Bayesian Modeling of Traffic Accident Rates in Poland Based on Weather Conditions

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This manuscript is well-written and clearly explains the research. 

Following are my comments:

• To be consistent with common scientific citation practices and to improve readability, please modify the in-text citation format to use author names followed by citation numbers.
  For example, on page 2, line 81:
  "Park et al. employed a generalized … increased in urban areas [24]"
  should be revised to:
  "Park et al. [24] employed a generalized … increased in urban areas."
• On page 5, line 188, the reference to "Table 5" appears to be incorrect and should instead be "Table 1."
• On page 6, line 203, the reference to "Table 5" should be corrected to "Table 2."
• Tables 4 and 5 are not cited or discussed in the text. Please include appropriate references to these tables and explain their relevance.
• Figures are currently not referred to in the text. Each figure should be cited and briefly discussed to clarify its role in the study.
• In Figure 2, please clarify why the plots related to yearly_population_density, yearly_passenger_cars, and yearly_road_density were not included. Their absence should be explained or the figures should be added if relevant.
• In Figures 11b, 13b, 15b, and 17b, please include the coefficient of determination (R²) to quantify the fit between observed and predicted values.

Author Response

We sincerely thank the Reviewer for their time and thoughtful feedback on our manuscript. Their comments have been very helpful in improving the clarity and quality of the paper. Below, we provide detailed responses to each point along with a summary of the corresponding changes made in the revised version.

Comment 1. To be consistent with common scientific citation practices and to improve readability, please modify the in-text citation format to use author names followed by citation numbers.

For example, on page 2, line 81:

"Park et al. employed a generalized … increased in urban areas [24]"

should be revised to:

"Park et al. [24] employed a generalized … increased in urban areas."

Response 1. We appreciate the suggestion. In response, we have revised the in-text citation format throughout the manuscript to follow the recommended style.

Comment 2. On page 5, line 188, the reference to "Table 5" appears to be incorrect and should instead be "Table 1."

️Response 2. We thank the reviewer for noting this incorrect reference. This was an oversight on our part and has been corrected in the revised manuscript.

Revised Manuscript Section. [Section 2.1.1. County-level modeling data, Paragraph 1] For county-level yearly analysis, […] their descriptions are presented in Table 1.

Comment 3. On page 6, line 203, the reference to "Table 5" should be corrected to "Table 2."

Response 3. We thank the reviewer for noting this incorrect reference. This was an oversight on our part and has been corrected in the revised manuscript.

Revised Manuscript Section. [Section 2.1.2. Nationwide modeling data, paragraph 1] For the nationwide weekly analysis, […] their descriptions are provided in Table 2.

Comment 4. Tables 4 and 5 are not cited or discussed in the text. Please include appropriate references to these tables and explain their relevance.

Response 4. We agree that all tables should be appropriately cited and discussed within the text. In response, we have added references to Tables 4 and 5 and included an appropriate discussion of their relevance in the revised manuscript.

Revised Manuscript Section. [Section 3.2.1. County-level models, lines 451–471] The county-level analysis revealed differences in predictive performance between the two competing models. As shown in Figure [19] and Table 4, Model 2 performed better according to the PSIS-LOO criterion. It achieved a higher expected log pointwise predictive density (ELPD-LOO) of –12788.50, compared to –12872.12 for Model 1. The ELPD difference of 83.61 suggests a modest improvement in predictive accuracy. Model 2 also had a higher effective number of parameters (p-LOO = 152.42 vs. 92.02), correctly reflecting its greater complexity. It received a slightly higher model weight (0.54 vs. 0.46), suggesting a mild preference, but not strong evidence of superiority.

[Section 3.2.2. Nationwide models, lines 472– 489] The weekly nationwide analysis revealed a clearer distinction in predictive performance between the two models compared to the previous county-level analysis. As shown in Figure [20] and Table 5, Model 4 outperformed Model 3 based on the PSIS-LOO criterion, achieving a higher expected log pointwise predictive density (ELPD-LOO) of –1940.79, compared to –2019.78 for Model 3. The ELPD difference of 78.99 suggests a meaningful improvement in predictive accuracy. Model 4 also received a higher model weight (0.64 vs. 0.36), providing moderate evidence in its favor.

Comment 5. Figures are currently not referred to in the text. Each figure should be cited and briefly discussed to clarify its role in the study.

Response 5. We appreciate this suggestion. In response, we thoroughly reviewed the manuscript and added all missing in-text citations for the figures. Where appropriate, we also included brief discussions to clarify the role of each figure within the context of the surrounding text.

