2.1. Research on the statistical analysis of crash frequency

Numerous empirical research has evaluated the effect of various attributes on crash frequency in a non-spatial statistical framework, such as Generalized Linear models (GLM), Poisson regression models (PR), Negative Binomial models (NB). These studies have investigated the influence of built environmental and transportation infrastructure on the probability of crash occurrences [13–18], as well as the role of the road characteristics such as arterial roads, rural roads, freeways [19–22], and specific locations like junctions and controlled crossroads [23–25]. Additionally, the impact of climatic attributes including wind strength, visibility, precipitation, light, and annual wet days on crash frequency has also been a subject of research [26–29]. The inclusion of the number of rainy days as a variable has helped in understanding the influence of factors like exposure and pavement condition on crash occurrences [28,30]. However, there has been a lack of comprehensive exploration into spatial variations of these explanatory factors affecting road crash frequency.

Although such non-spatial global regression models are useful to provide information about road crashes, they often assume the independence of variables. Therefore, they fail to adequately address the crucial aspect of spatial autocorrelation, potentially leading to biased results and reduced accuracy when examining the relationship between explanatory and dependent variables. Spatial dependence and spatial heterogeneity are two common spatial elements when considering spatial autocorrelation [6]. Spatial dependence refers to the possibility that the intensity of events at one location may affect the intensity of events at surrounding locations. Spatial heterogeneity means that events may not be evenly distributed across different spaces, such as the incidence of traffic accidents at different locations. These two issues often arise when collecting data by geographical units such as provinces, districts/cities, or villages. Scholars have also shown that the relationship between independent and dependent variables in collision accidents may change with spatial changes. Therefore, analyzing without considering spatial differences may lead to misleading conclusions [31]. Hence, there exists a critical need for further investigation utilizing spatially methodologies to obtain more precise insights into the relationship between the explanatory variables and overall crash occurrences [32].

2.2. Research on spatial analysis methodology of crash frequency

Previous studies have introduced various methodologies to assess the spatial impact of explanatory variables on crash frequency. Among these, geographically weighted regression (GWR) stands out as an effective approach, rooted in the fundamental principles of geography [33,34]. GWR enhances analysis by applying weights to nearby observations based on their proximity to a focal point, allowing for the adjustment of regression coefficients to reflect local contexts. This approach enables the exploration of location-specific relationships between dependent and independent variables, offering subtle insights into spatial dependence and heterogeneity. The application of GWR in road safety research has gained considerable recognition in recent years [35,36]. For instance, Mathew et al. [37] demonstrated GWR’s superiority in identifying spatial dependencies and variations in risk factors affecting teen crash frequency, compared to traditional generalized linear models. Similarly, Pirdavani et al. [38] also found that GWR outperforms the OLSR model in spatial analyses, underscoring its value in road safety studies.

Previous research has developed various types of GWR models, including Geographically Weighted Lasso regression (GWLR), Geographically Weighted Poisson regression (GWPR), Geographically Weighted Poisson Quantile regression (GWPQR), and Geographically Weighted Negative Binomial regression (GWNBR) [37,39–41]. These models are particularly useful in handling spatial effects while suitable for various types of data and problems. In dealing with the spatial effect of collision accident data, the GWPR and GWNBR models have shown better results than global non-spatial models [39,42,43]. However, Poisson distribution’s premise is that the mean is equal to the variance, which is not always true for crash data. To address this issue, GWNBR was introduced as a method for modeling counting data with spatial heterogeneity and excessive dispersion [44–45]. It has been widely used in some regional studies to explore the spatial changes of the relationship between explanatory variables and dependent variables [46–48]. In the field of accident analysis, several scholars have carried out research using GWNBR models. Gomes et al. [10] investigated the impact of spatial factors on crash frequency. Their findings indicated that the GWNBR model was better at capturing the spatial heterogeneity of crash frequency than GWPR model. Mathew et al. [37] used GWNBR to study the impact of road environmental factors such as road network, population characteristics and land factors on teen crash frequency. Oluwajana et al. [49] used seven goodness-of-fit indicators to compare the GWPR and GWNBR models, and the results showed that the GWNBR model with fixed bandwidth had the best predictive performance.

