Numerous studies have examined the impact of road characteristics on accident frequency [12] - [21]. Most studies used the traditional statistical models of Multiple Regression, Poisson, and Negative Binomial Regression [19] [22] - [30].

The problem with traditional models is that they assume that the residuals between observations are independent. Disregarding this hierarchical structure, when present, may result in models with biased estimates of parameters and biased standard errors. When working with longitudinal data (samples measured more than once over time) or grouped, this assumption of independence between variables may not make sense. There are several methodologies available to solve this problem, with perhaps the best known, in the non-Gaussian context, being the Generalized Estimating Equations (GEE) methodology. The GEE model showed better results, however, for horizontal curves, stretches in which accident causes have been poorly studied [31]. One of the principal characteristics of GEE is its ability to unify several statistical techniques that are usually studied separately. This makes it possible to increase the number of assumptions admitted and to examine more than just the linear relationships between the explanatory variables and the response. This type of model allows the potential interactions between variables to be evaluated and is capable of modeling databases with longitudinal, spatial, or multilevel structures.

Various models have outlined accidents on horizontal curves based on variables that include the length of the curve, degree of curvature, and grade. Almost all models used the traffic volume for each segment, based on estimated AADT counts. These studies indicate that a greater central angle [32] [33] a greater slope [21] [34] and greater superelevation increases accident frequency [33], while greater radius reduces it [33] [35]. Considering the same variables and the influence of geometric parameters on accident severity, studies indicate that greater slope increases severity [36] [37] while greater radius [36] and greater superelevation reduce it [38].

The inclusion of spatial relationships in a safety analysis can be an important consideration for a more accurate and comprehensive approach. The spatial relationship of a curve to adjacent curves, including distance to adjacent curves, direction of rotation of adjacent curves, radius of adjacent curves, and length of adjacent curves, as well as the vertical curvature, are also important characteristics that can influence the safety of a horizontal curve or a series of curves [3].

Based on the variables found in the literature and their influence on accident frequency, the following variables were selected a priori for this study: horizontal curvature (radius, degree of curve, deflection angle, curve length), lane width, shoulder width and type, traffic volume, and grade. Qualitative spatial variables, such as land use (rural or urban), road-track profile (flat, wavy, or mountainous), layout (straight or flat), day of the week, climatic conditions, and accident type will be used to assist in the selection of homogeneous sections.

The study methodology was developed using three principal steps: 1) construction of a database from data collection and semi-automatic extraction of highways from vector bases and/or satellite images, 2) homogeneous segmentation of highways, and 3) accident frequency modeling.

The roads and all associated information were divided into homogeneous segments in three different ways: two by the methodology proposed by HSM [39] and another based on kernel density. They are listed below:

• Segmentation method 1: Based on HSM, segments are between intersections with a minimum length of 160 m.

• Segmentation method 2: Variation of the HSM method, with 50 m before and 50 m after curves, avoiding short segments and minimizing the problem of incorrect location of accidents.

• Segmentation method 3: Division of segments based on Kernel density and all variables used in the stepwise procedure are explanatory within each segment with their original values.

The proposed model is classified as a Generalized Estimating Equations model, which can be interpreted as an extension of the Generalized Linear Models for panel data and incorporates a variety of variables in addition to just traffic volumes. The initial function was proposed by Liang and Zeger [40]:

μ i = β 0 ∗ ( β 1 X 1 i + β 2 X 2 i + ⋯ + β n X n ) + ε (1)

Em que: μ i = predicted annual rate of accidents; β 0 , β 1 , ⋯ , β n = regression parameters; X 1 i , X 2 i , ⋯ , X q i = the variables of interest; ε = specification error

The choice of method is mainly due to the possibility of combining quantitative and categorical variables, not only as dummy variables (binary - 0 or 1), but as multinomial variables (having more than two ordinal categorical variables). The dependent variable is of the count type (number of accidents that occur in a given segment) and the linking function is a negative binomial.

