category: literaturenote citekey: caimacrolevelpedestrianbicycle2016 title: "Macro-level pedestrian and bicycle crash analysis: Incorporating spatial spillover effects in dual state count models" authors: "Cai, Qing; Lee, Jaeyoung; Eluru, Naveen; Abdel-Aty, Mohamed" year: 2016 date: 2016-00-00 2016 doi: 10.1016/j.aap.2016.04.018 publication: "Accident Analysis & Prevention" url: "https://doi.org/10.1016/j.aap.2016.04.018" zotero_key: PZ57WG5S zotero_storage: GU7D4WI8 collections: transpordi_planeerimine folder: 001_artiklid/Road Safety Indicators firstAuthor: "Cai, Qing"

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Accident Analysis and Prevention

jour nal home page: www.elsevier.com/locate/aap

Macro-level pedestrian and bicycle crash analysis: Incorporating spatial spillover effects in dual state count models

Qing Cai, Jaeyoung Lee∗, Naveen Eluru, Mohamed Abdel-Aty

Department of Civil, Environment and Construction Engineering, University of Central Florida,Orlando, FL 32816, USA

a r t i c l e i n f o

Article history: Received 6 August 2015 Received in revised form 9 December 2015 Accepted 15 April 2016 Available online 3 May 2016

Keywords:

Macro-level crash analysis Pedestrian and bicycle crashes Dual-state models Spatial independent variables Zero-inflated negative binomial Hurdle negative binomial models

a b s t r a c t

This study attempts to explore the viability of dual-state models (i.e., zero-inflated and hurdle models)for traffic analysis zones (TAZs) based pedestrian and bicycle crash frequency analysis. Additionally, spatial spillover effects are explored in the models by employing exogenous variables from neighboring zones. The dual-state models such as zero-inflated negative binomial and hurdle negative binomial models (with and without spatial effects) are compared with the conventional single-state model (i.e., negative binomial). The model comparison for pedestrian and bicycle crashes revealed that the models that considered observed spatial effects perform better than the models that did not consider the observed spatial effects. Across the models with spatial spillover effects, the dual-state models especially zero-inflated negative binomialmodel offeredbetterperformance comparedto single-statemodels.Moreover,themodel results clearly highlighted the importance of various traffic, roadway, and sociodemographic characteristics of the TAZ as well as neighboring TAZs on pedestrian and bicycle crash frequency.

1. Introduction

Active forms of transportation such as walking and bicycling have the lowest impact on the environment and improve the physical health of pedestrians and bicyclists. With growing concern of worsening global climate change and increasing obesity among adults in developed countries, it is hardly surprising that transportation decision makers are proactively encouraging the adoption of active forms of transportation for short distance trips. However, transportation safety concerns related to active transportation users form one of the biggest impediments to their adoption as a preferred alternative to private vehicle use for shorter trips. According to the National Highway Traffic Safety Administration (NHTSA), from 2004 to 2013, the proportion of pedestrian fatalities has steadily increased from 11% to 14% (NHSTA, 2013a) while the proportion of bicyclist fatalities has increased from 1.7% to 2.3% (NHTSA, 2013b). Thus, traffic crashes and the consequent injury and fatality remain a deterrent for active modes of transportation, specifically in North American communities (Wei and Lovegrove, 2013). Any effort to reduce the social burden of these crashes would necessitate the implementation of policies that enhance safety for active transportation users. An importanttoolto identify the critical factors affecting occurrence of bicycle crashes is the application of planning level crash prediction models.

Traditionally, transportation crash prediction models are developed for two levels: micro and macro-level. At the micro-level, crashes on a segment or intersection are analyzed to identify the influence of geometric design, lighting and traffic flow characteristics with the objective of offering engineering solutions (such as installing sidewalk and bike lane, adding lighting). On the other hand, the macro-level crashes from a spatial aggregation (such as traffic analysis zone (TAZ) or county) are considered to quantify the impact of socioeconomic and demographic characteristics, transportation demand and network attributes so as to provide countermeasures from a planning perspective. The current research effort contributes to burgeoning literature on active transportation user safety by examining pedestrian and bicycle crashes in the state of Florida at a macro-level. Specifically, in this study, a comprehensive analysis of pedestrian and bicycle crashes is conducted at the macro-level by employing several crash frequency models. A host of exogenous variables including socio-economic and demographic characteristics, transportation network characteristics, and traffic flow characteristics are considered in the model development. In addition, exogenous variables from neighboring zones are also considered in the analysis to accountfor spatial proximity effects on crash frequency. The overall model development exercise will allow us to identify important determinants of pedestrian and bicycle crashes in Florida while also providing valuable

E-mail address: jaeyoung@knights.ucf.edu (J. Lee).

^∗ Corresponding Author.

insight on appropriate model frameworks for macro-level crash analysis.

1.1. Literature review

A number of research efforts have examined transportation (vehicle, pedestrian and bicycle) related crash frequency (see (Lord and Mannering, 2010) for a detailed review). These studies have been conducted for different modes #XPS##x2212; vehicle (automobiles and motorbikes), pedestrian and bicycle and for different scales − micro (such as intersection and segment) and macro-level (such as census tract, traffic analysis zone, county). The model structures considered in earlier literature include Poisson, Poisson-Lognormal, Poisson-Gamma regression (also known as negative binomial (NB)), Poisson-Weibull, and Generalized Waring models (Abdel-Aty and Radwan, 2000; Miaou et al., 2003; Aguero-Valverde and Jovanis, 2008; Lord and Miranda-Moreno, 2008; Maher and Mountain, 2009; Cheng et al., 2013; Peng et al., 2014). Among these model structures, the NB model offers a closed form expression while relaxing the equal mean variance equality constraint and serves as the workhorse for crash count modeling.

