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1
1
Research highlights 2
This paper investigates factors associated with crash occurrences by different 3
transportation modes. 4
Zonal factors have significant effects on intersection crashes. 5
Zonal factors contribute to tracking the source of heterogeneity of risk factors. 6
Zonal factors play a more important role in elevating the model performance 7
for non-motorized than motor-vehicle crashes. 8
A relatively smaller buffer width to extract zonal factors yields better 9
estimations. 10
11
12
2
The Effect of Zonal Factors in Estimating Crash Risks by Transportation 1
Modes: Motor Vehicle, Bicycle and Pedestrian 2
3
Jie WANG1, Helai HUANG1*, Qiang ZENG2 4
5
1 School of Traffic and Transportation Engineering, Central South University, Changsha, Hunan, China 6
2 School of Civil Engineering and Transportation, South China University of Technology, Guangzhou, 7
Guangdong, China 8
*Correspondence 9
E-mail address: jie_wang@csu.edu.cn (J. Wang), huanghelai@csu.edu.cn (H. Huang), 10
zengqiang@scut.edu.cn (Q. Zeng) 11
12
Abstract: Objectives: This paper aimed to (i) differentiate the effects of contributory factors on 13
crash risks related to different transportation modes, i.e., motor vehicle, bicycle and pedestrian; (ii) 14
explore the potential contribution of zone-level factors which are traditionally excluded or 15
omitted, so as to track the source of heterogeneous effects of certain risk factors in 16
crash-frequency models by different modes. Methods: Two analytical methods, i.e. negative 17
binomial models (NB) and random parameters negative binomial models (RPNB), were 18
employed to relate crash frequencies of different transportation modes to a variety of risk factors 19
at intersections. Five years of crash data, traffic volume, geometric design as well as macroscopic 20
variables at traffic analysis zone (TAZ) level for 279 intersections were used for analysis as a case 21
study. Results: Among the findings are: (1) the sets of significant variables in crash-frequency 22
analysis differed for different transportation modes; (2) omission of macroscopic variables would 23
result in biased parameters estimation and incorrect inferences; (3) the zonal factors (macroscopic 24
factors) considered played a more important role in elevating the model performance for 25
non-motorized than motor-vehicle crashes; (4) a relatively smaller buffer width to extract 26
macroscopic factors surrounding the intersection yielded better estimations. 2 7
Keywords: transportation modes; macroscopic variables; unobserved heterogeneity; buffer 28
width; intersection safety 29
30
1 Introduction 31
Many communities have increased their interest in the implementation of multimodal 32
transportation and advocated for the shift from motor vehicles to non-motorized modes of 33
transportation, i.e., walking and cycling. In spite of the health and environmental benefits, an 34
increasing number of crashes involving pedestrians and bicyclists has become a major concern in 35
improving traffic safety. For example in 2013, the United States had 4735 pedestrian and 743 36
bicyclist deaths, accounting for 18% of all U.S. highway fatalities (NHTSA, 2013). The Federal 3 7
Highway Administration’s office of safety has established pedestrian and bicyclist safety as one 38
of its top priorities. Thus, it is essential for traffic safety engineers to provide appropriate 39
countermeasures or policies to achieve friendly and safe multimodal transportation. 40
A comprehensive understanding of contributing factors associated with crash occurrences by 41
different modes is a prerequisite for developing safety improvement programs to effectively 42
reduce traffic crashes. For a given road entity (e.g. road segments or intersections), the potential 4 3
factors associated with multimodal crashes could be summarized as in Figure 1, according with 44
Miranda-Moreno et al. (2011), Mitra and Washington (2012), Ukkusuri et al. (2012) and Strauss et 45
al. (2003; 2014). The factors influencing road-entity-level crash frequency by modes include 46
macroscopic factors related to built environment of the road entities - such as population and 47
economic characteristics, land use characteristics and travel behaviors - as well as road features 48
3
and traffic characteristics of the road entities. In addition, crash occurrence is also associated with 1
individual characteristics such as gender, age, education, alcohol consumption, and other driver 2
and pedestrian behaviors (Ryb et al., 2007). Although discrete individual-level factors are not 3
available to be integrated into the crash-frequency model, individual characteristics are always 4
influenced by macroscopic factors (Christoffel and Gallagher, 1999). Therefore, macroscopic 5
factors could serve as a surrogate for individual behaviors. 6
7
8
Figure 1. Factors associated with multimodal crashes 9
10
The choice of appropriate analytical method and the selection of representative explanatory 11
variables are two important considerations for obtaining accurate model predictions. Over the 1 2
past three decades, considerable research efforts have been devoted to developing and applying 1 3
sophisticated methodological approaches associated with the analysis of crash frequency. 1 4
Detailed descriptions and assessments of crash-frequency models can be found in the review 15
papers by Lord and Mannering (2010) and Mannering and Bhat (2014). However, relatively few 1 6
studies have focused on the identification and inclusion of traditionally excluded or omitted 17
variables in crash-frequency analysis. In particular, variables related to macroscopic factors 18
previously described (in Figure 1) are normally unavailable in crash databases and as a result 1 9
have rarely been examined in great detail. Mitra and Washington (2012) is one of a few studies 20
exploring the omitted variables in crash-frequency modeling. The authors developed two 21
different models of estimating intersection crash frequency, one with traffic volume as the only 22
independent variable, and the other with several spatial factors in addition to commonly 23
included geometric design and traffic factors. Through contrastive analysis of the two models, 2 4
results indicated that some spatial factors, such as local influences of weather, sun glare, 2 5
