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category: literaturenote citekey: ferreiratrafficflowaccidentsrelationship2013 title: Traffic flow-accidents relationship for urban intersections on the basis of the translog function authors: "Ferreira, Sara; Couto, António" year: 2013 date: 2013-00-00 2013 doi: 10.1016/j.ssci.2013.07.007 publication: Safety Science url: "https://doi.org/10.1016/j.ssci.2013.07.007" zotero_key: 8UAD2F9T zotero_storage: PQSCE4II collections: transpordi_planeerimine folder: 001_artiklid/Road Safety Indicators firstAuthor: "Ferreira, Sara"

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Contents lists available at SciVerse ScienceDirect

Safety Science

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

Traffic flow-accidents relationship for urban intersections on the basis of the translog function

Sara Ferreira ⇑ , António Couto

Civil Engineering Department, School of Engineering, Porto University, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal

article info

Article history: Received 20 February 2013 Received in revised form 30 May 2013 Accepted 8 July 2013 Available online 30 July 2013

Keywords: Translog function Negative binomial Urban intersections Random effects model Elasticity

abstract

In this paper, we use a flexible function to represent the traffic flow-accidents relationship for urban intersections, called translog function. Considering the common use of predictive models and given the complexity of the traffic movements that usually occur at intersections, this flexible function provides a richer interpretation of traffic flow-accidents relationship. Therefore, five functional forms commonly used are compared to the translog function by modeling accidents using data of four-legged signalized intersections. Better results in terms of goodness of fit are obtained for the translog model. In addition, sensitivity analysis shows that the translog function is distinct from the log-linear functions, especially at the boundary values, revealing the potential to capture the accident risk complexity usually existent at intersections as a result of several traffic movements. Moreover, to analyze if the omitted variables cause variable bias, and thus, affecting the previous model assessment and comparison, panel count data models are applied, namely, random effects models. The results obtained are consistent with those previously obtained, which proves that the translog model may be an alternative providing a richer interpretation of the accident occurrence at intersections.

-2013 Elsevier Ltd. All rights reserved.

1. Introduction

Intersections are complex road entities because numerous conflicting traffic movements occur at these sites and are influenced by signalization and visibility conditions. Thus, it is not surprising that a great number of road accidents occur at intersections (Guo et al., 2010; Haleem et al., 2010a; Mitra and Washington, 2012). Identification of hazardous intersections, diagnosis of safety problems at hazardous intersections, prescription and prioritization of safety treatments, and safety evaluation of treatments applied are crucial aspects of road safety management (Persaud and Nguyen, 1998).

In order to improve the collective understanding of accident causes and contributing factors, the research community has focused considerable attention on analytical methods development (negative binomial models, zero inflated models, etc.), on better experimental designs (before-after studies, etc.), on improving exposure measures, and on model specification improvements (non-linear relationships, etc.) (Mitra and Washington, 2012). There have been considerable analytical improvements in the past few decades rejecting linear regression models in favor of count and modified count models. A considerable number of studies have focused on specification issues with count regression models as can be seen in a comprehensive and recent review provided by Lord and Mannering (2010). Moreover, and according to Hauer and Bamfo (1997), ''in the course of modeling, the modeler will make two major decisions: (a) what explanatory variables to include in the model equation; and, (b) what its functional form should be.'' The first decision should be a balance between the number of explanatory variables and a reasonable goodness-of-fit (Reurings et al., 2005). Nevertheless, it depends on the type of roadway element and the availability of data whether or not the variables should be included in the model. As pointed out by Reurings et al. (2005), explanatory variables should include variables that have been found in previous studies to wield a major influence on accident frequency; can be measured in a valid and reliable way; and are not very highly correlated with other variables included in the model.

Several authors have shown that the traffic flow is the most powerful variable, particularly for intersection accident models, which in time has a relatively poor explanatory power compared to the accident prediction models for road segments (Greibe, 2003). In fact, accident prediction models using the traffic flow variable as the only input have been commonly used (Lord, 2000; Vieira Gomes et al., 2012).

