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The Effect of Road Environment Factors on Freeway Traffic Crash Frequency during Daylight, Twilight, and Night Conditions
Sungmin Hong Ph.D Student Division of Engineering and Policy for Sustainable Environment Graduate School of Engineering, Hokkaido University Kita 13, Nishi 8, Kita-ku, Sapporo Hokkaido, Japan, 060-8628 Tel: +81-11-706-6822 Fax: +81-11-706-6822 E-mail:
[email protected]
Joonki Kim, D.Sc. Associate Research Fellow Center for Road Policy Research Korea Research Institute for Human Settlements 254 Simin-daero, Dongan-gu, Anyang-si Gyeonggi-do, Korea, 431-712 Tel: +82-31-380-0285 Fax: +82-31-380-0484 E-mail:
[email protected]
*Cheol Oh, Ph.D. Associate Professor Dept. of Transportation and Logistics Engineering Hanyang University 55 Hanyangdaehak-ro, Sangnok-gu, Ansan-si Gyeonggi-do, Korea, 426-791 Tel: +82-31-400-5158 Fax: +82-31-436-8147 E-mail:
[email protected]
Gudmundur F. Ulfarsson, Ph.D. Professor Faculty of Civil and Environmental Engineering University of Iceland Hjardarhagi 6 IS-107 Reykjavik, Iceland Tel: +354-525-4907 Fax: +354-525-4362 E-mail:
[email protected] Word Count: 5,908 + 5*250 = 7,158 Submitted for Presentation at the 93rd Annual Meeting of the Transportation Research Board and for Publication in the Transportation Research Record * Corresponding author
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ABSTRACT This research investigated the effect of road environment factors on freeway traffic crashes in Korea during the years 2007 through 2010 under the light conditions: daylight, twilight, night, and for comparison the whole 24 hour day. Both random and fixed parameter crash frequency models were estimated and applied in the analysis. Several factors had statistically significant effects only during certain light conditions, notably, number of interchanges/junctions and number of bridges during daylight; traffic share of heavy vehicles during night and the whole 24 hour period; short tangent ( 0, the expected rate of crashes, is the Poisson parameter. The Poisson regression is developed by expressing the Poisson parameter as a log-linear-inparameters function: ln ππ = π½π₯π , (2) where π₯π is a vector of observable explanatory variables and Ξ² is a vector of estimable parameters (22). The Poisson regression is based on the assumption that the variance and mean are equal. If the variance in π¦π is greater than the mean, there may be problems of over-dispersion. In this case, the negative binomial model can be used. In the negative binomial model, an error term has been added to (2) (2): ln Ξ»π = π½π₯π + ππ . (3) The exponentiated error term is assumed to be gamma-distributed with the mean of exp(Ξ΅π ) being 1 and the variance Ξ± > 0 . Then VAR[π¦π ] = πΈ[π¦π ][1 + πΌπΈ[π¦π ]] = πΈ[π¦π ] + πΌπΈ[π¦π ]2 . In other words, when Ξ± of the negative binomial model is not statistically significantly different from 0, the negative binomial model reduces to the Poisson regression model. Therefore, if Ξ± is statistically significantly larger than 0, the negative binomial model is appropriate. Here the frequency data were based on long, fixed-length sections which include varying geometrics and traffic. In order to capture this heterogeneity, this research followed the work of (12) and (13) by developing random parameter Poisson and negative binomial regressions. The estimable random parameter is expressed: Ξ²π = π½ + ππ , (4) where ππ is a randomly distributed term. An alternative specification allowing a correlation across random parameters (23), Ξ²π = π½ + Ξππ , using a lower triangular covariance matrix, Ξ, is also tested. Recommendations for distributions of random parameters have been provided and were used in this research to develop the models (24-25). For continuous variables, the normal distribution was used; and the uniform distribution was used for indicator variables. The expected rate, Ξ»π , expressed conditioned on ππ in the Poisson and negative binomial equations becomes: Poisson regression model: ππ |ππ = exp(π½π₯π ), (5) Negative binomial model: ππ |ππ = exp(π½π₯π + Ξ΅π ), (6) where ππ |ππ is the expected rate of crashes on freeway section π during a time period conditional on the random parameter ππ . As a result, the log-likelihood (πΏπΏ) of the model becomes: πΏπΏ = ββπ ln β«π π(ππ )π(π¦π |ππ )πππ , (7)
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where g(β) is the probability density function of ππ . The simulated maximum likelihood method was used to estimate the random parameters Poisson and negative binomial models. Halton draws were used to simulate the integral in (7). This method has been commonly used because it samples a distribution more efficiently than equally distributed draws (24, 26, 27). The goodness-of-fit (GOF) values for the models were calculated using the log-likelihood ratio, π2 : πΏπΏ(π½)βπ π2 = 1 β πΏπΏ(0) , (8)