1.2. Handling excess zeros

One methodological challenge often faced in analyzing count variables is the presence of a large number of zeros. The classical count models (such as Poisson and NB) allocate a probability to observe zero counts, which is often insufficient to account for the preponderance of zeros in a count data distribution. In crash count variable models,the presence of excess zeros may resultfrom two underlying processes or states of crash frequency likelihoods: crash-free state (or zero crash state) and crash state (see (Shankar et al., 1997) for more explanation). The zero crash state can be a mixture oftrue zeros (where the zones are inherently safe (Shankar et al., 1997)) and sampling zeros (where excess zeros are results of potential underreporting of crash data (Miaou, 1994)). In presence of such dual-state, application of single-state model (Poisson and NB) may result in biased and inconsistent parameter estimates.

In econometric literature, two potential relaxations of the single-state count models are proposed for addressing the issue of excess zeros. The first approach − the zero inflated (ZI) model − is typically used for accommodating the effect of both true and sampling zeros, and has been employed in severaltransportation safety studies (Shankar et al., 1997; Chin and Quddus, 2003). The second approach − the Hurdle model − is typically used in the presence of sampling zeros and has seldom been used in transportation safety literature. The two approaches differ in the approach employed to address the excess zeros. The appropriate framework for analysis might depend on the actual empirical dataset under consideration. Table 1 presents a summary of previous studies that have considered zero-inflated and hurdle models to analyze crashes. The table provides information on type and severity of crash analyzed, spatial and temporal unit of analysis and the data collection duration. From the table, it is evident that all the existing zero-inflated and hurdle studies are conducted at a micro-level such as segment and intersection exceptfor Brijs et al.(2006), which conducted crash analysis at macro-level by assigning crashes to the closest weather station. Second, with the exception of study (Hu et al., 2011; Hosseinpour et al., 2013; Hosseinpour et al., 2014), the range of observation of the studyperiodis one year or less;thatmay explainthepreponderance of zeros in the data (Lord et al., 2005). Third, the zero-inflated model always offers better statistical fit to crash data.

1.3. Issues with dual-state models

To be sure, several research studies have criticized the application of zero-inflated model for traffic safety analysis (Lord et al., 2005; Lord et al., 2007; Kweon, 2011). The authors question the basic dual-state assumption for crash occurrence and have conducted extensive analysis at the micro-level and indicated that the development of models with dual-state process is inconsistent with crash data at the micro-level. While the reasoning behind the "non-applicability" is plausible for micro-level the reasoning does not necessarily carry over to the macro-level crash counts. At the macro-level it is possible to visualize dual-state data generation with some macro-level units having zero pedestrian and bicyclist crashes − possibly because these spatial units have no pedestrian and bicycle demand (because of lack of walking and cycling infrastructure). In such cases the dual-state representation will allow us to identify spatial units that are likely to have zero cases as a function of exogenous variables (for example very low walking and cycling infrastructure might result in the higher probability of a zero state). Hence, we have considered the possible existence of dual-state models for pedestrian and bicycle crashes at the macro level in our research. If the data generation does support the dualstate models, ignoring the excess zeros and estimating traditional NB models will result in biased estimates.

1.4. Spatial spillover effects

In macro-level analysis, crashes occurring in a spatial unit are aggregated to obtain the crash frequency. The aggregation process might introduce errors in identifying the exogenous variables for the spatial unit. For example, a crash occurring closer to the boundary of the unit might be strongly related to the neighboring zone than the actual zone where the crash occurred. This is a result of arbitrarily demarcating space. To accommodate for such spatial unit induced bias, two approaches to incorporate spatial dependency are considered: (1) spatial error correlation effects (unobserved exogenous variables at one location affect dependent variable at the targeted and neighboring locations) and (2) spatial spillover effects (observed exogenous variables at one location having impacts on the dependent variable at both the targeted and neighboring locations) (Narayanamoorthy et al., 2013). Several research efforts have accommodated for spatial random error in safety literature (for example see (Huang et al., 2010; Siddiqui et al., 2012; Lee et al., 2015)). On the other hand, researchers have considered a spatially lagged dependent variable at neighboring units for the spatial spillover effects (Lascala et al., 2000; Quddus, 2008; Ha and Thill, 2011). However, the utility of such spatially lagged dependent variable models, particularly for prediction, is limited since developing prediction frameworks for spatially lagged models is involved. In our analysis, to accommodate for spatial effects, we propose the consideration of exogenous variables from neighboring zones for accounting for spatial dependency. The approach, referred to as spatial spillover model, is easy to implement and allows practitioners to understand and quantify the influence of neighboring units on crash frequency.

In summary, the current study contributes to non-motorized macro-level crash analysis along two directions: (1) evaluate the viability of dual-state models for non-motorized crash analysis at macro-level; and (2) introduction of spatial independent variables accounting for spatial spillover effects on crash frequency. Towards this end, conventional single-state model (i.e., NB) and two dual-state models (i.e., zero-inflated NB (ZINB) and hurdle NB (HNB)) with and without spatial independent variables are developed for both pedestrian and bicycle crashes at a TAZ level in Florida. Overall, both pedestrian and bicycle crashes have 6 model structures estimated − NB model without/with spatial effects (aspatial/spatial NB), ZINB model without/with spatial effects (aspatial/spatial ZINB), and HNB model without/with spatial effects (aspatial/spatial HNB). The model development process considers a sample for model calibration and a hold-out sample

Table 1 Summary of Previous Traffic Safety Studies Using Zero-Inflated Models.