However, as recently Lord and Mannering (2010) pointed out, leaving out important explanatory variables results in biased parameter estimates that can produce erroneous inferences and accident-frequency forecasts. In order to minimize the effect of unobserved heterogeneity from potentially omitted variables

⇑ Corresponding author. Tel.: +351 225 081 968. E-mail addresses: sara@fe.up.pt (S. Ferreira), fcouto@fe.up.pt (A. Couto).

within subjects (sites, intersection, etc.), random panel data models have been used (Anastasopoulos and Mannering, 2009; Mitra and Washington, 2012).

In terms of statistical relationship between road accidents and explanatory variables, the second decision referred by Hauer and Bamfo (1997), the functional form used may affect the performance of the models, especially in the case of the variables representing traffic flow at intersection models given the complexity of traffic interaction to be represented. Although the statistical relationship between road accidents and traffic flows has been modeled extensively, a lack of consensus exists on the functional form that best represents that relationship (Miaou and Lord, 2003). In the case of the models that include several variables, which characterize the geometry and functionality of the intersection, traffic flows are usually represented by one variable, the sum of the entering traffic flows (which Turner and Nicholson (1997) call the direct approach), along with variables such as the presence of left and right turns. In contrast, in the case of the models, which only include traffic flow variables, a variety of functional forms have been employed to describe the conflicting flows entering the intersection (the indirect approach (Turner and Nicholson, 1997)).

Several statistical techniques, such as generalized linear models (GLM) and empirical Bayes methods, have been applied to modeling the relationship between road accidents and traffic flow at intersections (Chin and Quddus, 2003; Miaou and Lord, 2003; Lord and Mannering, 2010). Regardless of the statistical technique used, it remains a question what functional form best represents the crucial relationship between accidents and traffic flows at intersections, given the associated complexity of the traffic movements that usually occurs at intersections. As Miaou and Lord (2003) noted, defining a model that produces a good fit to the data set, which is easily accomplished using numerous smoothing techniques and associated software, is no longer a challenge; rather, criteria based on logic, (e.g., reason, consistency, and coherency), flexibility, extensibility, and interpretability of the functional form should be considered in model improvements.

GLM and other statistical techniques typically used to model accidents rely on a logarithmic transformation to linearize the multiplicative equation (herein referred as a log-linear model). It is well known that log-linear models assume constant elasticity for the estimation parameters. Indeed, this assumption can be a limitation in representing the relationships between the traffic flow and accident frequency at an intersection. Recently, Couto and Ferreira (2011) have studied this possible limitation in the case of road accidents at urban road segments using a flexible functional form called translog function, modeled by a generalized linear model with a negative binomial (NB) distribution. The results suggest that the translog function has considerable potential for modeling the occurrence of accidents, not only in terms of goodness of fit statistics and residual analysis (the NB distribution with the translog function yields better results than the traditional NB distribution) but also, providing proper boundary values and thus contributing to a well-defined mean structure that fits the heterogeneous data usually exhibited by accident samples.

The translog function was first described by Christensen et al. (1973) and has been applied extensively in transportation research, especially in logistics and policy studies based on transportation costs (Karlaftis and McCarthy, 2002; Link, 2006; Couto and Graham, 2009; Satar et al., 2010). The translog function can generalize NB and Poisson distributions through the estimation of quadratic cross-product terms, thereby allowing elasticity to vary with the values of other variables.

The objectives of this study are to analyze the efficiency of the translog function to represent the relationship between accidents and traffic flows at intersections while keeping a good performance. In order to best comprehend the potential of the translog function to represent the accident risk complexity due to the several traffic movements usually present at intersections, a comparison of this functional form with others commonly used is provided applying a GLM technique. Goodness of fit as well as other performance measures are applied. In addition, flexibility and interpretability are considered in the model analysis. Moreover, in order to deal with the potential bias in parameter estimates due to omitted variables in the traffic flow-only model, panel data models are also applied. These models allow each estimated parameter of the model to vary across each individual observation in the data set. Therefore, the unobserved heterogeneity from one intersection to another is accounted. Therefore, two different statistical models were developed, one applying the negative binomial (NB) distribution to the function forms, which include variables related to traffic flows from major and minorroads, and a second one applying a panel data model. This latter intends to ensure that the omitted variables as well as the unobserved heterogeneity from one intersection to another do not affect the evaluation and comparison of the function forms.