proximity to drinking establishment, proximity to school and demographic attributes near 26
intersections, have significant explanatory power and their exclusion leads to biased estimates. 27
Statistical methods such as spatial and temporal correlation, multilevel, random effect, 28
random parameter, and latent class approaches have been developed to address this issue of 29
unobserved heterogeneity (Anastasopoulos and Mannering, 2009; Dong et al., 2016; Mannering et 3 0
al., 2016; Quddus, 2008; Wang and Huang, 2016; Xu and Huang, 2015; Xu et al. 2016), as these 3 1
omitted explanatory variables can be regarded as part of the unobserved heterogeneity. 3 2
Unobserved heterogeneity impacts traffic safety analysis in two ways: the first problem is that the 33
selected explanatory variables cannot fully account for the cross-section or longitudinal-section 3 4
variations in crash counts due to unobserved road geometrics, environmental factors, driver 35
behavior and other confounding factors, which lead to impaired predictive performance of the 36
model (called heterogeneity in model prediction); the second problem is that these unobserved 37
factors are always correlated with observed factors and thus biased parameters will be estimated 3 8
and incorrect inferences could be drawn (called heterogeneity in the coefficient estimator). While 3 9
these approaches will mitigate the adverse impacts of omitting significant explanatory variables, 40
the resulting model estimates still fail to track the original source of heterogeneity and quantify 41
the safety effect of omitted variables (such as macroscopic factors shown in Figure 1). Omission of 42
4
important explanatory variables still remains a problem even with advanced statistical 1
approaches to capture unobserved heterogeneity (Mannering et al., 2016). 2
The study by Mitra and Washington (2012) attempted to investigate the safety effect of some 3
important omitted variables on total crash frequency and their contribution on model estimation. 4
As Venkataraman et al. (2013) stated, frequency models of crash outcome type can provide 5
substantial insights into the effect of explanatory variables and assist in examining the 6
heterogeneity effects in roadway geometric features. This paper aims to extend previous research 7
(Mitra and Washington, 2012) and investigate how macroscopic factors affect the crash-frequency 8
analysis for different transportation modes. This is because there may be some inconsistent 9
impacts of some macroscopic variables on motor vehicle and non-motorized (including b icycle 10
and pedestrian) crashes. For example, Lee et al. (2015) utilized a multivariate model for 11
investigating motor vehicle and non-motorized crashes at the macroscopic level. Results for the 12
parameter estimation suggested that some zonal variables related to demographics and road 1 3
characteristics have different directional effects on motor vehicle and non-motorized crashes. 14
Meanwhile, the most appropriate width of buffer to extract macroscopic factors may be 15
inconsistent between modeling motor vehicle and non-motorized crashes. Therefore, it is 16
advisable to model the crash frequency by separate transportation modes to examine the effects 1 7
of macroscopic factors. 18
In summary the objective of this paper is twofold: (1) to examine the effects of a host of 19
contributing factors including both macroscopic and microscopic factors on crash occurrence 2 0
with respect to different transportation modes; (2) to shed further light on the contribution of the 21
macroscopic factors which are traditionally excluded or omitted variables, to tracking the source 22
of heterogeneity effects in coefficient estimators of regularly used variables and improving the 2 3
model performance in crash-frequency analysis related to different modes. 24
2 Data preparation 25
In this study, data collected for 279 intersections located in Hillsborough County, Florida, 26
USA were used to develop the intersection crash-frequency models for different transportation 27
modes. The data for the analysis was mainly divided into four types: traffic crash data, traffic 28
characteristics, road characteristics related to geometric design, traffic control/regulatory of the 2 9
intersection, macroscopic factors including trip production/attraction, demographic and 30
socio-economic characteristics surrounding the intersection. The derivation and processing of 31
these data sources are described next. 32
2.1 Crash data 33
Crash data for the intersections in a five-year period (2005–2009) were obtained from the 3 4
Florida Department of Transportation (FDOT) Crash Analysis Reporting (CAR) system. Crashes 35
were categorized as intersection related crashes if they occurred within the curb-line limits of 36
the intersection or if they occurred within the influence area of the intersection, which is 250 feet 37
away from the stop line. Intersection-level crash data was disaggregated into motor vehicle, 3 8
bicycle, and pedestrian crashes. A motor vehicle crash was defined as a collision between two or 39
more motor vehicles or between a motor vehicle and an object. A bicycle crash referred to a 40
collision between a motor vehicle and a bicycle. Likewise, a pedestrian crash denoted a collision 41
between a motor vehicle and a pedestrian. 42
2.2 Traffic characteristics 43
Previous researches suggest that traffic characteristics such as motor vehicle, pedestrian and 4 4
bicycle volume are the most important factors influencing crash occurrences. Motor vehicle 45
5
volume represented by average annual daily traffic (AADT) can be collected from the FDOT 1
Roadway Characteristics Inventory. Two motor vehicle volume variables including AADT from 2
major road and AADT from minor road of 5-year (2005–2009) average were also obtained. Actual 3
pedestrian and bicycle volume are not regularly available. The collected macroscopic data such as 4
population were used to serve as a surrogate for pedestrian and bicycle volume as suggested by 5
Jacobsen (2003) and Miranda-Moreno et al.(2011). 6
2.3 Road features 7
Road features related to geometric design and regulatory/control attributes of the road entity 8
were collected from the FDOT Roadway Characteristics Inventory. The road factors considered in 9
