Introduction

The Highway Safety Manual (HSM) provides a predictive method and analytical procedures to support road projects in various phases, including planning, preliminary design, final design, and construction ¹. The HSM procedure aids in identifying locations with a potential for reducing crashes or their severity, diagnosing issues, selecting countermeasures, conducting economic appraisals, prioritizing projects, and evaluating safety effectiveness ². The HSM primarily describes the predictive method and crash modification factors (CMF). The predictive model includes the safety performance functions (SPFs) used to forecast crash counts at different roadway elements ³. These SPFs should be adjusted to local conditions for optimal results ⁴. Calibration is essential because the overall crash frequencies can vary significantly from one location to another due to various factors, such as crash reporting thresholds and crash reporting system procedures ¹. To adapt the predictive model or SPF to local conditions, there are two options: calibrating the SPF or developing a new one.

The choice between calibration and development depends on the resources available to the institution and its objectives. The FHWA (Federal Highway Administration) issued report FHWA-SA-14-004, which evaluates these two options ⁵. Calibration maintains the structure of the original equations, allowing to compare the results across different areas. On the other hand, during development, the equations are more likely to change, leading to differences in the comparisons. SPF calibration can be achieved by either calculating the calibration factor or estimating the calibration function.

The calibration factor procedure involves adjusting the original model proposed by the HSM to fit the local conditions. The calibration factor is calculated as the ratio of the total number of observed crashes to the total number of predicted crashes. At times, the model may either overpredict or underpredict the number of accidents. For instance, in Ecuador, the calibration factor for two-lane rural roads was reported to range between 0,12 and 0,25 ⁶. In this case, the model overpredicted the crash frequency. Here, it is important to note that rural roads are typically susceptible to more severe accidents due to topographical limitations. Similar studies have been conducted in numerous countries and for various types of roads.

Other works have concentrated on the calibration of SPFs, which are models used to predict the number of crashes. In the HSM, SPFs are negative binomial regression models that establish a log-linear relationship between crash frequency and various site characteristics ⁷. These models invariably include traffic volume (AADT) and can also incorporate site-specific factors such as section length and lane width, among others. It is important to note that the calibration of SPFs should be conducted separately for different crash types ⁸ rather than combining all crash types into one model. Models specific to crash types tend to provide a better fit in comparison with using a single model that estimates the total crashes together with the overall proportion of each crash type ⁹.

There is a substantial amount of research worldwide that involves calibrating SPFs for two- lane rural roads. For instance, the original HSM model consistently underpredicts crashes on two-lane rural roads in Utah ¹⁰ and Illinois ⁴. In Pennsylvania, the findings suggest significant variations in safety performance across engineering districts, a factor that should be considered by other state transportation agencies ¹¹. Similar results have been identified in research conducted in Virginia ¹², Oregon, and Georgia ¹³. The performance of SPFs may also differ depending on the type of pavement, as indicated by results obtained in Michigan ¹⁴. Additionally, in flat terrain in Spain, the HSM model exhibits both underprediction and overprediction tendencies in response to variations in AADT ¹⁵. Note that it is necessary to adjust SPF models to local conditions in order to achieve more accurate results. These equations are widely used worldwide because they are fundamental for road safety analysis. However, in Ecuador, there has been limited progress on this issue. Hence, the primary objective of this study is to calibrate SPFs for two-lane rural highways in the mountainous regions of Ecuador, considering various types of accidents and different traffic volume ranges. 25 roads in the Andes mountains were selected for evaluation, totaling approximately 630 km in length. Geometric and operational data were collected for each of these roads and compared to recorded road crashes.

To present this calibration, this document is structured as follows. Firstly, it describes the predictive model of the HSM. Secondly, it provides detailed information on the materials and methods employed, including an analysis of the data collection process, calibration procedure, and validation phase. Subsequently, it presents the results obtained from the models and an analysis of prediction errors. Finally, the document highlights the main findings and conclusions.