2. Data

This research is conducted using data from four-legged signalized urban intersections in Fortaleza, Brazil, for the years of 2007 and 2009 collected by the Sistema de Informação de Acidentes de Trânsito de Fortaleza (SIAT-FOR), which includes property damage only accidents and accidents with victims. A total of 120 intersections were used, which correspond to a total of 1123 accidents.

The signalized intersections were selected from a heavily congested region in Fortaleza city which is adjacent to its central business district. The annual average daily traffic – AADT information was obtained from inductive loop detectors individually for every intersection approach for the years of 2007 and 2009. The vast majority of intersections consists of arterial and collector streets with no dedicated left (or right) turn lanes or left-turn signal phase. It is also worth notice that in Brazil, right-turns on red as well as left-turns without specific signal phase are not allowed. A full description of the Fortaleza intersections as well as accident data and traffic volume is provided elsewhere (Cunto et al., 2012).

Table 1 presents a summary of AADT for major and minor approaches as well as the total observed crash frequency for four-legged signalized data set.

3. Background

Because accident data are discrete and non-negative, it may be natural to model accident occurrence using regression to fit a model to a Poisson or NB distribution form. The Poisson distribution form has some limitations. One important constraint is that the mean must be equal to the variance. Many researchers have reported a significant overdispersion in accident data, i.e., a variance much greater than the mean (Lord and Mannering, 2010). The NB distribution may be more suitable for describing discrete and non-negative events such as accident occurrence because it does not have the constraint that the mean and variance must be equal.

Table 1 Descriptive statistics of the variables.

Variables
Mean		 Max.
Total Accidents
6.9
AADT major
23,271
AADT minor
12,115		 6.7
7230
5906		 0
4629
3527		 48
44,312
38,431

In addition, the mean is linked to a vector of explanatory variables by a log-linear function:

$$\mu_i = \exp(\beta X_i + \varepsilon_i)$$ or, equivalently $\ln(\mu_i) = \beta X_i + \varepsilon_i$ (1)

where Xi is a vector of explanatory variables, b is a vector of estimable parameters and exp(ei) is a gamma-distributed error term with mean 1 and variance a. The addition of this term allows the variance to differ from the means as Var[yi] = E[yi][1 + a E[yi]] = E[yi] + a E[yi] 2 . In this formulation, the expected number of events per period for roadway entity is given by E[yi] = li = exp(bXi + ei). The estimated parameters b are used to make inferences about the explanatory variables thought to influence the accident process. To analyze the parameter estimation results, elasticities are determined to provide an estimate of the impact of a variable on the expected frequency. These elasticities can be interpreted as the percentage effects of relative changes in the variables on the expected frequency li (Washington et al., 2003). The elasticity of frequency li is defined as

$$E{\mathbf{x}{ik}}^{\mu_i} = \frac{\partial \mu_i}{\mui} \times \frac{\mathbf{x}{ik}}{\partial \mathbf{x}_{ik}} = \betak \mathbf{x}{ik} \tag{2}$$

where E represents the elasticity, xik is the value of the kth explanatory variable for observation i, bk is the estimated parameter for the kth explanatory variable and li is the expected frequency for observation i.

For dummy variables, the bks approximately measure the relative increase in li as the dummy variable changes from 0 to 1 (Washington et al., 2003):

$$E{x{ik}}^{\mu_i} = \frac{e^{\beta_k} - 1}{e^{\beta_k}}$$ (3)

However, the traditional NB model (with a log-linear function) assumes constant elasticity. This limitation can bias the interpretation of the relationship between the explanatory variables and the expected accident frequency. Thus, we have considered in this study a flexible functional form to model second-order effects, namely translog function (the reader is referred to Couto and Ferreira (2011) for more detailed description of this function):

$$\ln y = \beta0 + \sum{k=1}^{K} \beta_k \ln xk + \frac{1}{2} \sum{k=1}^{K} \sum{i=1}^{K} \gamma{ki} \ln x_k \ln x_i$$ (4)

Indeed, the second-order terms of the translog function allow for possible interactions among the variables and for possible non-linearities in the relationships.

