Spatial econometric models to understand factors affecting older drivers at accident hotspots

1. Introduction

Older drivers accounted for 25% of all car driver fatalities despite representing a smaller proportion of the driving population in the year 2023 (Department for Transport 2024). The killed or seriously injured (KSI) casualties involving at least one older driver in road accidents accounted for 36% of all KSI casualties (Department for Transport 2024). Several factors, such as declining vision, mental or physical illness and disabilities, and age-related decline, contribute to the accident risk of older drivers. The Older Driver Factsheet 2023 of the Reported Road Casualties Great Britain identified that failure to look properly, mental or physical illness, disabilities, and age-related decline were responsible for KSI casualties among 28%, 13%, and 11% older drivers, compared to 19%, 8%, and 2% other drivers, respectively (Department for Transport 2024). A total of 47% KSI casualties among older drivers, involved in road accidents, were located at T, Y, or staggered junctions (28%), crossroads (9%), roundabouts (5%), and other junctions (5%) in the UK (Department for Transport 2024). There are certain areas in the UK identified as accident hotspots for older drivers.

Older drivers are disproportionately involved in accidents in certain geographical areas, especially when turning across oncoming traffic, due to difficulties in judging the speed and distance of other vehicles and the complexity of navigating junctions. In addition, the complex road networks and varying traffic conditions present challenges for older drivers, emphasizing the need for heightened awareness and caution. The spatial distribution of road accidents involving older drivers has received growing attention due to demographic shifts and the increasing number of older people retaining driving privileges. As aging affects physical and cognitive functions, understanding the spatial patterns and contributing factors of road accidents involving older drivers is essential for developing effective policy and planning interventions. This paper applies spatial lag and spatial error models to develop the spatio-temporal accident causation models for older drivers in the West Midlands region in the UK.

2. Materials and Methods

This paper analyzed a total of 95,092 road accident data in the West Midlands region of the UK during the period of 2006 to 2016. The West Midlands region observed a steady and sustained reduction in deaths and serious injuries between 2007 and 2013 due to improved vehicle safety, driver training, and road safety education programs and campaigns (Transport for West Midlands 2023). However, road accidents and casualties started to increase on the region’s road network since 2013 (Transport for West Midlands 2023). The increase in road accidents and casualties causes significant risk to the health, safety, and well-being of road users. The road accident data from the Department for Transport showed that 11.24% of total accidents involved older drivers (age 60 years and above) during this period, of which 8.12% were male and 3.11% were female drivers. Most of the road accidents occurred in some specific areas in the West Midlands. The car insurance experts analyzed the road accident data from 2013 to 2020 in all local authorities across the West Midlands (Birmingham World 2022). Warwickshire area had the highest rate of accidents (2.28 accidents per 1000 people that was 34% higher than the West Midlands average) followed by the Birmingham city (2.26 accidents per 1000 people), Stoke on Trent (1.96 accidents per 1000 people), Herefordshire (1.84 accidents per 1000 people), Sandwell (1.8 accidents per 1000 people), and Wolverhampton (1.8 accidents per 1000 people) (Birmingham World 2022). Telford, Solihull, and Dudley were the top three safest areas in the West Midlands with approximately 35% fewer road accidents than the average of the West Midlands region (Birmingham World 2022).

The risk levels of gender, journey purposes, junction location, human and physical interventions at pedestrian crossing, lighting condition, weather condition, road surface condition, existence of hazards on carriageway, time severity, and type of junctions are the determinants of road accidents involving older drivers (Fig. 1). These categorical determinants of road accidents involving older drivers were numerically represented using dummy variables in the spatial econometric models. For example, the journey purpose affects road accident risk, with commuting and work-related trips contributing a significant proportion of road accidents. Fig. 1 shows that the ‘work’ category of journey purpose was represented by 1, followed by ‘social and leisure’ trip as 2, and ‘taking pupil from and to school’ as 3, based on their contribution to road accidents (Febres et al. 2019).

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Fig. 1.

Factors affecting road accidents involving older drivers.

