category: literaturenote citekey: coutonotemodelingroadaccident2011 title: "A note on modeling road accident frequency: A flexible elasticity model" authors: "Couto, António; Ferreira, Sara" year: 2011 date: 2011-00-00 2011 doi: 10.1016/j.aap.2011.05.033 publication: "Accident Analysis & Prevention" url: "https://doi.org/10.1016/j.aap.2011.05.033" zotero_key: F4KGU2P8 zotero_storage: DHB3GHBT collections: transpordi_planeerimine folder: 001_artiklid/Road Safety Indicators firstAuthor: "Couto, António"

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Accident Analysis and Prevention

j ou rnal home page: www.elsevier.com/locate/aap

A note on modeling road accident frequency: A flexible elasticity model

António Couto∗, Sara Ferreira

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

a r t i c l e i n f o

Article history: Received 24 January 2011 Received in revised form 24 May 2011 Accepted 25 May 2011

Keywords: Translog function Negative binomial Road accident models Regression models

a b s t r a c t

Count data models and their variants have been widely applied in accident modeling. The traditional log-linear function is used to represent the relationship between explanatory variables and the dependent variable (accident frequency). However, this function assumes constant elasticity for the estimation parameters, which is a limitation in the analysis of the effects of explanatory variables on accident risk. Although interaction effects between explanatory variables have been studied in the road safety context (where they are normally assessed by logistic regression), no one has yet examined the possibility of using a flexible function form allowing non-constant elasticity values. This paper seeks to explore the use of the translog function usually used in the economics context to allow the elasticity to vary with the values of other explanatory variables. Therefore, the objective of this study was to evaluate the application of the translog function to accident modeling and to compare the results with those of the traditional log-linear function negative binomial (NB) model. The results show that, in terms of goodness-of-fit statistics and residual analysis, the NB model with the translog function performs better than the traditional NB model. Additional evaluations in terms of predictive performance, hotspot identification and uncertainty associated with the estimated values were taken into account. Although this study is exploratory in nature, it suggests that the translog function has considerable potential for modeling accident observations. It is hoped that this novel accident modeling methodology will open the door to the reliable interpretation and evaluation of the influence of explanatory variables on accident frequency.

1. Introduction

In recent decades, there has been considerable research conducted on the development of accident prediction models, usually using generalized linear models (GLMs) to describe the influence of explanatory variables on road accidents (Fridstrom et al., 1995; Poch and Mannering, 1996; Sawalha and Sayed, 2001; Greibe, 2003; Lord and Persaud, 2004). To better understand the factors determining accidentfrequencies, additionaltechniques have been applied recently to the modeling of accidents, such as randomparameters count models. These particular models account for the presence of heterogeneity across observations by allowing some or all parameters to vary according to a random distribution (Anastasopoulos and Mannering, 2009). Other studies using a modeling framework, such as zero-inflated negative binomial models (Carson and Mannering, 2001; Lee and Mannering, 2002; Kumara and Chin, 2003), Conway–Maxwell–Poisson generalized linear models (Lord et al., 2008) and generalized negative binomial models (Mitra and Washington, 2007; Lord and Park, 2008), have focused on the performance of the models.

The GLM usually used to model accidents applies a logarithmic transformation to linearize the multiplicative equation and can be referred to as a log-linear function model. As is well known, these log-linear function models assume constant elasticity for the estimation parameters, and this can be a limitation to represent the relationships between explanatory variables and accident frequency. Moreover, why has traffic flow elasticity been assumed to be the same for both long and short segments? Additionally, why must the effect on accident risk of a percentage increase in traffic flow be the same in contexts of both high and low traffic flows?

Infact, studies (Lord, 2000; LordandPark, 2008;Hadayeghi et al., 2010) that have applied the CURE method proposed by Hauer and Bamfo (1997)to analyze the cumulative residuals for each explanatory variable have revealed that for high values of traffic flow the model did not perform well (i.e., the cumulative residual curves were larger than the +2.0 boundary and/or did not end near the zero-residual line). Moreover, Lord (2000) suggested that separate models could be developed when the cumulative residuals indicate a change from a possible underestimation of accidents to an overestimation of accidents. This solution assumed different elasticities for the traffic volume. Nonetheless, if the model has more than one variable, then this is not an achievable solution.

