)	0	36	5.58	4.04
z12, Right shoulder width (ft)	0	40	9.25	7.38
z13, Rural area (rural: 1; urban: 0)	0	1	0.81	–
z14, Vertical curve (grade percentage)	0	9.87	1.60	1.96

^a Dependent variable; Min: minimum; Max: maximum; SD: standard deviation.

between the logarithm of the expectation of crash rate (number of crashes per mile) andAADT are illustrated in Fig. 2. The solid marker line shows the crash rate from 10 highways in Washington State in 5 years (2002–2006), classified by AADT. As we can see, the logarithm of the average crash rate tends to increase when the AADT increases at a variable rate. The increase rate becomes smaller with the higher AADT, which indicates the inappropriateness of using a linear contribution function. To address this issue, we adopted a logarithmic calculation as the nonlinear predictor to approximate the impacts of AADT on crash rate:

$$U_A(x) = 2.346 \ln(x) - 3.4057 \tag{14}$$

Compared to the linear predictor L(x) = 0.3008x−1.6054, the value of R² increases from 0.8452 to 0.9673 when the nonlinear predictor, U(x), is utilized as shown in Fig. 2. Therefore, the nonlinear predictor is more suitable to describe the relationship between crash rate and AADT, and is employed in this study. The logarithm of AADT and its impacts on rear-end crash frequencies have been found significant and this results are consistent with many previous studies (e.g. Wong et al., 2007; Abdel-Aty and Haleem, 2011).

Similar studies can be conducted to other causal factors. In addition to the nonlinear predictor for the variable, grade, is established in Eq. (7), the other contribution function for the variables, truck percentage, and right shoulder width are developed in Eqs. (15) and (16), respectively:

$$U_{tr}(x) = \begin{cases} 1.2 + \frac{1.8(x-1)}{2} & x \le 3\ -1.786 \ln\left(\frac{x}{2} - 0.5\right) + 2.8438 & x > 3 \end{cases}$$ (15)

$$U_{TW}(x) = \begin{cases} 0 & x \le 1\ 1 & x > 1 \end{cases}$$ (16)

For the variables, right shoulder width, the step functions are verified appropriate by using different thresholds. For the other variables, no strong nonlinear associations are observed and linear predictors are sufficient to characterize their impacts on crash rate.

4.2. Parameter estimation

For comparison purposes, various GLMs (Eqs. (1) and (2)) and GNMs (Eqs. (8) and (9)) are established to model and analyze rear-end crash data. The parameters for both GLMs and GNMs are estimated based on maximum-likelihood estimation (MLE) methods. The dispersion parameter, , in Eq. (3) is obtained based on the method of moments estimator (MME). To illustrate their significantly different performance between GLM and GNM, a specific Quasi-Poisson regression model with a logarithm link function is estimated for both GLM and GNM. An open source statistical analysis package, R (Institute for Statistics and Mathematics, 2011), was used for model estimation and calibration in this study.

The explanatory variables were analyzed based their significance in the models. Residual deviance is utilized to examine and measure the model's goodness-of-fit. Table 2 shows the coefficients of significant variables and statistical test results of the Quasi-Poisson GLM. The calculated elasticity for each variable listed in the last column. Similarly, the coefficients of the significant variables and their significance are shown for the Quasi-Poisson GNM in Table 3. The variable, grade is identified significant in the Quasi-Poisson GNMadditionto the other variables recognized by the GLM. Rather than the deterministic elasticity values calculated for the GLM significant variables, the sign of the elasticity values for the nonlinear predictors of the GNM variables are changeable depending on their corresponding values and function specification. Eqs. (17) and (18) detailtheir values which are further interpreted in the following section. Table 4 shows the F-test for both Quasi-Poisson GLM and GNM. The F-value and its significant level of p = 0.000 indicate that in terms of the goodness-of-fit measure of residual deviance, the difference between the Quasi-Poisson GLM and GNM is statistically significant, and the Quasi-Poisson GNM outperforms its GLM counterpart.

$$EG = \begin{cases} 0 & -7 \le x{ij} \le -5.5, x{ij} = -4.4, -3.5 \le x{ij} \le 3.5, x{ij} = 4, 4.5 \le x{ij} < 7 \ 2.23x{ij} & -5.5 < x{ij} < -4.4 \ -1.25x{ij} & -4.4 < x{ij} < -3.5 \ 1.28x{ij} & 3.5 < x{ij} < 4 \ -86x{ij} & 4 < x{ij} < 4.5 \end{cases}$$

$$(17)$$

$$E{T} = \begin{cases} 0.22x{ij} & x{ij} \le 3\ -\frac{0.88x{ij}}{x{ij} - 1} & x{ij} > 3 \end{cases}$$ (18)

