Predicting Operating Speeds of Passenger Cars on Single-Carriageway Road Tangents

Abstract

This research addresses the challenge of predicting operating vehicles’ speeds (V85) on single-carriageway road tangents. While most previous models rely on preceding segment speeds or focus on curves, this research develops an independent prediction model specifically for road tangents, based on empirical data collected in Croatia. A total of 46 locations across 23 road cross-sections were analysed, with operating speeds measured using field radar surveys and fixed traffic counters. Following a comprehensive correlation and multicollinearity analysis of 24 geometric, environmental, and traffic-related variables, a multiple linear regression model was developed using a training dataset (36 locations) and validated on a separate test set (10 locations). The model includes nine statistically significant predictors: shoulder type (gravel), edge line quality (excellent and satisfactory), pavement quality (excellent), average summer daily traffic (ASDT), crash ratio, edge lane presence, overtaking allowed, and heavy goods vehicle share. The model demonstrated strong predictive performance (R² = 0.89, RMSE = 5.24), with validation results showing an average absolute deviation of 2.43%. These results confirm the model’s reliability and practical applicability in road design and traffic safety assessments.

1. Introduction

Traffic safety remains a pressing issue globally, affecting numerous countries. The World Health Organization (WHO) estimates that approximately 1.19 million people lose their lives in road crashes each year, with an even greater number sustaining injuries [1]. As a result, road traffic collisions rank as the eighth leading cause of death worldwide and are the most common cause of fatalities among individuals aged 5 to 29 [2]. The highest proportion of fatalities is recorded in low- and middle-income nations, where road infrastructure, vehicle safety standards, and traffic education are often inadequate. Given these trends, traffic accidents have emerged as one of the most serious public health concerns in contemporary society, and their impact is projected to grow in the future. According to OECD estimates (2018), road traffic accidents impose a significant financial burden on the European Union, amounting to approximately EUR 500 billion annually, or about 3% of its total GDP [3]. This financial impact accounts for medical costs, productivity losses, expenses for road repairs and reconstruction, and other associated expenditures. Research suggests that nearly one-third of fatal road accidents worldwide are linked to excessive or inappropriate speeds [4]. Numerous studies have confirmed a direct relationship between speed and accident probability, indicating that higher speeds not only increase the likelihood of crashes but also exacerbate their severity. Specifically, a 1% rise in average vehicle speed is associated with a 4% increase in fatal accident risk and a 3% increase in the likelihood of serious injuries [1]. This finding is consistent with earlier research demonstrating that excess speed is a key risk factor on rural single-carriageway roads, with higher operating speeds linked to both increased crash rates and greater injury severity [5,6,7].

Tangents, or straight road sections, allow vehicles to reach higher speeds due to the absence of geometric constraints, such as curves or intersections. While these segments improve traffic flow efficiency, they also contribute to higher crash rates and increased severity of collisions. Research indicates that excessive speeding is a primary factor in road traffic accidents, particularly on road tangents. Data from the National Highway Traffic Safety Administration (NHTSA) reveal that speeding was responsible for 29% of all fatal traffic accidents in 2022, highlighting the strong correlation between high-speed travel and accident severity [8]. Furthermore, statistics indicate that a significant proportion of fatal crashes on rural (68%) and urban (81%) roads occur on tangents, where drivers frequently exceed posted speed limits due to the absence of natural speed-calming features [9]. One possible explanation for the high proportion observed is that tangents often facilitate higher operating speeds and longer periods of unvaried driving, which may contribute to lapses in attention and delayed hazard perception. Additionally, tangents are frequently the location of overtaking manoeuvres, which, in mixed traffic conditions, can increase the risk of severe head-on collisions. These behavioural and operational characteristics may partly offset the geometric safety advantages that tangents typically provide in terms of visibility and stability. The physics of high-speed driving further amplifies the risk, as greater speeds reduce the time available for driver reactions and increase stopping distances, thereby raising the probability of crashes [10]. Moreover, collisions occurring at high speeds tend to result in more severe injuries or fatalities due to the greater impact forces involved [11]. A key challenge in road safety management is the discrepancy between regulatory speed limits and actual vehicle speeds, a concern also emphasized in the PIARC report Setting Credible Speed Limits [12], which stresses the importance of aligning posted limits with road function, design consistency, and driver expectations to ensure compliance. Research shows that on straight road segments, drivers commonly exceed posted limits, leading to a heightened risk of severe crashes [13]. The term “operating speed” refers to these actual vehicle speeds observed under free-flow conditions. It can be viewed as the actual speed a vehicle travels at under free-flow conditions [14], or as the 85th percentile speed distribution used to assess operating speeds associated with specific locations or road designs [15]. Research has indicated that operating speeds under free-flow conditions, especially on roads lacking speed monitoring or enforcement, often exceed the posted speed limits [16,17]. The highest operating speeds are usually recorded under free-flow conditions, where road design is the main factor influencing speed, rather than interactions with other vehicles. Therefore, it can be assumed that vehicle interaction has little impact on operating speeds under these conditions [18].

Numerous studies have investigated operating speeds on single-carriageway roads, encompassing various geometric elements such as horizontal and vertical curves, tangents, and their combinations. Among these, extensive attention has been given to horizontal curves, where the curve radius consistently emerges as the most statistically significant parameter influencing operating speeds across diverse road types [19,20,21,22,23,24]. For example, research [20] observed that vehicle speeds drop notably when the curve radius is under 300 m, whereas larger radii tend to stabilize speeds. In line with this, research [25] reported that radii exceeding 400 m exert minimal influence on speed behaviour. Compared to horizontal curves, research focused exclusively on tangent segments remains relatively sparse. Notwithstanding this relative scarcity, several European investigations have developed tangent-specific models and empirical analyses that directly address operating speeds on rural two-lane roads [26,27,28], providing methodological and contextual insights relevant to the Croatian road environment. A study by research [29], conducted in both urban and rural contexts, established a strong link between posted speed limits and observed operating speeds on tangents. The study also identified several additional factors, such as driveway density, type of median, and the presence of parked vehicles and pedestrians, that affect operating speed. Specifically, the presence of pedestrians was associated with lower speeds. Complementary European field studies that exploit GPS or continuous speed records corroborate the influence of the roadside environment and geometric subtleties on speed choice [30], reinforcing the role of localized factors such as driveway density, pavement quality, and roadside furniture in shaping V85 on tangents.

In several operating-speed models for road tangent segments, researchers have experimented with including the curvature change ratio (CCR), a measure of how sharply curvature varies over a given distance, as an explanatory variable. An analysis of floating-car data from 828 rural tangents found that CCR contributed negligibly to model performance, with corresponding R² values remaining below 0.01 [31]. On the other hand, on two-lane rural roads in Italy, the research incorporated the curvature change ratio (CCR) as a variable in developing an operating speed prediction model. The CCR was used to quantify the variation in horizontal curvature along the roadway and served as an indicator of alignment consistency. The authors found that higher CCR values (indicating more frequent or sharper curvature transitions) were associated with lower operating speeds, as drivers tend to adjust their speed more frequently in response to abrupt geometric changes. Conversely, lower CCR values, typical of smoother and more uniform alignments, correlated with higher and more stable operating speeds [32]. Although the curvature change ratio (CCR) has been employed in some operating speed studies, its role is closely intertwined with broader geometric and environmental factors, particularly terrain type. The CCR essentially quantifies the rate at which road curvature varies along a given alignment, which, in turn, reflects the alignment constraints imposed by the surrounding topography. In mountainous terrain, where space limitations and elevation changes necessitate frequent transitions between curves and tangents, CCR values tend to be high; in flat terrain, where alignment is generally straighter and curvature changes are minimal, CCR values are correspondingly low. Because terrain type already exerts a strong, direct influence on operating speed, through both driver perception and geometric design constraints, the CCR can act as a derived proxy for terrain conditions rather than an independent geometric characteristic.