Although interaction effects between some explanatory variables have been used in many road safety studies (Yannis et al., 2005; Hill and Boyle, 2006; Liu, 2007; Sze and Wong, 2007; Rhodes

^∗ Corresponding author. Tel.: +351 225 082 110. E-mail addresses: fcouto@fe.up.pt (A. Couto), sara@fe.up.pt (S. Ferreira).

and Pivik, 2011), the purpose was only to analyze the difference in risk for two or more populations as they move from one condition to another. For instance, in the case of segment length, it is understandable that this factor can influence the impact of traffic flow on accident frequency. Indeed, traffic flow can have a different impact on accident frequency depending on whether the segment is longer or shorter. This interaction can be modeled by a flexible functional form instead of the constant elasticity model usually used to represent the effects of explanatory variables. Thus, even though the interaction effects of explanatory variables were not the main focus of this study, the flexible function form takes them into account. Because the main objective of this study is to assess the possibility that explanatory variables have non-constant elasticity, it is expected that the assumption adopted in this study will outperform the mathematical limitations of some popular function forms because the flexible function can allow for the free variation of elasticity and thus provides a well-structured function and better fitting performance.

Flexible function forms are frequently used in econometrics (Caves et al., 1980; Diewert and Wales, 1987; Oum and Tretheway, 1989; Couto and Graham, 2009) because they allow analysts to model second-order effects, such as elasticities of substitution, which are functions of the second derivatives of production, cost or utility functions (Greene, 2008). The most popular flexible functional form is the translog function originated in Christensen et al. (1973). This function has been extensively applied in transportation research, especially in logistics and policy studies based on transportation costs (Karlaftis and Mccarthy, 2002; Link, 2006; Satar and Peoples, 2010).

The translog function form can generalize negative binomial (NB) and Poisson models through the estimation of quadratic "cross variables," thereby allowing the elasticity to vary with the values of other explanatory variables.

By using a flexible function, this paper provides a new interpretation of the relationship between explanatory variables and the dependent variable, and it tests the potential of this function form by applying it to accident modeling. To accomplish this objective, an NB model with a traditional log-linear functional form and an NB model with a translog function were developed and compared using a segment dataset. This dataset contains accident data collected at segments in Porto, Portugal, over a 5-year period. The models developed in this study consider two of the most widely used explanatory variables besides traffic flow (often referred to as general annual average daily traffic or AADT): segment length and the number of minor intersections per segment length. These explanatory variables are usually used to homogenize the segment and are easy to determine at all road safety study stages. The study will show that the model with a translog function performs better than the traditional log-linear function, producing statistically significant "cross variable" parameters.

The next section presents a brief background related to the traditional log-linear functional form and the translog function. The third section describes the methodology used for estimating and comparing the NB model with and withoutthe second-order terms. The data and estimation results are summarized in the fourth and fifth sections, respectively. The last section presents a summary and conclusions of the analysis performed in this work.

2. Background

Count-data modeling techniques are commonly used for accident frequency analysis because the number of accidents on roadway segments per unit oftime is a non-negative integer. Count data are generally modeled with a Poisson regression. One requirement of the Poisson distribution is that the mean of the count process equals its variance.When the variance is significantly larger than the mean, the data are said to be overdispersed. Accident data have been shown to exhibit over-dispersion (Sawalha and Sayed, 2001; Lord et al., 2005; Caliendo et al., 2007). In many cases, overdispersion count data are successfully modeled using an NB model (Washington et al., 2003).