the study are number of legs, presence of traffic signal, speed limit on major approach, and speed 10
limit on minor approach. 11
2.4 Macroscopic factors 12
Considerable previous studies on zonal-level crash-frequency models suggests that various 13
macroscopic factors such as trip production/attraction, demographic and socio-economic 14
characteristics affect area-wide traffic crashes (Quddus., 2008; Huang et al., 2010, 2016; Abdel-Aty 15
et al., 2011; Xu et al., 2014; Dong et al., 2015, 2016). It is hypothesized that the number of crashes 16
occurring at an intersection is also associated with these macroscopic factors surrounding the 17
intersection. 18
Trip production/attraction factors such as total trip productions/attraction, home-based work 19
productions/attraction, college productions/attraction at the TAZ level were collected from the 20
Intermodal Systems Development Unit of District 7 of the FDOT. Demographic and 21
socio-economic characteristics were examined including the geographical area of each TAZ, 2 2
population, income and commuting, which were downloaded from the United States Census 23
report. 24
ArcGIS 10.0 was used to generate a buffer around each selected intersection and conduct a 25
spatial analysis to extract macroscopic data from the TAZ layers. The process is described in 26
detail as follows. First, a spatial overlay of TAZ layers on a specified width buffer (including 27
0.25 mile, 0.5 mile and 1 mile buffer) was generated around each intersection. Then, spatial 28
analysis operators such as “intersect” and “join” available in the GIS environment were used to 29
intersect layers, join tables and extract selected trip production/attraction, demographic and 30
socio-economic characteristics within the generated buffer. Macroscopic factors were distributed 3 1
in proportion to the area of TAZ within the generated buffer. The ArcGIS procedure adopted to 32
extract and estimate macroscopic factors in this study was similar to the one discussed in detail 33
by Pulugurtha and Sambhara (2011) to develop pedestrian crash estimated models. 34
Table 1 provides descriptive statistics of crash data, traffic variables, road variables and 35
macroscopic variables located in 0.5 mile buffer. The values of macroscopic variables located at 36
0.25 and 1 mile buffer are not listed for compactness of the table. 37
38
39
40
41
42
43
44
45
46
47
6
Table 1 Summary of variable and descriptive statistics 1
Variable Definition Meana SDa Mina Maxa
Crash data
Motor vehicle crash Motor vehicle crash per intersection in 2005-2009 65.219 56.545 2.000 293.00
Bicycle crash Bicycle crash per intersection in 2005-2009 1.018 1.340 0.000 8.000
Pedestrian crash Pedestrian crash per int ersection in 2005-2009 1.276 1.889 0.000 11.000
Traffic and road variables
AADT-major AADT on major approach (103pcu) 28.364 17.684 2.600 71.300
AADT-minor AADT on minor approach (103pcu) 9.150 8.879 1.000 43.000
Leg-number Number of legs (4 legs=1, 3 legs=0) 0.670 0.471 0.000 1.000
traffic signal Presence of traffic signal (yes=1,no=0) 0.498 0.542 0.000 4.000
Speed-Major Speed limit on major approach (mph) 40.502 6.154 6.154 60.000
Speed-Minor Speed limit on minor approach (mph) 35.323 6.479 6.479 55.000
Macroscopic variables
PA_density Density of productions and attractions (per acre) 49.272 27.369 2.514 184.06
HB_prop Proportion of home-based productions and
attractions 0.680 0.084 0.317 0.840
Col_prop Proportion of college productions and attractions 0.025 0.059 0.000 0.415
Pop_density Density of total population (per acre) 5.631 2.554 0.342 11.798
Age 0 to 15_ prop Proportion of population between age 0 and 15 0.225 0.050 0.069 0.348
Age 16 to 64_prop Proportion of population between age 16 and 64 0.652 0.054 0.561 0.917
Pub_prop Proportion of workers commuting by public
transportation 0.030 0.031 0.000 0.149
Wal_ prop Proportion of workers commuting by walking 0.025 0.017 0.000 0.080
MHINC Median household income (in thousands) 36.728 15.199 4.300 89.035
a These values relating to macroscopic variables are only for 0.5 mile buffer. 2
3 Methodology 3
In previous crash-frequency analyses, Poisson and Negative binomial model (NB), along
4
with their variants (such as Poisson-lognormal model), are commonly used and proven to be
5
successful as they effectively model the rare, random, sporadic, and non-negative crash data. As
6
crash data exhibit over-dispersion (i.e., variance greater than mean), NB is superior to the Poisson
7
model. Compared with the basic Poisson model, NB includes a gamma-distributed error term in
8
Poisson mean to account for the over-dispersion due to omission of relevant variables or
9
measurement error in crash data. The formulation for NB can be presented as follows:
10
~ Poisson
i i
Y
(1)
11
0
ln
i i i
X β (2)
12
Where
i
Y
is the crash frequency by modes (i.e., motor vehicle, bicycle and pedestrian) at
13
intersection
i
, and
i
is the expectation of
i
Y
.
i
X
is a vector of explanatory variables.
0
is the
14
intercept,
β
is a vector of estimable parameters.
i
is the error term that is assumed to be
15
independent
X
and has a two-parameter gamma distribution.
16
The NB model presented in Eq.(2) could control for unobserved heterogeneity by omitted
17
variables. However, this model assumes that the unobserved variables are uncorrelated with the
18
observed exploratory variables. If this correlation exists, unobserved factors can introduce
19
variation in the effect of observed variables on crash likelihood. Random parameters approaches
20
are able to address this issue by allowing non-constant estimable parameters to vary across
21
7
observations (Mannering et al., 2016). In random parameters negative binomial model (RPNB),
1
estimable parameters (
β
) in Eq.(2) can be written as:
2
i i
(3)
3
Where
is the mean of the random parameter
i
,
i
is a randomly distributed term (e.g.,
4
a normally distributed term with mean 0 and variance
2
) that capture heterogeneity across
5
observations. The analyst can test for random parameters with Eq.(3), across all observations i for
6
each included explanatory variable. If the variance of the chosen distribution is not significantly
7
different from zero, it suggests that a conventional fixed parameter is statistically appropriate.
8
Thus the model always combines fixed and random parameters across the included explanatory
9
variables.