5. Model interpretations and discussions

The statistical analysis results are shown in Tables 2–4. Classical statisticaltheory indicates that residual deviance has an asymptotic chi-squared distribution and significantly larger values of residual deviance indicate a model deficiency. The residual deviances are 54,593 (13,253 degrees of freedom) and 59,499 (13,254 degrees of freedom) indicating a good fit of Quasi-Poisson GNM outperforms its GLM counterpart. The mean-variance relationship for the

Table 2 Description of explanatory variables in the GLMs.

Explanatory variables	Coeffa		st. errb	t-value	Ec
Constant	2.074		0.160	12.989
Right shoulder width (ft)	−0.016		0.004	−3.812	−0.148
AADT (in thousands)	0.022		0.001	22.985	0.291
Rural area (rural: 1; urban: 0)	−1.450		0.092	−15.809	−3.263
Speed limit (mph)	−0.022		0.004	−6.283	−1.212
Truck percentage	−0.032		0.005	−6.855	−0.440
Model evaluation
Dispersion parameter for Quasi-Poisson GLM		15.362
Null deviance on 13,259 degrees of freedom		94,888
Residual deviance on 13,254 degrees of freedom		59,499

^a Coefficients in the model.

Table 3 Description of explanatory variables in the GNMs.

Explanatory variables	Coeffa		st.errb	t-value	Ec
Constant	1.281		0.146	8.780
Uj for right shoulder width	−0.584		0.072	−8.069	−0.540
Uj for AADT	0.766		0.030	25.739	1.797
Uj for grade percentage	0.535		0.080	6.718	EG
Rural area (rural: 1; urban: 0)	−0.622		0.093	−6.674	−0.334
Speed limit (mph)	−0.010		0.003	−3.259	−0.551
Uj for truck percentage	0.245		0.033	7.449	ET
Model evaluation
Dispersion parameter for Quasi-Poisson GNM		14.196
Null deviance on 13,259 degrees of freedom		94,888
Residual deviance on 13,253 degrees of freedom		54,593

^a Coefficients in the model.

Poisson GLM and GNM (mean/variance = 1) is not valid for rearend crash data. The dispersion parameters for the GLM and GNM are 15.362 and 14.196, respectively, which are significantly different from one and verify the appropriateness of the Quasi-Poisson regression for both GLM and GNM. Apparently,the sample variance is larger than the sample mean for the rear-end crash data used in this study.

A total of five explanatory variables are identified significant, including right shoulder width, AADT, Rural Area, Speed Limit and Truck Percentage, for both Quasi-Poisson GLM and GNM. One more variables, grade is also identified significant in the Quasi-Poisson GNM. Due to more reasonable model specifications, the GNM includes more explicitly significant variables and performs better in extracting data patterns. The detailed model interpretations and discussions will be mostly based on the Quasi-Poisson GNM.

Three traffic flow-related variables are found significant for rear-end crash occurrence formulated in the GLM and GNM. The variable, Speed Limit, has a negative coefficient. Its estimation results indicate that the highway sections with higher speed limits are associated with lower rear-end crash frequency, which may result from the facts that high speed limits are normally set up for relatively desirable roadway conditions and geometric characteristics and the less rear-end crashes are observed along such roadways. The coefficients ofthe variable, AADT, are 0.022 and 0.584 for the GLMand GNM, respectively, whichillustrate increased traffic volumes may lead to more rear-end crashes. Furthermore, the AADT nonlinear predictor represented by Eq. (14) in the GNM method further indicates that a higher value of AADT is associated with a lower increasing rate of crash occurrence. The truck percentage is found significant to have a negative effect on rear-end crash occurrence in the GLM method. Its nonlinear function (Eq.(15)) and its elasticity (Eq. (18) in the GNM further indicates that the highest rear-end crash frequency is associated with the traffic composition with trucks: 3% and passenger cars: 97%. The probability of rearend crash occurrence increases when truck percentage increased before it reach the turning point of 3%. Then increased truck percentage will have a negative impact on rear-end crash occurrence. The GNMresults could provide better explanations thatthe number of rear-end crashes tend to increase when more trucks are observed due to its inferior operation performance compared to passenger cars. However, when truck percentage increase to a certain level, driver tend to pay more attention and drive cautiously to lead to a reduction in crash frequency.

