Analytic Methods in Accident Research 2 (2014) 12–20
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Analytic Methods in Accident Research journal homepage: www.elsevier.com/locate/amar
A heterogeneity-in-means count model for evaluating the effects of interchange type on heterogeneous influences of interstate geometrics on crash frequencies Narayan Venkataraman a,1, Venky Shankar b,n, Gudmundur F. Ulfarsson a,2, Damian Deptuch b,3 a b
Civil and Environmental Engineering, University of Iceland, VR-II, Hjardarhagi 2-6, IS-107 Reykjavik, Iceland Civil and Environmental Engineering, The Pennsylvania State University, 226C Sackett Building, University Park, PA 16802, United States
a r t i c l e in f o
abstract
Article history: Received 15 September 2013 Received in revised form 4 January 2014 Accepted 4 January 2014
This paper presents a heterogeneity-in-means, random-parameter negative binomial (HMRPNB) model of interchange type effects on interchange and non-interchange segment crash frequencies. For non-interchange segments, upstream and downstream type combinations were evaluated. Eight interchange types, namely, directional, semidirectional, clover, partclover, diamond, part diamond, single-point-urban-interchange (SPUI), and other were studied on the Washington State interstate system. A total of 575 interchange and 578 noninterchange segments were analyzed for the period 1999–2007. In interchange segments, semidirectional, partclover and other (excluding directional, diamond, SPUI, or fullclover) types significantly contributed to heterogeneity in the random parameter effects of average daily traffic, median continuous lighting proportion, minimum and maximum vertical gradients. Full and partial diamond types contributed to heterogeneity in the random parameter effects of median continuous lighting proportion, maximum horizontal degree of curvature, minimum and maximum vertical gradients. In non-interchange segments, the upstream type set including directional, semidirectional, and clover/collector-distributor type, and downstream set of directional, semidirectional, diamond, partclover, partdiamond and other type significantly contributed to heterogeneity in the random parameter means of total length of adjacent interchanges, two foot left shoulder width and two foot right shoulder width proportions. Statistically significant fixed parametric effects included urban/rural location, right continuous lighting proportion, proportion by length of three lane cross section, four lane cross section, three-to-four foot left shoulder, five-to-nine foot left shoulder, ten foot left shoulder, three-tofour foot right shoulder, five-to-nine foot right shoulder, and ten foot right shoulder, as well as number of horizontal curves in segment, and shortest horizontal curve length. Published by Elsevier Ltd.
Keywords: Heterogeneity-in-means Random-parameter Interchange Non-interchange
n
Corresponding author. Tel.: þ814 865 9434; fax: þ 814 863 7304. E-mail addresses:
[email protected] (N. Venkataraman),
[email protected],
[email protected] (V. Shankar),
[email protected] (G.F. Ulfarsson),
[email protected] (D. Deptuch). 1 Tel.: þ 354 525 4907. 2 Tel.: þ 354 525 4907; fax: þ 354 525 4632. 3 Tel.: þ 814 865 9434. 2213-6657/$ - see front matter Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.amar.2014.01.001
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1. Introduction Current literature on interchange type and its impact on crash occurrence is limited to standard negative binomial models, which have occupied the state of practice in crash history modeling over the past several years. Recent examples of interchange type and its relationship to crash occurrence include for example (Bared et al., 2005; Kim et al., 2007; Torbic et al., 2009; Kobelo, 2010). Recent studies by Chen et al. (2009) explored the safety impacts of lanes on freeway exit right side off ramps. A greater likelihood of accidents at interchanges is observed in the deceleration lanes (Lord and Bonneson, 2005). McCartt et al. (2004) show that run off the road crashes frequently occurred more on off ramps at night. Twomey et al. (1993) discuss the safety aspect with changes to geometric layout and interchange spacing. Moon and Hummer (2009) used generalized linear main effect models, main effect and interaction models to study the safety effectiveness of left hand ramps. Lee and Abdel-Aty (2008) in their recent study used a two level nested logit model to estimate crash occurrence for different ramp configurations (see also Lee and Abdel-Aty, 2009). Rakha et al. (2008) show that increasing access point spacing on interchanges would reduce crash rates. In the above mentioned work, the focus has been two fold: (a) on the impact of interchange