Although there are numerous studies on geographic risk factors related to crashes on the regional level, Lee et al. [50] applied Negative Binomial regression models to crash data from the United States and Italy to assess the transferability of models between these two countries. Their findings indicated that models for crashes were not transferable between the two nations, despite sharing several significant variables [51]. Considering this non-transferability between crash frequency analysis in various countries, it is important to explore the motorcycle crash frequency in developing countries like Cambodia.

Overall, research on spatial analysis of motorcycle crash frequency is very limited. Given the challenges in transferring crash frequency analysis findings across different countries, this study seeks to fill the existing research gaps by examining the spatial influence of built environment and climatic factors on motorcycle crash frequency in Cambodia, employing GWR-based methodologies.

3.1. Data preparation

In this section, a spatial motorcycle crash database is compiled with the terms and conditions for the source of the data. The motorcycle crash frequency data, which includes property-damage-only, injury, and fatal crashes, were extracted from Road Crash Victim and Information System (RCVIS) by Cambodian National Road Safety Committee, and the data were analyzed in aggregated, district-level form only. No personal, identifiable, or sensitive information was accessed or processed. For this study, we focused on crashes that occurred in 2019. The frequency of motorcycle crashes within each district was the dependent variable in our analysis. The selection of explanatory variables was guided by prior research and their anticipated influence on motorcycle crashes. This study adopts the “3Ds” framework of built environments – density, diversity, and design [52]. Density encompasses variables like population density, male population proportion, and household density, all of which are commonly used in zonal level traffic safety analyses [37,53–55]. Diversity is represented by land use proportions, residential areas, tourist attractions, green spaces, and cultural and sports facilities, which are identified as significant crash pattern influencers by previous research [56]. The design component includes road network characteristics such as road density, number of intersections, intersection density, and the lengths of major and minor roads, which are key factors affecting crash frequency [57]. Climate and built environment data were obtained from publicly available sources and used in accordance with their respective usage policies.

In this research, the primary spatial unit is the district level, which represents the second-tier administrative division in Cambodia, encompassing a total of 197 districts nationwide. Data regarding the road network, bus stops, administrative boundaries, land use types, and Points of Interest (POI) were sourced from OpenStreetMap (OSM). Using this OSM data, the road length, number of intersections were calculated with ArcGIS software. Population information was obtained from the 2019 General Population Census of the Kingdom of Cambodia, conducted by the National Institute of Statistics under the Ministry of Planning. Furthermore, climate-related data in 2019 were obtained from the VisualCrossing Weather API [58], which provides station-based historical weather observations. To spatially associate climate information with traffic analysis units, climate station data were matched to each district using the nearest-distance principle, whereby each district is assigned the observations from the geographically closest weather station.

Then, the dataset for this study underwent a rigorous data cleaning process for each variable, where observations with missing or omitted values were removed. The final dataset contained 3182 motorcycle crashes in total. The minimum, maximum, mean, and standard deviation of this dependent variable and other 17 explanatory variables are shown in Table 1. The motorcycle crash frequency ranges from 0 to 162 with an average of 16.152 and a standard deviation of 25.908. Demographic data show a male population portion averaging 0.488 with a low standard deviation, suggesting minor variation across districts. Population and household densities vary widely among districts. Built environment characteristics provide details on road network, bus station, residential areas, and amenities like schools and public services, while road network attributes with high stand deviation indicate significant variation among districts. Due to the fact that commercial area and industrial area are not available in many districts in our database, we do not include the commercial area and industrial area in our research. Finally, climate variables like average precipitation, annual rainy days, wind speed, temperature and humidity are reported, both of which could potentially impact crash frequency.

Fig 4 shows the spatial distribution of the built environments and climate characteristics. We could conclude that road density, population density, residential area proportion, and intersection density show large numbers in areas around the capital Phnom Penh, Krong Siem Reap in the northwest area, and the southwest Kampong Som area.

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Fig 4. Spatial distribution of built environments and climate characteristics.

(Base map data © OpenStreetMap contributors (ODbL); Analytical layers were created by the authors.).

https://doi.org/10.1371/journal.pone.0346916.g004

3.2. The measurement of spatial dependency - Moran’s Index

In this section, we use Moran’s Index to measure the spatial dependency of each potential explanatory variable. The Moran’s Index is calculated as follows:

(1)

where denotes the spatial weight of sample i and j, representing the spatial proximity or connection between them. N denotes the number of spatial units indexed by i and j. W is the sum of all spatial weights .