The probability density function of the NB distribution can be defined as:

$$P(Y = y_i | \mu_i, \alpha) = \frac{\Gamma(y_i + (1/\alpha))}{\Gamma(1/\alpha) \cdot y_i!} \cdot \left(\frac{1/\alpha}{(1/\alpha) + \mu_i}\right)^{1/\alpha} \times \left(\frac{\mu_i}{(1/\alpha) + \mu_i}\right)^{y_i}$$ (1)

where ˛ is the dispersion parameter (when ˛⁼ 1/ is equal to zero that is →∞, the NB distribution reverts back to the Poisson distribution) and (.) is ^a gamma function. Thus, the mean ⁱ ⁼ ^E{Y} can be smaller than the variance Var{Y} ⁼ ⁱ ⁺˛ⁱ 2.

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

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

where Xi is ^a vector of explanatory variables and ^ˇ is ^a vector of estimable parameters. In this formulation, the expected number of events per period is given by ^E[yi] ⁼ ⁱ ⁼ exp(ˇXi). The estimated parameters ˇ are used to make inferences about the explanatory variables thought to impact 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, and they can be interpreted as the effect of a 10% change in the variable on the expected frequency ⁱ (Washington et al., 2003). The elasticity of frequency ⁱ is defined as

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

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

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. Hence, a flexible function form has been considered to model second-order effects.

The translog function originated in Christensen et al. (1973) has become enormously popular in the economics context probably because itis easier to estimate and interpret as itis a quadratic function with all arguments in natural logarithms (Oum and Waters, 1996). As described in Greene (2008), one way to derive it is first to write ^y ⁼ ^g(x1, . . ., xk) of an unknown function form and then, ln ^y ⁼ ln ^g(x1, . . ., xk) ⁼ ^f(x1, . . ., xk). Hence, by the trivial transformation xk = exp(ln xk), one interprets the function as a function of the logarithms of ^x's. Thus, ln ^y ⁼ ^f(ln ^x1, . . ., ln xk). Now, expand this function in a second-order Taylor series around the point x = [1,1, . . ., 1] so that at the expansion point, the log of each variable is ^a convenient zero, then:

$$\ln y = f(0) + \sum_{k=1}^{K} [\partial f(.)/\partial \ln xk]{\ln x = 0} \ln x_k$$

$$+ \frac{1}{2} \sum{k=1}^{K} \sum{l=1}^{K} [\partial^2 f(.)/\partial \ln x_k \partial \ln xl]{\ln x = 0} \ln x_k \ln x_l$$ (3)

Because the function and its derivates evaluated at the fixed value 0 are constants, one can interpret them as the parameters and they can be expressed as 1:

$$\ln y = \beta0 + \sum{k=1}^K \beta_k \ln xk + \frac{1}{2} \sum{k=1}^K \sum{l=1}^K \gamma{kl} \ln x_k \ln x_l$$ (4)

It is interesting to note that if kl ⁼ 0, Eq. (4) becomes ^a function of the commonly used log-linear model. The second-order terms allow for possible cross-relationships among the variables and for possible non-linearities in the relationships (Oum and Waters, 1996).

The explanatory variable elasticity can be found by taking the differential of the function:

$$\frac{\partial \ln y}{\partial \ln x_k} = \betak + \sum{l=1}^K \gamma_{kl} \ln x_l \tag{5}$$

From this, it is clear that, in general, the elasticity varies with the proportions of the explanatory variables and with their individual levels.

The next section describes the methodology used to estimate and compare the NB model with a traditional log-linear function and the NB model with a translog function.

3. Methodology

Many different functional forms have been proposed to represent the relationship between accident frequency and explanatory variables. Two different types of functional form usually used by transportation safety analysts can be distinguished. One type is models that include several covariates that are related to the functional and geometrical characteristics of road networks (Poch and Mannering, 1996; Sawalha and Sayed, 2001; Greibe, 2003). Such models can be used to analyze the factors affecting road safety, their influence and how they are related to the dependent variable. The other type of functional form (often referred to as AADT models) uses only a traffic flow variable. Authors (Lord et al., 2008) have mentioned that these types of models are preferable to the former because they can be easily re-calibrated when they are applied in a jurisdiction other than the one in which it was developed.