The explanatory variable elasticity is defined as:

$$\frac{\partial \ln y}{\partial \ln x_k} = \betak + \sum{i=1}^K \gamma_{ki} \ln x_i \tag{5}$$

From Eq. (5), it is clear that, in general, the elasticity varies with the proportions of the explanatory variables as well as with their individual levels.

Given the potential heterogeneity in the data set used, random parameters and random effects models are applied to verify if heterogeneity affects the proposal model results and conclusions. These models assume that the unobserved intersection effects are distributed across the population of intersections according to some pre-determined distribution (Mitra and Washington, 2012). To allow for such random parameters in count-data models, estimable parameters can be written as bk = b + ui where ui is a randomly distributed term (for example a normally distributed term with mean zero and variance r2 ). With this equation, the Poisson parameter becomes li|ui = exp(bXi + ei). A random effects model is a special case of a random parameters model where the intercept is only modeled as a random parameter as opposed to all parameters in mean function.

To test if the main inferences provided by the panel data models are consistent with those of the NB models, the results of both models are presented below in two separate sections – Sections 4 and 5.

4. Functional form—NB models

Many functional forms have been postulated to represent the relationship between road accidents and traffic flows (Turner and Nicholson, 1997; Persaud et al., 2002; Kumara and Chin, 2003; Miaou and Lord, 2003). The functional forms proposed differ in the types and number of covariates included, the forms of variable transformation applied, and the parameterization schemes used (Hauer and Bamfo, 1997). The various parameterization schemes available are well developed, but they are of little use if an inappropriate functional form is used. This study is carried out using five commonly used functional forms (Turner and Nicholson, 1997; Miaou and Lord, 2003), in addition to the translog function, applied to traffic flow modeling of the four-legged signalized intersection accident data. The six models and their functional forms are presented in the equations above. The first five equations are traditional log-linear functions of the following forms:

FF(1):

$$\ln \mu_{it} = \beta_0 + \beta1 \ln(F{1{it}} + F{2_{it}}) \tag{6}$$

FF(2):

$$\ln \mu_{it} = \beta_0 + \beta1 \ln(F{1_{it}}) + \beta2 \ln(F{2_{it}})$$ (7)

FF(3):

$$\ln \mu_{it} = \beta_0 + \beta1 \ln(F{1{it}} F{2_{it}}) \tag{8}$$

FF(4):

$$\ln \mu_{it} = \beta_0 + \beta1 \ln(F{1{it}} + F{2_{it}}) + \beta2 \ln(F{2{it}}/F{1_{it}})$$ (9)

FF(5):

$$\ln \mu_{it} = \beta_0 + \beta1 \ln(F{1_{it}}) + \beta2 \ln(F{2_{it}}) + \beta3 F{2_{it}}$$ (10)

The translog function is expressed as follows – FF(6):

$$\ln \mu_{it} = \beta_0 + \beta1 \ln(F{1:it}) + \beta2 \ln(F{2:it}) + \beta3 [\ln(F{1:it})]^2 + \beta4 \ln(F{1:it}) \ln(F_{2:it}) + \beta5 [\ln(F{2:it})]^2$$ (11)

These six functional forms are tested in terms of goodness of fit analysis, and criteria such as flexibility and interpretability are used to compare the performance of the translog function with that of the other functional forms examined.

Subsection 4.1 summarizes the results obtained with the six functional forms, and subsection 4.2 presents a discussion of the capabilities and limitations of these functional forms.

4.1. Results

Model fitting is conducted for the six functional forms with a NB error structure. The explanatory variables of the five log-linear function NB models and the first-order variables of the translog function NB model are previously ''normalized'' by dividing every variable observation by its respective sample mean to facilitate model comparisons and interpretation of the results. The results for the FF(5) model show that the F2 term is found to be not statistically significant. Because the exclusion of this variable transforms the FF(5) model into the FF(2) model, the former is excluded from this study. In addition, the results for the FF(6) model show that the cross-product term [ln(F2,it)]2 is not statistically significant; therefore, this variable is excluded from the FF(6) model. Also, the correlations among the variables of FF(6) model were analyzed showing values less than 0.5. The final results are shown in Table 2 for the FF(1), FF(2), FF(3), FF(4) and FF(6) models.