Methodology	Study	Crash types	Spatial Unit	Temporal Unit	Number of Study Years
Zero-inflated	Shankar et al. (1997)	Total crashes	Road segment	2 years	2 years
	Miaou (1994)	Truck crashes	Road segment	1 year	5 years
	Chin and Quddus (2003)	Total/pedestrian/motorcycle crashes	Signalized intersection	1 year	1 year
	Brijs et al. (2006)	Total crashes	Weather station	1 h	1 year
	Hu et al. (2011)	Total crashes	Railroad-grade crossing	3 years	3 years
	Carson and Mannering (2001)	Crashes in ice condition	Road segment	1 year	3 years
	Lee and Mannering (2002)	Run-off-roadway crashes	Road segment	1 month	3 years
	Mitra et al. (2002)	Head-to-side/head-to-rear crashes	Signalized intersection	1 year	8 years
	Kumara and Chin (2003)	Total crashes	Signalized intersection	1 year	9 years
	Shankar et al. (2004)	Pedestrian crashes	Road segment	1 year	1 year
	Qin et al. (2004)	Single-vehicle/multi-vehicle crashes	Road segment	1 year	4 years
	Huang and Chin (2010)	Total crashes	Signalized intersection	1 year	8 years
	Jang et al. (2010)	Total crashes	Road segment	1 year	1 year
	Dong et al. (2014a)	Truck/Car crashes	Intersection	1 year	5 years
	Dong et al. (2014b)	Crashes by severity	Intersection	1 year	5 years
Hurdle	Hosseinpour et al. (2013)	Pedestrian crashes	Road segment	4 years	4 years
	Hosseinpour et al. (2014)	Head-on crashes	Road Segment	4 years	4 years
	Kweon (2011)	Total crashes	Road segment	<1 h	6 years

for validation. A comparison exercise is undertaken to identify the superior model in model estimation and validation. Finally, average marginal effects are computed for the best model to assess the effect of different factors, including the spatial variables on crash occurrence.

2. Methodology

2.1. Single-state models

The Poisson model is the traditional starting model for crash frequency analysis (Lord and Mannering, 2010). The Poisson model assumes that the mean and variance of the distribution are the same. Thus,the Poissonmodel cannot deal withthe over-dispersion (i.e. variance exceeds the mean). The NB model relaxes the equal mean variance assumption of Poisson model and allows for overdispersion parameter by adding an error term,i,to the mean ofthe Poisson model as:

$$\lambda_i = \exp\left(\beta_i x_i + \varepsilon_i\right) \tag{1}$$

where ⁱ is the expected number of Poisson distribution for entity i, xi is a set of explanatory variables, and ⁱ is the corresponding parameter. Usually, exp (i) is assumed to be gamma-distributed with mean 1 and variance so that the variance of the crash frequency distribution becomes ⁱ 1 + ⁱ and different from the mean i. The NB model for the crash count yi of entity iis given by

$$P(y_i) = \frac{\Gamma\left(y_i + \frac{1}{a}\right)}{\Gamma\left(y_i + 1\right)\Gamma\left(\frac{1}{a}\right)} \left(\frac{\alpha\lambda_i}{1 + \alpha\lambda_i}\right)^{y_i} \left(\frac{1}{1 + \alpha\lambda_i}\right)^{\frac{1}{\alpha}} \tag{2}$$

where yi is the number of crashes yi of entity i and refers (·) to the gamma function. The NB model can generally account overdispersion resulting from unobserved heterogeneity and temporal dependency, but may be improper for accounting for the overdispersion caused by excess zero counts (Rose et al., 2006).

2.2. Dual-state models

2.2.1. Zero-inflated model

The zero-inflated models assume that the data have a mixture with a degenerate distribution whose mass is concentrated at zero (Lambert, 1992). Thefirstpart ofthemixture is the extra zero counts and the second part is for the usual single state model conditional on the excess zeros. The zero-inflated NB model can be regarded as an extension of the traditional NB specification as:

$$y_i \sim \begin{cases} 0, with probability p_i \ NB, with probability 1 - p_i \end{cases}$$ (3)

The logistic regression model is employed to estimatepi,

$$p{i} = \frac{\exp\left(\beta{i}'x{i}\right)}{1 + \exp\left(\beta{i}'x_{i}\right)} \tag{4}$$

where ' ⁱ is the corresponding parameter.

Substituting Eq. (2) into Eq. (3) we can define ZINB model for crash counts yi of entity i as

$$P(y{i}) = \begin{cases} p{i} + (1 - p{i})(\frac{1}{1 + \alpha \lambda{i}})^{\frac{1}{\alpha}}, & y{i} = 0\ (1 - p{i})\frac{\Gamma(y{i} + \frac{1}{\alpha})}{\Gamma(y{i} + 1)\Gamma(\frac{1}{\alpha})} \frac{(\alpha \lambda{i})^{y{i}}}{(1 + \alpha \lambda{i})^{(y{i} + \frac{1}{\alpha})}}, & y_{i} > 0 \end{cases}$$ (5)

2.2.2. Hurdle models

The Hurdle models, proposed by Mullahy (1986), can be regarded as two-part models. The first part is a binary model dealing with whether the response crosses the "hurdle", and the second part is a truncated-at-zero count model. Assume that the first hurdle part of process is governed by function f1 and the second count process follows a truncated-at-hurdle function f2. The Hurdle models are defined as follows:

$$P(y_i) = \begin{cases} f_1(0) = p_i, y_i = 0 \ (1 - f_1(0)) \frac{f_2(j)}{1 - f_2(0)}, y_i > 0 \end{cases}$$ (6)

Hurdle NB model is obtained by specifying f² (·) as the NB distribution. Substitution Eq. (2) into Eq. (6) will result in ZINB model as follows:

$$P(y_i) = \begin{cases} p_i, & y_i = 0 \ (1 - p_i)(1 - \frac{1}{(1 + \alpha \lambda_i)^{\frac{1}{\alpha}}}) \frac{\Gamma(y_i + \frac{1}{\alpha})}{\Gamma(y_i + 1)\Gamma(\frac{1}{\alpha})} \frac{(\alpha \lambda_i)^{y_i}}{(1 + \alpha \lambda_i)^{0_i + \frac{1}{\alpha}}}. & y_i > 0 \end{cases}$$ (7)

As in the zero-inflated model, logistic regression will be applied for modeling pi.

Fig. 1. Pedestrian and bicycle crashes based on TAZs.