On rural tangents, wider lanes have been shown to encourage higher operating speeds. According to the Federal Highway Administration (FHWA), increases in lane and shoulder widths lead to faster speeds; for instance, the Highway Capacity Manual indicates that speeds on two-lane rural highways can increase by approximately 0.64 to 1.77 km/h for every 0.3 m increase in lane width [33]. Nonetheless, some studies, such as that by research [29], suggest that lane width does not universally exert a significant statistical effect on speed, implying the presence of additional influencing factors. In a broader study conducted by research [19] involving 251 locations in Oklahoma, three operating speed prediction models were developed, each tailored to specific speed limit categories: high-speed (80–105 km/h), low-speed (56–72 km/h), and variable-speed roads. The model developed for variable-speed conditions demonstrated the greatest predictive accuracy. Among the most consistently significant predictors were the posted speed limits and pavement friction. Notably, on roads with lower speed limits, only the posted limit emerged as statistically relevant. Differences in operating speed prediction across contexts can also be attributed to national and cultural variations in driver behaviour. Research [34], for instance, compared driving aggressiveness among drivers from Finland, the UK, Greece, Iran, the Netherlands, and Turkey. Greek drivers exhibited the highest incidence of aggressive behaviour and impatience, while British, Dutch, and Finnish drivers were among the least aggressive. Iranian and Turkish drivers displayed intermediate levels of such behaviour. These findings support the assertion that cultural differences can meaningfully affect speed choices [25]. Studies conducted in Spain and other European countries further indicate that the succession of road elements (tangent–curve sequences) and local geometric consistency strongly modulate driver speed adaptation, an effect documented in several European models of operating speed and transition behaviour [22,35].

While various operating speed prediction models for road tangents have been developed, many of them rely on the operating speed of the preceding horizontal curve or previous segment as an input variable [26,35,36,37,38]. However, this approach introduces a compounded source of uncertainty, as the speed on the preceding curve often needs to be estimated itself, especially in the context of newly designed roads, thereby amplifying cumulative prediction errors. In contrast, models that exclude this variable have shown limited accuracy when applied to real-world conditions in Croatia. Recent comparisons between predicted and measured operating speeds on single-carriageway road tangents within the country have revealed notable discrepancies, with average absolute prediction errors ranging from 11.5 km/h [39] to as much as 18 km/h [30]. The observed average absolute prediction errors of 11.5 km/h and 18 km/h are particularly significant within the context of operating speed prediction on road tangents, especially during the road design phase prior to construction. Accurate estimation of operating speeds on tangents is essential for informing geometric design parameters. Large discrepancies between predicted and actual operating speeds may result in designs that fail to accommodate real driving conditions: underestimations can lead to insufficient safety margins, increasing the likelihood of speed-related road traffic accidents, while overestimations may cause overly conservative designs that elevate construction costs without proportional safety benefits. Given that tangents typically allow higher speeds due to the absence of lateral constraints, precise speed prediction is critical for optimizing infrastructure to balance safety and efficiency. Consequently, reducing prediction errors is fundamental to enhancing the reliability of tangent-specific speed models, thereby supporting evidence-based design decisions and contributing to the mitigation of speeding-related risks on single-carriageway road tangents.

These findings highlight the need for more precise, independent models that do not rely on operating speed data from adjacent horizontal curves, particularly for application in the planning stages of road infrastructure. This paper builds upon the earlier research [40] that analysed both single- and dual-carriageway roads. In this research, the focus is narrowed to single-carriageway road tangents, and the underlying database is significantly expanded. This refined scope allows for a more detailed assessment of how geometric, infrastructural, and traffic variables influence operating speeds, thereby enabling the development of a more accurate and context-specific prediction model.

Following the above, this research investigates operating speeds specifically on tangent segments of single-carriageway roads in Croatia. The principal objective is to establish and validate a predictive model for operating speed, grounded in statistically determined relationships between speed and various influencing variables. Tangent sections are chosen for their unique impact on speed behaviour, as they lack the physical constraints typical of horizontal curves, such as lateral acceleration. This allows drivers to select speeds primarily based on geometric alignment, sight distance, and environmental cues [19,29]. Due to the absence of such geometric limitations, operating speeds on tangents are often higher, making them a critical element in understanding traffic flow variability and driver response. By focusing on tangents, this study seeks to uncover how specific roadway and contextual factors interact to shape operating speed. Through detailed data collection and correlation analysis, this research aims to generate empirical knowledge that can inform improved speed prediction and contribute to safer and more effective road design.

2. Methodology

A simplified overview of the applied methodology is presented in Figure 1, while a detailed explanation of each step is provided in the subsequent subsections.

The methodology of this research involved six key steps to develop a reliable operating speed prediction model for single-carriageway road tangents. First, location selection focused on representative sites across various Croatian regions, speed limits (70, 80, and 90 km/h), and traffic conditions to ensure diverse and relevant data. Next, data collection was carried out through two primary sources: field surveys using radar devices at 24 locations and fixed traffic counters managed by Croatian Roads Ltd. (Zagreb, Croatia), covering a total of 46 measurement points under consistent, favourable weather conditions. Following data gathering, data filtering and cleaning was performed to exclude outliers, non-passenger vehicles, and peak-hour traffic, ensuring only free-flow operating speeds were analysed. Then, variable identification was conducted, selecting 24 variables related to roadway geometry, traffic flow, and the roadside environment based on their potential influence on vehicle speeds. Subsequently, statistical and correlation analyses were applied to examine relationships between operating speeds and the identified variables, using normality tests, correlation coefficients, ANOVA, and multicollinearity assessments to select significant predictors. The final step involved stepwise multiple linear regression to develop an accurate operating speed prediction model, which was validated through a dataset split into training and testing subsets.

2.1. Data Collection

To ensure the relevance of the results, this study focused on road tangents on single-carriageway roads (state and county roads). The selection of representative locations was based on the following parameters:

• Geographical region: Central Croatia, Slavonia and Baranja, Lika, Istria, and Dalmatia.
• Existing speed limits: 70, 80, and 90 km/h. It is important to note that in Croatia, the default speed limit outside urban areas is 90 km/h unless otherwise specified by a posted speed limit sign.
• Additional factors: Road design characteristics, vehicle composition, road capacity, and other relevant influencing elements.

In selecting locations, an effort was made to achieve a relatively even geographical distribution across the country while ensuring that each of the specified speed limits was represented. The Croatian regions of Central Croatia, Slavonia and Baranja, Lika, Istria, and Dalmatia differ in topography, which influences road design and driver behaviour. Central Croatia and Slavonia are mostly flat to gently rolling, featuring relatively straight roads. In contrast, Lika and Dalmatia are more hilly regions with winding roads, resulting in more complex driving conditions. Istria, on the other hand, combines hilly terrain and coastal features, leading to varied road geometries and traffic patterns. These topographical differences affect road infrastructure and may shape local driving behaviour, as drivers adapt to the physical and environmental characteristics unique to each region.

Additionally, measurement site selection considered traffic volume variations, as reflected in both Average Annual Daily Traffic (AADT) and average summer daily traffic (ASDT), to capture a diverse range of traffic conditions.