Moran’s Index value is a measure that ranges from −1 to +1, which indicates perfect dispersion and perfect correlation, respectively. A value of 0 suggests that the spatial pattern is random, with no apparent clustering or dispersion. Positive values suggest that similar values occur near each other (spatial autocorrelation), while negative values suggest that dissimilar values occur near each other (spatial dispersion). After checking the value of Moran’s Index, the bus station number, school number, public service number and tourism place number variables were removed from our explanatory dataset due to their non-significant variation among districts. The Moran’s Index test results of the remaining variables are shown in Table 2. The positive z-score values indicate that the data are spatially clustered.

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Table 2. Moran’s Index and VIF of each variable.

https://doi.org/10.1371/journal.pone.0346916.t002

3.3. Geographically weighted negative binomial regression

The Geographically Weighted Negative Binomial Regression (GWNBR) is a statistical model developed by de Silva for analyzing spatial data where the response variable is count-based and potentially over-dispersed (i.e., the variance exceeds the mean) [59]. For the over dispersed data, the GWNBR can model data in a non-stationary manner. The negative binomial distribution has an overdispersion coefficient α. The formula for negative binomial regression is as follows:

(2)

where is an offset variable, , α is an excessive dispersion coefficient, is a parameter related to the explanatory variable , and , is the dependent variable, NB represents negative binomial. This model considers a logarithm link function

GWNBR allows parameters and α to vary with spatial variation. The formula is as follows:

(3)

where are the locations (coordinates) of the data points j.

Considering the weight of the data near point i, the local log-likelihood of GWNBR at i has the following formula:

(4)

where is the weight matrix and is an integral term, which can be calculated by the following equation.

(5)

When converging, the following equation can be used to estimate :

(6)

where is the coefficient vector estimated from region i, is a matrix composed of explanatory variables, is the GLM diagonal weighting matrix at convergence, is the adjusted dependent variable vector, calculated by the following equation:

(7)

The best fit of the model was assessed using Akaike Information Criterion (AIC). The lower value of AIC indicates a better fit of the model [60].

Another important parameter in evaluating the GWNBR model is the bandwidth. The selection of bandwidth and corresponding kernel function will affect the performance of the model. The rate at which bandwidth control data point weights decrease with increasing distance regression points. When the bandwidth is large, the weight slowly decays, and vice versa. The AIC and Cross-validation (CV) are two common evaluation indicators for finding the optimal bandwidth. The optimum AIC aims to identify the most suitable model by selecting a bandwidth that minimizes both the discrepancies between observed and predicted values and the complexity of the model [61]. Concurrently, the optimal CV method aims to pinpoint the best model by choosing a bandwidth that reduces the divergence between observed and fitted values through the cross-validation technique [61]. Therefore, when searching for the optimal bandwidth, it is necessary to reference the values of both indicators.

3.4. Measurement of goodness of fit

The three commonly used indicators for evaluating and comparing the performance of GWNBR models are AIC, Mean Absolute Deviation (MAD), and Root Mean Squared Error (RMSE). The AIC, MAD and RMSE indicators are defined as follows:

(8)

where is the maximum estimate of log likelihood and k is the degree of freedom.

(9)

(10)

where n is the number of observations, is the observed number of crashes, is the predicted number of crashes. The lower the values of AIC, MAD, and RMSE, the better the model performance.

4.1. The overall results of the global model and GWNBR models

In our research, non-spatial Ordinary Least Squares regression (OLSR), Poisson regression (PR), and Negative Binomial Regression (NBR) models were established as foundational comparisons for GWNBR models. Key variables identified in the NBR model informed the development of GWNBR models. These GWNBR models were particularly effective in assessing how various factors differently influence motorcycle crash frequency across Cambodia’s districts, highlighting the spatial variability of the relationship between the dependent variable and key explanatory variables.

Selecting the appropriate bandwidth and kernel function is pivotal in the calibration of GWNBR models [38]. GWNBR offers two kernel options: fixed or adaptive. With a fixed kernel, neighboring data points are selected based on a predefined distance criterion, such as including all neighbors within a 50 km radius. Conversely, the adaptive kernel’s selection of neighbors is guided by optimal values determined by the AIC or CV [61]. In this study, we opted for the adaptive kernel due to its flexibility in adjusting the spatial extent to encompass a broader geographic area, ensuring a consistent number of observations within the kernel’s scope.