2.3. Data preparation

About 16,240 pedestrian and 15,307 bicycle involved crashes that occurred in Florida in the period of 2010–2012 were compiled for the analysis. The State of Florida has 8518 TAZs, with an average area of 6.472 square miles. This TAZ system used in this paper is developed and used by the Florida Department of TransportationCentral Office for statewide leveltransportationplanning. Among the TAZs, as shown in Fig. 1, 46.18% of them have zero pedestrian crashes while 49.86% of them did not have any bicycle crashes. The explanatory variables considered for the analysis can be grouped into three categories: traffic (such as VMT (Vehicle-Miles-Traveled), proportion of heavy vehicle in VMT), roadway (such as signalized intersection density, length of bike lanes and sidewalks, etc.), and socio-demographic characteristics (such as population density, proportion of families without vehicle, etc.).

As highlighted earlier, the current analysis focuses on accommodating the impact of neighboring TAZs on the crash frequency models. Towards this end, for every TAZ, the TAZs that are adjacent are identified. Based on the identified neighbors, a new variable based on the value of the each exogenous variable from surrounding TAZs is computed. The variables thus created capture the spatial spillover effects of the neighboring TAZs on crash frequency. The descriptive statistics of the crash counts and independent variables are summarized in Table 2. Specifically,the table provides the values at a TAZ level as well as for the neighboring TAZ variables.

2.4. Modeling results and discussion

2.4.1. Goodness of fit

In this study, from the 8518 TAZs, 80% of the zones were randomly selected for models calibration and 20% were used for validation of the estimated models. The overall model estimation process involved estimating six models − 3 model types (NB, ZINB, and HNB models) with and without spatial independent variables of neighboring TAZs for pedestrian and bicycle crashes. Prior to discussing the model results, we present the goodness of fit measures of the estimated models in Table 3. The table presents the Log-likelihood, Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) − for the 6 models for estimation and validation samples. Several observations can be made from the results presented in Table 3. First, across pedestrian and bicycle crash models, the models with spatial independent variables offer substantially better fit compared to models without spatial independent variables. The results validate our hypothesis that characteristics of adjacent TAZs improve our understanding of crash frequency in the target TAZ. Second, the exact ordering alters between ZINB and HNB in some cases based on log-likelihood and AIC. However, the ZINB model offers the best fit across all model structures based on the BIC. Among aspatial and spatial models,the ZINB model always has the lowest BIC value indicating strong difference between ZINB and other models. The ZINB improves data fit with only a small increase in number of parameters. Hence, in terms of our results, we can conclude that the ZINB offers the best statistical fit for pedestrian and bicycle crashes. Third, in validation exercise, it is further reinforced that ZINB offers the best data fit.

3. Results

The results of six models (3 model types with and without spatial independent variables of neighboring TAZs) for pedestrian and bicycle crashes each are displayed in Table 4 and Table 5 separately. The results for NB models only have the count frequency component. For zero-inflated and hurdle models, the modeling results consist of two components: (1) logistic model component for zero state and (2) the count frequency component. Across the 6 models for either pedestrian or bicycle crashes, the significant variables are different. Some of the explanatory variables such as VMT, population density are transformed into the natural logarithmic scale. Generally, a log link between dependent and independent variables is specified in the modeling regression. Thus, with the transformation of the independent variables, the relationship of power function between explanatory variables and crash counts can be obtained which was widely adopted in previous research (Greibe, 2003; Abbas, 2004). Also, this transformation reduces variance and minimize the heteroscedasticity among the variables (Quddus, 2008; Gujarati, 2012). Meanwhile, with a log transformation the parameter of the explanatory variable results in a linear elasticity which is easy to interpret.While the results for allmodels for pedestrians and bicycle crashes are presented, the discussion focuses on the ZINB model with spatial independent variables that offers the best fit.

3.1. Pedestrian crash models for TAZs

For ZINB model with spatial independent variables,twelve independent variables of targeted TAZs and four spatial independent variables are significant in the count component.

The VMT variable is a measure of vehicle exposure and as expected increases the propensity for pedestrian crashes. However, with increase in heavy vehicle VMT, TAZs are likely to have

Table 2 Descriptive statistics of collected data.

Variables name	Targeted TAZs			Neighboring TAZs
	Mean	S.D.	Maxa	Mean	S.D.	Maxa
Crash variables
Pedestrian crash	1.907	3.315	39.000	–	–	–
Bicycle crash	1.797	3.309	88.000	–	–	–
Traffic & roadway variables
VMT	31381.0	41852.3	684742.8	195519.7	169120.3	2103376.3
Proportion of heavy vehicle in VMT	0.067	0.052	0.519	0.070	0.045	0.350
Proportion of length of arterial roads	0.221	0.275	1.000	0.144	0.125	1.000
Proportion of length of collectors	0.191	0.246	1.000	0.156	0.136	1.000
Proportion of length local roads	0.572	0.329	1.000	0.680	0.200	1.000
Signalized intersection density (number of signalized	0.227	0.578	8.756	0.378	5.552	495.032
intersections per mile)
Length of bike lanes	0.303	1.096	28.637	1.909	3.847	38.901
Length of sidewalks	0.993	1.750	25.683	6.304	6.745	77.720
Socio-demographic variables
Population density	2520.3	4043.3	63069.0	2330.2	3489.7	57181.9
Proportion of families without vehicle	0.095	0.123	1.000	0.095	0.108	1.000
School enrollments density	775.02	5983.05	255147.24	684.22	2900.54	102285.73
Proportion of urban area	0.722	0.430	1.000	0.650	0.434	1.000
Distance to the nearest urban area	2.140	5.441	44.101	–	–	–
Hotels, motels, and timeshare rooms density	172.49	941.71	32609.84	121.678	528.078	11397.148
No of total employment	1140.10	1722.45	31932.15	6917.245	6725.135	76533.000
Proportion of industry employment	0.176	0.232	1.000	0.183	0.177	1.000
Proportion of commercial employment	0.299	0.235	1.000	0.305	0.177	1.000
Proportion of service employment	0.525	0.257	1.000	0.495	0.186	1.000
No of commuters by public transportation	18.813	54.273	934.000	119.582	246.299	3559.985
No of commuters by cycling	5.894	19.804	775.000	90.869	128.399	1902.135
No of commuters by walking	14.354	34.680	1288.000	37.566	74.484	1634.530

^a The minimum values for all variables are zero.