For the purpose of developing the model, it was first necessary to collect data on operating speeds. These speed measurements were obtained using two sources (Figure 2):

• Through field surveys and radar-based speed measurements [40];
• Using data from fixed traffic counters operated by Croatian Roads Ltd. [41].

Data were gathered in both directions of travel, resulting in a total of 46 unique measurement locations. It is also important to note that there is no permanent speed enforcement in the broader areas surrounding the selected locations, which could influence the operating vehicles’ speeds.

Figure 2. Map showing selected locations (cross-sections) on single-carriageway road tangents in Croatia [40,42].

Figure 2. Map showing selected locations (cross-sections) on single-carriageway road tangents in Croatia [40,42].

2.1.1. Data Collected Through Field Survey

Speed measurements were obtained through field surveys conducted across 12 cross-sections, encompassing 24 distinct measurement locations [40]. A radar device, discretely mounted on vertical road signs to minimize driver awareness, was employed for data collection (Figure 3). Measurements were carried out under favourable weather conditions, characterized by moderate temperatures, minimal cloud cover, and the absence of precipitation, wind, or fog, ensuring optimal visibility. Prior to the fieldwork, an assessment of current road conditions was performed to mitigate potential disruptions caused by temporary road closures, traffic diversions, roadworks, accidents, or other unforeseen incidents.

This study recorded a total of 37,093 vehicle samples. After data collection, the dataset underwent a systematic pre-processing procedure aimed at eliminating anomalies and extreme outliers to ensure analytical strength. Anomalies were first detected through visual inspection of scatterplots, histograms, and boxplots for each variable (including speed values), which allowed the identification of implausible or inconsistent values relative to the expected engineering and physical ranges. For example, operating speeds below 30 km/h or above 200 km/h under free-flow conditions or negative traffic volumes were flagged for verification. Where original field documentation confirmed an error, the value was corrected; otherwise, the record was removed. These criteria ensured a balance between data quality control and preservation of valid variability within the dataset. Following the data cleaning process, 11,653 samples remained for analysis [40]. This research exclusively focused on passenger cars, while motorcycles were excluded because of their low share in the traffic stream, which would have resulted in a non-representative and statistically insignificant sample. Additionally, heavy goods vehicles and buses were not considered, as they are subject to lower speed limits than passenger cars. It is also noteworthy that most heavy goods vehicles are equipped with tachographs, which can detect speed violations and, in turn, contribute to speed regulation. Furthermore, only off-peak traffic data were analysed to eliminate the influence of peak-hour driving behaviours. To ensure that only operating speeds under free-flow conditions were considered, a minimum time gap of five seconds between consecutive vehicles was applied. This criterion was implemented to prevent the influence of a preceding vehicle on the speed of the following vehicle [29]. After filtering the dataset accordingly, the 85th percentile speed was calculated for each travel direction at each cross-section.

2.1.2. Data from Fixed Traffic Counters Operated by Croatian Roads Ltd.

In addition to speed measurements obtained through field surveys, this research utilized data from fixed traffic counters operated by Croatian Roads Ltd., installed on state roads (and some county roads) in Croatia. These fixed counters are inductive loops that continuously monitor traffic flow (including traffic volumes, traffic flow structure, speeds, etc.) over the years (Figure 4). This approach provided data on operating speeds (V85) under the same conditions as those recorded through field surveys—namely, favourable weather conditions characterized by moderate temperatures, minimal cloud cover, and the absence of precipitation, wind, or fog.

From all available counter locations on state roads, a total of 11 cross-sections, corresponding to 22 distinct measurement locations, were selected. These locations were deemed the most suitable for this research as they are situated on tangent sections outside urban areas, evenly distributed across the country, and representative of various traffic volumes and speed limits (70, 80, and 90 km/h). It is important to note that raw data were not directly obtained; instead, this study relied on pre-analysed data provided by Prometis Ltd. [41], meaning that details about the original sample composition remain unknown. Additionally, the dataset used in this research corresponds to the most recent available year: 2021.

As shown in Appendix A, the locations encompass various speed limits as well as different values for AADT and ASDT. It should, however, be noted that AADT and ASDT data were only available for the entire cross-section (both directions of travel) [42]. Therefore, the AADT and ASDT values for each direction of travel were assumed to be half of the AADT and ASDT values. Data were gathered in both directions of travel, resulting in a total of 46 unique measurement locations. It is also important to note that there is no permanent speed enforcement in the broader areas surrounding the selected locations, which could influence the operating speeds of vehicles within the zones of the selected locations.

2.2. Variable Identification

For the purpose of this research, 24 variables were identified to evaluate their influence on vehicle operating speeds. These variables were selected based on their relevance to traffic flow dynamics, roadway geometry, and their potential impact on driver behaviour, ultimately affecting vehicle speeds. The selection of these variables was based on the influencing factors identified in research paper [40].

The analysis encompassed both roadway characteristics and traffic flow parameters. Roadway characteristics included elements such as lane width, speed limits, and longitudinal slope, which are fundamental in defining the physical constraints and perceived driving conditions on road tangents. Furthermore, traffic flow variables, such as AADT, ASDT, and the heavy goods vehicle share, were examined to determine their influence on speed variations. Each variable was assessed for its individual effect on operating speed for single-carriageway road tangents (Appendix B):

• Passenger side (3 categories: cut sections, guardrails, and flat area);
• Shoulder type (5 categories: no shoulder, extremely narrow, unpaved (earth/grass), gravel, and paved (asphalt/concrete));
• Edge line quality (4 categories: absent, poor, satisfactory, and excellent);
• Visibility (3 categories: poor, good, and excellent);
• Pavement quality;
• Lateral access density;
• Tangent length (up to measurement point);
• Tangent length (from measurement point);
• Speed limit;
• AADT;
• ASDT;
• Lane width;
• Longitudinal slope;
• Crash ratio;
• Design speed;
• Terrain type;
• Guardrail presence;
• Edge lane presence;
• Radius of previous curve;
• Radius of following curve;
• Road category;
• Overtaking allowed;
• Heavy goods vehicles share;
• Disruptive factor presence.