The SAS/IML© macros [65] were employed in our study to develop the GWNBR models. The GWNBR models require the determination of the optimum bandwidth for model calibration. Thus, in this study, we determined the optimum bandwidth of GWNBR by adopting both the optimum AIC and CV indicators. The golden section search method was employed to find the optimal bandwidth in our study.

Table 3 shows the performance metrics for both non-spatial and spatial models, focusing on Likelihood, AIC, RMSE, and MAD. The results highlight the GWNBR model with adaptive bandwidth and optimized AIC as superior compared to other models, as evidenced by its lower MAD, RMSE, and AIC values. This confirms the necessity of considering the spatial variation and over dispersion that exists in the motorcycle crash data. The MAD values increase from 3.031 in the GWNBR model with adaptive optimized CV indicator to 4.224 in the NBR model, which further marks a notable enhancement in the model’s ability to account for variability. Additionally, the AIC values for the GWNBR model with adaptive AIC are markedly lower compared to those of the global models, highlighting the superior efficiency of the GWNBR approach. This suggests that incorporating spatial variation into the analysis can notably enhance the explanatory power of the model. The detailed outcomes derived from the GWNBR model with adaptive AIC will be thoroughly examined in the subsequent section.

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Table 3. Performance comparison of each model.

https://doi.org/10.1371/journal.pone.0346916.t003

Diagnostic statistics further justify the progressive transition from PR to NBR framework. For the PR model, the dispersion statistic (χ²/df = 12.3) substantially exceeds the conventional threshold of unity, indicating severe overdispersion and rendering the Poisson assumption inappropriate for the motorcycle crash data. To address this issue, a NBR model was estimated; however, Moran’s I test on the global NBR residuals showed significant positive spatial autocorrelation (Moran’s I = 0.5007, z = 12.0523, p < 0.01), suggesting that spatial heterogeneity remained inadequately captured. In contrast, the GWNBR model with an adaptive bandwidth selected by the AIC criterion effectively mitigated residual spatial dependence, with Moran’s I reduced to 0.0301 and no longer statistically significant (p = 0.265). This indicates that allowing regression parameters to vary spatially substantially improves model adequacy. Moreover, relative to the global NBR model, the GWNBR model achieved a 44.2% reduction in RMSE (from 14.872 to 7.795) and a marked decrease in AIC (from 1394.088 to 965.323). These results demonstrate that the incorporation of both overdispersion and spatial non-stationarity yields improvements that are not only statistically significant but also practically meaningful, thereby providing strong empirical support for the adoption of the GWNBR framework.

4.2. Local estimates

This section explores the spatial association variations between various built environments, climate characteristics variables and motorcycle crash frequency are analyzed and presented in Table 4 and Fig 5. Only the local estimates of each explanatory variable by the best GWNBR with adaptive bandwidth and optimal AIC model will be presented. Five statistical measures are utilized to show the non-stationary spatial effect of each explanatory variable: the minimum value (MIN), 25th percentile lower quartile (LQ), median, 75th percentile upper quartile (UQ), and maximum value (MAX), as detailed in Table 4. The local coefficients across various districts exhibit diverse and sometimes surprising directional tendencies, demonstrating the GWNBR model’s capability at capturing spatial non-stationarity. In some districts, a particular independent variable may show a statistically significant relationship with motorcycle crash frequency, whereas in others, the relationship may not be significant. Additionally, the LQ, Median, and UQ values reveal that some built environment and climate characteristic estimates fluctuate from negative to positive, a complexity not observable in non-spatial models such as the OLSR, PR, and NBR models. This phenomenon is consistently observed in the outcomes of the GWNBR models, where both positive and negative coefficients were estimated across all zones, indicating the complex and variable nature of the spatial relationships being analyzed.

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Table 4. Estimation results of the GWNBR model for local variables.

https://doi.org/10.1371/journal.pone.0346916.t004

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Fig 5. Local estimates of built environments and climate from GWNBR model.

(Base map data © OpenStreetMap contributors (ODbL); Analytical layers were created by the authors.).

https://doi.org/10.1371/journal.pone.0346916.g005