Safety performance functions

El HSM ¹ states that, for rural two-lane and two-way roadway segments, the predictive model

takes the form of Eq. (1).

Where:

Eq. (1) in the model predicts either the total average crash frequency or that of specific crash severity or collision types. This model addresses the regression-to-the-mean bias by focusing on the long-term expected average crash frequency, rather than relying solely on the short-term observed crash frequency ¹. To account for variations in segment characteristics when compared to base conditions, Crash Modification Factors (CMFs) are employed with the aim of adjusting the number of crashes. While the HSM provides these CMFs, there is also a website known as the Crash Modification Factors Clearinghouse¹⁶, which compiles CMFs from various studies, categorized by country or region. In Eq. (1), the model incorporates the safety performance function, the expression of which can be found in Eq. (2).

Where:

Eq. (2) is based on a negative binomial distribution, which is better suited for modeling the high natural variability of crash data compared to traditional modeling techniques ¹. Moreover, this equation involves only two variables, i.e., the average annual daily traffic and the segment length, making it easier to calibrate the model in comparison to other SPFs with more variables. The SPF results can be calculated with a high level of accuracy even when there are only 1-3 years of recorded crashes for the studied site ¹⁷. SPFs are developed using statistical multiple regression techniques, employing historical crash data collected over several years at sites with similar characteristics and encompassing a wide range of AADT values ¹. These SPFs provide estimates for the average number of crashes when the segments meet the base conditions outlined in Table I. When a segment does not meet the base conditions, it should be adjusted using the CMFs proposed by the HSM model.

Table I Base conditions for roadway segments on rural two-lane, two-way roads 

Source: Highway Safety Manual (1).

The base conditions shown in Table I may not necessarily be directly applicable or valid in the context of Ecuador. The suitability of these factors for calibration in the country would depend on several factors, namely.

Materials and methods

To calibrate SPFs, it is essential to gather data on the AADT, the length of the segment, and the number of crashes recorded in it. This section provides an overview of the materials and methods employed for data collection and processing. Additionally, it outlines the procedures for calibrating and validating the models.

Segment selection

Initially, 25 two-lane roads were selected, with lengths ranging from 7,4 to 40 miles, as indicated in Table II. The AADT values, calculated or measured in 2017, have been included in this table. It is worth noting that the AADT estimation process was primarily carried out by students participating in the Road Projects course at Universidad Técnica Particular de Loja, along with the OSEVI-UTPL traffic counter ¹⁸.

Table II Selected two-lane rural roads in this study 

Source: Master’s students of the Road Projects course at Universidad Técnica Particular de Loja and the OSEVI-UTPL traffic counter ¹⁸

The features of every road listed in Table II were analyzed. These roads have a width between 11 and 12 ft, which is similar to the base condition. Additionally, these roads have a shoulder width between 2 and 4 ft, which is less than the base condition. However, it is important to note that these roads were constructed using older standards, where the shoulder width was narrower. The entire shoulder is paved, in accordance with national guidelines and base conditions. The roadside hazard rating (RHR) ranged between 3 and 7, as these are mountain roads. It is common for Ecuadorian mountain roads to have this level of risk, and it can be challenging to find RHR values equal to 3, which is the base condition. The driveway density was less than 5 driveways per mile, which is also common in mountain roads. Despite these differences, the remaining road features were consistent with the base conditions. Even though it may not be possible to meet all base conditions, efforts were made to align with the majority of them.

Crash data

Crash data were sourced from the National Transport Agency of Ecuador’s website ¹⁹. For each segment listed in Table II, three types of data were collected: total crashes (T), fatal and injury crashes (FI), and property damage only (PDO). It is important to note that these data were not directly available on the website and had to be estimated based on the causes of the accidents. The HSM recommends certain guidelines for data collection. It suggests a minimum sample size of 30-50 sites, with the entire group of calibration sites representing at least 100 crashes per year. It may take one to three years to accumulate a sufficient number of crash events to meet these crash frequency criteria ¹.