Table 3 Comparison of goodness-of-fits between different models.

| Pedestrian Crash | | | | | | | | |------------------------|----------|---------|----------|---------|----------|---------|--| | | NB | | ZINB | | HNB | | | | Calibration (N = 6815) | Aspatial | Spatial | Aspatial | Spatial | Aspatial | Spatial | | | No of parameters | 15 | 17 | 20 | 22 | 24 | 28 | | | Log-likelihood | −9972.4 | −9926.6 | −9944.3 | −9890 | −9964.4 | −9912.5 | | | AIC | 19974.7 | 19887.3 | 19928.5 | 19824 | 19976.8 | 19881 | | | BIC | 20077.1 | 20003.3 | 20065.1 | 19974.2 | 20140.7 | 20072.2 | | | Validation (N = 1703) | Aspatial | Spatial | Aspatial | Spatial | Aspatial | Spatial | | | No of parameters | 15 | 17 | 20 | 22 | 24 | 28 | | | Log-likelihood | −2680.5 | −2662.4 | −2449.9 | −2437.8 | −2464.3 | −2459.4 | | | AIC | 5391 | 5358.8 | 4939.7 | 4919.5 | 4976.5 | 4974.8 | | | BIC | 5472.6 | 5451.2 | 5048.5 | 5039.2 | 5107.1 | 5127.1 | | | Bicycle Crash | | | | | | | | | | NB | | ZINB | | HNB | | | | Calibration (N = 6815) | Aspatial | Spatial | Aspatial | Spatial | Aspatial | Spatial | | | No of parameters | 14 | 19 | 18 | 22 | 25 | 33 | | | Log-likelihood | −9412.4 | −9326.0 | −9385.6 | −9309.0 | −9387.2 | −9286.3 | | | AIC | 18852.9 | 18689.9 | 18807.2 | 18662.1 | 18824.3 | 18638.6 | | | BIC | 18948.5 | 18819.6 | 18930.1 | 18812.3 | 18995 | 18863.9 | | | Validation (N = 1703) | Aspatial | Spatial | Aspatial | Spatial | Aspatial | Spatial | | | No of parameters | 14 | 19 | 18 | 22 | 25 | 33 | | | Log-likelihood | −2771.6 | −2785.9 | −2393.4 | −2355.6 | −2396.4 | −2364.8 | | | AIC | 5571.2 | 5609.8 | 4822.8 | 4755.2 | 4842.8 | 4795.7 | | | BIC | 5647.4 | 5713.2 | 4920.7 | 4874.9 | 4978.8 | 4975.2 | |

lower pedestrian exposure resulting in lower probability of vehiclepedestrian interactions.

Population density and total employment variables are surrogate measures of pedestrian exposure (Siddiqui et al., 2012). Hence, it is expected that these variables have positive impacts on crash frequency. The variables proportion of local roads by length, signalized intersection density, and length of sidewalks are reflections of pedestrian access. Increased local roads, signalized intersections, and sidewalks may attract more pedestrians and are likely to increase crash frequency. The positive estimate of the number of hotels, motels and timeshare rooms' variable reflects land use characteristics that are likely to encourage walking in the vicinity increasing pedestrian exposure. It is observed that in TAZs with higher number of commuters by walking and public transportation, the propensity for pedestrian crashes is higher. The commuters by walking and public transportation reflect zones with higher pedestrian activity resulting in increased crash risk (Abdel-Aty et al., 2013). As the distance from a TAZ geometric centroid to the nearest

Table 4 Models results for pedestrian crash of TAZs.

	NB				ZINB				HNB
Count Model	Aspatial		Spatial		Aspatial		Spatial		Aspatial		Spatial
Parameter	Est.	S.E.	Est.	S.E.	Est.	S.E.	Est.	S.E.	Est.	S.E.	Est.	S.E.
Intercept	−4.513	0.139	−4.632	0.142	−4.202	0.159	−4.323	0.162	−3.504	0.187	−3.745	0.198
TAZ independent variables
Log (VMT)	0.145	0.009	0.142	0.009	0.155	0.009	0.154	0.009	0.112	0.011	0.103	0.011
Proportion of heavy vehicle mileage in VMT	−1.108	0.416	−1.123	0.413	−1.424	0.422	−1.522	0.416	−1.890	0.556	−1.656	0.547
Log (population density)	0.124	0.011	0.105	0.011	0.102	0.011	0.093	0.011	0.115	0.014	0.097	0.014
Log (number of total employment)	0.235	0.013	0.225	0.013	0.205	0.015	0.195	0.015	0.186	0.017	0.186	0.017
Proportion of length of local roads	0.467	0.059	0.471	0.058	0.504	0.060	0.508	0.059	0.480	0.080	0.454	0.080
Log (signalized intersection density)	0.291	0.028	0.267	0.028	0.256	0.030	0.267	0.031	0.274	0.038	0.286	0.040
Log (length of sidewalks)	0.272	0.025	0.277	0.024	0.244	0.025	0.255	0.025	0.271	0.028	0.273	0.028
Log (hotels, motels, and timeshare rooms density)	0.022	0.006	0.026	0.006	0.021	0.006	0.030	0.006	0.030	0.007	0.037	0.007
Log (number of commuters by public transportation)	0.194	0.009	0.129	0.012	0.189	0.009	0.125	0.012	0.205	0.011	0.134	0.014
Log (number of commuters by walking)	0.067	0.011	0.065	0.011	0.052	0.012	0.056	0.012	0.057	0.013	0.060	0.013
Log (number of commuters by cycling)	0.027	0.011	0.031	0.011	0.027	0.011	0.030	0.011	–	–	–	–
Log (distance to nearest urban area)	−0.027	0.006	−0.024	0.006	−0.028	0.006	−0.025	0.006	–	–	–	–
Proportion of families without vehicle	–	–	–	–	0.717	0.136	–	–	–	–	–	–
Proportion of service employment	0.314	0.062	0.221	0.068	0.296	0.062	–	–	–	–	–	–
Spatial Independent Variables
Proportion of service employment of neighboring TAZs	–	–	0.253	0.091	–	–	0.301	0.083	–	–	0.376	0.103
Log (signalized intersection density of neighboring TAZs)	–	–	–	–	–	–	−0.291	0.063	–	–	−0.211	0.073
Proportion of families without vehicle of neighboring TAZs	–	–	–	–	–	–	1.29	0.172	–	–	–	–
Log (number of commuters by public transportation of	–	–	0.099	0.011	–	–	0.091	0.011	–	–	0.108	0.014
neighboring TAZs)
Dispersion	0.445	0.020	0.423	0.020	0.393	0.022	0.367	0.021	0.419	0.028	0.386	0.026