Although the core of the study is not to compare the models based on fit performance, some GOF measures are presented in Table 2, namely, Akaike's information criterion (AIC), Elvik's index ðR2 PEÞ and McFadden's pseudo R2 . As shown in Table 2, the GOF measures are similar, although suggesting that the FF(6) model has the best performance, mainly taking into account the R2 PE value. Furthermore, the mean absolute deviation (MAD) and the mean squared predictive error (MSPE) criteria were also used in the study done by Lord and Mahlawat (2009) and Haleem et al. (2010b) for assessing the GOF of the fitted models. The overall prediction accuracy is estimated by dividing the total predicted accident by the total observed accident at the collected intersections (Haleem et al., 2010b). The values for the five models of these three measures of effectiveness are shown in Table 3. From this table, the best model in terms of the three mentioned measures is the FF(6) model.

Another method used in this evaluation is the cumulative residual (CURE) analysis method, proposed by Hauer and Bamfo (1997), in which the cumulative residuals (the sum of the differences between the actual and fitted values) are plotted in increasing order for each explanatory variable separately. The plot reveals how well the functional form fits the data with respect to each explanatory variable. The CURE plots for the five models are shown in Fig. 1 (constructed for AADT major). As can be seen by this figure, the residual analysis supports the conclusion stated above that the fit performance is similar for all models, although the translog FF(6) model fits the data better because it is the only one that presents a cumulative residual line between the both boundary lines for all range of AADT values.

In summary, this comparative analysis indicates that the FF(6) model performs better than the other four models, showing superior fits capacity of the translog function for all AADT range values.

An important issue in developing and using regression models is parameter interpretation (sensitivity analysis). To analyze and compare the influence of each explanatory variable on accident frequency in each of the five models, elasticity values are analyzed. The elasticity values for the four log-linear function NB models are consistent with expectations as all variable elasticities indicate an increase of the accident risk when the traffic flow increases as well (Persaud et al., 2002). It should be noted that the two traffic flow variables of the FF(2) model are the same of the two first traffic flow variables of the FF(6) model and therefore, the elasticities are comparable. Hereupon, a significant changed is observed on

Table 3 Measures of effectiveness for FF(1), FF(2), FF(3), FF(4) and FF(6) functional forms—NB models.

Measures FF(1) FF(2) FF(3) FF(4) FF(6)
MAD 3.60 3.49 3.46 3.54 3.42
MSPE 34.08 31.35 31.57 33.80 27.89
Overall prediction accuracy 0.986 0.990 0.987 0.996 1.000

the FF(2) model variables when the cross variables are included, the FF(6) model, suggesting the importance of the cross variables to capture the interaction of the traffic conflicts. Indeed, the essential trait of the translog function is that it adds a new interpretation of the effects of the explanatory variables on accident risk because the cross-product terms may change the magnitude of those effects, depending on the traffic flows, and thus allow the logic of differential risks.

4.2. Capabilities and limitations of functional forms

Many authors have investigated functional forms to represent the relationship between accidents and traffic flow. Nonetheless, a lack of consensus exists on the best functional form for representing that relationship (Turner and Nicholson, 1997; Persaud et al., 2002; Kumara and Chin, 2003; Miaou and Lord, 2003). The FF(2) functional form is the most popular, particularly in the case of models with only traffic flow variables. The FF(6) model used a flexible functional form that allows the estimation of non-constant elasticity by the introducing of cross-product terms. In that sense, FF(6) is more flexible than FF(2) simply because the latter is a special case of the former, which includes more unknown parameters.