Among the geometric characteristics and design elements, two variables, including right shoulder width, and grade, are found

Table 4 F-test between the GLM and GNM.

	Resid. df	Resid. Dev	df	Deviance	F-value	Pr (>F)
GLM GNM	13,254 13,253	59,499 54,593	1	4906.4	345.61	<2.20E − 16

^b Standard error.

^c Average elasticity value.

^b Standard error.

^c Average elasticity value.

significant for rear-end crash occurrence in the GNM. On the other hand, only one variable, right shoulder width, is significant in the GLM. The GLM results show that the crash frequency tend to decrease along the roadway section with wider right shoulders. The GNM results indicate that rear-end crash frequency significantly decrease when roadway sections are characterized with right shoulders compared to ones without right shoulders. However, no significant difference is observed after the right shoulders are bigger than certain value (in this research the critical right shoulder width is equal to 1 ft). The variable, grade, is not found significant in the GLM but its significant attributes are identified by the GNM. Its nonlinear predictor (Eq.(7)) and its elasticity (Eq.(17)) in the GNMmethod indicates that rear-end crash frequency reaches its two peak values when grade increases to the certain thresholds (4.4%for the downhill and 4%for the uphill in this study).Its impacts on crash occurrence decrease after its exceed these critical values because drivers may pay more attention on abnormal geometric characteristics and roadway conditions. The nonlinear predictor also indicates its minor impacts on crash occurrence when the grade is small (from −3.5% to 3.5%) or extremely large (from −7% to −5.5% for downhill sections and from 4.5% to 7% for uphill sections).

In terms of area types, only one variable is found significant: rural areas in both the GLM and the GNM. The estimation results indicate that compared tourbanhighways, ruralhighways are associated with lower rear-end crash frequency.

In summary,the GNM outperforms the GLM by illustrating more reasonable model specifications and flexible result interpretations. More sensible and applicable explanations for the relationship between crash frequency and its contribution factors are supported and justified by the GNM. The GNM-based approaches overcome the constraints associated with the previous GLMs in describing the monotonous relationship between significant causal factors and crash occurrence. The findings are useful to identify significant causalfactors and developmore robust and applicable countermeasures against rear-end crash occurrence.

6. Conclusion

To improve highway safety, many studies have been conducted to identify the relationship between causal factors and rear-end crash occurrence by using various modeling techniques. Most of these studies are GLM-based approaches, which assume that the crash frequency and its contributing factors can be formulated through a linearizing link function. However, this assumption is violated for in many situations. This paper focused on GNM-based approach development enabling a nonlinear, like non-monotonic, relationship linking crash frequency and its associated factors. In the GNM-based approach, nonlinear predictors are used to replace the GLM linear predictors. If the nonlinear predictors are identical with the correspondent variable, the GNMs will degrade to GLMs.

The GNMs can be used to model more complicated relationships between any count data and its associated factors. In this paper, the GNMs were applied to 5-year (2002–2006) crash data, rear-end crash data in Washington State. One variable, grade, were found not significant in GLMs, but significant in GNMs. Results from the GNMs indicates that the crash risks tend to increase when the values of grade increase initially, then after certain point, the crash risks tend to decrease when grade continues increasing. The nonmonotonic relationship is also found between the crash frequency and truck percentage by GNMs. This non-monotonic relationship between the crash frequency and its associated factors could not be described in the original GLMs. This indicates that GNMs are more powerful on modeling the relationship between crash frequency and its associated factors. Moreover, the calculation of the elasticity in the GNMs also can reveal the non-monotonic relationship depends on the predictor format and the values ofthe variables. The elasticity from GNMs can provide more accurate information for transportation agency to improve traffic safety. As a new modeling approach, the GNM may better extract the non-linear relationships among variables andprovidenew perspectives ofinterpreting crash causal factors in a more complete and comprehensive manner.

Acknowledgments

Authors would like to acknowledge the Natural Science Foundation of China (Project No. 51138003) for their partial funding support to this research. Appreciation also goes to the staff of Highway Safety Information System (HSIS) for providing the accident data. The authors also would like to give thanks to Sherry Kim from University of Washington for her English editing effort.

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