Probabilistic Model
Intercept	–	–	–	–	0.070	0.413	-0.047	0.431	5.733	0.237	5.791	0.238
TAZ independent variables
Log (VMT)	–	–	–	–	–	–	–	–	−0.188	0.015	−0.184	0.015
Log (length of sidewalks)	–	–	–	–	−2.143	0.729	−1.995	0.715	−0.500	0.064	−0.502	0.064
Log (number of total employment)	–	–	–	–	−0.240	0.070	−0.232	0.072	−0.299	0.023	−0.295	0.023
Log (number of commuters by walking)	–	–	–	–	−0.527	0.153	−0.501	0.148	−0.138	0.027	−0.136	0.027
Proportion of length of local roads	–	–	–	–	–	–	–	–	−0.510	0.104	−0.516	0.104
Log (signalized intersection density)	–	–	–	–	–	–	–	–	−0.331	0.054	−0.319	0.054
Log (population density)	–	–	–	–	–	–	–	–	−0.164	0.019	−0.155	0.019
Proportion of service employment	–	–	–	–	–	–	–	–	−0.405	0.126	−0.413	0.127
Log (number of commuters by public transportation)	–	–	–	–					−0.247	0.025	−0.192	0.030
Log (number of commuters by cycling)	–	–	–	–	–	–	–	–	−0.074	0.032	−0.074	0.032
Log (distance to nearest urban area)	–	–	–	–	–	–	–	–	0.030	0.008	0.027	0.008
Spatial Independent Variables
Log (number of commuters by public transportation of neighboring TAZs)	–	–	–	–	–	–	–	–	–	–	−0.075	0.022

All explanatory variables are significant at 95% confidence level.

urban region increases, pedestrian crash risk in the TAZ reduces − a sign of low pedestrian activity in the suburban regions.

Among the significant spatial spillover variables, the proportion of service employment corresponds to surrounding land use characteristics that attract pedestrians and therefore increases the propensity of pedestrian crashes. Interestingly, the impact of signalized intersection density of neighboring TAZs is found to be negatively associated with pedestrian crash frequency. This result is in contrast to the impact of the same variable for the targeted TAZ. The number of signalized intersections reflects the increase in exposure for pedestrians thus resulting in an increase in crashes. However, the influence of signalized intersections in neighboring TAZs has an ameliorating impact on crash frequency i.e., TAZs surrounded by zones with higher signalized intersection density have a lower propensity for crash occurrence because the higher density of signalized intersections is likely to increase driver awareness of pedestrians offsetting the exposure effect marginally (Zajac and Ivan, 2003; Eluru et al., 2008). The proportion of families without vehicles in the vicinity of TAZ represents captive individuals that are forced to use public transit and pedestrian/bicycle modes. Thus increased presence of such families is likely to increase pedestrian exposure, leading to more pedestrian crashes. Higher number of commuters by public transportation in the neighboring TAZs also results in increased pedestrian crash frequency.

In the probabilistic component, only the length of sidewalks, number of total employment, and number of commuters by public transportation of the targeted TAZs are significant. As expected, these three variables are negatively associated with the propensity of zero pedestrian crashes. As these variables serve as surrogates for pedestrian activity, it is expected that TAZs with higher levels of these variables are unlikely to be assigned to the zero crash state. Interestingly, no spatial spillover effects are found to be significant in the probabilistic part.

3.2. Bicycle crash models for TAZs

In the ZINB model with spatial variables presented in Table 5 eleven variables for the TAZs and five variables of neighboring TAZs affect bicycle crash frequency. The impacts of exogenous variables in the bicycle crash frequency model are very similar to the impact of these variables in the pedestrian crash frequency model. This is not surprising because, TAZs that are likely to experience high pedestrian activity are also likely to experience high bicyclist activity.

For the count component, the exogenous variables for the TAZ that increase the crash propensity are VMT, population density, total employment, proportion of local roads by length, signalized intersection density, length of sidewalks, proportion of commuters

Table 5 Models results for bicycle crash of TAZs.