Comment 6. In Figure 2, please clarify why the plots related to yearly_population_density, yearly_passenger_cars, and yearly_road_density were not included. Their absence should be explained or the figures should be added if relevant.

Response 6. We appreciate this suggestion. In response, we have clarified in the caption of Figure 2 that the distributions of yearly variables were excluded, as they offered little additional insight for the analysis.

[Section 2.1.2. Nationwide modeling data, Figure 2 caption] Normalized distributions of weather factors […] during the 2020-2023 study period. Distributions of yearly variables were excluded from the figure as they offered little additional insight for the analysis.

Comment 7. In Figures 11b, 13b, 15b, and 17b, please include the coefficient of determination (R²) to quantify the fit between observed and predicted values.

Response 7. We fully agree with this suggestion. In response, we have added the coefficient of determination (R²) to the aforementioned Figures to quantify the predictive accuracy between observed and predicted values.

[Section 3.1.1. Model 1, Figure 12b caption] Scatter plot of observed versus predicted yearly accident counts with the identity line (red dashed). Points clustering around this line indicate accurate predictions, with increasing variance at higher accident counts. The coefficient of determination (R² = 0.810) indicates good model fit.

[Section 3.1.2. Model 2, Figure 15b caption] Scatter plot of observed versus predicted yearly accident counts with the identity line (red dashed). The coefficient of determination (R² = 0.813) indicates a slightly higher model fit compared to Model 1.

[Section 3.1.3. Model 3, Figure 18b caption] Scatter plot of observed versus predicted weekly accident counts with the identity line (red dashed). Points clustering around this line indicate accurate predictions, with increasing variance at both lower and higher accident counts, suggesting areas for potential model improvement. The coefficient of determination (R² = 0.607) indicates moderate model fit.

[Section 3.1.4. Model 4, Figure 20b caption] Scatter plot of observed versus predicted weekly accident counts with the identity line (red dashed). The reduced scatter around the line indicates improved predictive performance across most of the data range, particularly in the 300-500 accident range. The coefficient of determination (R² = 0.643) indicates improved model fit compared to Model 3.

Reviewer 2 Report

Comments and Suggestions for Authors

1. The authors have conflated the terms “accident” and “incident.” I recommend replacing every occurrence of “incident” with “accident” throughout the manuscript.
2. The Bayesian Poisson model adopted is too simple, more advanced methods such as Bayesian spatial and temporal models are suggested, please refer to the following works for the detailed introduction and formulations:

Bayesian spatial-temporal model for the main and interaction effects of roadway and weather characteristics on freeway crash incidence. Accident Analysis and Prevention, 2019, 132: 105249.

Spatial joint analysis for zonal daytime and nighttime crash frequencies using a Bayesian bivariate conditional autoregressive model. Journal of Transportation Safety and Security, 2020, 12(4): 566-585.

Applying a Bayesian multivariate spatio-temporal interaction model based approach to rank sites with promise using severity-weighted decision parameters [J]. Accident Analysis and Prevention, 2021, 157: 106190.

3. When citing previous studies, please integrate the citation directly with the author name rather than clustering all reference numbers at the end of the sentence. For example: Qiu et al. [19].
4. At the end of the paragraph corresponding to lines 75–114, please include a concise summary of the limitations of the existing methods to strengthen the rationale for employing a Bayesian statistical framework in this work.
5. Please revise the title of the final section to “Conclusion”.

Author Response

We sincerely thank the Reviewer for their time and thoughtful feedback on our manuscript. Their comments have been very helpful in improving the clarity and quality of the paper. Below, we provide detailed responses to each point along with a summary of the corresponding changes made in the revised version.

Comment 1. The authors have conflated the terms “accident” and “incident.” I recommend replacing every occurrence of “incident” with “accident” throughout the manuscript.

Response 1. We appreciate this suggestion and agree with the reviewer’s observation. While the terms “accident” and “incident” are sometimes used interchangeably in road traffic literature, we acknowledge the importance of maintaining clear and consistent terminology. In response, we have reviewed the manuscript and replaced all occurrences of “incident” with “accident”.

Comment 2. The Bayesian Poisson model adopted is too simple, more advanced methods such as Bayesian spatial and temporal models are suggested, please refer to the following works for the detailed introduction and formulations:

Bayesian spatial-temporal model for the main and interaction effects of roadway and weather characteristics on freeway crash incidence. Accident Analysis and Prevention, 2019, 132: 105249.

Spatial joint analysis for zonal daytime and nighttime crash frequencies using a Bayesian bivariate conditional autoregressive model. Journal of Transportation Safety and Security, 2020, 12(4): 566-585.