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The ‘junction location’ variable was categorized based on the locational vulnerability to road accidents. The ‘junction location’ variable is categorized as 0 for ‘no physical crossing facilities within 50 meters,’ 1 for ‘pelican, puffin, toucan or similar non-junction pedestrian light crossing,’ 2 for ‘pedestrian phase at traffic signal junction,’ 3 for ‘footbridge or subway,’ 4 for ‘Zebra crossing’ and 5 for ‘Central refuge’ (Fig. 1). Similarly, human and physical interventions at pedestrian crossings were numerically represented using dummy variables (Fig. 1). Poor lighting conditions reduce contrast sensitivity in older drivers making it harder for them to spot complete urban road environments such as stopping sight distances, vehicle’s speed and stationary cars or breakdown vehicles at junctions (Amin 2020). Sometimes, poor lighting conditions combined with extreme weather, poor road surface, and hazardous carriageways increase the accident risk of older drivers (Amin 2020). Fig. 1 shows the numerical values of the dummy variables for lighting condition (light), weather condition (weather), road surface condition (road surface), level of carriage hazard (carriage hazard), time severity (time severity), and road or junction type (road junction type) depending on the accident severity.

Traditional statistical models overlook the spatial dependence that naturally exists in geographically referenced data, potentially leading to biased or inefficient estimates. In contrast, spatial econometric models such as spatial lag and spatial error models incorporate the spatial autocorrelation among accident records, acknowledging that accidents in one location may be influenced by conditions in neighboring areas. This is particularly important in road accident studies, where road condition, environment, and driver’s behavior exhibit strong spatial patterns. By portraying these spatial interactions and heterogeneities, spatial econometric models enhance the accuracy of accident risk assessments and contribute to a comprehensive understanding of the underlying mechanisms for accident occurrences involving older drivers. The Spatial Autoregressive (SAR) model is a fundamental spatial econometric technique used to include spatial dependence in cross-sectional or panel data. In the context of road accidents, the SAR model accounts for the possibility that the number or severity of accidents involving older drivers in one geographical unit may be influenced by accident occurrences involving older drivers in neighboring units. This spatial spillover effect is introduced through a spatially lagged dependent variable that incorporates a weighted average of accident outcomes in adjacent areas (Equation 1). The spatial lag and spatial error models were used in spatial statistics to estimate the spatial dependency of independent variables using maximum likelihood methods and to ensure that errors are not spatially autocorrelated, respectively.

Where Y is the number of accidents involving older drivers, ρ is the SAR parameter, $W_y$is the spatial weights matrix of accidents involving older drivers, β is a vector of coefficients of factors affecting road accidents involving older drivers, and ε is the random error term. The term ρW_y represents the spatially lagged dependent variable (road accidents involving older drivers), incorporating the influence of neighboring accidents on the accidents involving older drivers. The vector of error term (ε) has been shown in Equation (2), where $u$ is a vector of errors with variance ${\sigma }^2$ and $\lambda$is the SAR parameter.

The spatial autocorrelation, ρ, is the correlation among accidents in one location and of adjacent accidents. A high number of accidents in one location geographically clusters near a high number of accidents, medium near medium, and low near a low number of accidents. An n-by-n binary geographic connectivity/weights matrix ($W_y$) can identify the neighboring values of road accidents involving older drivers. Thiessen polygons are created as a polygon shape file derived from a point shape file of the road accident database. Each polygon surrounds the location of road accident so that every accident location within it is closer to that point than to any other accident location. Thiessen polygons provide a basis for calculating contiguity-based spatial weights for point data, as the shared boundaries of Thiessen polygons of road accident locations define the spatial adjacency of road accidents. The spatial weight matrix was created using queen contiguity, defining the neighbors of Thiessen polygons of accident locations as spatial units sharing a common edge or a common vertex.