10
As previously stated, heterogeneity effects of certain risk factors mainly derive from the 11
combined effects of unobserved variables that have been omitted from the model. Although the 12
random parameters approaches could mitigate the adverse impacts of omitting variables, the 1 3
original source of unobserved heterogeneity (or what are the major factors that lead to 14
unobserved heterogeneity) still fails to be well understood. Thus one of the aims of this study is 1 5
to test the potential role of macroscopic variables (referring to as the influential omitted variables) 16
in tracking the source of heterogeneity effects related to commonly used traffic and road 17
variables. 18
To this end, two different model specifications are estimated and compared, both with 19
random parameters approaches. In the base model only traffic and road variables are included, 20
while the second model traffic and road variables as well as macroscopic variables at TAZ level 2 1
surrounding the intersections are included. If the test results for parameter estimation on a 22
variable appear as random in base model (variance of the chosen distribution is significantly 23
different from zero), while this variable has the fixed effect in the second model. Then we could 24
infer that the heterogeneity effect of this variable is mainly caused by these macroscopic variables. 25
For another case, the safety effect of this variable is still random in second model; the source of 26
heterogeneity effect of this variable still cannot be clearly distinguished, maybe due to other 27
important omitted variables. 28
Apart from the potential role in accounting for the heterogeneity effects in parameters
29
estimation, integrating the macroscopic variables could also decrease the variance of the random
30
error (i.e. overdispersion) and thus improve the model performance in crash-frequency
31
prediction. The proportion of reduction in variance (PRV), also called explained variance,
32
proposed by Raudenbush and Bryk (2002) can be used to assess the overall explanatory power of
33
macroscopic factors for modeling the crashes by different modes. In this case, the PRV of
34
macroscopic variables is defined as:
35
2 2
0 1
2
0
-
PRV
(4)
36
Where
2
0
is the variance of the error term in the base model without macroscopic variables.
37
2
1
is the variance of error term in the full model with macroscopic variables. The value of PRV is
38
bounded by 0 and 1, and a higher value indicates a stronger explanatory power of macroscopic
39
factors on the crash occurrence.
40
Furthermore, two goodness-of-fit statistics are used for model comparisons: Akaike
41
Information Criterion (AIC) and log-likelihood ratio (LR).
42
The AIC is calculated as follows:
43
2 2
AIC LL p
(5)
44
Where LL is the log-likelihoods at convergence for the estimated model, and
p
is the
45
number of parameters in the statistical model. The model with the lower AIC is considered to
46
8
have the better goodness of fit.
1
The LR value is the chi-squared value in the log-likelihood ratio test for the null hypothesis
2
test that reveals whether or not the equivalence of two models should be rejected. The likelihood
3
ratio statistic is,
4
2( )
N A
LR LL LL
(6)
5
which is
2
distributed with J degrees of freedom, where J = KA − KN (KA and KN are the
6
number of coefficients for the alternative model and the null model, respectively ), LLN and LLA are
7
the log-likelihoods at convergence for the null model and the alternative model, respectively. The
8
null hypothesis for Eq. (6) is that the alternative model does not have a significantly lower
9
log-likelihood than the null models, indicating a lack of significant difference between the null
10
model and the alternative model.
11
4 Results and discussion 12
Three types of crash-frequency models for motor vehicles, bicycles and pedestrians were 13
developed. Each type of model involved eight separate models based on four model 1 4
specifications (one with only traffic volume and road features, the other three with macroscopic 1 5
factors overlaid on 0.25, 0.5, 1 mile buffer respectively in addition to commonly included traffic 1 6
volume and road features ) and two analytical methods (NB and RPNB). 17
LIMDEP econometric software was used to develop the statistical models described above. 18
To enable focus on the most significant variables, variables that were not found to significantly 19
different from zero at the 0.1 level of significant using a t-test were removed. Meanwhile, the 2 0
likelihood ratio test was used to guarantee that each added variable significantly improved the 2 1
overall model performance. In the RPNB, if the variance of a random parameter was not 22
statistically different from zero, the random parameter was simplified to be fixed across 23
intersections. Thus, the results in NB were in accordance with that in RPBN when no estimate 24
parameters of explanatory variable were statistically random. 25
This analysis below will emphasize testing effects of macroscopic factors on model 26
performance in crash-frequency analysis for three transportation modes, and then comparison 27
results of the parameter estimates and marginal effects between the base model and the full 28
model with macroscopic variables will be presented and interpreted. 29
4.1 Effects of macroscopic factors on model performance 30
Tables 2-4 show goodness-of-fit measures for motor vehicle, bicycle and pedestrian 3 1
crash-frequency models, respectively. As shown in Table 2, only five of eight motor vehicle 32
crash-frequency models, including two base models (NB model and RANB model) and three 33
fully specified NB models, were presented since there was no significant random parameter as 3 4
measured by the t-statistics in all three models with macroscopic variables. Although there was 35
no substantial difference in goodness-of-fit as reflected by likelihood ratio test between the base 36
NB and RPNB model, the present of significant random parameters ( e.g. the variable of ‘presence 37
of traffic signal’ in this case study) demonstrated the existent of heterogeneity of risk factors in 38