Applying a Bayesian multivariate spatio-temporal interaction model based approach to rank sites with promise using severity-weighted decision parameters [J]. Accident Analysis and Prevention, 2021, 157: 106190.

Response 2. We appreciate the reviewer’s thoughtful suggestion. As this study represents our initial work on this topic, we opted to begin with relatively simple Bayesian Poisson models to establish a clear and interpretable foundation. Nevertheless, we have carefully reviewed the cited works and found them both insightful and highly relevant. Accordingly, we have expanded the Introduction to cite and briefly discuss these studies in order to better situate our approach within the broader literature. We have also revised the section on future research directions to acknowledge the potential benefits of more advanced spatial and spatio-temporal Bayesian modeling frameworks, as highlighted by the reviewer.

[Section 1. Introduction, lines 19–162] Recent studies reinforce the growing […] geographical regions or time periods [39]. Zeng et al. [40] proposed a Bayesian bivariate conditional autoregressive model that jointly analyzes daytime and nighttime crash frequencies while accounting for spatial correlations across traffic analysis zones. Their findings reveal significant spatial effects and high correlation between crash periods, with key predictors including land use and traffic exposure. Wen et al.[41] expanded this perspective by developing a Bayesian spatio-temporal model that captures not only the main effects but also interactions between road geometry and weather factors, such as wind-speed–slope and precipitation–curve combinations, demonstrating their impact on crash risk. Meanwhile, Zeng et al.[42] introduced a Bayesian multivariate spatio-temporal interaction framework for ranking hazardous sites based on severity-weighted crash frequencies, showing that models incorporating spatial and temporal dependencies outperform simpler alternatives.

[Section 4. Conclusions, lines 490-534] Future research should pursue several extensions. First, the Bayesian Poisson model adopted here could be expanded to include spatial and spatio-temporal components, enabling more refined estimation of regional and temporal patterns. […]

Comment 3. When citing previous studies, please integrate the citation directly with the author name rather than clustering all reference numbers at the end of the sentence. For example: Qiu et al. [19].

Response 3. We appreciate this suggestion. In response, we have revised the in-text citation format throughout the manuscript to follow the recommended style.

Comment 4. At the end of the paragraph corresponding to lines 75–114, please include a concise summary of the limitations of the existing methods to strengthen the rationale for employing a Bayesian statistical framework in this work.

Response 4. We appreciate this suggestion. As the limitations of existing methods were already briefly mentioned in the following paragraph, we chose to revise that paragraph to place greater emphasis on them. This approach avoids redundancy and more clearly motivates the use of a Bayesian statistical framework in our study.

Revised Manuscript Section. [Section 1. Introduction, lines 19–162] Many of the described modeling approaches face limitations in their ability to fully account for uncertainty, incorporate prior knowledge, and represent complex hierarchical or spatio-temporal structures. They often rely on restrictive assumptions and lack the flexibility needed to capture nonlinear interactions. These limitations underscore the rationale for employing Bayesian statistical frameworks, which offer a coherent probabilistic foundation for addressing such complexities.  By incorporating prior information, quantifying uncertainty, and updating beliefs as new data become available, they effectively capture hierarchical structures, spatial and temporal dependencies, and intricate parameter interactions [36].

Comment 5. Please revise the title of the final section to “Conclusion”.

Response 5. We agree that the title suggested by the reviewer better reflects the content of the final section. Accordingly, we have revised the section title from “Discussion” to “Conclusions” in the updated manuscript.

Reviewer 3 Report

Comments and Suggestions for Authors

1.This study addresses an important transportation safety issue by analyzing the relationships between traffic incident rates, weather conditions, and socioeconomic factors in Poland. The attempt to integrate multiple explanatory variables using statistical models is good, and the use of WAIC and PSIS-LOO for model comparison reflects a good understanding of model evaluation.

2. While the dataset employed in this study appears to be extensive and well-structured, its description within the current manuscript is insufficient. As it stands, the dataset is largely referenced from prior work [42], with only a brief overview provided in the text. Given that the validity and interpretability of the modeling results are directly dependent on the dataset’s structure, granularity, and preprocessing steps, the paper should provide a comprehensive and self-contained description of the data.

3. The text states: “Our approach to modeling traffic incident rates in Poland employs generalized linear models based on the poisson distribution”. But the Poisson distribution models counts, not rates. This could mislead readers unless clarified.

4. Please specify whether they are any missing data handling procedures, Describe data quality checks, if any, and the rationale for selecting 2015–2023 as the study window.