spacing, and (b) on interchange type. In both cases, the main points of inquiry relate to a traffic flow integration perspective along the lines of analyses done using microsimulation and highway capacity manual methods. Without question, this integration perspective is very valuable for the development of multifaceted insights into interchange design. However, the models used are limited in specification; further, they are fixed parameter models which compounds the heterogeneity effect that is missing from constraining the parameters to be the same across observations. Therefore, it is entirely plausible that interchange type effects on crash propensities may be inferred inaccurately. Pande and Abdel-Aty (2006) used randomly selected non-crash data to analyze the crashes by first harmful event initiating the crash such as rear end, sideswipe. This analysis was conducted in a freeway setting with a particular focus on lane changing related crashes. The authors claim in their paper that lane changing related crashes were not influenced by freeway geometrics. This finding seems to be a premature finding, considering that a comprehensive analysis of freeway geometrics would indicate a relationship with interchange type, and the potential for interchange type to therefore influence crashes involving lane changing types as well. Our goal in this paper is to evaluate the impact of interchange type on geometric random parameter means while accounting for heterogeneity across segments, and incorporating a comprehensive set of geometric elements to minimize specification bias. While it is clear that interchange type and density play a role in crash occurrence on freeways, it is not evident as to how variations in interchange design induce variations in crash occurrence. We address this question via the examination of the marginal effects of geometric parameters of interstates as they relate to interstate crash occurrence. Recent evidence of stochasticity in the parameters associated with interstate geometric effects (see for example, Venkataraman et al., 2011) indicates that variations in interchange type may be a source worthy of further exploration. Random parameter models have been used in the safety literature recently to addresses both frequency and severity aspects of crashes. In the frequency-severity context, as examples, Anastasopoulos and Mannering (2009, 2011) and Milton et al. (2008) used random parameter negative binomial (RPNB) regression to account for heterogeneity via road geometrics, pavement and traffic characteristics and vehicle types. Other studies relating to heterogeneity effects include those by Chin and Quddus (2003) where panel count data models, such as the random effect negative binomial, were used to deal with unobserved explanatory variables that affect frequencies and severities. Abdel-Aty et al. (2007) analyzed the conditions that affect crash occurrence on freeway ramps by type and configurations and uses ITS strategies to reduce crash risk on freeways (see also Bauer and Harwood, 1998). They indicate that the inclusion of corridor effects in the mean function could explain enough variation that some of the model covariates would be rendered non-significant and thereby the inference would be affected as well. Dhindsa (2006) investigated variable speed limits and ramp metering to reduce crash potential on congested freeways using micro simulation. A recent study (Wang et al., 2009) explores factors that significantly influence injury severity at freeway diverge areas. Karlaftis and Tarko (1998) used clustering to account for heterogeneity which in turn resulted in improved fits over traditional negative binomial models. Sittikariya et al. (2009) underscore the importance of heterogeneity due to time varying effects. Collectively, it is clear that the sources of heterogeneity affecting the mean rate function of crashes are varied, and that a comprehensive specification of geometrics is a basic requirement that should potentially include interchange type as a potential source of this heterogeneity. This highlights the importance of our attempt in this paper to focus on heterogeneity-in-the-means as a potentially constructive pathway for a detailed evaluation of factors affecting the marginal effects of any geometrics parameter in a random parameter framework. In the frequency context in safety analysis, factors could include interchange type on interstates and divided roadways with access control via grade separated crossings; or roadway configurations such as roundabouts and traffic signal control at grade crossings. The usefulness of a heterogeneity-in-means approach lies in the ability of the model to tease out type related differences in model parameter means via interactions between road geometry and configuration type. If there are differences, then, a basic random parameter framework may not be sufficient in exposing the influence of interchange type. We apply this philosophy to the analysis of interstate crashes via a discussion of eight major interchange types observed on the Washington State interstate system in the United States. The remaining sections of this paper discuss the types and their impact on random parameters associated with roadway geometrics in the following manner: first, we present a brief discussion of the modeling methodology, followed by a