To adjust a generalized linear model, the vector (β) of parameter estimates was determined. These coefficients were estimated from the observed data.

In this study, the first step was to verify whether the estimated coefficients were significant, that is, whether there was a statistically significant association between the explanatory variables and the response variable. Wald’s chi-square statistical test was used to assess the adherence of the accident distribution between the actual and predicted data. The χ² calc value was obtained from experimental data, taking into account both observed and expected values.

As this is an alternative hypothesis, in which the observed accident frequencies are different from the predicted frequencies, there was a need to verify the association between groups by comparing the calculated χ² data with the tabulated χ² data. The tabulated χ² depends on the number of degrees of freedom and the level of significance adopted.

The hypothesis that the model fits the data is rejected if the p-value associated with the test statistic is less than the level of significance α. Thus, for level of significance α, a decision is made by comparing the two χ² values:

If χ² calculated ≥ χ² tabulated → the model is rejected

If χ² calculated ≤ χ² tabulated → the model is accepted

The higher the χ² value, the more significant the relationship between the dependent variable and the independent variable.

The quality-of-fit indications are based on the Wald Hypothesis Test values in the different models. The Wald test is used to test the null hypothesis that the estimated β_j parameter is equal to zero.

Two statistical elements were considered when analyzing the quality of the fit of each model generated: 1) the Quasi-likelihood Information Criterion (QIC) and 2) the accumulated residue test (CURE Plot).

The QIC is a modification of the Akaike information criterion (AIC) in the GEE procedure. The comparison of the models is done using the maximum likelihood logarithm, which is the one that best fits the observed data. The QIC is expressed by Equation (2).

QIC = 2 ∗ LIK + 2 K (2)

where: LIK = is the maximized likelihood log, k = is the number of regression coefficients, and r = number of parameters estimated for the calculation of E_i.

According to this criterion, the best model is the one having the lowest QIC value. Several other information criteria are available in the spatial statistics tools, most of which are variations of the QIC, with changes in the way they penalize parameters or observations.

The CURE method to assess the quality of the fit is based on the study of residuals, that is, the difference between the number of accidents observed in a location and the value expected for the same location in the same time period, considering that residuals assume an abnormal distribution. The CURE Plot graph is used to examine residuals after estimating the parameters of the model and assessing whether the chosen function fits each explanatory variable over the entire range of values represented. The trend of residuals with respect to AADT (or other variables) can be assessed in relation to variance. An upward or downward deviation is a sign that the model consistently predicts fewer or more accidents, respectively, than were counted. It is therefore desirable that the cumulative graph of residuals oscillate close to zero or between two additional curves formed by the acceptable limits (±2ρ*) for cumulative residuals.

To validate the model, the Root Mean Square Error (RMSE) was used. RMSE is commonly used to express the accuracy of numerical results with the advantage that it presents error values in the same dimensions as the variable analyzed.

The scope of the analysis was highway BR 232, between km 141 and km 356, latitudes 8˚02'30"S and 8˚39'27"S and longitudes 36˚11'56"W and 37˚48'57"W (Figure 1). The 255 km stretch of rural federal highway runs through the municipalities of São Caetano, Pesqueira, Arcoverde, Cruzeiro do Nordeste, and Custódia, in northeastern Brazil.

Information was obtained from the Federal Highway Patrol Database for the years between 2007 and 2016, which contains the Incident Records and Police Reports, as well as from the DNIT highway base, the OSM cartographic base, and the digital terrain model provided by the Condepe/Fidem Agency.

The AADT values considered for the years 2014, 2015, and 2016 were obtained from the National Department of Transport Infrastructure (DNIT), including both volumetric and classificatory traffic counts. For previous years (2007 to 2013), as there was no active collection point in the study area, the AADT from the ANTT Annual Report (2015) was considered, as shown in Table 1.