In terms of flexibility and interpretability of the explanatory variables' elasticities, all of the functional forms examined indicate that accident frequency increases with increasing traffic volume. However, in the case of the FF(6), predicted accident frequency increases, in general, more than with other model forms for higher values of the major and minor traffic flows. This effect is revealed by the positive values of the parameter estimate of the cross-product term [ln(F1,it)]2 and the cross-product term [ln(F1,it)] [ln(F2,it)]. This fact may explain the better fit of the FF(6) model for superior values of the AADT major shown by the CURE plot of Fig. 1.

To analyze the flexibility and interpretability of the FF(1), FF(2), FF(3), FF(4) and FF(6) functional forms estimated accident frequencies are determined and plotted as shows in Fig. 2. The graphs (a–c)

Table 2 Model estimation results for the five functional forms—NB models (t statistics in parentheses).

Parameter FF(1) FF(2) FF(3) FF(4) FF(6)
b0 1.8641
(32.991)		 1.8429
(33.175)		 1.8468
(34.027)		 1.8481
(32.195)		 1.7300
(25.972)
b1 1.1988
(9.488)		 0.4286
(3.361)		 0.6590
(10.532)		 1.1934
(9.338)		 0.6050
(3.661)
b2 – 0.8160
(6.444)		 – 0.3777
(3.781)		 0.7768
(6.254)
b3 – – – – 0.6900
(2.572)
b4 – – – – 0.7877
(2.647)
a
(overdispersion)		 0.3233
(5.984)		 0.2810
(5.790)		 0.2881
(5.887)		 0.2926
(6.131)		 0.2520
(5.782)
Number of observations
Log-likelihood at convergence
AIC
McFadden Pseudo R2
R2
PE (%)		 162
451.098
5.60614
0.1846
30.5		 162
443.266
5.52180
0.1641
39.6		 162
444.976
5.53056
0.1654
38.1		 162
444.930
5.54235
0.1749
37.1		 162
437.176
5.47131
0.1456
45.8

Fig. 1. Cumulative residual plots for the FF(1), FF(2), FF(3), FF(4) and FF(6) models (parameter F1).

of this figure compare the number of accidents estimated by the log-linear function NB models, FF(1), FF(2), FF(3) and FF(4), and the translog function NB model, FF(6) for three minor traffic flow values (a minimum, mean and maximum, respectively), and the graphs (d–f) compare the same but for three major traffic flow values (a minimum, mean and maximum, respectively).

These graphs of Fig. 2 clearly demonstrate that the translog function behaves differently from the log-linear functions, especially at the boundary values (minimum and maximum values). This behavior is evident in graphs (c and d), i.e., these graphs show that setting F2 at its maximum, accidents increase with the increase of F1, and setting F1 at its minimum, accidents decrease with the increase of F2, respectively. In the former cases, the number of accidents estimated by FF(6) is much higher than that estimated by the other functional forms, suggesting that lower differences between the major and minor traffic flows (F2/F1 approaching 1) produce a high number of conflicts in the intersection that may be reflected in a higher accident risk. In the latter case, the number of accidents estimated by FF(6) is much higher than that estimated by the other functional forms, almost without variation of the accident risk when the values of F2 increase probably because of the few number of observations in this situation. Here it is worth mentioned the fact that when the F2/F1 approaching 0 the accident risk is high by the FF(6) model in contrast to the other models, which show an accident risk near zero. In addition, the graph (f) shows a faster accident risk increase when the F1 is maximum and F2 increases. Interestingly, the graph (a) presents a marked variation of the accident risk due to F1 variation when setting F2 at its minimum value. In fact, when the F1 range values are inferior to 10,000 approximately, the estimated accident frequency by the FF(6) model is very high compared to the other estimated model values, but when the F1 value is higher than 10,000 approximately, a marked decrease is observed approaching the estimated accident frequency to zero. In contrast, the estimated accident frequency behavior of the other four models are very similar with particular differences at the boundary AADT values in the FF(4) model. This analysis suggests that the translog function may provide proper boundary values, which are capable to capture the accident risk complexity of four-legged signalized intersections.