The roadside environment could influence driver behaviour and, consequently, operating speed. Therefore, three distinct passenger-side conditions were identified, namely, cut sections, guardrails, and flat areas. Similarly, for shoulder type, five categories were determined, namely, no shoulder, extremely narrow, unpaved (earth/grass), gravel, and paved (asphalt/concrete). Edge line quality was assessed subjectively in four categories: absent, poor, satisfactory, and excellent. Visibility, evaluated through forward sight distance based on terrain, slope, and vegetation, was classified as poor, good, or excellent. Lower visibility may have an impact on lower speeds due to perceived risk. Lateral access density was measured as the number of accesses (side approaches) in the previous 1 kilometre prior to the measurement location. Pavement condition, categorized as poor, satisfactory, or excellent, was evaluated through visible distress features, such as rutting and potholes; deteriorated surfaces tend to lower speeds. The lengths of tangent sections before and after the measurement point were precisely measured using QGIS 3.32.3 software. Speed limits were determined on-site or via Google Street View. Traffic volume and structure were examined through AADT and ASDT data from Croatian Roads Ltd. [42]. Lane width was measured on-site or via Google Earth Pro. Longitudinal slope was measured using field instruments on-site or via Google Earth Pro. On the other hand, the crash ratio indicator was derived by calculating the ratio of registered vehicles to speed-related crashes within the county where each site is located. This factor was defined as the ratio between the number of recorded speed-related crashes and the number of registered vehicles within a specific county in Croatia, both measured over a one-year period. This annualized metric served as a normalized indicator of speed-related crash frequency relative to the size of the vehicle fleet in each county. By accounting for the number of registered vehicles, the crash ratio enabled a more accurate comparison of road safety conditions across counties with differing vehicle ownership levels. The variable was introduced to capture the potential regional variations in driver behaviour, enforcement intensity, and road infrastructure quality, which may influence the prevalence of speed-related incidents. A higher accident ratio may suggest an increased likelihood of speed-related incidents, influencing driver behaviour and risk perception [40]. Design speed, as defined by Croatian regulations (Official Gazette 110/2001 and 90/2022) [43], is the highest speed at which complete driving safety is guaranteed in free-flowing traffic conditions along the entire section of a road, under optimal weather conditions and with proper maintenance of the road. Terrain type (flat, hilly, mountainous) was considered because of its effect on geometry and speed; operating speeds tend to decrease in more challenging terrain [44]. The presence of guardrails was also determined on-site or via Google Street. The same was performed for edge lane presence. Road category—state or county roads—was also considered, with state roads generally having better quality, equipment, and maintenance. The share of heavy goods vehicles, obtained from Croatian Roads Ltd. [42], was included. Finally, the presence of roadside features (disruptive factor presence), such as bus stops, lay-bys, rest areas, and pedestrian crossings, may prompt drivers to reduce speed due to the increased likelihood of stopping and pedestrian activity; it was measured on-site and via Google Street.

It should also be noted that although tangent lengths (up to and from the measurement point) and the radii of preceding and following curves were analysed as separate variables, these aspects are closely interrelated. The influence of previous curve radius on operating speed is likely to diminish as the distance between the curve and the measurement point increases. Conversely, a very short tangent after a sharp curve may still constrain speed due to the lingering effect of deceleration and acceleration phases. In this study, these variables were treated individually to isolate their direct statistical association with V85, but their combined effect warrants further investigation, potentially through interaction terms in future modelling efforts.

2.3. Data Analysis

The dataset utilized in this research underwent a thorough statistical examination using SAS JMP Pro 18 software. The aim of the analysis was to explore the relationships between the operating speeds (V85) and a range of geometric-, environmental-, and traffic-related variables along single-carriageway road tangents. This process was intended to uncover significant relationships that could enhance the understanding of speed patterns on such road segments.

The analytical procedure commenced with data pre-processing, during which anomalies and extreme outliers were identified and excluded to reduce the influence of potential measurement inaccuracies. To maintain consistency and eliminate the effects of vehicular interactions, only speed observations of passenger cars recorded under free-flow traffic conditions were included in the final analysis.

For continuous variables, the Shapiro–Wilk test was first conducted to assess the normality of their distributions. In cases where the assumption of normality was satisfied, the Pearson correlation coefficient was applied to evaluate the linear relationship between operating speed and the respective variable. For variables that deviated from normal distribution, Spearman’s rank-order correlation was employed, as it is suitable for identifying monotonic associations without requiring normality. A significance threshold of 0.05 was adopted for determining statistically significant correlations.

When analysing categorical variables, either a one-way analysis of variance (ANOVA) or an independent samples t-test was conducted to examine differences in operating speeds across variable categories. In instances where significant differences were detected, the Tukey–Kramer post hoc test was used to perform pairwise comparisons, enabling identification and quantification of specific category-level differences in operating speed. It should be noted that for the purpose of further model development, categorical variables (e.g., terrain type, visibility) were converted into dummy variables to enable correlation analysis.

Following the identification of the relationship between operating speed and the determined variables, multicollinearity analysis was conducted to examine interrelationships among the variables themselves. This step was essential for selecting appropriate variables to develop the prediction model. Subsequently, a model for predicting operating speeds on single-carriageway road tangents was developed using SAS JMP Pro 18 software. To develop the operating speed prediction model, a stepwise multiple linear regression analysis employing the forward selection method was conducted. This approach enabled the systematic inclusion of independent variables based on their statistical significance and correlation strength with the dependent variable (V85). At each iteration, one predictor was added to the model, beginning with the variable exhibiting the highest correlation with the operating speed. Only parameters meeting the 95% confidence interval threshold (p < 0.05) were retained, ensuring the inclusion of variables that contribute meaningfully to the model’s explanatory power. This method balances model simplicity with predictive accuracy by avoiding the inclusion of non-significant predictors. The general mathematical structure of the multiple linear regression model used in this study is presented in Equation (1).

$$Y=a_0+\sum b_i{X}_i$$

(1)

where Y = dependent variable, a₀ = regression constant, b_i = regression coefficient, and X_i = explanatory variables.

Although modern variable selection and predictive techniques, such as LASSO, Ridge regression, Elastic Net, and machine learning algorithms (e.g., random forests, gradient boosting), offer strong tools for handling high-dimensional data, their advantages are less pronounced in the context of this study. The dataset analysed comprised a relatively small number of observations (n = 36) and a moderate set of predictors, all of which were carefully pre-screened through statistical testing (Spearman’s ρ, t-tests, ANOVA) and multicollinearity diagnostics (correlation matrix, VIF values). This pre-selection ensured that only theoretically relevant and statistically stable variables entered the modelling stage, thereby reducing the risk of overfitting and minimising the need for penalization methods. Furthermore, while machine learning techniques can capture complex non-linear relationships, they often operate as “black box” models, providing limited insight into the explicit magnitude and direction of each predictor’s effect. In contrast, multiple linear regression yields easily interpretable coefficients expressed in physical units, allowing for direct quantification of the impact of each road and traffic characteristic on operating speed. Since the aim of this research is not solely accurate predictions but also an in-depth understanding of the causal and practical influence of each factor, the transparency, simplicity, and well-established inferential capabilities of multiple linear regression make it more appropriate than opaque, data-driven algorithms.

Furthermore, to assess the reliability of the model, validation was performed using a dataset split approach (randomly): ≈80% of the total dataset (18 cross-sections; 36 locations) was used for model training, while the remaining ≈20% (5 cross-sections; 10 locations) was reserved for model testing.

3. Results and Discussion

3.1. Relationship Between Operating Speed and Determined Variables

This section explores the relationship between operating speed (V85) and various variables on single-carriageway road tangents (using the training set—36 locations). The analytical methods were selected based on the nature of each variable type, as summarized in

Table 1. Statistical approach for determining variables on the training set (36 locations).

Variable	Type	Normal Distribution	Test Used
Passenger side	Categorical—Multi-level	---	ANOVA
Shoulder type	Categorical—Multi-level	---	ANOVA
Edge line quality	Categorical—Multi-level	---	ANOVA
Visibility	Categorical—Multi-level	---	ANOVA
Pavement quality	Categorical—Multi-level	---	ANOVA
Lateral access density	Continuous	NO	Spearman
Tangent length (up to measurement point)	Continuous	NO	Spearman
Tangent length (from measurement point)	Continuous	NO	Spearman
Speed limit	Categorical—Multi-level	---	ANOVA
AADT	Continuous	NO	Spearman
ASDT	Continuous	NO	Spearman
Lane width	Categorical—Multi-level	---	ANOVA
Longitudinal slope	Continuous	NO	Spearman
Crash ratio	Continuous	NO	Spearman
Design speed	Categorical—Multi-level	---	ANOVA
Terrain type	Categorical—Multi-level	---	ANOVA
Guardrail presence	Binary	---	t-test
Edge lane presence	Binary	---	t-test
Radius of previous curve	Continuous	NO	Spearman
Radius of following curve	Continuous	NO	Spearman
Road category	Binary	---	t-test
Overtaking allowed	Binary	---	t-test
Heavy goods vehicles share	Continuous	NO	Spearman
Disruptive factor presence	Binary	---	t-test

.