base model without considering macroscopic factors. More interestingly, no significant random 39
parameters were found in all three full models with macroscopic variables. This implied that the 40
heterogeneous effects of risk factors on motor vehicle crash frequency could be mostly captured 4 1
by these macroscopic variables, at least for the Hillsborough dataset examined here. Frequency 42
analysis models for bicycle crashes presented a similar result to motor vehicles crashes, as shown 43
in Table 3. However, this was not the case for the pedestrian crash-frequency models (Table 4). 44
Significant random parameters, such as ‘presence of traffic signal’, existed both in pedestrian 45
crash models with and without macroscopic variables, suggesting that the heterogeneity effect in 46
9
parameters estimation cannot be completely picked up by these macroscopic variables. 1
Apart from the potential effect in tracking the heterogeneity, results also revealed that 2
incorporating the macroscopic variables in crash-frequency analysis leaded to an increasing 3
model complexity but a considerable improvement in overall fit as measured by log likelihood at 4
convergence. As shown in Table2, the likelihood ratio test comparing the full NB models and the 5
base NB models indicated that we were more than 99.99 % confident that the full models with 6
macroscopic variables (except the full model with 1.0 mile buffer-width macroscopic variables) 7
were statistically superior. This comparison suggested that the macroscopic variables explained a 8
portion of variability in crash occurrences and should not be omitted in motor vehicle 9
crash-frequency model. In regard to bicycle and pedestrian models, omission of macroscopic 10
variables will also lead to a significant decrease in goodness-of-fit, as shown in Tables 3-4. 11
Comparing model outputs developed based on 0.25 mile, 0.5 mile and 1 mile buffer width 12
data, these models with macroscopic variables of 0.25 mile buffer width had the lowest AIC, 1 3
conversely, the models with macroscopic variables of 1 mile buffer width had the highest AIC in 14
all three types of crash-frequency models by modes. Thus, a relatively smaller buffer width in 15
extracting macroscopic factors surrounding the intersection would yield a better estimate. 16
To further assess and compare the overall explanatory power of macroscopic variables, the 17
values of PRVs were calculated. As shown in Table 2, the motor vehicle crash-frequency model 1 8
with macroscopic variables of 0.25 mile buffer width had the highest PRV of 7.98%. This meant 19
that 7.98% of unexplained variation resulted from those omitted macroscopic variables, which 20
also suggested the usefulness of the motor vehicle crash-frequency analysis by integrating 21
macroscopic factors. Accordingly, the highest values of PRV were 33.02% and 26.37% in bicycle 22
and pedestrian crash-frequency models respectively, as shown in Tables 3-4. 23
Table 2 Goodness-of-fit measures for motor vehicle crash-frequency models 24
Model statistics Base model 0.25 mile 0.50 mile 1 mile
NB RPNB NB NB NB
Number of observers 279 279 279 279 279
Number of parameters 6 7 10 10 10
Log likelihood at convergence -1280.07 -1279.42 -1269.25 -1270.86 -1276.28
AIC 2572.14 2572.84 2558.50 2561.73 2572.56
Log-likelihood ratio test
2 = -2(LLN-LLA)
1.300 21.645 18.416 7.588
Degrees of freedom
1 4 4 4
P-value
0.26 <0.01 <0.01 0.11
Explanatory power of macroscopic factors
Variance of the error t erm, 0.250 0.246 0.230 0.232 0.243
Proportion of reduction in variance, PRV
1.23% 7.98% 6.85% 2.67%
Note: LLN denotes the log likelihood at convergence for Base + NB model. 25
26
By comparing PRVs in models by different transportation modes, the PRVs in bicycle and 27
pedestrian crash-frequency model were much higher than that in motor vehicle models. In other 28
words, integrating macroscopic factors in non-motorized crash-frequency model was more vital 29
than that in developing motor vehicle model. This result was in line with the expectation. One 30
possible reason for this distinct effect is that pedestrian/bicycle volume (or pedestrian/bicycle 31
activity) which is commonly identified as the main determinants of pedestrian/ bicycle crash 32
frequency has been omitted in the base pedestrian/bicycle crash-frequency models. Integrating 33
macroscopic factors for pedestrian/bicycle crash-frequency analysis made up the absence of 34
10
pedestrian/bicycle volume in predicting pedestrian/bicycle crash frequencies to some extent as 1
demonstrated by previous study (Jacobsen, 2003; Miranda-Moreno et al., 2011) that macroscopic 2
data can serve as a surrogate for pedestrian and bicycle volume. Another reason, maybe even 3
more importantly, originates from the differences in the travel distance between non-motorized 4
and motor vehicle modes. As walking and bicycle are short-distance transportation modes, crash 5
victims of pedestrians and bicyclists generally reside near the crash intersection, and thus, the 6
macroscopic factors extracted surrounding the intersection can probably better reflect pedestrian 7
and/or bicyclist behaviors than that for motor drivers. 8
Table 3 Goodness-of-fit measures for bicycle crash-frequency models 9
Model statistics Base model 0.25 mile 0.50 mile 1 mile
NB RPNB NB NB NB
Number of observers 279 279 279 279 279
Number of parameters 6 7 8 8 8
Log likelihood at convergence -362.53 -361.61 -350.42 -351.14 -351.62
AIC 737.06 737.22 716.85 718.28 719.24
Log-likelihood ratio test
2 = -2(LLN-LLA)
1.837 24.209 22.777 21.821
Degrees of freedom
1 2 2 2
P-value
0.17 <0.01 <0.01 <0.01
Explanatory power of macroscopic factors
Variance of the error term 0.402 0.335 0.269 0.274 0.280
Proportion of reduction in variance, PRV
16.48% 33.02% 31.66% 30.27%
Note: LLN denotes the log likelihood at convergence for Base + NB model. 10
11
Table 4 Goodness-of-fit measures for pedestrian crash-frequency models 12
Model statistics Base model 0.25 mile 0.50 mile 1 mile
NB RPNB
NB RPNB
NB RPNB NB RPNB
Number of observers 279 279 279 279 279 279 279 279
Number of parameters 4 5 7 8 7 8 7 8
Log likelihood at convergence -399.51
-397.89
-386.34
-386.26