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description of the empirical setting. We then present results of the modeling process followed by interpretations via marginal effects and conclude with implications for safety inference and future research. 2. Modeling methodology We begin with a general random parameter framework where the probability density of an observed count ni for interchange segment “i” is given by pðni Þ ¼ gðni ; X i ; Z i ; β1i ; β2i ; θÞ
ð1Þ
where Xi and Zi are covariate vectors, and β1i is a vector of nonrandom parameters with dimension K1, β2i is a vector of random parameters of dimension K2, and θ being an ancillary parameter for the overdispersion factor in the negative binomial. Further if, β2i ¼ β2 þΔZ i þ Σνi þΠνi
ð2Þ
where Δ is a K2 M matrix with M variables contributing to variation in the fixed mean β2, Zi is a M 1 matrix, ∑ is a diagonal matrix of K2 scale parameters (standard deviations of the parameter means), vi is a latent error term matrix that is based on the distribution of each random parameter, and Π is a lower diagonal matrix specifying the correlation elements for the random parameters. Given this type of partitioning into nonrandom and random parameter components, heterogeneity-in-the-means of the random parameter vector of dimension K2 is represented by the second term in Eq. (2), thereby indicating that interchange type related effects are captured via the vector of measurements Zi. In our case, Zi includes dummies for each interchange type ranging from directional, semidirectional, clover, diamond, part clover, part diamond, single point urban, to other, the last category including types that do not fall into the prior seven definitions. (In the case of non-interchange segments, the dummies would include upstream and downstream types.) In the absence of significant elements of Δ, heterogeneity in the random parameter mean is latent, although the random parameter β2i can still be specified by a distribution as long as the mean and standard deviation of the mean are significant. If we define a rate of occurrence function λi such that ln λi ¼ β0 X i þ εi
ð3Þ
where β ¼ ðβ1 ; β2 Þ, and expðεi ) is gamma distributed (1, 1/θ), where θ is the inverse of the overdispersion parameter, then, θ ni Γðni þ θÞ θ λi ð4Þ pðni jνi Þ ¼ ΓðθÞΓðni þ 1Þ θ þλi θ þλi Z pðni Þ ¼
θ ni Γðni þ θÞ θ λi hðνi Þdνi ΓðθÞΓðni þ 1Þ θ þλi θ þλi
Likelihood across the sample is given by θ ni Z N Γðni þ θÞ θ λi hðνi Þdνi ∏ ΓðθÞΓðni þ 1Þ θ þλi θ þ λi i¼1
ð5Þ
ð6Þ
The density for segment “i” is computed via simulation of the integral using Halton draws (see for example Train, 2009 for a detailed discussion of the Halton sequence). The simulation based likelihood computation is conducted over a set of R. Halton draws for each segment, and then, the approximated probabilities P(ni) are multiplied over the sample for each iteration, followed by optimization of the likelihood until convergence. We use 30,000 Halton draws in our estimation, with each P(ni) approximated by Pðni Þ ¼
30;000 1 ∑ Pðnir Þjνir 30; 000 1
ð7Þ
In our case, in addition to the above considerations, since we have a panel of 9 consecutive years comprising the period 1999–2007, the likelihoods for each segment are composed of not one density, but the product of 9 densities involving one for each year. From Eqs. (2) and (3), we can see that ln λi ¼ β01 X 1i þβ02i X 2i þ εi
ð8Þ
ln λi ¼ β01 X 1i þðβ2 þ ΔZ i þ Σνi þ Πνi Þ0 X 2i þεi
ð9Þ
where X 1i is the covariate vector for the fixed parameters X 1i and X2i is the covariate vector for the random parameters X1i. The heterogeneity in means term given by ΔZ i captures interactions between interchange type and the vector of roadway geometrics contained in X2i through the net effect ΔZ i X 2i . It is this interaction that influences the stochastic effect of X2i explicitly thereby resulting in heterogeneous parameter means. Depending on the interaction between a segment's road geometry and the type of the interchange it is located in, the parameter means vary from segment to segment. For model estimation purposes, since we have interchange and non-interchange datasets, separate estimations were conducted.