The normality of the numerical variables was initially evaluated using the Shapiro–Wilk test to assess whether the data conformed to a normal distribution. Since none of the continuous variables satisfied the normality assumption, the relationships between operating speed and these factors were examined exclusively using Spearman’s rank correlation, which is suitable for non-normally distributed data. The detailed results of the normality assessment conducted using the Shapiro–Wilk test are presented in Figure 5.

The Shapiro–Wilk test was applied to assess the distributional properties of the continuous variables considered in the analysis. The results indicate that none of the examined variables exhibited a normal distribution at the 0.05 significance level. Although some variables, such as longitudinal slope (W = 0.9147, p = 0.0089) and tangent lengths (W = 0.8848 and 0.8830, respectively), showed test statistics relatively close to the threshold of normality, the presence of statistically significant p-values suggests deviations from the normal distribution across all variables. This outcome is not uncommon in empirical transportation datasets, where variability in road geometry, traffic characteristics, and environmental conditions can lead to non-normal distributions. Accordingly, non-parametric methods, specifically Spearman’s rank correlation, were deemed more appropriate for examining the relationships between operating speed and the selected variables. This approach ensured the robustness and reliability of the statistical analysis, given the characteristics of the data. The results of the Spearman correlation analysis are presented in

Table 2. Spearman’s rank correlation for continuous variables (values marked with * indicate statistically significant results at the 0.05 level).

Continuous Variables	Spearman’s ρ	Significance Level (p-Value)
Lateral access density	−0.1119	0.5159
Tangent length (up to measurement point)	0.2278	0.1816
Tangent length (from measurement point)	0.1878	0.2728
AADT	−0.5669	0.0003 *
ASDT	−0.4713	0.0037 *
Longitudinal slope	−0.1246	0.4689
Crash ratio	0.3065	0.0690
Radius of previous curve	0.3099	0.0659
Radius of following curve	0.3313	0.0484 *
Heavy goods vehicle share	0.2645	0.1190

.

The results of the Spearman’s rank correlation analysis provide an overview of the relationships between operating speeds and the selected continuous variables. Several statistically significant correlations were identified at the 0.05 significance level, highlighting the influence of specific traffic and geometric factors on speed behaviour.

The AADT and ASDT show statistically significant negative correlations with operating speed, with Spearman’s ρ values of −0.5669 (p-value = 0.0003) and −0.4713 (p-value = 0.0037), respectively. These findings suggest that higher traffic volumes, particularly during peak summer periods, are associated with reduced operating speeds. This is consistent with expectations, as increased vehicle flow can contribute to congestion and reduced driver flexibility, thereby limiting the ability to maintain higher speeds.

A significant positive correlation was also observed between the radius of the following curve and operating speed (ρ = 0.3313, p-value = 0.0484), indicating that vehicles tend to maintain higher speeds when approaching tangents followed by curves with larger radii. This can be attributed to the increased comfort and perceived safety when navigating through smoother, less abrupt horizontal curvature.

Other variables such as tangent lengths, crash ratio, and radius of the previous curve showed moderate correlation coefficients (ρ between 0.18 and 0.31), but their associations were not statistically significant at the 0.05 level. While these results suggest potential trends, they should be interpreted with caution due to the lack of statistical confirmation. Similarly, variables such as lateral access density and longitudinal slope exhibited weak correlations with operating speed and were also not statistically significant. The positive coefficient for heavy goods vehicle share (HVS) may appear counterintuitive, since higher proportions of slower-moving vehicles are generally expected to reduce operating speeds. However, several context-specific factors may explain this outcome:

• Free-flow measurement conditions: Data were collected only under free-flow conditions, with passenger cars having no immediate leading vehicle within five seconds. In this situation, higher HVS values might reflect lower overall traffic density for passenger cars, enabling them to travel faster.
• Driver behaviour adaptation: In mixed traffic, where heavy goods vehicles are common, passenger car drivers may accelerate more aggressively during overtaking opportunities, especially on long tangents with overtaking permitted, thereby raising the observed V85.

Comparable anomalies have been reported in some European operating speed models [22,35], where high commercial traffic was linked to higher free-flow car speeds due to road alignment quality and overtaking opportunities.

In conclusion, the analysis identified AADT, ASDT, and radius of the following curve as the most relevant continuous predictors of operating speed among those considered. The use of Spearman’s rank correlation ensured that the relationships were evaluated reliably, given the non-normal distribution of the data. These results reinforce the need to account for both traffic volumes and road geometry in operating speed assessments.

To evaluate whether operating speeds significantly differed between the two groups of each binary categorical variable, an independent samples t-test was performed. The results of the t-test analysis are presented in Figure 6.

The t-test analysis did not reveal any statistically significant differences in operating speeds for the tested variables at the 0.05 significance level. For instance, the presence of a guardrail showed a mean speed difference of −6.135 km/h (t = −0.735, p-value = 0.4970), which was not statistically significant. Similarly, for edge lane presence, the observed mean difference was 4.556 km/h (t = 0.730, p-value = 0.4805), and for road category (county road vs. state road), the difference was 4.446 km/h (t = 0.767, p-value = 0.4597), both falling short of statistical significance.

Additionally, the variables overtaking allowed and presence of a disruptive factor yielded negligible mean differences of −0.606 km/h and −0.656 km/h, respectively, with extremely low t-ratios and p-values far above the conventional threshold (p-value = 0.9306 and p-value = 0.9523, respectively). These results suggest that, within the observed sample, these factors did not exert a measurable or consistent influence on operating speed.

It should be noted that for some binary variables, the number of observations within one of the categories was relatively small. For example, only a few sites were found where a disruptive factor was present in the dataset. This imbalance reduces the statistical power of the t-tests, meaning that non-significant results should be interpreted with caution. While the absence of significance may reflect a true lack of effect, it is also possible that some relationships could emerge as significant with a more balanced sample across categories.

Although none of the tested binary variables demonstrated statistically significant effects, these findings remain informative. The absence of significant differences may be attributed to the relatively small sample size within certain subgroups, limiting the statistical power of the t-tests. Furthermore, it is possible that the influence of these categorical features on operating speed is more nuanced and interacts with other variables not captured in this binary classification.

In summary, while the t-test results did not yield statistically significant differences in operating speed across the tested binary variables, the analysis still contributes to a more comprehensive understanding of their individual roles. These findings underscore the importance of considering multivariable interactions in future modelling and suggest that certain categorical elements may influence speed behaviour indirectly or in combination with other geometric and traffic characteristics.

For the categorical variables with more than two levels, the association between operating speeds and the various categories was evaluated using analysis of variance (ANOVA). This method enabled the identification of statistically significant differences in operating speeds among the different groups. The results of the ANOVA are presented in Figure 7.

The ANOVA results revealed that edge line quality, lane width, and design speed (Vp) were significantly associated with variations in operating speeds. Specifically, edge line quality showed a statistically significant effect (F = 4.297, p-value = 0.0118), indicating that better or more visible edge markings may influence drivers’ speed choices. Similarly, lane width and design speed both exhibited statistically significant effects (F = 4.785, p-value = 0.0073), which aligns with expectations, as wider lanes and higher design speeds typically promote higher operating speeds due to enhanced comfort and perceived safety.