-387.27
-387.16 -388.52
-388.13
AIC 807.03
805.77
786.68
788.51
788.54
790.32 791.05
792.27
Log-likelihood ratio test
2 = -2(LLN-LLA)
3.256 26.352
26.515
24.487
24.712 21.983
22.761
Degrees of freedom
1 3 4 3 4 3 4
P-value
0.08 <0.01 <0.01 <0.01 <0.01 <0.01 <0.01
Explanatory power of macroscopic factors
Variance of the error term 0.785
0.613 0.585 0.578 0.595 0.587 0.601 0.595
Proportion of reduction in variance, PRV
21.96%
25.57%
26.37%
24.26%
25.21% 23.48%
24.18%
Note: LLN denotes the log likelihood at convergence for Base + NB model. 13
14
15
16
11
4.2 Parameter estimates and marginal effects 1
Three types of crash-frequency models for motor vehicles, bicycles and pedestrians were 2
estimated and each type involved eight separate models based on four model specifications and 3
two analytical methods, yielding a total of 24 models. Three full models which have a better 4
goodness-of-fit, including motor vehicle NB model with 0.25 mile buffer width macroscopic 5
variables, bicycle NB model with 0.25 mile buffer width macroscopic variables and pedestrian NB 6
model with 0.25 mile buffer width macroscopic variables, were selected as recommended models 7
for reasons outlined. Meanwhile, the parameter estimates of three base NB models for motor 8
vehicles, bicycles and pedestrians were presented for comparison. Tables 5-7 show the parameter 9
estimates and their t-statistics for motor vehicle, bicycles and pedestrians crash-frequency models, 1 0
respectively. Since the model has nonlinear coefficients, direct parameters will not show a unit 11
effect on the number of crashes. Table 8 thus summarizes the results for marginal effects, which 12
could be interpreted as the average impact of a unit change in an explanatory variable on crash 13
frequency. 14
Comparing marginal effects of microscopic variables (traffic volume and road variables) 15
between the full NB models and the base NB models revealed some important differences. The 16
major differences were in marginal estimates of the variable ‘presence of traffic signal.’ The 17
results in the base model using NB approach showed that ‘presence of traffic signal’ had positive 18
association with the crash frequency for all three transportation modes; while this variable 19
became no statistically significant for motor vehicle and bicycle crashes in the full NB models. 20
Meanwhile, the present of macroscopic variables also modified the marginal effects of 21
microscopic variables (see Table 8). For example, the marginal effects of ln(AADT-minor) were 22
6.38 and 0.14 respectively for motor vehicle and bicycle crashes in the base models, while these 23
values were 7.67 for motor vehicle crashes and 0.17 for bicycle crashes in models with 24
macroscopic variables. This difference clearly showed that in the absence of important 25
macroscopic variables, the marginal effects of ln(AADT-minor) for motor vehicle and bicycle 26
crashes were biased downwards by 16.8% and 17.6% respectively. These results agreed with the 27
safety research by Mitra and Washington (2012) that the exclusion of important variables may 2 8
cause bias in coefficient estimates and incorrect inferences. 29
According to the results of parameter estimates (Tables 5-7) and their marginal effects (Table 30
8), significant variable sets for crashes were not consistent for different transportation modes. 31
‘AADT on major approach’ and ‘density of total population’ were two contributing factors that 3 2
had statistically significant effects on the three response variables (i.e., motor vehicle, bicycle and 33
pedestrian crash frequency). Four variables including ’AADT on minor approach,’ ‘number of 34
legs’, ‘proportion of college productions and attractions’ and ‘proportion of workers commuting 35
by public transportation’ were significant for two response variables. Six variables were solely 36
associated with one response variable: ‘presence of traffic signal’, ‘speed limit on major 37
approach,’ ‘speed limit on minor approach,’ ‘proportion of home-based productions and 3 8
attractions,’ ‘proportion of population between age 16 and 64’ and ‘proportion of workers 39
commuting by walking’. The detailed interpretations for these significant risk factors are offered 40
in the following. 41
42
43
44
45
46
47
12
Table 5 Parameter estimates for motor vehicle crash-frequency models 1
Variables Base NB model Full NB model
Mean Standard Error
t-Statistic
Mean Standard Error t-Statistic
Ln(AADT-major)
0.728 0.038 19.13 0.728 0.042 17.30
Ln(AADT-minor)
0.097 0.041 2.35 0.117 0.043 2.74
Leg-number 0.451 0.066 6.83 0.414 0.068 6.11
traffic signal 0.159 0.058 2.75
Speed-Minor 0 .031 0.004 8.30 0.028 0.006 4.49
HB_prop
-2.015 0.513 -3.93
Col_prop
-2.324 0.673 -2.64
Pop_density
0.039 0.015 2.70
Pub_prop
-0.761 0.349 -2.14
Intercept 1.377 0.467 2.95
Note: all parameters are significant at the 0.1 level or better. 2
3
Table 6 Parameter estimates for bicycle crash-frequency models 4
Variables Base NB model Full NB model
Mean Standard Error
t-Statistic
Mean Standard Error t-Statistic
Ln(AADT-major)
0.490 0.130 3.77 0.580 0.127 4.58
Ln(AADT-minor)
0.132 0.085 1.55 0.165 0.086 1.92
Leg-number 0.369 0.171 2.15 0.464 0.173 2.69
traffic signal 0.312 0.135 2.32
Col_prop
-2.791 1.669 -1.67
Pop_density
0.082 0.031 2.64
Wal_ prop
10.638 3.782 2.81
intercept -2.304 0.412 -5.59 -3.283 0.447 -7.35
Note: all parameters are significant at the 0.1 level or better. 5
6
Table 7 Parameter estimates for pedestrian crash-frequency models 7
Variables Base NB model Full NB model
Mean Standard Error
t-Statistic
Mean Standard Error
t-Statistic
Ln(AADT-major) 0.877 0.161 5.45 0.924 0.162 5.69
traffic signal 0.732 0.154 4.75 0.522 0.156 3.35
Speed-Major -0.076 0.013 -5.91 -0.049 0.018 -2.76
Pop_density
0.090 0.031 2.88
Age 16 to 64_prop
-3.089 1.112 -2.78
Wal_ prop
11.211 3.809 2.94
Note: all parameters are significant at the 0.1 level or better. 8
9
10
11
13
Table 8 Estimate results for marginal effects of risk factors 1
Variables Motor vehicle Bicycle Pedestrian
Base Full Base Full Base Full
Ln(AADT-major) 48.00 47.65 0.50 0.59 1.15 1.19
Ln(AADT-minor) 6.38 7.67 0.14 0.17
Leg-number 26.60 24.48 0.35 0.43