The lack of standardization of police reports and the lack of rigor in filling them out reduce their reliability and their usefulness for studies. An analysis therefore had to be carried out to identify any absences or inconsistencies in the information recorded in the reports. Tables that did not contain all of the necessary information, such as location, type, and accident date, were excluded from the sample.

A database was created that grouped detailed information on lane widths, shoulder conditions, road curvature, grade, and AADT on the 215 km stretch of

rural highway in Pernambuco. This was achieved using geoprocessing tools to extract relevant attributes from the road network, spatial characteristics of the surroundings, and traffic flow, which were then combined with the accident database created for the study. The accident data included in the database contained all accidents registered over a 10-year period, from January 2007 to December 2016.

Two groups of variables were considered, one related to spatial variables (group 1) and the other to roadway geometry (group 2). The spatial variables considered were: accident cause, age group, accident type, day of the week, time, layout, condition, cause 1 (with injured victims, without victims, with fatal victims, ignored), road type, land use, period of the day (full daytime, full nighttime). The second group of variables included: lane width, shoulder width and type, segment length, grade and superelevation, curve radius and curve length, including the length of the transition spiral, if any.

For segmentation 1 (Figure 2), the results were verified using a sample of homogeneous road sections, selected to have a minimum length of 160 m. According to this criterion, of the 253 straight sections identified, 200 were selected. For the curved stretches, 88 out of a total of 226 were selected, meeting the criterion of a minimum radius of 100 m. Because of this significant reduction in the number of curves, tests were also carried out using a minimum radius of 50 m, totaling 115 stretches.

For segmentation 2, a variation of the homogeneous HSM segmentation of the HSM, the homogeneous stretches contiguous to the curves were excluded to a distance of at least 50 meters from the curve start and end points (Figure 3). This partial exclusion of the sections contiguous to the curves was done to isolate the influences of the curves when considering the accident history for the calibration procedure. It is necessary to differentiate accidents into those occurring on curves and on straight sections, however, this differentiation is performed in an approximate manner based on the km where the accident occurred. The accidents that occurred within these areas were added to their respective curved sections.

For homogeneous segmentation considering spatial criteria, grouping was performed by sub-sections according to the road surface type, land use, terrain type, roadway layout, and grade. Through the Query Builder tool, a consultation

was made to identify the accidents associated with each group and where they occurred, over the entire period of analysis.

At first, to ensure that segmentation was carried out according to the spatial characteristics without considering accident frequency, a Risk Index was created. According to the characteristics most often presented in the literature and their respective ranges, values were established ranging from 1 to 3, where 1 is low risk, 2 is medium risk, and 3 is high risk for accidents (Table 2). Table 3 shows the estimated risk index values for the category variables Day of the Week and Age Group.

The risk index ranges from 3 to 8, with 3 having the lowest risk and 8 the highest. For example, a stretch 1880 m long with an AADT of 4800 vpd on a downward slope has a risk index of 5, whereas a stretch with an AADT of 4800 vpd on a downward slope with a 500 m radius curve has a risk index of 7, according to the composition presented in Table 4.

The Kernel estimating technique was applied, based on the index, in order to identify the areas with similar spatial characteristics, as shown in Figure 4. When

crossing the spatial variables with the geometric variables, for example, the “road layout” and “grade” variables, the Kernel estimating technique was also applied to identify and verify the differences in concentrations between the road layout and the presence of a rising or descending slope. The procedure was repeated for various combinations of clusters.

After segmentation of the homogeneous stretches, the accidents that fit within the selected segments of highway were associated with them.

With the database structured in this manner, it was possible to compare the distribution of accident severity and accident frequency on curved stretches, considering the slope of the terrain. The results show that approximately 68% of accidents occur on straight stretches and 32% on curved stretches, however, attention is drawn to the accident severity. Of the accidents that occurred on straight stretches (220), 29% (64) were serious and 9% (9) were fatal, compared to 35% (37) and 18% (19), respectively, for curved stretches, which had a total of 103 accidents (Figure 5). The analyses also show that approximately 41% of accidents in curved stretches occurred on a descending slope, with 40% of the total being serious accidents and 19% fatal, while the percentage was less than 1% for all cases on straight stretches (Figure 6).