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where πΏπΏ(π½) is the log-likelihood at convergence for the full model, πΏπΏ(0) is the value for the initial model with all parameters set to zero, and π is the number of estimated parameters. In order to determine whether or not the fixed parameter model and random parameter model for the same light condition were statistically significantly different, a likelihood ratio π 2 -test was performed. The likelihood ratio π 2 value is expressed: π 2 = β2[LL(π½)π β LL(π½)π ], (9) where πΏπΏ(π½)π , πΏπΏ(π½)π are the log-likelihood at convergence values for the two models being compared, model π and π , with degrees of freedom equal to the difference in the number of parameters in the two models. Marginal effects were used to explain the size of the relationship between independent variables and the dependent variable. In the case of a crash frequency model, the marginal effect measures changes in the expected number of crashes through the partial derivative βππ / βπ₯ of some independent variable π₯. ππ is defined through equations (2) and (3) for the Poisson and negative binomial respectively. The marginal effect is a point value, calculated for each section and must therefore be aggregated, here by taking the average for all sections (23). ANALYSIS The dependent variables of the four estimated models were the number of traffic crashes occurring in each section during 1) daylight, 2) night, 3) twilight (sunset or sunrise), and 4) the whole 24 hour day for comparison. The investigation was performed using the variables presented in Table 1. The random parameter models were estimated using 1000 Halton draws in the simulation, and parameters significant at the 0.1 level of significance were kept, less significant variables were constrained out of the models. This level of significance was chosen rather than the more stringent 0.05 level in order to provide better statistical power for the analysis, which was useful as the number of observations is 281. Poisson and negative binomial models were estimated using both fixed and random parameter models. The results show the over-dispersion parameters of the fixed parameter negative binomial models were always statistically significantly larger than zero. Hence the fixed parameter negative binomial models are presented in Table 2. For the random parameter negative binomial models, the over-dispersion was not statistically significantly larger than zero for any of the models. The random parameter Poisson models are therefore presented in Table 2. A test of the alternative specification of equation (4), allowing correlation across random parameters, showed that constraining this correlation out resulted in better fitting models. The marginal effects of the fixed parameter negative binomial and random parameter Poisson models are presented in Table 3. At this point the question becomes which of these models is preferred, the fixed parameter negative binomial or the random parameter Poisson? The results of the π 2 -test in (9) comparing the random parameter Poisson model and fixed parameter negative binomial model are presented at the end of Table 2. If the calculated π 2 value exceeds the 90% critical value given the degrees of freedom, the fixed parameter model and random parameter model are statistically significantly different (π 2 random-fixed > π 2 critical ). The analysis found that for the models of daylight, night, and whole day, the fixed parameter negative binomial model and the random parameter Poisson model were statistically significantly different, while for the twilight models the fixed
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parameter negative binomial model and the random parameter Poisson model were not statistically significantly different. The results for random parameter estimates are summarized in Table 4, which also presents the type of distribution for each parameter along with the positive and negative parameter densities. This information aids the interpretation of results as this shows the percentage share of roadway sections where the random parameter takes a positive or negative value.
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TABLE 2 Estimation results for random and fixed parameter models of freeway crash frequency
Constant Traffic Characteristics Traffic share of light vehicles Standard deviation of parameter distribution (N) Traffic share of heavy vehicles Geometric Characteristics Number of lanes Standard deviation of parameter distribution (N) Number of interchanges/junctions Short tangents (L