Conversely, variables such as passenger side environment, shoulder type, visibility conditions, pavement quality, posted speed limit, and terrain type did not show statistically significant differences in operating speeds across their respective categories. For instance, shoulder type (F = 1.629, p-value = 0.1919) and terrain type (F = 2.305, p-value = 0.1156) approached the threshold of significance, suggesting a possible trend that may warrant further investigation with a larger dataset.

Interestingly, speed limit did not significantly influence operating speeds (F = 0.199, p-value = 0.8205), which may point to a divergence between regulatory limits and actual driver behaviour. This observation reinforces the relevance of design consistency and perceived road characteristics over formal signage in shaping speed behaviour. Overall, the ANOVA analysis supports the conclusion that certain infrastructure-related factors, particularly lane width, design speed, and edge line quality, have a measurable impact on operating vehicle speeds.

To further investigate the statistically significant effects identified through ANOVA, a Tukey–Kramer post hoc test was performed for the relevant variables: edge line quality, lane width, and design speed (Figure 8). This procedure enabled detailed pairwise comparisons between the levels within each factor, providing insight into where significant differences in operating speeds occurred. Importantly, the Tukey–Kramer method accounts for the increased risk of Type I error, that is, the likelihood of incorrectly identifying a significant difference when none exists that arises from conducting multiple comparisons. By controlling the family-wise error rate, the test ensures that the conclusions drawn about pairwise differences are statistically sound and not due to random variation, thereby strengthening the reliability of the findings.

The results revealed noteworthy pairwise distinctions within specific variable levels, offering more nuanced insights into how these infrastructure elements influence driver behaviour. For edge line quality, the results indicate that road sections with excellent markings were associated with the highest mean operating speed (108.04 km/h), followed by satisfactory, absence, and poor categories. A statistically significant difference (p = 0.0358) was observed between the excellent and absence categories, suggesting that the presence and visibility of high-quality edge lines may positively influence driving speed by improving perceived roadway guidance and confidence. Although no other pairwise comparisons reached statistical significance, some (e.g., excellent vs. poor) showed notable numerical differences.

Regarding lane width, roads with a 3.00 m lane width were associated with significantly higher operating speeds compared to 3.25 m lanes (difference = 18.84 km/h, p = 0.0039). Interestingly, this result may seem counterintuitive, as wider lanes are typically associated with higher speeds. One possible explanation is that 3.00 m lanes may appear visually narrower, prompting more cautious driving environments to be built around them (e.g., better alignment or less traffic complexity).

A similar trend was observed in the analysis of design speed (Vp) categories, where a statistically significant difference was again found between 70 km/h and 80 km/h road sections (difference = 18.84 km/h, p = 0.0039). Interestingly, the 70 km/h sections exhibited the highest mean operating speed, suggesting that posted or designed speeds may not always reflect actual driver behaviour, potentially due to factors like road alignment or perceived safety. These results also highlight the need to consider the difference between what a road is in objective terms and how it appears to the driver. While geometric and roadside characteristics can be quantified precisely, their influence on speed choice depends on whether drivers perceive them as relevant to safety. For example, a tangent section may contain numerous concealed driveways or minor intersections that do not register as potential hazards, leading to higher speeds despite objectively increased risk. Likewise, drivers may overestimate forward visibility (due to alignment or roadside vegetation), maintaining speed even when actual sight distance is limited. This aligns with the Perception of Possible Interaction (PPI) concept, which emphasizes that drivers adapt their speeds primarily when they perceive a tangible likelihood of interaction with other road users [45]. In this research, the disruptive factor presence variable captures a related aspect, representing visible roadside elements such as bus stops or pedestrian crossings that are more likely to trigger speed adjustments. However, when such elements are not perceived, the operating speed may remain unaffected, helping explain why some objectively important variables show weaker statistical associations with operating speed.

3.2. Operating Speed Prediction Model Development

The selection of independent variables for inclusion in the multiple linear regression model was carried out through a structured and methodologically sound process. First, the potential relevance of each variable was evaluated based on its observed statistical association with the dependent variable, operating speed (V85), using correlation analysis, t-tests, and ANOVA, as presented in the previous section. Variables that showed either statistically significant relationships or strong theoretical justification grounded in traffic engineering principles were retained for further consideration.

To ensure that the final model met the statistical assumptions of linear regression, especially with regard to multicollinearity, all categorical variables were first converted into dummy (binary) variables. This step was essential, as multicollinearity diagnostics, such as correlation matrices and variance inflation factor (VIF) analysis, require all input variables to be expressed numerically. The transformation enabled the inclusion of qualitative information (e.g., edge line quality, shoulder type) in a statistically appropriate manner while preserving the interpretability of categorical effects. The coding procedure, along with the descriptive statistics of all continuous and dummy variables, is presented in Appendix B. Variables—Descriptive Statistics.

Following the numeration process, a correlation matrix (Appendix C. Correlation Matrix) was generated to examine potential multicollinearity among the full set of candidate variables. Variables exhibiting strong pairwise correlations (|r| > 0.5) were critically assessed, and where conceptual or statistical overlap was identified, the less explanatory or redundant variable was excluded. This approach allowed for the construction of a final model that avoids inflated standard errors, ensures stability of coefficient estimates, and retains theoretical coherence. Through this sequence, transformation, diagnostic screening, and informed selection, a balanced and statistically sound foundation for model development was established. Several notable associations were observed, including a very strong correlation between average summer daily traffic (ASDT) and Average Annual Daily Traffic (AADT) (r = 0.94), which measure similar aspects of traffic volume; only ASDT was included in the model to avoid redundancy. Moderate correlations were also found between crash ratio (CR) and AADT (r = 0.46), as well as between heavy goods vehicle share (HVS) and terrain-related or pavement condition variables, reflecting expected relationships in real-world road environments. Edge line quality categories “excellent” and “satisfactory” were negatively correlated (r = −0.65) due to their mutual exclusivity, but their inclusion as separate dummy variables was methodologically acceptable, as the stepwise procedure ensured only one was retained in any specific model iteration. Overall, no pairwise correlations among the final predictors exceeded levels that would indicate harmful multicollinearity, supporting the stability of the estimated coefficients.

To further assess multicollinearity, variance inflation factor (VIF) values were calculated for each variable included in the final model. All VIF values were well below the commonly accepted threshold of 5, with the highest being 3.14, confirming the absence of harmful multicollinearity and supporting model stability. Special consideration was given to the two dummy variables representing different categories of edge line quality: “excellent” (VIF = 3.14) and “satisfactory” (VIF = 2.95). These variables are, by definition, mutually exclusive and were found to be negatively correlated during the initial multicollinearity analysis, indicating conceptual overlap. However, this issue did not compromise the model’s validity, as the stepwise regression procedure systematically selected only one of these mutually exclusive dummy variables in any given model iteration. Therefore, both categories could be safely included in the pool of candidate predictors without risking multicollinearity-induced overfitting. Furthermore, each variable was retained based on its high statistical significance (p-value < 0.001 for “excellent”; p-value = 0.0002 for “satisfactory”) and its distinct practical interpretation in the real-world road environment. Their inclusion enhanced the model’s explanatory capacity while preserving statistical reliability and conceptual coherence.