Count Model	Aspatial NB		Spatial		Aspatial ZINB		Spatial		Aspatial HNB		Spatial
Parameter	Est.	S.E.	Est.	S.E.	Est.	S.E.	Est.	S.E.	Est.	S.E.	Est.	S.E.
Intercept	−4.650	0.154	−4.672	0.167	−4.090	0.181	−4.673	0.190	−3.620	0.220	−4.031	0.237
TAZ independent variables
Log (VMT)	0.190	0.009	0.162	0.010	0.186	0.010	0.164	0.010	0.168	0.013	0.148	0.013
Proportion of heavy vehicle mileage in VMT	−4.260	0.485	−3.306	0.490	−4.244	0.487	−2.787	0.496	−4.115	0.665	−2.949	0.660
Log (population density)	0.152	0.013	0.130	0.013	0.133	0.014	0.087	0.015	0.131	0.018	0.084	0.020
Log (number of total employment)	0.193	0.014	0.194	0.014	0.157	0.016	0.161	0.016	0.142	0.018	0.134	0.018
Proportion of length of local roads	0.535	0.062	0.441	0.064	0.517	0.063	0.525	0.063	0.422	0.086	0.401	0.085
Log (signalized intersection density)	0.196	0.030	0.234	0.032	0.172	0.031	0.203	0.033	0.125	0.041	0.184	0.044
Log (length of sidewalks)	0.284	0.026	0.271	0.025	0.214	0.027	0.228	0.026	0.219	0.030	0.217	0.029
Log (number of commuters by public transportation)	0.106	0.010	0.086	0.012	0.107	0.010	–	–	0.096	0.012	0.084	0.012
Log (number of commuters by walking)	0.087	0.012	0.085	0.012	0.090	0.012	0.104	0.012	0.101	0.014	0.099	0.014
Log (number of commuters by cycling)	0.109	0.011	0.070	0.012	0.110	0.011	0.088	0.012	0.108	0.012	0.071	0.013
Log (distance to nearest urban area)	-0.103	0.011	−0.098	0.011	-0.097	0.011	-0.074	0.011	-0.092	0.024	-0.065	0.023
Proportion of service employment	0.205	0.066	0.153	0.067	0.192	0.066	0.173	0.067	–	–	–	–
Spatial Independent Variables
Proportion of industry employment of neighboring TAZs	–	–		0.106	–	–		0.106	–	–	–	–
Log (signalized intersection density of neighboring TAZs)	–	–	−0.361	0.075	–	–	−0.242	0.069	–	–		0.095
Log (population density of neighboring TAZs)	–	–	−0.319 –	–	–	–	−0.473 0.113	0.018	–	–	−0.545 0.109	0.023
Log (number of commuters by public transportation of	–	–	0.035	0.012	–	–	0.068	0.010	–	–	–	–
neighboring TAZs)
Log (number of commuters by cycling of neighboring TAZs)	–	–	0.093	0.012	–	–	0.073	0.012	–	–	0.098	0.014
Proportion of length of local roads of neighboring TAZs	–	–	0.354	0.125	–	–	–	–	–	–	–	–
Dispersion	0.481	0.022	0.443	0.021	0.425	0.022	0.397	0.021	0.454	0.031	0.406	0.028
Probabilistic Model
Intercept	–	–	–	–	1.565	0.489	1.296	0.509	5.452	0.241	5.700	0.279
TAZ independent variables
Log (VMT)	–	–	–	–	–	–	–	–	−0.222	0.016	−0.217	0.017
Log (length of sidewalks)	–	–	–	–	−4.455	1.272	−4.819	1.563	−0.676	0.066	−0.681	0.066
Log (population density)	–	–	–	–	−0.149	0.05	−0.135	0.053	−0.177	0.021	−0.102	0.024
Log (number of total employment)	–	–	–	–	−0.328	0.058	−0.313	0.060	−0.236	0.023	−0.216	0.024
Proportion of heavy vehicle mileage in VMT	–	–	–	–	–	–	–	–	5.347	0.836	4.258	0.861
Proportion of length of local roads	–	–	–	–	–	–	–	–	−0.709	0.109	−0.696	0.112
Log (signalized intersection density)	–	–	–	–	–	–	–	–	−0.286	0.054	−0.243	0.056
Log (number of commuters by public transportation)	–	–	–	–	–	–	–	–	−0.210	0.025	−0.147	0.031
Log (number of commuters by walking)	–	–	–	–	–	–	–	–	−0.081	0.028	−0.079	0.028
Log (number of commuters by cycling)	–	–	–	–	–	–	–	–	−0.158	0.032	−0.099	0.035
Spatial Independent Varriables
Proportion of length of arterial of neighboring TAZs	–	–	–	–	–	–	–	–	–	–	1.337	0.290
Log (population density of neighboring TAZs)	–	–	–	–	–	k	–	–	–	–	−0.096	0.033
Log (hotels, motels, and timeshare rooms density of	–	–	–	–	–	–	–	–	–	–	−0.041	0.018
neighboring TAZs)
Log (number of commuters by public transportation of	–	–	–	–	–	–	–	–	–	–	−0.069	0.026
neighboring TAZs)
Log (number of commuters by cycling of neighboring TAZs)	–	–	–	–	–	–	–	–	–	–	−0.082	0.025

All explanatory variables are significant at 95% confidence level.

by walking as well as cycling, and proportion of service employment. The exogenous variables for the TAZ that reduce crash propensity are proportion of heavy vehicle mileage and the distance of the TAZ centroid from the nearest urban region. There are three main difference in the TAZ variable impacts between pedestrian and bicyclist crash frequency. First, the number of commuters by public transportation does not have significant impacts on crash frequency as it is possible that public transportation and bicycling are not as strongly correlated as is the case with public transportation and pedestrians. Second, the density of hotel, motel and time share rooms does notimpact bicycle crash frequency as tourists are less likely to be bicyclists. Third, the service employment count in the TAZ affects bicycle crash frequency while affecting pedestrian crash frequency as a spillover effect. While, the exact reason for the result is unclear, it could be a manifestation of differences of how land-use affects pedestrians and bicyclists.

In terms of spatial spillover effects,the significant variables vary between pedestrian and bicyclists. Specifically,the high proportion of industry employment in neighboring TAZs has a negative association with crash propensity indicating that surrounding regions especially the targeted TAZs are unlikely to have significant bicyclist exposure. The signalizedintersectiondensity exhibits the same relationship as described for pedestrian crashes. On the other hand, from the neighboring TAZs, population density, number of commuters by public transit and cycling are surrogates for bicyclist exposure and are found positively associated with bicycle crashes.

In the probabilistic component, only three explanatory variables of targeted TAZs variables are significant. The length of sidewalks, population density and total employment variables, as expected, have negative influence on assigning a TAZ to a zero crash state. The bicycle crash probabilistic component also does not have any statistically significant spatial variables.

3.3. Marginal effects

The ZINB has two components, the probabilistic and the count component with exogenous variables possibly affecting both components. Thus, it is not straight-forward to identify the exact magnitude of the variable impact. Hence, to facilitate a quantitative comparison of variable impacts, marginal effects for the ZINB for pedestrians and bicyclists are computed. The marginal effects capture the change in the dependent variable in response to a small

Table 6 Average marginal effect for ZINB model with spatial independent variables.