The models developed were calibrated using the GEE technique, assuming errors with a Negative Binomial distribution because of the presence of a large number of observations with zero value and, therefore, high dispersion. In the SPSS software, version 23.0.0, this analysis can be found in the procedures: Analyze >> Generalized Linear Models >> Generalized Estimating Equations.

There was insufficient data to build a model for varying shoulder width values and traffic volume. The lane width was also constant throughout the study section. Although there was a single point in the entire section studied where traffic volume was counted, AADT was considered in the model, because its importance

is consolidated in the literature. Two adjusted models were then prepared, with three variations corresponding to the homogeneous segmentations. (1, 2, 3) for each model. They are:

Model 1—dependent variable (frequency); age group, day of the week, AADT (categorical variables); radius, grade, length (covariables); lane width and shoulder width (only for type 1 and 2 segmentation).

Model 2—dependent variable (frequency); grade (categorical variable); radius, length, AADT (covariables); lane width and shoulder width (only for type 1 and 2 segmentation).

To evaluate the effect of constant traffic, two variations of models 1 and 2, called models 3 and 4, were considered, using the same segmentations 1, 2, and 3, with AADT included as an offset variable. The model terms were factorially combined so that all combinations between variables could be evaluated. A summary of the estimated models is described in Table 5.

The significance of the parameter coefficients and the deviance were observed, in order to analyze whether the variables were significant for the model. With QIC, the correlation structures were evaluated and the best global model was selected with the CURE Plot. A significance level of 5% was used, meaning that

variables with a p-value greater than 5% were not considered to be significant. In the analysis of deviance, a Chi-square distribution of 5% significance was used. Therefore, if the contribution of the variable to the deviance was less than 1.96, the variable should not be included in the model.

When adding the lane width and shoulder width variables, neither model obtained satisfactory results. In both cases of the model tested, the parameter associated with the lane width and shoulder width variables was not statistically significant for α = 5%.

This result might be related to the constant values for all of the elements of the sample. The calibration results for models 1, 2, 3, and 4 are shown in Tables 6-9. Differences in the signs of the coefficients may indicate, depending on the segmentation, an opposite influence of the variable on the expected number of accidents estimated by the model.

The choice of working correlation matrix represents intra-individual dependency. The best structure should be sought, using the lowest QIC as a criterion. The QIC values found by adjusting models 1, 2, 3, and 4 with other correlation matrices are shown in Tables 10-13.

It can be seen that, according to the QIC parameter, the exchangeable correlation structure was the one that best fit the longitudinal data for the models generated.

Number of observations in the database = 428.

Number of observations in the database = 428.

Number of observations in the database = 428.

Number of observations in the database = 428.

With this correlation structure, it can be said that the correlation between any two observations within a group are constant. The adjusted Segmentation 3 offered the best result for all models, however, most parameters were not statistically significant (p > 0.05).

The CURE Plot graphs of the models are presented in Figure 7 and Figure 8. For Models 1 and 2, it is possible to observe that the curve for cumulative residuals oscillates around 0 and does not cross the upper nor the lower acceptable limit. For models 3 and 4, the cumulative residual curve oscillates around 0 but exceeds the upper limit. Therefore, the best accident prediction model is model 2, because it presented the lowest QIC value (600.30).

The results obtained from the validation demonstrate that the best model for accident prevention is Model 2, because the root mean square error of the model adjustment (ΔRMSE) is closest to zero, with a value of −0.082 (Table 14).