Previous operating speed models for tangent sections [19,29,36,38] have frequently identified posted speed limit, lane width, pavement condition, and roadside environment as significant variables. In contrast, the present model excluded posted speed limit and lane width (despite their significance in ANOVA) because their effects were not consistently strong once other factors were controlled for in the regression. Instead, the final model included variables such as shoulder type—gravel, edge line quality, and edge lane presence, factors less frequently emphasized in earlier tangent-specific models but consistent with some European studies that highlight pavement markings and cross-section delineation as important speed cues [22,30]. The inclusion of ASDT and crash ratio aligns with recent Italian and Spanish models that integrate traffic exposure and regional safety performance into speed prediction [27,32]. The heavy goods vehicle share predictor is relatively uncommon in tangent-specific models, suggesting a contribution unique to the Croatian context that is promising to be applicable in other countries as well.

The final model (

Table 3. Regression results—model.

Variable	Estimate	Std Error	p-Value	Lower 95%	Upper 95%	VIF
Intercept	78.46	4.18	<0.0001	69.86	87.06	---
Shoulder type—Gravel	12.32	2.43	<0.0001	7.32	17.32	1.46
Edge line quality—Excellent	26.99	3.22	<0.0001	20.36	33.61	3.14
Edge line quality—Satisfactory	16.24	3.79	0.0002	8.46	24.02	2.95
Pavement quality—Excellent	−5.25	2.32	0.0325	−10.02	−0.47	1.75
ASDT	−0.0007	0.00	0.0016	−0.00104	−0.00027	1.32
Crash ratio	1795.02	467.25	0.0007	834.57	2755.47	1.20
Edge lane presence	−14.63	3.10	<0.0001	−21.00	−8.26	2.36
Overtaking allowed	7.59	2.70	0.0093	2.04	13.14	1.50
Heavy goods vehicle share	80.11	11.08	<0.0001	57.34	102.88	1.47
R2	0.89
R2Adj	0.86
RMSE	5.24
Mean of response	103.08
SSE	713.20
DFE	26
Cp	10
Observations	36

) achieved a high level of explanatory power, with a coefficient of determination R² of 0.89 and an adjusted R² (R²_Adj) of 0.86, indicating that the model accounts for 89% of the variance in observed operating speeds. The Root Mean Square Error (RMSE) of 5.24 and a relatively low Sum of Squares for Error (SSE) of 713.20 further affirm the model’s accuracy. All included variables were statistically significant at the 95% confidence level (p < 0.05), supporting the stability of their relationship with the dependent variable.

The Degrees of Freedom for Error (DFE), calculated as the difference between the number of observations (36) and the number of estimated parameters (including the intercept), equals 26, indicating a balanced model with sufficient residual degrees of freedom. Furthermore, the model’s Mallows’ Cp statistic is 10, which closely matches the number of parameters in the model. This suggests that the model strikes a good balance between complexity and explanatory power, avoiding both underfitting and overfitting.

3.3. Operating Speed Prediction Model Validation

Residual analysis (Figure 9) was conducted to assess the model’s validity and confirm compliance with linear regression assumptions. The residuals were found to be normally distributed, supported visually by histogram and Q-Q plots, and statistically confirmed by the Shapiro–Wilk test (

Table 4. Shapiro–Wilk test results (residual distribution).

	W	p-Value
Shapiro–Wilk	0.95	0.12

), suggesting no significant deviation from normality. Moreover, scatter plots of residuals versus fitted values demonstrated a uniform dispersion around the zero line, indicating homoscedasticity and the absence of systematic error patterns. The predicted V85 values were consistently close to the observed values, reinforcing the model’s reliability.

To further verify the assumption of constant variance of residuals, a Breusch–Pagan test was conducted. In this procedure, the squared residuals from the fitted operating speed regression model were regressed on the same set of predictor variables used in the main model. The overall F-test (

Table 5. Auxiliary regression for heteroscedasticity testing—Breusch–Pagan test (dependent variable: squared residuals from the main regression model; predictors: same as in main model).

Source	Sum of Squares	Mean Square	F Ratio
Model	2630.03	292.23	0.77
Error	9931.24	381.97	Prob > F
C. Total	12,561.28		0.65

) from this auxiliary regression yielded a p-value of 0.65, indicating no statistically significant relationship between the residual variance and the predictors. This result confirms that the residuals exhibit homoscedasticity, meaning the spread of prediction errors is approximately constant across the range of fitted values. Therefore, the standard errors, confidence intervals, and p-values reported for the model coefficients can be considered reliable.

To evaluate the predictive capability and generalizability of the developed operating speed prediction model, a validation process was carried out using data from 10 additional locations (corresponding to 5 road cross-sections) that were not included in the previous model development phase. This external validation was essential to assess the reliability and applicability of the model under new conditions.

As previously described, the model was initially developed using 36 locations (from 18 cross-sections), while validation was performed on 10 locations (5 additional cross-sections), comprising a representative set of road tangent conditions. For each validation site, the observed operating speed (V85-Test) was compared to the model-predicted operating speed (V85-Predicted). The validation results are shown in

Table 6. Model validation results.

Data Collected from	Location Name	Direction	V85-Test [km/h]	V85-Predicted [km/h]	Absolute Difference [km/h]	Absolute Deviation [%]
Field survey	Dubrovnik	towards Dubrovnik	76	80.99	4.99	6.56%
Field survey	Varaždin	from Varaždin	99	101.23	2.23	2.25%
Field survey	Virovitica	from Novaki	97	96.24	0.76	0.78%
Field survey	Virovitica	towards Novaki	97	96.24	0.76	0.78%
Field survey	Zadar	towards Zadar	96	97.24	1.24	1.29%
Fixed traffic counter	Paka	towards Novi Marof	90	94.20	4.20	4.67%
Fixed traffic counter	Brlenić	towards Draganić	95	92.28	2.72	2.86%
Fixed traffic counter	Kašina	towards Soblinec (DC3)	88	89.85	1.85	2.11%
Fixed traffic counter	Trema	towards Sveti Ivan Žabno	100	100.63	0.63	0.63%
Fixed traffic counter	Sibinj Krmpotski	towards Senj	97	99.32	2.32	2.39%
Average absolute deviation						2.43%

.

To better reflect prediction accuracy regardless of direction, the absolute deviation was also computed. The average absolute deviation across all validation sites was 2.43%, indicating that, on average, the model’s predictions deviated by less than 3% from the actual observed speeds. The largest deviation observed was 6.56%, and the most accurate prediction had a deviation as low as 0.63%. Importantly, the deviations were balanced on both sides of the zero line, with no systematic overestimation or underestimation, which further confirms the neutrality and reliability of the model across different locations. The validation outcomes confirm the predictive strength and stability of the model under different conditions. An average absolute deviation of 2.43% supports the model’s practical usability for real-world applications, such as road design evaluation, speed management, and safety assessments. It also suggests that the variables included in the model are well aligned with the underlying factors influencing operating speed on single-carriageway road tangents. Minor deviations observed in some locations (especially those exceeding 5%) may be attributed to local features not fully captured by the model, such as temporary environmental influences, driver behaviour variability, or micro-geometric irregularities. Nevertheless, the overall validation confirms that the model generalizes well beyond the development dataset.

4. Conclusions

This research successfully developed a multiple linear regression model for predicting vehicles’ operating speeds (V85) on single-carriageway road tangents, based on comprehensive field and sensor-based data collected from 23 cross-sections (46 locations) in Croatia. The primary objective was to identify and determine the influence of road geometry and infrastructure elements, as well as environmental and traffic characteristics, on actual driving speeds under free-flow conditions.