Because the purpose of the study is not to predict the number of accidents but rather to provide a methodological approach to gain new insights into the ways that explanatory variables influence accident frequency, a functional form was considered with more than the traffic flow variable. In addition to the traffic flow variable, explanatory variables usually used to characterize the segment and that have been reported by researchers to be significantly important in modeling accidents were included (Fridstrom et al., 1995; Mountain and Fawaz, 1996; OCDE, 1997; Lord, 2006). Therefore, variables describing the number of minor intersections and segment lengths have been included in this segments model. These explanatory variables are easy to determine and consequently can be used at all road safety study stages.

To select the proper relationship between the dependent variable and explanatory variables, a method proposed by Hauer and Bamfo (1997) was applied. Basically, the Integrate-Differentiate method consists of drawing an Empirical Integral Function for each explanatory variable and then comparing the shape with the pre-established graphs of well-known functions (power, gamma, exponential or polynomial functions). The function with the most similar shape is selected to represent the relationship between the dependent variable and the explanatory variable being investigated. By applying this method to the explanatory variables being used in this study, it was found that the relationships between accident frequency and both AADT and segment length could be described by the power function. In the case of the minor intersection density variable, the exponential function was found to be more suitable to describe the aforementioned relationship.

A time trend variable has also been included in this study to reflect potential changes in the overall accident level over time. A number of studies have included a time trend variable that allows for changes in terms of risk over time (Fridstrom et al., 1995; Mountain et al., 1998). It has been previously demonstrated that this variable has a negative effect on the frequency of accidents, thus leading to overall improvement due to such factors as safety education, enforcement and vehicles.

Thus, the NB model with a traditional log-linear function takes the form:

$$\ln \mu_{it} = \beta_0 + \beta1 \ln(F{it}) + \beta_2 \ln(Leng_i) + \beta_3 IntDen_i + \ln(\gamma) TimeT_t$$ (6)

where it is the mean number of accidents in segment ⁱ at time period t, Fit is the annual average daily traffic in both directions (segment i; time period t), Lengi represents the length of the segment i, TimeTt is the time trend variable, IntDensi is the minor intersection density (segment ⁱ) and ^ˇ^k (^k ⁼ 0–3) and are the estimated parameters.

The translog functional form is expressed as:

$$\ln \mu_{it} = \beta_0 + \beta1 \ln(F{it}) + \beta_2 \ln(Leng_i) + \beta_3 IntDen_i + \ln(\gamma) TimeT_t$$

$$+ \beta4 [\ln(F{it})]^2 + \beta_5 [\ln(Leng_i)]^2 + \beta_6 (IntDen_i)^2 + \beta_7$$

$$\times \ln(F_{it}) \ln(Leng_i) + \beta8 \ln(F{it}) IntDen_i$$

$$+ \beta_9 \ln(Leng_i) IntDen_i$$ (7)

where it is the mean number of accidents per year, the explanatory variables are the same as in Eq. (6) and ^ˇ^k (^k ⁼ 0–9) and are the estimated parameters. Note that the time trend variable does not take the form of a "cross variable," hence no interaction is expected between this variable and the other explanatory variables on accident frequency.

To compare the two types of model applied in this study, several methods were used to estimate their goodness-of-fit (GOF). Four different GOF statisticaltests –Akaike's Information Criterion (AIC), Freeman–Tuckey ^R² (R² FT), Elvik index (R² PE) and McFadden Pseudo R² as described in Bozdogan (1987), Fridstrom et al. (1995) and Greene (2008) were employed.

The model with the lowest AIC is considered the model with the best statistical fit. On the other hand, the model with the largest McFadden Pseudo R² value indicates the best fitting model. The Elvik index and Freeman–Tuckey R² are two measures of GOF that describe how much "variation" in the dependent variable is "explained" by the model estimate as compared to the total variation present in the data set.