The spatial autocorrelation affects the statistical analysis by altering the variance of variables. For example, positive spatial autocorrelation increases the likelihood of wrongly rejecting the null hypothesis. This reflects the presence of unnecessary information in geo-referenced data, reducing the contribution of each observed data to statistical calculations. In addition, spatial autocorrelation describes overall spatial patterns supporting geographic predictions and enabling the identification of significant anomalies. A significant ρ suggests that accidents involving older drivers in one area are not independent of those in proximate areas, and thus spatial dependence must be explicitly accounted for to avoid biased or inefficient estimates (Anselin 1995). Moran’s I and Gary’s C are commonly used spatial statistics to calculate the spatial autocorrelation. Geary’s C measures the local spatial autocorrelation, and Moran’s I calculates the global spatial autocorrelation (Ceadserv1.nku.edu, 2019). Moran’s I ranges from -1 to +1, where the values between 0 and -1 represent a negative spatial autocorrelation and between 0 and +1 represents a positive spatial autocorrelation. This paper used Moran’s I to estimate the spatial autocorrelation of accidents involving older drivers (Equation 3).

Where N is the number of accidents neighboring at i and j locations, x is the accident records, $\overline{x}$ is the average of accident records, $w_{ij}$ is the spatial weight and $W$ is the sum of all $w_{ij}$.

2.1 Data analysis

The spatial pattern of accident data involving older drivers was analyzed using the LISA map (Fig. 2a) and Moran’s I. The X-axis and Y-axis of the LISA scatter plot represent the standardized value of accidents (accidentED) and spatial lag of accidents (lagged accidentED) (i.e., average accident rate in neighboring areas) (Fig. 2b). The LISA analysis measures local spatial autocorrelation and identifies areas where high (or low) accident rates are clustered with similar neighboring values categorizing as High-High (HH) or Low-Low (LL) clusters and High-Low (HL) and Low-High (LH) outliers (Anselin, 1995; Eustace et al., 2018). The HH zone represents accident hotspots, while LL, HL, and LH zones represent safe, spatial outliers and resilient areas of spatial outliers, respectively. The value of Moran’s I is 0.341, representing positive spatial autocorrelation among accidents involving older drivers (Fig. 2b). The Moran’s I scatter plot is split into four sections representing four types of spatial autocorrelation. The Moran’s I scatter plot shows a good representation of spatial clustering of accidents involving older drivers and a low number of spatial outliers. Accidents involving older drivers are mostly clustered in major cities and junctions of West Midlands regions.

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Fig. 2.

Spatial analysis of accidents involving older drivers. (a) LISA cluster map, (b) LISA scatterplot (Moran’s I = 0.341).

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Since the accidents involving older drivers have a spatial pattern, this spatial relationship disrupts the independent assumption of conventional regression models. The SAR models are designed to account for this dependence by incorporating the influence of neighboring observations into the regression model. To understand the existence of spatial dependence in the accidents involving older drivers and the appropriateness of spatial lag and spatial error models, the spatial diagnosis tests were performed using R-square, log likelihood, Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and likelihood ratio test probability (Table 1).

Table 1. Spatial diagnosis test of SAR models for road accidents involving older drivers.

Criteria of evaluation	OLS	Spatial lag	Spatial error
R-square	0.1908	0.1908	0.1909
Log likelihood	-24422	-24421.9	-24421.837
AIC	48868	48869.8	48867.7
BIC	48981.6	48992.8	48981.2
Likelihood ratio Test Probability		(0.1997) 0.65	(0.3348) 0.05

The R-square is around 0.19 for all three models, representing variance of the dependent variable (accidents involving older drivers) is 19.09% for the three models. However, R-square does not confirm the accuracy, causality, and appropriateness of the models. It should be interpreted alongside other metrics and diagnostic tools to avoid misleading conclusions, especially in the presence of overfitting or irrelevant predictors.

The log-likelihood is usually a negative number that shows how well the models explain the observed data in comparing different models and fitting model parameters. The maximum log-likelihood value of the spatial error model shows that the model is more likely to make observed data (Table 1). Sometimes, log-likelihood is applied as part of AIC and BIC to add a penalty for complexity (number of parameters) and overfitting. AIC and BIC are very effective in comparing two or more models fitted to the same accident data. The lowest values of AIC and BIC indicate that the spatial error model provides the best balance between model fitness and complexity to the observed data compared to OLS and the spatial lag model (Table 1). The lower values of AIC and BIC for the spatial error model indicate that the accident data involving older drivers have a spatial pattern of unobserved factors, and the Thiessen polygons of accident records share common unobserved influences. The AIC and BIC values for spatial lag and OLS models are higher as the models are influenced by the outcome spillover effects and observed variables, respectively. Lower values of AIC and BIC for the spatial error model indicate fewer explanatory variables, higher accuracy, and better fitness.