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3. Empirical setting The interstate interchange network in Washington State in the United States is modeled in our study. A total of 575 interchanges and 578 non-interchanges were scanned comprising of all interchanges in the network. Of the 575 interchanges, 29 (5.044%) were of the directional type, 33 were semi directional (5.739%), 22 (3.826%) were clover type, 281 were diamond (48.870%), 9 were single point, urban interchanges (1.565%), 128 were part clover (22.260%), 28 were part diamond (4.870%), and 45 were other (7.826%). An interchange segment was defined to be composed of mainline cross sections between the farthest ramp termini on either side of the overpass or underpass. As such, ramp elements are excluded from this analysis, although their impacts on mainline operation and safety are not mutually exclusive from mainline effects. The heterogeneity-in-means approach helps capture some of these unobserved effects due to ramp configurations without explicit accommodation of ramp characteristics. A detailed description of the dataset used in this study is provided in prior research (see for example, Venkataraman et al., 2011, 2013). The dataset is built on interchange-to-interchange scales, where an interchange is defined as the section of freeway between the farthest ramp endpoints on either side of the interchange. The freeway portion in between two interchanges (defined previously) is considered a non-interchange segment. Roadway geometrics were then measured in a percent-by-length of segment manner, where variations in any geometric measure were weighted by its proportion of the length of the segment. Traffic volume was also measured in a weighted manner by proportion of length. Therefore, shoulder widths and number of lanes variables were proportion by length measurements. Horizontal and vertical curvature measurements were indicated in the form of smallest and largest absolute values in the segment. Lighting type was a variable that was introduced in the dataset on the basis of location on the roadside. Median side lighting indicated the proportion by length of fixed roadway lighting only on the median side of the freeway, while right side lighting type indicated the proportion by length of fixed roadway lighting on the outside shoulder side in the direction of travel. We briefly describe key characteristics of the variables used in this study. The mean number of total crashes was 10.69, while, the maximum was 388 and the minimum was zero. In terms of roadway characteristics, mean median side continuous roadway lighting proportion by length of interchange segment was 12%, while mean right side continuous lighting proportion by length was 9.7%. Two lane cross section proportion by length was 52%, three lane proportion 34%, four lane proportion 13% and five-or-more lane proportions 0.2%. Two foot right shoulder width proportions by length were 17.9%, three to four right shoulder width proportions 20.65%, Two foot left shoulder width proportions by length were 17%, three-to-four left shoulder width proportions 19.9%. The mean number of horizontal curves in a segment was 1.805, while, shortest in segment horizontal curve was 0.182 miles long, and longest in segment horizontal curve was 0.275 miles in length. The smallest central angle mean was 12.1761. The mean number of vertical curves in a segment was 3.12, while the smallest and largest absolute gradients were 1.23 and 2.78 percent respectively on average. 4. Model estimation results We baseline our heterogeneity-in-means random parameter negative binomial models against fixed parameter negative binomial models. As shown in Table 2, for interchange segments, the fixed parameter negative binomial model resulted in a log likelihood of 13,754.78 with 20 parameters, while the random parameter showed a substantial improvement in loglikelihood to 12,610.30. The non-interchange model improvement as shown in Table 3 is from 15,396.41 to 13,636.77. Likelihood ratio tests for both models indicate strong statistical support based on a chi-squared level at p o0.05. Five geometric variable parameters were treated as random, with heterogeneity in their means influenced by several interchange type categories with the exception of “other” type. The types vary in how they influence each random parameter. For example, the ADT random parameter is influenced by clover-CD, SPUI and part-diamond types, whereas, the median lighting parameter is influenced by directional, full-diamond, and part-clover types. Largest degree of curvature random parameter mean is influenced by SPUI, part-diamond, and part-clover types. The smallest vertical gradient random parameter mean is influenced by directional, SPUI, and part-diamond, whereas the largest vertical gradient is influenced by semi-directional, SPUI, part-clover, and part-diamond types. The noted repeated significance of the SPUI and the partdiamond effects on parameter heterogeneity is underscored by conditional mean analysis shown in the model interpretation section. The non-interchange