The road segments were associated with the closest crash record. Subsequently, the database was organized into two time periods: 2017-2019 and 2020-2021. This non-probability sampling division allowed calibrating predictive models via the crash data from the earlier period (2017-2019) and validating them using the more recent data (2020-2021). Initially, an analysis was conducted on individual segments, where 50 sites were randomly selected from those available. Between 2017 and 2019, 1.042 sites had no accidents, while 197 did. In total, 246 crashes were recorded during this period. Due to the low number of records and the high number of sites with no accidents, the random selection did not yield 100 crashes, despite a three-year analysis period. Consequently, we decided to use all 25 road sections as if they were individual segments, as shown in Table III.

Table III Number of crashes in every road for the period of time analyzed (based on ANT data) 

The data from 2019-2021 were then used for model calibration. It is worth noting that, if more than three years of data were available, they could be useful for assessing potential time trends in anual calibration factor estimates ¹⁷.

Table III shows that the highest number of observed accidents took place on road No. 4, which also has the highest AADT. Surprisingly, the longest road ¹⁴ does not have the highest number of accidents, possibly due to its low AADT value. This pattern is consistent with previously estimated SPFs. Interestingly, several roads had no recorded accidents in the analyzed period, while others had no records in the first period but did in the second. It is worth noting that the total crashes (T) value is approximately double that of fatal and injury crashes (FI) or property damage only (PDO). However, this is coincidental, as this pattern does not hold when analyzing each site separately. These variations and trends suggest that the selected sites may be representative since they encompass various situations encountered in engineering practice.

Results and discusión

Continuing with the procedure outlined in the previous section, the results of the calibration and validation of the SPFs are presented.

Model calibration

Table IVpresents the calibration results, beginning with the general model for all traffic, and then the AADT ranges resulting from the cluster analysis, which achieved a 95 % similarity. The histogram in Fig. 1 confirms the ranges obtained in the cluster analysis. However, special attention must be paid to the last range, as there is a lack of AADT values until reaching nearly 15.000 vehicles/day. This stratification of AADT for the calibration of the SPF equations is in line with previous findings (¹⁵, ²⁰).

Table IV Safety performance function proposed by this study 

T: total crashes; FI: fatal and injury crashes; PDO: property damage only; N _spf-rs: predicted average crash frequency for base conditions using a statistical regression model (crashes/year); AADT: average annual daily traffic (vehicles per day); L: length of the road segment (miles).

Figure 1 Histogram of AADT 

Table IVshows that the exponents for total crashes are lower than those for fatal and injury crashes, while the exponents for the latter are very similar to those for property damage only. Additionally, as the number of vehicles increases, the absolute value of these exponents increases, and this trend applies to all three categories of accidents. Only in the range of up to 1.000 vehicles per day do positive exponents appear, whereas, in the other models, the exponents are negative, as seen in the original equation.

Given these observed trends, using a single model, such as Eqs. (3), (4), and (5), would likely result in higher prediction errors when compared to using Eqs. (6) through ⁽¹⁸⁾. In light of this, the general model will not undergo validation in the following section.

Validation of the models

For validation purposes, we utilized the adjusted AADT, segment length, and crash data from 2020-2021. Eqs. (6)-(18) in Table IV were employed to predict the number of crashes for 2020-2021, as summarized in Table V. The selection of specific equations was based on the corresponding AADT values for each road segment. Table V shows that the total number of observed crashes in the period was 140, while the predicted count was 166,00. For fatal and injury crashes (F+I), there were 78 observed crashes, compared to a predicted count of 106,50. Property damage only (PDO) crashes had 62 observed crashes, with a predicted count of 102,15.