In sum, comparing the five functional forms in terms of interpretability, the functional form of FF(6) model may provides a richer interpretation of the data than the other forms because it differentiates risk depending on the relative magnitudes of the major and minor traffic flows. For instance, when a vehicle is entering an intersection, it is exposed to accident risk involving the vehicle itself and vehicles approach by the same leg; however, in addition, its risk is influenced by the number of vehicles turning to the right or to the left, which may be captured by the cross-product terms.

Note that all of these model forms, with the exception of FF(1), suffer from the common limitation of restricting the minor traffic flow (F2) to nonzero values.

Fig. 2. Estimated number of accidents per year according to the FF(1), FF(2), FF(3), FF(4) and FF(6) models for four-legged signalized intersections: (a) F2 equals to minimum value, (b) F2 equals to mean value, (c) F2 equals to maximum value, (d) F1 equals a minimum value, (e) F1 equals a mean value, and (f) F1 equals a maximum value.

5. Functional form and data heterogeneity—Panel data models

In order to minimize the effect of unobserved heterogeneity from potentially omitted variables, panel models are used to capture unobserved intersection-specific effects considering the five functional forms previously described. The main objective of this application is to ensure that the previous results and analysis are not affected by estimation bias and therefore leading to an incorrect inference. In fact, constraining the parameters to be constant when they actually vary across observations could lead to inconsistent and bias parameter estimates (Washington et al., 2003). Given the potential heterogeneity in accident-frequency data, a panel count data model such as random parameters or random effects models may be appropriate.

Therefore, based on previous works (Anastasopoulos and Mannering, 2009; El-Basyouny and Sayed, 2009; Mitra and Washington, 2012), Poisson and NB random parameters and random effects models are applied to account for individual heterogeneity. Although NB models are usually more suitable because data are often over dispersed (i.e. variance > mean), in the presence of a well-defined mean function the extra-variance structure of NB generally becomes insignificant (Mitra and Washington, 2007). Therefore, the panel data models are applied using both hypothesis, namely the Poisson and NB distributions. In addition, because random effects model is a special case of random parameter model where the intercept is only modeled as random parameter as opposed to all parameters in mean function, the random parameters models are first applied. For the functional form of the random parameters density functions the several hypothesis, which are normal, uniform and triangular distributions, are also considered. Because maximum likelihood estimation of the random parameters Poisson and NB models is computationally cumbersome, a simulation-based maximum likelihood method is used assuming the Halton draws approach (please see Greene (2008) and Anastasopoulos and Mannering (2009) for further details on panel count data models).

A random parameter model is chosen when the standard deviation of the parameter density is statistically significant. If its estimated standard deviation is not statistically different from 0, the parameter is fixed across the population of intersections. In this study, the estimation results indicate that for all function forms analyzed the standard deviations of the parameter's distribution are not statistically significant different from 0 with the exception of the intercept. In addition, the overdispersion parameters of NB distribution for all models are not statistically significant, suggesting, in accordance to the aforementioned, that the NB structure be-

Table 4 Model estimation results for the five functional forms—random effects Poisson models (t statistics in parentheses).

Parameter FF(1) FF(2) FF(3) FF(4) FF(6)
b0 1.6998 1.6992 1.6989 1.7000 1.6060
(45.417) (45.529) (45.563) (45.332) (36.617)
Standard deviation of parameter distribution 0.5693 0.5221 0.5303 0.5364 0.5114
(19.347) (17.917) (18.556) (18.758) (18.613)
b1 1.2589 0.4664 0.6679 1.2128 0.6572
(13.671) (5.419) (14.983) (13.239) (6.944)
b2 – 0.7890 – 0.3170 0.7516
(11.287) (5.833) (10.756)
b3 – – – – 0.7037
(4.666)
b4 – – – – 0.7744
(5.004)
Number of observations 162 162 162 162 162
Log-likelihood at convergence 445.3639 438.9492 440.0952 440.8071 433.9064
AIC 5.53536 5.46851 5.47031 5.49145 5.43094
McFadden Pseudo R2 0.8024 0.8032 0.8030 0.8032 0.8038

comes insignificant. Therefore, the random effects Poisson structure is presented in this study for all models. Also, the normal distribution was found to provide the best statistical fit. In this sense, Table 4 shows the results of the best fit models for all functional forms that are random effects Poisson models with normal distribution for the density function of the intercept parameter. Simulation-based maximum likelihood with 200 Halton draws is used.