model is influenced by both upstream and downstream interchange types. Therefore, the vector of statistically significant types included upstream directional interchange, semi directional, and clover collector distributor, and downstream the set of directional, semi directional, diamond, part clover, part-diamond and other type significantly contributed to heterogeneity in the random parameter means of total length of adjacent interchanges, two foot left shoulder width and two foot right shoulder width proportions. Statistically significant fixed parametric effects in the interchange and non-interchange models included urban/rural location, right continuous lighting proportion, proportion by length of three lane cross section, four lane cross section length, three-to-four foot left shoulder widths, five-to-nine foot left shoulder widths, ten foot left shoulder widths, three-tofour foot right shoulder widths, five-to-nine foot right shoulder widths, and ten foot right shoulder widths, as well as number of horizontal curves, and shortest horizontal curve length. The structural parameters associated with the fixed and random effects are all significant, as are the standard deviations of the random parameters, the off diagonal elements of the
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Cholesky matrix and the heterogeneity-in-means estimates for the interchange types in interchange segments, as well as upstream and downstream interchange types in non-interchange segments. 5. Model interpretation and findings In order to tease out the heterogeneous influences of roadway geometrics under varying interchange types, it is necessary to examine the sub population behavior of random parameters for interchange and non-interchange segments. There are eight sub populations of interchange types for interchange segments, whereas the sub populations for noninterchange segments is defined by the upstream downstream combination of built interchange types. The theoretical combination of upstream downstream combinations is much larger than the observed combinations, so discussion of the subpopulations for non-interchange segments is restricted to combinations where data is available from the Washington State system. It is to be noted that the statistical significance of geometric effects is discussed on the basis of the entire parameter vector including the structural parameters (shown as fixed effects with significant t-statistics, and random parameter means and standard deviations with significant t-statistics in Tables 1 and 2 below), heterogeneity-in-the-means and off diagonal Table 1 Model results for heterogeneity in means random parameter negative binomial estimation of interchange crash frequency in Washington State. Variable
Constant Logarithm of length of segment in miles Logarithm of ADT
Fixed parameter
Normally distributed random parameter
Mean
Mean
6.675 44.225 0.669 36.361
Urban indicator (1 if urban location; 0 else) 0.047 Lighting type by length proportion Right side lighting segment prop. Median continuous segment prop.
Number of lanes by length proportion Three lane prop. Four lane prop.
t-Stat.
0.129 N/A
0.488 1.196
N/A 0.931
t-Stat.
N/A 62.084
2.508
N/A
N/A
3.702 N/A
N/A 0.190
N/A 5.165
Heterogeneity in random parameter mean
Std. dev.
t-Stat
Variable
N/A N/A 0.047 29.636 Clover/CD interchange Single point urban interchange Part diamond interchange N/A N/A
N/A N/A 0.319 13.546 Directional interchange Full diamond interchange Part clover interchange
Mean
t-Stat
0.019 0.185
5.866 10.282
0.028
4.717
N/A 1.198 0.441
N/A 16.806 8.062
0.639
10.866
21.341 45.363
N/A N/A
N/A N/A
N/A N/A
N/A N/A
N/A N/A
N/A N/A
Left shoulder width by length proportion Three to four foot left shoulder width prop. 0.303 9.277 Five to nine foot left shoulder width prop. 0.538 20.989 Ten foot left shoulder width prop. 0.617 27.682
N/A N/A N/A
N/A N/A N/A
N/A N/A N/A
N/A N/A N/A N/A N/A N/A
N/A N/A N/A
N/A N/A N/A
N/A
N/A
N/A
N/A N/A
N/A
N/A
N/A N/A
N/A N/A
N/A N/A
N/A N/A N/A N/A
N/A N/A
N/A N/A
N/A N/A 0.114
N/A N/A 14.451
N/A N/A 0.035
N/A N/A 0.353
N/A N/A 6.851
0.098 0.163
8.453 6.224
0.085
7.128
Right shoulder width by length proportion Three to four foot right shoulder width 0.171 5.162 prop. Five to nine foot right shoulder width prop. 0.433 16.681 Ten foot right shoulder width prop. 0.522 22.866 Horizontal curvature Number of horizontal curves in segment Shortest horizontal curve length Largest degree of curvature in segment
Vertical curvature Smallest vertical curve gradient in segment
0.048 0.126 N/A
N/A
6.933 2.834 N/A
N/A
0.041
5.309
0.006
N/A N/A N/A N/A 7.416 Single point urban interchange Part clover interchange Part diamond interchange
1.177 Directional interchange
N. Venkataraman et al. / Analytic Methods in Accident Research 2 (2014) 12–20
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Table 1 (continued ) Variable
Fixed parameter
Normally distributed random parameter
Mean
Mean
Largest vertical curve gradient in segment
Scale parameter for overdispersion Log likelihood for constant parameter negative binomial
t-Stat.