A GOF evaluation based on residual analysis was also used. This method is known as the CURE method and it was initially proposed by Hauer and Bamfo (1997). It can be used to evaluate the fit of models by examining the cumulative residuals for each explanatory variable separately. The cumulative residuals are plotted in increasing order and the closer the curve oscillates around the

¹ Note that kl ⁼ lk.

Fig. 1. Accident number distribution.

Table 1 Summary statistics for the Porto data (5-year).

Variablea	Min.	Max.	Mean	S.D.
Dependent variable (accident number)	0	27	2.85	3.90
AADT	142.37	64067.80	15240.38	11699.49
Segment length (in m)	20.71	3342.78	313.18	352.56
Number of minor intersections per kilometer	0.00	32.04	4.57	5.18

^a ¹⁹⁸⁰ observations (396 segments ^× ⁵ years).

zero-residual line, the better the model fits the data (the reader is referred to Hauer and Bamfo (1997) for a more detailed description ofthis method). The CURE plot reveals how wellthe functionalform fits the data with respect to each individual explanatory variable.

The next section describes the characteristics of the dataset.

4. Data

To illustrate the application of the translog function to model accidents, data from Porto, Portugal, were used. These data consist of accident data from urban segments classified by local and principal distributor roads collected over a 5-year period (January 1st 2001 to December 31st 2005). The accident data were obtained from the official Portuguese Police Security database covering all police-recorded accidents with local information (injury accidents and property damage only accidents).Allthe accidents were related to a specific locality by applying a Geographic Information System (GIS)tool. The data consist of 5650 police records of accidents, 1183 of which were personal injury accidents and 4467 were property damage only accidents, and they are related to 396 segments resulting from a road network characterization as a series of nodes and segments from a previous work (Ferreira, 2010). Fig. 1 shows the distribution of these accident numbers.

The daily traffic (defined as the Annual Average Daily Traffic or AADT) was estimated by the Porto "SATURN" traffic model. This model was developed in 2005 for the peak hourly flow (PHF) of the Porto road network and involved all the recommend development stages (specification; coding; error checking; calibration; validation) (Tavares, 2003). The PHF was expanded to AADT every five years by applying zone factors that relate the PHF with the daily traffic flow calculated using data provided by permanent counting stations located throughout the principal city zones belonging to the Urban Traffic Center.

In addition, GIS was used to find geometric characteristics of the segments, namely the number of minor intersections and the lengths. These data were processed in a doctoral study (Ferreira, 2010). Table 1 presents the summary statistics for the dataset.

5. Estimation results

Given all the "cross variables" required by the translog function model, the parameters were estimated. However, the results

Table 2 Model estimation results for the translog function and log-linear function negative binomial models (t-statistics in parentheses).

| Variable | Negative binomial parameter estimates | | | |------------------------------------|---------------------------------------|-------------------------|--| | | Log-linear function model | Translog function model | | | | 0.7350 (14.141) | 0.6766 (11.734) | | | ˇ0 | 0.3701 (13.623) | 0.4249 (15.037) | | | ˇ1 | 0.8910 (31.28) | 1.0026 (28.109) | | | ˇ2 | 0.0380 (8.738) | 0.0233 (3.259) | | | ˇ3
ln() | −0.0583 (−3.674) | −0.0572 (−3.658) | | | | – | −0.0690 (−3.311) | | | ˇ5 | – | 0.0018 (2.777) | | | ˇ6 | – | −0.0780 (−2.851) | | | ˇ7 | – | 0.0434 (5.672) | | | ˇ9
˛ (overdispersion parameter) | 0.4849 (15.263) | 0.4449 (14.687) | | | AIC | 3.73084 | 3.70226 | | | McFadden Pseudo R2 | 0.1096 | 0.0984 | | | R2 | 63.7% | 67.9% | | | FT
R2
PE | 69.3% | 71.8% | |

Fig. 2. Cumulative residual plots for the log-linear function and translog function NB models (parameter F).