The likelihood ratio test probability compares null and alternative hypotheses to explain the observed data, and the p-value for the test is the probability of observing a test statistic as extreme or more extreme than the observed data. The large value of the likelihood ratio for the spatial error model (0.3348) with a low p-value (<0.05) indicates that the alternative hypothesis fits the data significantly better than the null hypothesis (Table 1). On the contrary, the small value of the likelihood ratio with a high p-value (≥0.05) fails to reject the null hypothesis. Despite having a lower value of likelihood ratio, the spatial lag model has a high p-value (0.65), rejecting the alternative hypothesis and indicating that the added parameters didn’t improve the model (Table 1). Spatial lag and error models both show some improvement in the spatial diagnosis tests compared to the OLS model. However, the spatial error model is the best-fitted model of the observed accident data based on the selected criteria. Therefore, the parameter estimates of the spatial error model are taken as the final output for developing the factors affecting accidents involving older drivers (Table 2).

Table 2. Parameter estimation of road accidents involving older drivers by a spatial error model.

Variable	Coefficient	Std. Error	z-value	Probability
Constant	0.0745	0.006	12.042	0.000
Gender of drivers	-0.036	0.002	-19.438	0.000
Journey purpose	0.0641	0.002	28.764	0.000
Junction location	0.0041	0.001	12.648	0.000
Pedestrian crossing (human interventions)	0.0202	0.018	1.102	0.270
Pedestrian crossing (physical interventions)	-0.0004	0.002	-0.343	0.731
Lighting condition	-0.027	0.001	-21.885	0.000
Weather condition	0.0005	0.001	0.391	0.695
Road surface condition	-0.0068	0.002	-4.031	0.000
Level of carriage hazard	-0.0035	0.002	-1.927	0.053
Time severity	0.0019	0.001	1.847	0.064
Road or junction type	0.0019	0.001	2.233	0.025
Lambda	0.0516	0.008	0.576	0.000

Table 2 summarizes the parameters estimated by the spatial error model for accidents involving older drivers. The journey purpose, junction location, pedestrian crossing (human interventions), weather conditions, time severity, and road or junction types have a positive relationship with accidents involving older drivers (Table 2). The highest value of journey purpose was assigned to ‘taking pupils from and to school’, while the lowest value was assigned to ‘work.’ The positive correlation between journey purpose and accidents involving older drivers states that older people driving their grandchildren are more vulnerable to accident risks (Table 2). Grandparents do school runs to support the parents with increased work commitments, childcare expenses, and school transport costs, as well as having stronger intergenerational family involvement. In addition, parents trust and rely on the grandparents compared to professional child carers. However, grandparents might be at greater risk during the school drop-offs and pick-ups due to slower reaction times, vision and hearing decline, medication side effects, difficulty in navigating the busy and congested school zones, and confusion with modern traffic rules and regulations. Similarly, the cognitive decline, increased fragility, turning and merging errors, and confusion in unfamiliar areas might increase the accident risk of older drivers, especially in social trips, as those are less predictable.

Age-related cognitive and perceptual decline significantly increases accident risk for older drivers at road junctions where rapid information processing and complex decision-making are often required (Table 2). Urban road junctions, particularly T- and Y-junctions, and signalized junctions, pose significant hazards for older drivers due to their vulnerability in situations involving right-turning maneuvers, failure to yield, and misjudgment of oncoming traffic speeds. Older drivers are disproportionately involved in angle and side-impact collisions at junctions, often under daylight and favorable weather conditions, indicating that internal driver factors rather than environmental conditions are the primary contributors (Table 2). The statistical insignificance of weather conditions and the negative correlation of lighting conditions with accidents involving older drivers justify this argument, respectively (Table 2). In addition, visual and physical limitations such as reduced peripheral vision, diminished contrast sensitivity, decreased depth perception, and reduced neck mobility impair older adults’ ability to detect vehicles and pedestrians approaching from multiple directions at junctions. Older drivers may struggle with judging gaps in traffic and following rapidly changing patterns of movement at the roundabouts and complex multilane junctions.