t-Stat.
Std. dev.
N/A 0.045 3.164
N/A
16.846
Heterogeneity in random parameter mean t-Stat
0.007
Single point urban interchange Part diamond interchange 3.661 Semi directional interchange Single point urban interchange Part clover interchange Part diamond interchange
0.643 26.197 13,754.78 Log likelihood at random parameter negative binomial convergence 5.324 5.349 5.332 5175
AIC BIC HQIC Number of observations
Variable
Mean
t-Stat
0.540 10.380 0.158
6.363
0.045
5.983
0.258
8.169
0.060 0.286
8.142 12.671
12,610.30
4.893 4.958 4.916
Table 2 Model results for heterogeneity in means random parameter negative binomial estimation of non-interchange total crashes in Washington State Variable
Constant
Logarithm of length Logarithm of ADT Total length of adjacent interchanges
Lighting type by length proportion No roadway lighting segment prop.
Fixed parameter
Normally distributed random parameter
Mean
Mean
N/A
0.717 0.930 N/A
0.133
t-Stat.
N/A 6.616
65.987 73.945 N/A
N/A N/A 0.140
t-Stat.
Heterogeneity in random parameter mean
Std. Dev.
47.894
N/A N/A 4.254
0.756
N/A N/A 0.170
t-Stat.
Variable
46.426 Upstream directional interchange Upstream semi directional Upstream clover collector distrib. Downstream directional Downstream semi directional Downstream diamond Downstream part clover Downstream part diamond Downstream other type N/A N/A 51.471 Upstream directional interchange Upstream clover collector distrib. Upstream part clover Upstream part diamond Downstream semi directional Downstream diamond Downstream part clover Downstream part diamond
Mean
t-Stat.
0.453 6.084 0.059 2.105 0.637 4.972 0.390 0.719 0.392 0.322 0.673 0.406
8.824 8.185 5.616 4.316 6.645 5.056
0.333
8.886
0.252
3.788
0.040 3.988 0.169 6.827 0.181 4.079 0.172 4.778 0.061 1.547 0.453 7.335
6.629
N/A
N/A
N/A
N/A N/A
N/A
N/A
Number of lanes by length proportion Two lane prop. 0.854 40.822 Three lane prop. 0.630 32.000
N/A N/A
N/A N/A
N/A N/A
N/A N/A N/A N/A
N/A N/A
N/A N/A
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Table 2 (continued ) Variable
Fixed parameter
Normally distributed random parameter
Mean
Mean
Left shoulder width by length proportion Two foot left shoulder
N/A
t-Stat.
N/A
t-Stat.
0.528
Heterogeneity in random parameter mean
Std. Dev.
16.017
0.587
width prop. Six–nine foot left Shoulder width prop.
0.129
5.520
N/A
N/A
N/A
Right shoulder width by length proportion Two foot right shoulder width prop N/A
N/A
0.431
10.812
0.054
10.461 11.209 6.547 6.655
N/A N/A N/A N/A
N/A N/A N/A N/A
N/A N/A N/A N/A
Horizontal curvature Number of horizontal curves in segment 0.026 Shortest horizontal curve length in miles 0.692 Longest horizontal curve length in miles 0.248 Shortest horizontal curve radius in 0.052 segment
Vertical curvature Number of vertical curves in segment Smallest vertical rate of curvature Scale parameter for overdispersion Log likelihood for constant parameter negative binomial
0.0123 0.145 19.033
AIC BIC HQIC No. of observations
t-Stat.