showed that two "cross variables" were not statistically significant. These variables correspond to [ln(Fit)]2 and ln(Fit)(IntDeni). In fact, the correlation matrix analysis reveals a strong possibility of correlation between [ln(Fit)]2 and ln(Fit) and between ln(Fit)(IntDeni) and (IntDeni). Table 2 presents the estimationresults for both the traditional log-linear function NB model and the translog function NB model excluding the insignificant variables.2,3 As shown in Table 2, excluding the McFadden Pseudo R² values, the translog model provides the best fit according to the three different test statistics (i.e., AIC, R² FT, R² PE), although they are all very similar. The ^R² FT and ^R² PE measures indicate that the translog model explains 68% and 72% of the systematic component, respectively, compared with 64% and 69% explained by the traditional NB model. These findings may be an outgrowth of the traditional NB model's restriction that the estimated elasticity is constant.

In fact, the residual analysis stresses these conclusions. Comparing the CURE plotted for the two models (refer to Figs. 2–4), one can conclude that the translog function NB model, in general, fits the data better with respect to each individual explanatory variable than does the log-linear function NB model. The exception

Fig. 3. Cumulative residual plots for the log-linear function and translog function NB models (parameter Leng).

is Fig. 3b, because the cumulative residual line is slightly larger than the ⁻2.0 boundary for segment length values above approximately 1200 m.

An important issue in developing and using regression models is parameter interpretation. To analyze and to compare the influence of each explanatory variable on accident frequency in each of the two models, the elasticity values were analyzed. With regard

Fig. 4. Cumulative residual plots for the log-linear function and translog function NB models (parameter IntDen).

² The explanatory variables of the log-linear function NB model and the firstorder variables of the translog function NB model were previously "normalized" by dividing every variable observation by its respective sample mean to facilitate model comparisons and interpretation of the results.

³ Despite the panel nature of the data, measures to deal with panel data were not used because information on a range of exogenous variables has been incorporated that characterizes segment effects. These effects are quasi-invariant over time and the use of firm dummy variables was found to produce severe multicollinearity problems. Also, tests for possible temporal correlation and heteroskedasticity problems were conducted using a series of likelihood ratio tests as described by Poch and Mannering (1996), revealing that these issues do not significantly affect the resulting estimates.

to the elasticity results from the log-linear function NB model, the values are consistent with the findings of previous research (see for example Mountain and Fawaz, 1996; Mountain et al., 1998; Greibe, 2003). Nevertheless,the AADT variable has low elasticity compared with similar studies (accidents increase by about 4% when the traffic volume increases by 10%). This resultindicates reducedinfluence of this variable on accident frequency. Because the segment length parameter is not significantly different from 1, the accident frequency is almost proportional to road length, as was expected. Consistent with expectations, the model also reveals that accident frequency increases when the minor intersection density increases (Eit xik ⁼ ⁰.04(IntDeni)), although the impact on accident frequency is low (ˇ³ ⁼ 0.04). Concerning the effects of local and/or national road safety policies and programs on accident frequency, parameter was estimated with statistical significance to be 0.95, implying that for the period studied the accident risk declined annually by an average of 5%, thereby accumulating to a reduction of 23% over the whole period.

Comparing the parameters estimated by the traditional NB model with the translog function NB model (see Table 2), it is clear that the effects of the explanatory variables on accident frequency are similar for both models, although some improvement can be observed in the case of the values estimated by the latter model. Indeed, the elasticities of the AADT and the segment length variables have increased slightly to values more similar to those of past research.