traffic signal
10.47
0.32
0.96
0.67
Speed-Major
-0.10 -0.06
Speed-Minor
2.03
1.87
HB_prop
-131.94
Col_prop
-152.21
-2.85
Pop_density
2.58
0.08
0.12
Age 16 to 64_prop
-3.96
Pub_prop
-49.86
10.88
Wal_ prop
14.39
Note: all parameters are significant at the 0.1 level or better. 2
4.2.1 Traffic volume 3
Similar to numerous prior studies, traffic volumes are significant variables for intersection 4
crashes and are positively correlated with crash occurrence (Lee and Abdel-Aty, 2005; Mitra and 5
Washington, 2012; Xie et al., 2013). The marginal effects of ln(AADT-major) were 47.65 , 0.59 and 6
1.19 respectively for motor vehicle, bicycle and pedestrian crashes in the full models, indicating 7
that an average of thousand increase in major approach AADT will lead to a 2.31, 0.03 and 0.06 8
increase in motor vehicle ,bicycle and pedestrian crash frequency respectively. Similarly, an 9
average of thousand increase in minor approach AADT was associated with a 0.69 and 0.02 10
increase in motor vehicle and bicycle crashes. 11
4.2.2 Number of legs 12
The marginal effects of ‘number of legs’ on motor vehicle and bicycle crashes were 24.28 and 13
0.43. This result suggested that the four-legged intersection was associated with 24.28 more motor 14
vehicle crashes and 0.43 more bicycle crashes compared to the intersection with three legs. This 15
result was generally expected and agreed with the preliminary finding that a larger number of 16
legs may increase the likelihood of crash occurrence due to more potential conflicts (Zeng and 17
Huang, 2014). However, this variable did not found to have significant effects on pedestrian crash 18
frequency. This may be due to the higher design standards of facilities (such as marked 1 9
crosswalks) in the large-leg intersection that leads to mixed effects of ‘number of legs’ on 20
pedestrian safety. 21
4.2.3 Traffic signal 22
The effects of traffic signal on safety are very interesting. The results of parameter estimates 23
in the base model using NB approach showed that ‘presence of traffic signal’ had positive 24
association with all three target variables. This is not consistent with the empirical hypothesis that 25
the installation of traffic lights could improve intersection safety. In the base model using RPNB 26
approach, ‘presence of traffic signal’ had a positive effect on three target variables but with a 27
varying magnitude across intersections (the results for the RPNB model were not presented since 2 8
there was no significant improvement in goodness-of-fit compared to the NB model). More 29
interestingly, this variable became no statistically significant for both motor vehicle and bicycle 3 0
crashes in the models integrating macroscopic variables. The possible reason for this difference is 31
14
that the macroscopic variables could account for a positive and heterogeneity effect of the 1
‘presence of traffic signal’. This implies that the installation of traffic lights itself will not increase 2
the crash risk but traffic lights are always installed at relatively hazardous sites; however, more 3
work is needed to verify this conclusion. 4
4.2.4 Speed limit 5
The variables related to speed limit had inconsistent effects on crash occurrence by different 6
modes. ‘Speed limit on minor approach’ was positively correlated with the frequency of motor 7
vehicle crashes. An increase of 10 mph in speed limit on minor approach will increase motor 8
vehicle crashes by 18.7. This is generally expected since at high speeds the time to react to 9
changes in the environment is shorter, leading to higher crash frequency. However, ‘speed limit 10
on major approach’ was negatively associated with pedestrian crashes. The frequency of 11
pedestrian crashes will decrease by 0.6 with per 10 mph increase in speed limit on major 12
approach. The probable reason for this result is that a higher speed limit is always related with 13
higher design standards of facilities such as pedestrian overcrossing and underpass. The effect of 14
speed limit on bicycle crashes was not significant. 15
4.2.5 Trip characteristics 16
Results showed that the proportion of home-based productions and attractions near an 17
intersection was negatively associated with motor vehicle crashes. The frequency of motor 18
vehicle crashes will decrease by 1.32 with a percentage increase in proportion of home-based 19
trips. This is reasonable since drivers of a home-based trip are more familiar with the traffic 20
environment and have more cautious driving behaviors (Abdel-Aty et al., 2011). ‘Proportion of 21
college productions and attractions’ was negatively associated with motor vehicle and bicycle 22
crashes while was not statistically significant for pedestrian crashes. A percentage increase in 23
proportion of college trips will result in a 1.52 and 0.03 decrease in motor vehicle and bicycle 24
crashes. The decreased motor vehicle and bicycle crashes may be due to better traffic control 2 5
measures in these areas. However, higher proportion of college productions and attractions, 26
which is always related to better traffic control measures and high number of walking trips, leads 2 7
to mixed effects on pedestrian crashes. 28
4.2.6 Demographic characteristics 29
‘Density of total population’ was the influential macroscopic variable and was positively 30
associated with all three target variables. The marginal effects of population density near an 3 1
intersection showed that an increase in population density will increase the frequency of motor 32
vehicle, bicycle and pedestrian crashes by 2.48, 0.08, and 0.12. This agrees with previous studies 33
as a larger population is always consistent with more opportunities in terms of crash exposure 34
(Lee et al., 2015; Mitra and Washington, 2012; Pulugurtha and Sambhara, 2011). In addition, the 35
proportion of population between age 16 and 64 near an intersection was found to have negative 36
effects on pedestrian crashes. The frequency of pedestrian crashes will decrease by 0.04 with a 37
percentage increase in proportion of population aged 16-64. This may be due to middle-aged 38