5.The findings could be contextualized within the broader literature. Are the observed patterns  consistent with international studies? This would strengthen the generalizability of the results.

Author Response

We sincerely thank the Reviewer for their time and thoughtful feedback on our manuscript. Their comments have been very helpful in improving the clarity and quality of the paper. Below, we provide detailed responses to each point along with a summary of the corresponding changes made in the revised version.

Comment 1. This study addresses an important transportation safety issue by analyzing the relationships between traffic incident rates, weather conditions, and socioeconomic factors in Poland. The attempt to integrate multiple explanatory variables using statistical models is good, and the use of WAIC and PSIS-LOO for model comparison reflects a good understanding of model evaluation.

Response 1. We appreciate the reviewer’s positive feedback and are grateful that our analysis and approach to model evaluation were well received.

Comment 2. While the dataset employed in this study appears to be extensive and well-structured, its description within the current manuscript is insufficient. As it stands, the dataset is largely referenced from prior work [42], with only a brief overview provided in the text. Given that the validity and interpretability of the modeling results are directly dependent on the dataset’s structure, granularity, and preprocessing steps, the paper should provide a comprehensive and self-contained description of the data.

Response 2. We appreciate this comment. In response, we have expanded the Data section to provide a more comprehensive and self-contained description of the dataset. We also clarified in the text that additional information can be found in our previous work, which is appropriately referenced.

Revised Manuscript Section. [Section 2.1. Data, lines 164–218] The dataset employed in this study is a comprehensive aggregation of road traffic accidents recorded across Poland between 2015 and 2023, a period selected to ensure data integrity and completeness. Records prior to 2015 exhibited significant data quality issues, including a high incidence of missing values, while 2023 represented the most recent full year of data available during the dataset's development. The resulting dataset contains approximately 250,000 individual accident records, each integrating police-reported accident characteristics with two contextual layers. For each accident, the dataset includes location-specific weather conditions—temperature, relative humidity, hourly precipitation, and 24-hour cumulative precipitation—estimated using universal kriging with elevation drift. It also incorporates a suite of county-level socioeconomic indicators, such as population density, road infrastructure metrics, vehicle fleet composition, and historical accident statistics, providing a comprehensive profile for every recorded event. Additionally, […]

This dataset […] data from the Institute of Meteorology and Water Management (IMGW). The creation of this dataset involved extensive preprocessing to ensure its quality and analytical robustness. Key steps included the standardization of administrative unit names to resolve inconsistencies, the removal of physically implausible meteorological outliers, and the clipping of precipitation values below the minimum detectable threshold. As a result of these procedures, the data utilized in this study is clean, requiring no further imputation. Detailed preprocessing methodology […]

For our analysis, […], and road density). The selection of these variables is directly informed by established road safety literature and the preliminary findings presented in our foundational work [45]. Weather parameters such as temperature, humidity, and precipitation were chosen for their well-documented effects on driver behavior and road surface conditions. Our previous analysis confirmed these relationships, revealing a consistent positive correlation between temperature and accident frequency (correlation coefficient of 0.09) and a notable negative correlation for humidity (-0.10). The selected socioeconomic indicators serve as robust proxies for traffic exposure and infrastructure characteristics. Specifically, population density and passenger car registrations were included due to their strong positive correlation with total accident counts (0.45 and 0.82, respectively), while road density provides a crucial measure of infrastructure development, showing a meaningful correlation (0.31). Based on the original dataset, specific preprocessing was conducted for each analytical problem.

Comment 3. The text states: “Our approach to modeling traffic incident rates in Poland employs generalized linear models based on the poisson distribution”. But the Poisson distribution models counts, not rates. This could mislead readers unless clarified.

Response 3. We appreciate this correction and acknowledge the imprecise terminology. It is indeed correct that the Poisson distribution models counts, not rates, and this distinction is important for clarity. In response, we have corrected this problematic naming throughout the manuscript.

Comment 4. Please specify whether they are any missing data handling procedures, Describe data quality checks, if any, and the rationale for selecting 2015–2023 as the study window.

Response 4. We agree with this point. The dataset used in this study was carefully prepared to ensure its suitability for Bayesian analysis, thereby eliminating the need for additional missing data handling procedures. The selection of the study window was driven by data availability. In response, we have added explanatory comments in the relevant sections of the manuscript to clarify our data quality assurance procedures and the rationale for the chosen study window.