Regardless of the better fit of the translog function NB model, its essential trait is that it adds a new approach to defining the mean function structure of an NB model in terms of the flexibility and interpretability of the explanatory variables' elasticities. In fact, a closer look atthe parameters estimated by the translog function NB model presented in Table 2 reveals that, e.g., with increasing traffic volume, the accident frequency also increases, but when the segment length is higher, the magnitude of this effect is lower ceteris paribus To better illustrate the differences among these models, estimated accident frequencies from the two models are shown in Figs. 5 and 6. Fig. 5 compares the number of accidents estimated by the log-linear function NB model and the translog function NB model for three segment length values and considering a minor intersection density equal to the mean sample value. This figure shows that the estimated values are slightly different, with the translog function NB output being lower than the traditional NB output. The same can be illustrated with the minor intersection density variable interaction with segment length (see Fig. 6) and considering the AADT mean sample value. In this case, the numbers of accidents estimated by the two models are very distinct, especially when IntDen is equal to 10 minor intersections per kilometer, with the translog function NB output being lower than the log-linear function NB output. For IntDen equal to 0, the models' estimated accident frequency outputs are reversed, i.e., the loglinear function NB output is higher than the translog function NB output.

These figures clearly demonstrate that the translog function is particularly distinct from the log-linear function at the boundary values. This fact suggests that data heterogeneity can be considered in the translog function. As such, ignoring this variation can undermine the goodness of any estimate of individual sites as stated by Miaou and Lord (2003). Indeed, proper boundary values can be particularly important to modeling data with a large number of zeros (low samplemeanvalues) or withahighernumber of accidents. The latter can be referred to as a case of hotspot identification analysis. Due to the factthatthe residuals analysis ofthe translog function NB model showed lower values when the observations include a high number of accidents, this function is likely to be outperformed in the prediction of accidents at sites with a high probability of many accidents occurring.

Fig. 5. Estimated values for the Porto dataset (with IntDen = 4.57 mean sample value). Log-linear function versus translog function NB models.

Although the study was not motivated by hotspot identification methods, a comparative analysis between these two functions was considered. Using the model estimates from both functions as a criterion for the identification of potential hotspot, the top 10% of the 396 segments (considering a 5 year average) were selected. Then, the potential hotspots were compared with the hotspots identified through observed accident frequencies. As expected, the translog function NB model "correctly" identified more hotspots than the log-linear function NB (8% more sites). Moreover, following similar procedures as those applied in Geedipally and Lord (2010), outcome classification was used to compare the performance of the two models, using as a threshold value the mean of all the sites in the Porto data. These comparative results are presented in Table 3. As expected, the translog function NB model "correctly" identified more hotspots than the log-linear function NB. More-

Table 3 Evaluation of model performance at identifying the "true" hotspot.

Number of sites	Log	Translog
Number of false positives (Type I errors)a	92	89
Number of false negatives (Type II errors)a	66	58
Number of sites correctly classified as non-hotspots	155	158
Number of sites correctly classified as hotspots	83	91

^a As defined in Geedipally and Lord (2010).

Fig. 6. Estimated values for the Porto dataset (with F = 15240.38 mean sample value). Log-linear function versus translog function NB models.

over, as a measure of the capacity of the method to detect "true" hotspots, the proportion of sites that were correctly detected by the translog function as hotspots was 61%, whereas the log-linear function detected 56%, indicating that the former performs slightly better. Furthermore, the smaller value of the proportion of the total number of errors (Type I and II) to the number of sites under analysis obtained by the translog function (37% compared to 40% for the log-linear function) confirms the previous results.

In order to stress the potential of the translog function form, it wasdecidedto also evaluate itspredictiveperformance considering the methodology described in Lord et al. (2008). Thus, two samples consisting of 80% of the original data were randomly extracted to fit the models. Then, the estimated models were applied to the remaining 20%. The mean absolute deviance (MAD) and the mean-squared predictive error (MSPE) were computed to assess the predictive performance and are presented in Table 4 (for a detailed description of this method, the readers are referred to Lord et al. (2008)). As can be seen in Table 4, the results for both models and for the full data model estimation exhibit similar characteristics, although with slight differences in the MAD and MSPE measures suggesting that the translog function NB model performs better.