Variable
Mean
33.363 Upstream clover collector distrib. Upstream part clover Downstream directional Downstream diamond N/A Downstream other type
0.190
t-Stat.
2.637
0.119 2.708 0.379 5.389 0.355 8.506 0.109 1.93
3.433 Upstream clover collector 0.592 8.154 distributor Interchange Upstream part diamond 0.544 4.640 Downstream semi directional 0.358 5.986 Downstream diamond 0.213 4.511 Downstream part clover 0.254 4.790
N/A N/A N/A N/A
4.651 N/A N/A N/A N/A 3.161 N/A N/A N/A N/A 0.948 20.079 13,636.77 15,396.41 Log likelihood at random parameter negative binomial convergence 5.265 AIC 5.338 BIC 5.291 HQIC 5202
N/A N/A N/A
N/A N/A N/A
N/A N/A N/A
N/A N/A
N/A N/A
N/A N/A
4.893 4.958 4.916
Table 3 Heterogeneity in random parameter means of interchange segment geometrics.
Log-ADT Median continuous lighting prop. Largest horizontal degree Smallest vertical gradient Largest vertical gradient
Directional
Semi
Clover CD
Diamond
SPUI
Part clover
Part diamond
Other
0.934 1.555 0.109 0.062 0.015
0.941 0.187 0.111 0.047 0.022
0.958 0.148 0.111 0.043 0.015
0.936 0.234 0.110 0.044 0.017
1.139 0.155 0.492 0.532 0.300
0.937 0.733 0.011 0.045 0.041
0.918 0.166 0.078 0.144 0.272
0.941 0.189 0.110 0.047 0.020
effects resulting from the parameter correlation vector. Therefore, direct interpretation through the structural parameters alone does not present a complete picture of the segment specific geometric effects for random coefficients in the models. Conditional means at the segmental level are derived using Bayes rule (see Train, 2009 for details), and the parameter means are reported for the statistically significant interchange types in Table 4. hðβjni; X i; θÞ ¼ Z β¼
Pðni jX i; βÞgðβjθÞ Pðni jX i; θÞ
R βpðni; jX i; βÞgðβjθÞdβ βhðβjni; X i; θÞdβ ¼ R pðni; jX i; βÞgðβjθÞdβ
ð10Þ
ð11Þ
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Table 4 Heterogeneity in random parameter means of non-interchange segment geometrics.
Constant Two foot left shoulder width prop. Two foot right shoulder width prop. Total length of adjacent Interchanges
Part clover/part clover
Part clover/diamond
Diamond/part clover
Diamond/diamond
6.450 0.499 0.157 0.157
6.205 0.043 0.220 0.012
6.33 0.553 0.173 0.086
6.22 0.170 0.218 0.036
The means, β of the conditional distribution (shown as “h” in Eq. 10) of the parameter β (on the observed outcome n, X and the distribution parameter θ) are computed using non closed form simulations involving weighted average of the parameter draws. Table 3 shows the parameter means for interchange types. As seen in Table 3 below, the conditional means of the geometric effects vary by interchange type. With the exception of the ADT variable, the effects are substantially varied with some changes in sign as well. For example, median continuous lighting proportion is associated with increased crash likelihoods at all interchange types with the exception of full diamond types. This indicates the benefit of median continuous lighting types in full diamond settings, while the effect is expected to be the worst in full directional interchange settings. Horizontal degree of curvature tends to have counter-productive effects on crash occurrence in most interchange settings with the exception of part diamond types. The counter-productive effect is most pronounced in single point urban interchange settings, but it is to be mentioned here that this requires further research due to the sample size of single point urban interchanges. Smallest vertical gradient in an interchange segment is expected to decrease crash likelihoods in directional, SPUI, and part-diamond settings, whereas its counter-productive effect is relatively uniform in semi directional, clover-collector– distributor, full-diamond, part clover and other types. Largest vertical gradient is expected to decrease crash likelihoods in full directional, clover-collector–distributor, full-diamond, SPUI, and other type settings. Similar to the effect of smallest vertical gradient, its benefit is greatest in SPUI settings, albeit the sample size of SPUI interchanges. The counter-productive effect of largest vertical gradient is most pronounced in part-diamond settings. It is also to be noted that median continuous lighting type shows the widest variation in effect size across interchange types. In addition, the effects are not trivial, with the smallest effect resulting in an expected 16% increase in crash likelihoods. Table 4 shows heterogeneity in random parameter means for roadway geometrics in non-interchange segments. The number of possible upstream/ downstream interchange combinations far exceeds as-built combinations (as evidenced in the Washington State system). A total of 53 as-built combinations were observed on the interstate system, and out of those, the four major upstream downstream combinations are shown below in terms of their heterogeneous influences on roadway geometric parameters. The four major