Manual traffic control environments can be cognitively demanding for older drivers, as they require real-time interpretation of hand signals and situational awareness without the predictability of automated signals. Older drivers may struggle to interpret or notice hand gestures from traffic officers or school crossing patrol officers (also known as a lollipop person), especially in visually complex environments or under poor lighting conditions, increasing the likelihood of delayed or inappropriate responses. In addition, manual traffic control often involves dynamic changes in flow direction or temporary alterations in priority rules that may confuse older drivers who rely heavily on familiar and routine driving patterns. The situation can be aggravated in construction zones, event detours, pedestrian crossings near schools and hospitals, and emergency redirection, where manual control is common and creates unfamiliar road environments that can increase anxiety and risk for older drivers. However, older drivers are generally more cautious at pedestrian crossings than younger drivers, often slowing down and yielding consistently to pedestrians. Their experience and self-awareness of age-related limitations, such as slower reaction times and reduced mobility, make them deliberate and attentive in pedestrian-heavy areas. Their self-awareness and vigilance at the pedestrian crossings operated both by physical and human interventions result in insignificant influence on the accidents, despite older drivers being more vulnerable to accident risks (Table 2). Table 2 shows that pedestrian crossings with both physical and human interventions are statistically insignificant for the spatial error model of road accidents involving older drivers.

Older drivers are disproportionately affected by poor light conditions due to age-related deterioration in night vision, contrast sensitivity, and glare recovery, all of which impair hazard detection and reaction times. Driving at night or in dimly lit conditions reduces visibility of road signs, pedestrians, and other vehicles, which poses significant challenges for older drivers. In addition, glare from headlights or streetlights exacerbates visual strain in older drivers, leading to temporary blindness or discomfort that impairs decision-making at junctions, crosswalks, and turning maneuvers. Road accidents involving older drivers in low-light environments are more likely to occur at junctions, on undivided rural roads, and during left-turn maneuvers, where visual and spatial judgment is critical. Understanding their declining cognitive function, physical abilities, and night vision, older drivers self-regulate their driving habits and tend to avoid nighttime driving. They are more likely to be involved in road accidents during daylight hours and are particularly overrepresented in accidents at junctions during daylight hours. The negative coefficient of lighting condition (-0.027) explains that older drivers often drive during the daylight, leading to higher exposure during these times (Table 2).

Similarly, older drivers face increased accident risk during poor weather due to age-related declines in vision, reaction time, and cognitive flexibility. Adverse conditions such as rain, snow, fog, and ice reduce visibility and traction, making driving more complex. These challenges are amplified for older adults who may struggle with identifying hazards, judging distances, and reacting quickly. However, older drivers often have more experience and may drive more cautiously, which can help mitigate some risks. They also tend to avoid driving in extreme conditions (night, rain, snow), which reduces exposure. This is why the weather condition variable is insignificant (p = 0.695) to the road accidents involving older drivers in the West Midlands region (Table 2).

3. Discussion

The positive value of Moran’s I states that neighboring accident locations tend to have similar accident risks involving older drivers. The LISA map shows a clear spatial dependency between accidents, with most incidents involving older drivers concentrated on inner-city roads (Fig. 2). This pattern may be influenced by higher traffic volumes, increased pedestrian activity, and more complex road geometries in urban areas.

To determine the presence of spatial dependence in older driver accidents and to assess which SAR model best fits the data, spatial diagnostic tests were conducted. These tests compared three models, such as OLS, spatial lag, and spatial error models, using R-square, log-likelihood, AIC, BIC, and likelihood ratio tests. While both spatial lag and spatial error models account for spatial dependence, the spatial error model emerges as the best fit based on log-likelihood, AIC, BIC, and likelihood ratio test probability (Table 1). Consequently, the parameter estimations of the spatial error model were chosen as the definitive results for understanding factors influencing accidents involving older drivers.