people walking less in comparison to young and/or old people, as well as having better ability in 39
avoiding crash risk (Huang et al., 2010). 40
4.2.7 Commute behaviors 41
A percentage increase in proportion workers commuting by public transportation near an 42
intersection was associated with a 0.50 decrease in motor vehicles and a 0.10 increase in bicycle 43
crashes, indicating that the public transportation had opposite effects on motor vehicles and 44
bicycle crashes. In addition, ‘proportion of workers commuting by walking’ had significant and 45
positive associations with pedestrian crash occurrence. The number of pedestrian crashes will 46
15
increase by 0.14, with a percentage increase in proportion workers commuting by walking. This 1
result is not surprising since walking is always associated with the exposure of pedestrian 2
crashes. 3
5. Conclusions and recommendations 4
This paper sought to examine the effects of omitted macroscopic factors in crash-frequency 5
models by transportation modes at intersections. For this purpose, several separate 6
intersection-level crash-frequency model for motor vehicle, bicycle and pedestrian modes were 7
developed. Road characteristics related to geometric design and regulatory/control attributes and 8
traffic characteristics of the intersection entities, as well as macroscopic factors including trip 9
production/attraction, demographic and socio-economic characteristics and commute behaviors 10
at TAZ level surrounding the intersection, were used as explanatory variables. Those data 1 1
extracted for 279 intersections located in Hillsborough County, Florida, USA, were used for 12
model development. 13
The empirical analysis revealed a number of interesting findings. First, omission of 1 4
macroscopic variables would result in biased estimation of retained microscopic variables. 15
Results for marginal effects of traffic volumes and road features showed significant differences 1 6
between in the base model and the full model with macroscopic variables. For example, the safety 17
effect of minor approach AADT on motor vehicle and bicycle crashes are biased downwards by 18
16.8% and 17.6% in the absent of macroscopic variables. 19
Second, macroscopic variables had potential effects in tracking the heterogeneity of certain 20
risk factors. The results in the base model using RPNB approach showed that the safety effect of 21
‘presence of traffic signal’ was best fit with a normally distributed random parameter suggesting 22
‘presence of traffic signal’ had heterogeneous effects across intersections; while this variable 23
became no statistically significant in models with macroscopic variables. This implied that the 24
heterogeneous effects of ‘presence of traffic signal’ on motor vehicles crashes could be mostly 25
captured by these macroscopic variables, at least for the Hillsborough dataset. 26
Third, model comparison using log likelihood at convergence suggested that considering 27
macroscopic variables was vital in elevating the model performance. In addition, the values of 28
PRV were further calculated to assess the explanatory power of macroscopic variables. Results 29
showed the values of PRV in bicycle and pedestrian crash-frequency model were much higher 30
than in motor vehicle models, indicating that integrating macroscopic factors played a more 31
important role in developing non-motorized crash-frequency model than in developing motor 32
vehicle models. 33
Fourth, comparing model outputs developed based on 0.25 mile, 0.5 mile and 1 mile buffer 3 4
width data, models with macroscopic variables of 0.25 mile buffer width had the lowest AIC and 35
highest PRV, conversely, the models with macroscopic variables of 1 mile buffer width had the 36
highest AIC and lowest PRV in all three types of crash-frequency models by modes. Thus a 37
relatively smaller buffer width to extract macroscopic factors around the intersection would 38
provide a better estimate. 39
Finally, macroscopic factors of the surrounding zone of an intersection, such as ‘proportion 40
of home-based productions and attractions’, ‘proportion of college productions and attractions’, 41
‘density of total population’, ‘proportion of population between age 16 and 64’, proportion of 42
workers commuting by public transportation and ‘proportion of workers commuting by walking’, 4 3
were demonstrated to have significant effects on intersection crashes; while these variables are 44
always ignored in traditionally micro-level (e.g., intersections and segments) crash frequency 45
model. This indicated that not only traffic volumes and road features but also macroscopic factors 4 6
should be considered in estimating crash risk and identifying crash-prone locations. 47
The topic of integrating macroscopic factors in intersection/segment-level crash-frequency 4 8
16
model is emerging. This study has great research potential in pro-actively predicting crash risk 1
and identifying suitable countermeasures to reduce the crashes at “new” intersections/segments 2
as well as intersections/segments near “new” development. Nevertheless, several limitations 3
should be noted for this study. First, it is worthwhile to apply this model to other intersections 4
and regions in order to investigate spatial transferability of the calibrated models. In addition, 5
there may be some correlations among crash frequency by different transportation modes within 6
intersections, which is caused by some unobserved influential factors. Therefore, a simultaneous 7
model accounting for correlations of crashes among transportation modes, such as the 8
multivariate model, will be further explored to investigate this issue. 9
Acknowledgements 1 0
This work was jointly supported by: 1) Natural Science Foundation of China (No.71371192, 11
No. 71561167001); 2) the Research Fund for Fok Ying Tong Education Foundation of Hong Kong 1 2
(142005); and 3) Fundamental Research Funds for the Central Universities of CSU (No. 13
2016zzts050). We would like to thank Dr. Mohamed Abdel-Aty at the University of Central 14
Florida and the Florida Department of Transportation for providing the data. 15
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