Comparing Table 2 with Table 4, the results show that the loglikelihood at convergence of random effects Poisson models for all functional forms as well as the Mc Fadden Pseudo R2 are better as expected whereas these models allow to capture the data heterogeneity improving the mean function. In addition, the parameters estimated by the random effects models of the explanatory variables of the five functional forms are very similar to those of the NB models including the mean value of the parameters estimation of the intercept.

Note that the purpose of using random effects models is to ensure that the potential omitted variables in the intersection models do not affect the previous models assessed and comparison. Thus, the random effects models do not provide additional information about the interpretability and flexibility of the five functional forms.

6. Summary and conclusions

Building on the results of previous study by Couto and Ferreira (2011) on urban road segments, data from the city of Fortaleza, Brazil are used to investigate the ability of the translog function to model the relationship between road accidents and traffic flows at four-legged signalized intersections, in comparison to other functional forms that have been applied or proposed for this purpose by other authors. The translog function can generalize NB and Poisson distributions through the estimation of quadratic cross-product terms, thereby allowing elasticity to vary with the values of other variables. The objective of this study is to analyze if this translog function characteristic can represent the traffic flow movements/conflict existent at intersections that may not be very well represented by the commonly used functions, which are based on a constant elasticity.

We have considered two types of models, namely, count data models, and panel count data models, with NB distribution and Poisson distribution, respectively. The purpose of considering the panel data models is to analyze if the exclusion of variables in intersection sample cause omitted variable bias in model estimation compromising the results inference by the count data models. Given these caveats, the following conclusions are offered:

  • 1. In the case of count data models with NB distributions, the GOF and CURE analysis results indicate that the translog model performs better than the traditional log-linear functions.
  • 2. The effectiveness measures MAD and MSPE, and overall prediction accuracy reveal that the translog model outperforms the traditional log-linear functions.
  • 3. At the boundary values of the minor and major traffic flows, the configurations of the lines plotted for estimated accident frequency of each functional form exhibit differences, particularly in the case of the translog model. This result was expected due to the presence of the cross product terms that allows free variation of the elasticity, thereby allowing the elasticities of the major and minor traffic flow to vary according to the magnitudes of the major and minor traffic flow.
  • 4. The results obtained with the random effects models for the five functional forms are consistent with those obtained with the count data models suggesting that the previous analysis is not affected by potential omitted variables bias. Note that with the random effects model the overdispersion parameter for all models are statistically insignificant, suggesting a well-defined mean function, and thus a Poisson structure was used.

Although the purpose of this study is not to develop a prediction model, the two firsts conclusions are provided to ensure the performance capacity of the translog function. The main characteristic of this function, which may enhance the traffic flow-accident relationship, is the flexibility of the explanatory variables' elasticities that is well demonstrated by the graphs for the boundary values as outlined above in point 3. These findings support the idea that the major and minor traffic flows behave differently when the ratio of the minor approach volume to the major approach volume approaching 0 or 1. In fact, when the difference between major and minor traffic volume increases or decreases, it is expected that the accident risk changes significantly. Therefore, the translog function provides a richer interpretation of the data than the traditional log-linear functions because the flexible elasticity afforded by translog function allows the introduction of the logic of differential risks, i.e., the exposure to risk involved with both approaches.

Finally, it is hoped that this study will contribute to more reliable modeling of the relationship between road accidents and traffic flows at four-legged signalized intersections and thereby improve the accuracy of estimation of accident risks at urban intersections.

Nonetheless, additional research can be performed to examine the abilities of the translog function by extending the analysis to multiple data sets for both urban and rural intersections and applying the translog function in conjunction with other statistical techniques such as varying parameter dispersion and spatial modeling.

Acknowledgements

The authors wish to acknowledge the help of Prof. Flávio Cunto as well as the Autarquia Municipal de Trânsito de Fortaleza, Brazil for providing the valuable accident and traffic data used in this study.

References