Finally, to examine the uncertainties associated with the estimated values, an empirical Bayes (EB) method was used in this study as an indirect method of evaluating uncertainty. This

Table 4 Predictive performance evaluation measures.

| Measurea | Sample 1 | | Sample 2 | | | |----------|----------|----------|----------|----------|--| | | Log | Translog | Log | Translog | | | MADFit | 2.0 | 1.9 | 1.9 | 1.8 | | | MSPEFit | 9.9 | 8.3 | 8.7 | 7.2 | | | MADPred | 1.6 | 1.5 | 2.0 | 2.0 | | | MSPEPred | 5.8 | 5.7 | 11.1 | 12.1 | |

^a For a detailed description of these measures, the readers are referred to Lord et al. (2008).

approach is based on the fact that the EB method achieves a better estimate of the expected number of accidents, whereas the present procedure is a result of a combination accident counts with accident estimates. To provide a comparative measure, the EB values were computed and then compared with the estimated model (EM) values for both models analyzed. This comparison is based on the percentage difference calculated as (EM estimate − EB estimate) × 100/EB estimate. The averages ofthis percentage difference for the log-linear function NB model and the translog function NB model were found to be 22% and 19%, respectively.Although this particular formulation provides attractive statistical properties, an extensive investigation is needed using a full Bayesian treatment as a direct method of providing exact measures of uncertainty (Mitra and Washington, 2007).

6. Summary and conclusions

This paper introduces a novel accident modeling methodology using a flexible function named the translog function that allows future research to adopt a new interpretation of the elasticity of explanatory variables.

This paper demonstrated the potential of the translog function NB model to provide new insights into the ways that explanatory variables influence accident frequency. The translog model is commonly used in econometric studies wherein it provides a flexible function form to model second-order effects, thus allowing the elasticity to vary according to the magnitude of the explanatory variable. In the context of road accident analysis, this innovative methodology can be seen as an extension of the Poisson or NB model in the case of accident modeling.

Using 5 years of road accident data from urban segments (Porto, Portugal), the translog function NB model was compared with the log-linear function NB model commonly used to analyze road accidents. The results of this study show that the translog function NB model, in general, performs better than the log-linear NB model in terms of GOF statistics and residual analysis. Also, the estimation results provide some interesting findings. For instance, the traffic volume elasticity varied (although not substantially) with the segment length, revealing variable influences of these variables on accident frequency, especially for lower segment length values. In addition, it was found that the segment length elasticity changed in conjunction with the minor intersection density, with a higher effect when the number of intersections per segmentlength is high.

These results suggest that the translog function provides proper boundary values and thus contributes a well-defined mean structure that fits the heterogeneous data usually exhibited by accident samples. A simple comparison between the two models was carried out suggesting that the translog function NB model has the potential to provide better estimates of the expected number of accidents at high-risk sites. Nonetheless, further research should be conducted to address data characterized by many zeros and to optimize the identification of hotspots.

In addition, a preliminary study was conducted on the uncertainty associated with the estimates of the model parameters. Using an EB estimation analysis, the translog function NB model appears to offer positive results. Nonetheless, for the purpose of measuring estimation uncertainty, a full Bayesian modeling framework should be developed for the translog function form in future research.

Although the analysis conducted in this research was conducted for a segment dataset, the potential of flexible function forms to structure the mean function of intersection accident models should be analyzed as an alternative to the function forms usually postulated for the interaction between traffic flows in the minor and major approaches of intersections.

Although this study is exploratory in nature, it suggests that the translog function has considerable potential for modeling accident observations. Finally, it is hoped that the new function represented by the translog model described in this paper may be adopted by other researchers, opening a door to the reliable interpretation and evaluation of the influence of explanatory variables on accident frequency.

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António Couto Assistant Professor; Ph.D., Faculty of Engineering of Porto University, Porto, Portugal, specializes in the economics of transport, focusing particularly on modeling the implications of transport provisions and accessibility for productivity and economic growth.

Sara Ferreira Assistant Professor; Ph.D., Faculty of Engineering of Porto University, Porto, Portugal, is mainly engaged in road safety analysis, namely accident modeling, hotspot analysis and mitigating measures.

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