It is worth mentioning that the parameters obtained for the variables Day of the Week and Age Group agree with the values found in the simulations for variable selection. Taking the age group of those over 50 as a reference, young people between 18 and 30 are 22.7% more likely to be involved in fatal accidents while adults between 30 and 50 are 34% less likely to be involved in accidents. On weekends, the chance of accidents occurring is 67% higher than during the week.

For Segmentation 1, based on HSM, the selected variables have larger standard errors than those selected for other segmentation approaches. This is likely because, on highways, homogeneous segments change only at intersections, producing very long segments, where a great number of them have zero acidentes, and with considerable variation within individual segments in the other variables that cannot be adequately modeled.

For the model estimated for Segmentation 2, which includes 50 m of roadway on each end of a curve, the results were also significant. However, they tend to underestimate the number of accidents for low AADT values and overestimate accidents for higher AADT values.

Initially, the segmentation producing the worst results in the number of variables that can be included in the model was Segmentation 3, in which all variables are explanatory for each segment. Therefore, variable categories were created, based on fixed value ranges, to improve the statistical power of the model. These categories were defined by attempts to obtain the best fit of the model and statistical significance for the main parameter estimates.

Finally, the GEE model was defined in order to predict the occurrence of accidents in a segment considering the AADT, curve radius, segment length, and grade, as shown in Equation (3):

μ i = e ( β 0 + β 1 AADT + β 2 R + β 3 Greide + β 4 L + ε ) (3)

where: μ i = frequency of expected accidents per year; β 0 = intercept; β₁, β₂, β₃ and β₄ = parameters; R = Curve radius (m), L = segment length (m), Grade = Grade (negative, positive, or zero), and ε = error term.

Table 15 shows the model effects of all of the independent variables. The variable categories have no absolute values, but define the value of the parameter estimate (β_n in column 3). The exponent of the estimated parameter (and β_n in column 5) can be interpreted as a form of relative risk value for any declared variable category. This means that the following interpretations can be made based on each of the variables, considering that all other variables in the model have been kept constant.

The study showed that curves with a radius less than or equal to 600 m have a 3.2 times greater risk of accidents than curves with a radius greater than 2200 m (relatively straight). It also showed that sections with a downward slope have a risk of accidents 1.6 times greater than upward sloping or level road sections. Straight stretches longer than 1000 m on a downward slope, followed by a curve, have a risk of accident 2.2 times greater.

Equation (3) was solved for all variable categories in the model. The average value of the rate for accidents with victims in curved sections per kilometer was low: 0.048. This reflects the low frequency of accidents in these stretches. However, the causes may be related to the low traffic flow on rural roads, to the fact that there is a single point Where traffic data is collected in the studied section, or even the underreporting of this type of accident. Therefore, it was more significant to present the model’s results from the combination of road characteristics, including the radii of the curves. For the sample mean of 0.048 accidents per km, the value 1.0 was defined.

To visualize the data more easily, a color code was applied: green represents an expected value for accidents with victims below the sample average (less than 1.0), yellow represents scenarios with a risk between the average and double the average (between 1 and 2), and orange represents scenarios in which the risk is two to three times the average value (between 2 and 3). The red color represents an extreme risk condition in which the predicted accident value was more than three times the sample average. Table 16 shows changes in the expected risk level for accidents in curves based on the predicted value.

From these results, it can be concluded that radii between 600 and 1500 meters

should be preferred in all scenarios for the design of new roads in order to reduce the frequency of accidents in curves. The results also show that long downward-sloping stretches followed by curves with radii less than 600 m offer the greatest risk for accidents. If highways with a radius less than 600 m were converted into highways having radii greater than 600 m, accidents with victims in curves would decrease by about 18%. Roads with radii smaller than 600 m on a downward slope would see a reduction of 27%. With the model results and the historical accident numbers for the analyzed segments, the calibration procedure was carried out by dividing the actual total value by the calculated estimated value. The value obtained was 2.35 for Segmentation 1 and 1.75 for Segmentation 3.