	Pedestrian		Bicycle
Variables	dy/dx	S.E	dy/dx	S.E
TAZ independent variables
Log (VMT)	0.292	0.018	0.291	0.018
Proportion of heavy vehicle mileage in VMT	−2.888	0.791	−4.937	0.885
Log (population density)	0.176	0.021	0.162	0.027
Log (number of total employment)	0.382	0.027	0.302	0.027
Proportion of length of local roads	0.965	0.114	0.930	0.113
Log (signalized intersection density)	0.506	0.06	0.359	0.059
Log (length of sidewalks)	0.587	0.05	0.671	0.077
Log (hotels, motels, and timeshare rooms density)	0.056	0.011	–	–
Log (number of commuters by public transportation)	0.238	0.022	–	–
Log (number of commuters by walking)	0.131	0.021	0.184	0.021
Log (number of commuters by cycling)	0.057	0.02	0.156	0.021
Log (distance to nearest urban area)	−0.047	0.011	−0.132	0.019
Proportion of service employment	–	–	0.307	0.118
Spatial Independent Variables
Proportion of service employment of neighboring TAZs	0.572	0.158	–	–
Proportion of industry employment of neighboring TAZs	–	–	−0.428	0.189
Log (signalized intersection density of neighboring TAZs)	−0.552	0.119	−0.838	0.124
Proportion of families without vehicle of neighboring T AZs	2.447	0.329	–	–
Log (population density of neighboring TAZs)	–	–	0.200	0.033
Log (number of commuters by public transportation of neighboring TAZs)	0.173	0.021	0.120	0.019
Log (number of commuters by cycling of neighboring TAZs)	–	–	0.130	0.021

change in the independent variables. The results of the marginal effect calculation are presented in Table 6. As is expected, the sign of the marginal effects closely follows the sign from model results described in Table 4 and 5. The marginal effects represent the percentage change in the crash frequency variable for a 1% change in the exogenous variable. For example, for the first row in Table 6, a 1% change in Log(VMT) is likely to result in a 0.292% change in pedestrian crash frequency and 0.291% change in bicyclist crash frequency. The other parameters can also be interpreted in a similar fashion.

The following observations can be made based on the results presented. First, the impact of spatial spillover effects on the crash models is significant and is comparable to the influence of other exogenous variables. Hence, it is important that analysts consider such observed spatial spillover effects in crash frequency modeling. Second, the exogenous variable impacts on pedestrian and bicycle crash models are similar for a large number of variables including VMT, population density, total employment, number of commuters by walking, proportion oflocal road in length, and number of public transportation commuters in neighboring TAZs. All of these variables have marginal effects with positive values, indicating number of crashes (for both pedestrian and bicycle crashes) increase as these variables increase. Third, the exogenous variables such as proportion of heavy vehicle VMT, proportion of service employment, number of commuters by public transportation and cycling, proportion of families without vehicles in the neighboring TAZs, service employment and industry employment in neighboring TAZs have significantly different marginal impacts across the two models. Their negative marginal effect values show that crash counts will decrease if these variables increase. Finally, as indicated by the marginal effects of the signalized intersection density the exogenous variable for TAZ and neighboring TAZs could exhibit distinct effects both in sign and magnitude. The allowance of such non-linear impacts accommodates for heterogeneity in the data.

4. Conclusion

With growing concern of global warming and obesity concerns, active forms of transportation offer an environmentally friendly and physically active alternative for short distance trips. A strong impediment to universal adoption of active forms of transportation, particularly in North America, is the inherent safety risk for active modes of transportation. Towards developing counter measures to reduce safety risks, it is essential to study the influence of exogenous factors on pedestrian and bicycle crashes. This study contributes to safety literature by conducting a macro-level planning analysis for pedestrian and bicycle crashes at a Traffic Analysis Zone (TAZ) level in Florida. The study considers dual-state count models (zero-inflated negative binomial (ZINB) and hurdle negative binomial(HNB))for analysis by comparing with classical single state (negative binomial (NB)) and. In addition to the dual-state models,the research proposes the consideration of spatial spillover effects of exogenous variables from neighboring TAZs. The model development exercise involved estimating 6 model structures each for pedestrians and bicyclists. These include NB model with and without spatial effects, ZINB model with and without spatial effects and HNB with and without spatial effects. The estimated model performance was evaluated for the calibration sample and the validation sample using the following measures: Log-likelihood, Akaike Information Criterion and Bayesian Information Criterion.

The model comparison exercise for pedestrians and bicyclists highlighted that models with spatial spillover effects consistently outperformed the models that did not consider the spatial effects. Across the three models with spatial spillover effects, the ZINB model offered the best fit for pedestrian and bicyclists. The model results clearly highlighted the importance of several variables including traffic (such as VMT and heavy vehicle mileage), roadway (such as signalized intersection density, length of sidewalks and bike lanes, and etc.) and socio-demographic characteristics (such as population density, commuters by public transportation, walking and cycling) of the targeted and neighboring TAZs. To facilitate a quantitative comparison of variable impacts, marginal effects for the ZINB for pedestrians and bicyclists are computed. The results revealed the importance in sign andmagnitude ofthe spatial spillover effect relative to other exogenous variables. Further, the marginal effects computation allowed us to identify factors that substantially increase crash risk for pedestrians and bicyclists. In terms of actionable information, it is important to identify zones with high public transit, pedestrian and bicyclist commuters and undertake infrastructure improvements to improve safety.

To be sure, the study is not without limitations. While the influence of spatial spillover effects is considered, we do not consider the impact of spatial unobserved effects. Extending the current approachto accommodate forunobserved spatialterms will beuseful. In this study, we considered the distance from urban centers to the TAZ centroids determined purely based on the physical characteristics. It would be useful to consider TAZ centroids based on activity facilities. Also, itis possible to hypothesize thatthere might be common unobserved factors that affect pedestrian and bicyclists. Future research extensions might consider such unobserved effects in the model structure.