combinations are: upstream part clover/downstream part clover, upstream part clover/ downstream diamond, upstream diamond/downstream part clover and upstream diamond/downstream diamond types. As observed in Table 4, the interchange types uniformly contribute to heterogeneity in the random effect itself, but with a magnitude that is stable. The size of the magnitude indicates that even after accounting for geometric effects, unobserved heterogeneity still remains in substantial form, presumably due to environmental effects and driver behavior effects. Two foot left shoulder width proportions have a consistently counter-productive effect under all four major upstream downstream combinations, with the diamond/part clover effect being most pronounced. In comparison, two foot right shoulder width proportions while uniformly counter-productive peak at far less magnitudes (in part clover/diamond setting). The importance of total length of adjacent interchanges (upstream and downstream interchange lengths combined) is underscored via the heterogeneity in its means under the four interchange combination settings. It is evident that the setting with potentially largest footprint (diamond/diamond) contributes to a beneficial effect from a total crash likelihood standpoint. The setting with the most counter-productive effect is the part clover/part clover footprint. It is to be noted here that the partial impact of a diamond interchange (whether it is upstream or downstream) is almost benign, with magnitudes substantially less than the part clover/part clover footprint.
6. Conclusions and Recommendations This analysis has shown the importance of heterogeneous influences in the construct of random parameter models of interchange type effects on interstate safety. The variation in the parameter means across interchange types underscores the utility of considering alternative types from a safety standpoint, which in turn can lead to better safety conscious planning of highway design. For example, traffic engineers and highway designers can explore trade-offs between the geometric footprint and cost of an interchange and the resultant safety benefits that can arise from reduced heterogeneity in the impact of the interchange type on the relationship between segment geometrics and crash propensities. In this paper, the focus is on total crashes, whereas, this method can be extended to the evaluation of crashes by severity type. By doing so, designers can separate insights on trade-offs between interchange complexity due to type and safety benefits, and thereby adopt targeted design policies for freeway safety optimization. Second, the importance of upstream/downstream related heterogeneity is underscored from a safety conscious management of traffic flow as well. Clearly, the strength of the random effect in the non-interchange model underscores unobserved effects due to traffic flow and environmental factors. It may be
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a useful point of departure from this study to pursue the influence of traffic flow variables explicitly in the construct of safety models. To date, the address of flow variables in safety modeling has been limited primarily to the use of ADT level variables, or in rare cases, the use of peak hour level measurements. Substantial room exists for the refinement of flow variables in terms of the need for improved capture of unobserved effects due to flow in interchange settings. The use of interchange combinations was strategically guided by computational limitations in terms of the number of random parameters current computational software can address. With eight configuration types and 30,000 Halton draws, the estimation time of models shown in this paper regularly exceeded a day per run. The 30,000 Halton draw limit was determined empirically through an exhaustive search of critical parameters in order to ensure there were no empirical identification issues. It is entirely possible that other empirical contexts may require fewer or greater number of simulation draws. In the context of non-interchange models, the upstream-downstream interchange combinations in total (53) far exceeded the number of random parameters that can be computationally accommodated. If each parameter is to represent a combination, one would need 53 parameters to identify all sources of heterogeneity uniquely. The Limdep software package was used to estimate the models in this paper. In a broader context with interchange systems from multiple states, the combination number can start to escalate. Further research is required in this area in order to explore opportunities for scaling up the number of parameters for evaluation of all observed combinations explicitly. In this paper, the combination problem was addressed as a reduced form vector of sixteen type elements. 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