Table V Observed and expected crashes calculated using the equations 6-18 from Table IV 

The scatterplot of values from Table V is presented in Fig. 2 . In an ideal scenario, all the points would closely align around a 45° reference line. However, it is essential to remember that SPFs provide ‘long-term average results, not ‘short-term average’ ones ¹⁷. Although there are points where the equations overestimate or underestimate the observed crashes, it is necessary to conduct an error analysis to determine the validity of the equations.

Figure 2 Expected versus observed crash frequency 2020-2021 

Table VI presents the prediction errors for the crashes predicted using Eqs. (6)-(18). Regarding the MSPE, lower values are preferable, as they indicate a better fit of the model to the data. In this case, the values could be deemed acceptable, with the highest value being 27,8.

Table VI Prediction errors of the proposed safety performance function 

The MAD provides a measure of the average magnitude of prediction variability, and smaller values are preferred over larger ones. In this case, the maximum value was 3,6, which was deemed acceptable.

The MPB offers insights into whether the equations overpredict or underpredict. A positive value suggests underprediction, while a negative one implies overprediction. As observed in Fig. 2 , the equations are slightly underpredicting, consistent with the calculated positive values of MPB.

Lastly, the MAPE calculates the average error percentage between the observed and predicted values, relative to the former. Lower MAPE values indicate better predictions.

However, in this case, the values are relatively high due to their close relationship with the total recorded crash values.

Despite the higher MAPE values, the rest of the error metrics constitute noteworthy results, suggesting that the models can be regarded as validated.

The presented and validated models represent significant progress in the field of road safety for two-lane roads. Using separate equations for different types of accidents, as demonstrated here, addresses the limitation of a single equation, such as the one proposed by the HSM, which can lead to errors when applied without distinction among accident types. This finding is consistent with previous research ⁹. Furthermore, the utilization of models based on traffic ranges, as shown in this study, has also been recognized in previous studies, and it is confirmed to provide improved results ¹⁷. These advancements contribute to a more nuanced and accurate understanding of road safety, ultimately aiding in the development of more effective safety measures and policies for two-lane roads.

Conclusions

The objective of this study was to propose Safety Performance Functions (SPFs) for two- lane rural highways in mountainous environments in Ecuador. After analyzing 25 road sections and accident data from 2017 to 2021, the following conclusions can be drawn: It was possible to successfully calibrate and validate 12 SPFs for two-lane roads. Unlike the Highway Safety Manual (HSM), which provides a single model for all accident types and Average Annual Daily Traffic (AADT) ranges, this study offers four models for total crashes, four for fatal and injury accidents, and four for property damage only. These models cover various AADT ranges, thus enhancing their applicability.

The base conditions, as defined in the HSM, are primarily derived from road characteristics commonly found in the United States. Our study area in Ecuador includes roads that may have different characteristics, such as narrower shoulder widths, variations in roadside hazard ratings, and mountainous terrain, which are not fully aligned with the HSM’s base conditions. To address this limitation, we conducted our calibration and validation efforts using the available data, which reflect the local conditions of Ecuadorian mountain roads. While our approach provided valuable insights into road safety in our specific context, we recognize that it may have introduced variations compared to a strict adherence to HSM base conditions.

This study underscores the importance of calibrating models developed elsewhere before applying them to local conditions. The calibrated equations can be used for existing roads, alternative designs for existing ones, or new road projects, ultimately contributing to improved road safety for two-lane rural roads in Ecuador.

However, it is important to acknowledge the study’s limitations. Firstly, it focused exclusively on two-lane roads in mountainous environments, which may not fully represent all road types in the country. Additionally, there was limited representation of high-traffic sites, which could lead to inconsistencies in the models for the highest AADT range. Furthermore, the study could not verify whether all road accidents were accurately collected and categorized, relying solely on data provided by the National Transport Agency (ANT).

Despite these limitations, the study significantly contributes by introducing various SPF equations for three types of road accidents and several traffic volume ranges that were previously lacking in the country. These models serve as a valuable starting point for analyzing accident rates in Ecuador and other regions with similar road characteristics