TThe journey purpose, junction location, time severity, gender, and specific road or junction types have a positive correlation with road accidents involving older drivers in the West Midlands (Table 2). The positive relationship between journey purpose and accidents involving older drivers gives us an insight into the role of grandparents in school runs. The positive coefficients of location and types of junctions justify that age-related cognitive and perceptual declines significantly increase the accident risk for older drivers (Table 2). These locations require rapid information processing and complex decision-making, tasks that become more challenging with age. This leads to a higher involvement in angle and side-impact collisions, primarily in good weather and daylight conditions (Table 2). Overall, the spatial error model highlights that a combination of aging, traffic environment complexity, and road characteristics contributes to the accident risk for older drivers. This study provides evidence-based methods for evaluating a policy’s effectiveness, infrastructure risk, and demographic impact. The findings extended the scope of road safety to include intergenerational mobility patterns, risk exposure during routine and often overlooked trips of older drivers, support area-specific road safety programs, and prioritize infrastructure funding to high-risk areas.

4. Limitations

The spatial error model highlights that aging-related human factors combined with complex traffic environments contribute significantly to the higher accident risk for older drivers. However, this spatial analysis was limited only to the road accidents in the West Midlands, UK, and the spatial econometric models didn’t consider all nuanced interactions between aging drivers and environmental stimuli. Minor accidents or near-misses were not captured in police or insurance records, limiting the accident dataset. This study analyzed 10-year accident records, which may be affected by the evolving driving patterns, road infrastructure, and vehicle technology. Future studies can focus on cleaning and standardizing data, managing time-related complications with transformations and advanced models, applying appropriate time-series and machine learning techniques, incorporating external variables and event studies, and continuously updating spatial models to adapt to changing conditions. These strategies lead to more reliable and actionable insights from long-term data. The spatial error model, while effective at capturing spatial dependency, didn’t completely account for individual behavioral differences or real-time conditions. Future studies can consider in-vehicle and telematics to supplement spatial models with behavior metrics. There is a scope to improve the findings from spatial econometric models by applying machine learning models for pattern recognition and expanding the study to other regions for comparative analysis and broader policy implications.

5. Conclusions

Older drivers in the UK are more likely to be involved in road accidents due to factors such as age-related declines in vision, cognitive function, and physical health. They often struggle to assess speed and distance, especially at junctions. Certain regions are more prone to these accidents, highlighting the need for targeted safety measures as the older driver population continues to grow in the UK. This paper applies spatial econometric models to analyze accident patterns involving older drivers in the West Midlands to help develop better safety policies and strategies.

The spatial analysis reveals a positive spatial pattern in road accidents involving older drivers, with most incidents clustered in inner-city areas, likely due to higher traffic, more pedestrians, and complex road layouts. Spatial diagnostic tests comparing different models show that the spatial error model best explains the data by accounting for spatial dependence. The spatial error model estimated the journey purpose, junction types, gender, weather, and time of day as the most important factors of increasing accident risk for older drivers. Cognitive and perceptual declines make junctions particularly hazardous for older drivers, especially at T- and Y-junctions and signalized junctions where complex decisions and rapid information processing are needed. Manual traffic control methods, like hand signals, add challenges due to increased cognitive load and visual complexity, worsened by hearing impairments. Older drivers struggle more in poor light and bad weather due to age-related declines in vision, reaction time, and cognitive processing. Older drivers avoid driving in extreme weather and at nighttime due to decreased vision, slower reflexes, and cognitive decline. Physical limitations, fatigue, and health concerns also play a role, as well as a reduced sense of confidence in handling challenging driving conditions. These factors make older drivers more cautious and likely to avoid driving in less-than-ideal circumstances.

It is essential to create an environment that addresses their specific needs while providing the necessary tools and support systems to improve the safety of older drivers in urban areas. This includes redesigning infrastructure, offering tailored training, encouraging health assessments, utilizing new technologies, and creating alternative transportation options. By combining these efforts, we can ensure that older people remain safe and mobile, contributing to the broader goal of creating cities that are inclusive for all ages.