Before proceeding with model development, a comprehensive analysis was undertaken to explore the relationships between operating speed and a range of explanatory variables, using appropriate statistical tests based on variable type and distribution. Spearman’s rank correlation identified several continuous variables with significant or near-significant associations with operating speed (V85), including AADT and ASDT (both negatively correlated) and radius of adjacent curves (positively correlated). These findings suggest that higher traffic volumes tend to reduce operating speeds, while greater curvature radii are associated with increased speeds. Binary variables assessed through independent-sample t-tests did not yield statistically significant effects, although some, such as the presence of a guardrail or road category, showed notable mean differences that may reflect trends in larger samples. The absence of significance here highlights the complexity of interactions between driver behaviour and infrastructure features. ANOVA analysis of categorical variables with more than two levels revealed significant effects for edge line quality, lane width, and design speed, factors that consistently influence drivers’ speed perception and comfort. Follow-up Tukey–Kramer tests confirmed statistically significant pairwise differences, particularly between excellent and absent edge lines, and between narrower and wider lanes, providing a nuanced understanding of how specific infrastructure elements impact speed. These results collectively underscore the importance of incorporating both geometric and traffic-related variables into predictive modelling and demonstrate that certain road design features have measurable, interpretable effects on vehicle operating speed.

For model development, the dataset was partitioned into a training set (18 cross-sections, 36 locations) and a test/validation set (5 cross-sections, 10 locations). The training set was used for variable screening, multicollinearity testing, and stepwise regression analysis using SAS JMP 18 Pro. A total of nine variables were selected for inclusion in the final model:

• Shoulder type—gravel;
• Edge line quality—excellent;
• Edge line quality—satisfactory;
• Pavement quality—excellent;
• ASDT;
• Crash ratio;
• Edge lane presence;
• Overtaking allowed;
• Heavy goods vehicles share.

The model achieved a high level of predictive accuracy, with an R² of 0.89, R²_Adj of 0.86, and RMSE of 5.24, demonstrating excellent explanatory power. The residuals showed no violation of linear regression assumptions, as confirmed by the Shapiro–Wilk test and residual distribution analysis.

Validation using the reserved test set confirmed the model’s generalizability, with an average absolute deviation of just 2.43% between predicted and observed V85 values. This suggests strong consistency and practical reliability across different road environments, affirming the model’s usefulness in speed prediction for design assessment, safety evaluation, and policy formulation.

The developed model offers direct applicability in road design and traffic management. For instance, engineers could use it during the geometric design phase to assess whether a proposed tangent’s characteristics (e.g., shoulder type, edge line quality) align with desired operating speeds, reducing the need for postconstruction corrections. Additionally, transportation agencies might employ the model to identify high-risk segments where predicted V85 significantly exceeds design speeds, prioritizing safety interventions such as speed cameras or lane narrowing.

Accurate operating-speed prediction has direct economic value across the road asset life cycle. At the design stage, more reliable operating speed estimates reduce costly after-construction corrections that stem from a mismatch between design/posted speeds and actual operating speeds. Most importantly, at the safety/economic externality level, even small systematic reductions in operating speed have outsized impacts on crash harm: as previously stated, a 1% change in mean speed is associated with ~4% change in fatal crash risk and ~3% change in serious injury risk. Therefore, closing the gap between predicted and real operating speeds translates into meaningful reductions in expected crash losses. Indeed, from a project-delivery perspective, improved predictions also enable targeted treatments (e.g., high-contrast edge lines, shoulder type upgrades, etc.) precisely where predicted operating speed materially exceeds design intent, instead of blanket, network-wide measures. Such targeting reduces capital outlays and recurring maintenance/enforcement expenditure while supporting credible speed limits and self-explanatory roads.

Nevertheless, several limitations must be acknowledged:

• Geographic specificity: The model is based solely on Croatian road infrastructure and driver behaviour. While it reflects national conditions accurately, extrapolation to other countries or regions should be performed cautiously and with potential local calibration. Also, it should be taken into account that the share of foreign drivers within the scope of this research is unknown.
• Environmental constraints: The analysis only included data collected under favourable weather and daylight conditions. Variables such as nighttime visibility, rain, snow, or fog were not included, potentially limiting applicability to broader scenarios.
• Data resolution limitations: Some variables, such as the crash ratio, were defined at the county level, which may obscure local safety nuances. Additionally, heavy goods vehicle share and traffic data were averaged across cross-sections, potentially smoothing out micro-level effects. Therefore, more precise data on these variables is necessary for future research.
• Variable interpretability and borderline significance: Although tangent length (from the measurement point) had a p-value slightly above 0.05, it was retained to preserve model integrity. Nonetheless, its individual explanatory impact remains modest and may benefit from further investigation into future models.
• Model structure: The use of stepwise linear regression, though effective and statistically valid, has inherent limitations in capturing nonlinearities or interaction effects among predictors. More advanced techniques, such as generalized additive models, random forests, or quantile regression, could be explored in future research, especially with larger or more granular datasets.
• This study included a sample of only 46 locations; while sufficient for traditional statistical modelling, it raises the question of how the results might evolve with a substantially larger dataset. Future research could explore the development of predictive models based on big data sources, such as those provided by platforms like Google or TomTom, which may offer extensive, high-resolution traffic and speed data across time and space. Such an approach might enable more nuanced modelling and potentially reveal patterns and interactions that are not detectable through conventional data collection methods.
• Another direction involves testing the model’s transferability to other road categories and geometric contexts. While this study focused on single-carriageway tangents, applying and calibrating the methodology to dual-carriageways and tangent–curve sequences could reveal whether different lane configurations, median treatments, or curve proximities alter the relative influence of geometric and traffic variables on operating speed.
• In addition to the limitations already noted, it is important to recognize that certain findings from this study diverge from widely reported relationships in the operating speed literature. A notable example is the effect of lane width: while many previous studies have found that wider lanes are generally associated with higher operating speeds [29,46], this analysis indicated the opposite trend, with narrower lanes linked to slightly higher speeds on the sampled tangents. This result may reflect context-specific influences such as driver expectancy, the interaction between lane width and shoulder type, or site selection bias within the dataset. It may also indicate that in certain environments, narrower lanes occur on higher-standard tangents with favourable alignment and visibility, thus supporting higher speeds despite the reduced lane width. Such findings should be interpreted cautiously and verified through further research in other settings before being generalized. This reinforces the need for broader data collection across different road classes, traffic compositions, and environmental conditions to determine whether such inverse relationships represent local anomalies or point to a need to revisit widely accepted design speed assumptions in the literature.
• Finally, incorporating human factors in a more explicit and measurable way offers potential to improve predictive accuracy. Variables that reflect drivers’ perceived likelihood of interaction with other road users, such as roadside activity, overtaking frequency, or sight-line interruptions, could be operationalized through on-field observation or video analytics. By capturing not only the infrastructural and physical attributes of the road but also how drivers interpret them, future models could better explain deviations between posted limits, design speeds, and operating speeds. Integrating these enhanced models into practical decision support tools for engineers and planners would help bridge the gap between predictive research and everyday road design practice.

In conclusion, this study presents a methodologically sound, empirically validated, and highly predictive model for estimating operating vehicle speeds on single-carriageway road tangents. It bridges a notable research gap in speed behaviour modelling on tangent sections, offers actionable insights for roadway design and road safety planning, and establishes a solid foundation for further development of context-specific speed prediction tools. Future studies should aim to enhance the model’s breadth by incorporating temporal, environmental, and behavioural variables and expanding its validation to international datasets.