Revised Manuscript Section. [Section 2.1. Data, lines 164–218] The dataset employed in this study is a comprehensive aggregation of road traffic accidents recorded across Poland between 2015 and 2023, a period selected to ensure data integrity and completeness. Records prior to 2015 exhibited significant data quality issues, including a high incidence of missing values, while 2023 represented the most recent full year of data available during the dataset's development. [...]

This dataset […] data from the Institute of Meteorology and Water Management (IMGW). The creation of this dataset involved extensive preprocessing to ensure its quality and analytical robustness. Key steps included the standardization of administrative unit names to resolve inconsistencies, the removal of physically implausible meteorological outliers, and the clipping of precipitation values below the minimum detectable threshold. As a result of these procedures, the data utilized in this study is clean, requiring no further imputation. Detailed preprocessing methodology […]

 

Comment 5. The findings could be contextualized within the broader literature. Are the observed patterns consistent with international studies? This would strengthen the generalizability of the results.

Response 5. We appreciate this suggestion. In response, we have revised the Conclusions section to contextualize our findings within the broader international literature.

Revised Manuscript Section. [Section 4. Conclusions, lines 490–544] This study investigated […] modest positive effect. This aligns with the well-established principle of traffic exposure, a fundamental concept in traffic safety research~\citep{fridstrom_measuring_1995}. Road density exhibited a negative effect, suggesting that better-developed road networks may contribute to accident reduction. This finding is consistent with literature on the benefits of "forgiving" and thoughtfully designed infrastructure, which can mitigate driver error even in areas with high traffic volumes~\citep{budzynski_assessment_2021, kiso_development_2021}. Posterior predictive checks […] precipitation had negative effects. The seemingly counterintuitive negative effects of temperature and precipitation at the annual level may reflect broader climate patterns rather than, event-specific risks. Counties with higher average annual temperatures and more precipitation may experience fewer days with highly hazardous conditions, which are known to significantly increase collision risk~\citep{abohassan_effects_2022, eisenberg_effects_2005}. The positive association with humidity could be a proxy for conditions with reduced visibility, such as fog or mist, which are known hazards. The inclusion of […]

The weekly nationwide […] a slight negative effect. The positive effect of temperature at the weekly scale, in contrast to the yearly finding, is highly consistent with international studies that link short-term temperature increases and heatwaves to a higher risk of crashes~\citep{basagana_high_2015, park_heatwave_2021}. The slight negative effect of precipitation, while counterintuitive, can be explained by adaptive driver behavior; research has shown that drivers often reduce speeds and increase following distances during rainfall~\citep{billot_multilevel_2009}. While the model captured […]

Reviewer 4 Report

Comments and Suggestions for Authors

Road traffic accidents pose a substantial global public health burden, resulting in significant fatalities and economic costs. This study employs Bayesian Poisson regression to model traffic incident rates in Poland, focusing on the intricate relationships between weather conditions
and socioeconomic factors. Analyzing both yearly county-level and weekly nationwide data from 2020 to 2023, we created four distinct models examining the relationships between incident occurrence and predictors including temperature, humidity, precipitation, population density, passenger car registrations, and road infrastructure. Model selection, based on WAIC and PSIS-LOO criteria, demonstrated that integrating both weather and socioeconomic variables enhanced predictive accuracy, particularly at the county level. The analysis reveals a clear pattern of traffic incident predictors: socioeconomic factors emerge as critical determinants of accident rates over longer temporal scales and across granular regional contexts, while meteorological conditions prove most significant in capturing
short-term temporal variations. This research offers a data-driven approach to identify high-risk conditions and inform targeted interventions to reduce traffic incidents, contributing to Poland’s national road safety objectives of minimizing fatalities and injuries.

 

The manuscript presents a well-structured and engaging study that applies Bayesian Poisson regression to model road traffic accident (RTA) rates in Poland, considering weather conditions and socioeconomic factors. The research conducts a comprehensive data analysis at both the county and national levels, utilizing various models to investigate the relationships between RTA frequency and predictive variables. The use of Bayesian methods is a strong aspect, allowing for robust uncertainty analysis.

Strengths of the Manuscript

• The article addresses a highly relevant and crucial topic concerning road traffic accident (RTA) modeling and the impact of weather conditions. This is of significant importance for public health and safety.
• The justification for using Bayesian Poisson regression is well-founded and appropriate. This approach allows for a more comprehensive understanding of the relationships, including the crucial assessment of parameter uncertainty.
• The study demonstrates a thorough analytical approach by incorporating both annual data at the county level and weekly data at the national level. This broad scope enhances the robustness of the findings.
• The authors' use of WAIC and PSIS-LOO criteria for model selection is justified, highlighting their commitment to ensuring the reliability of their conclusions.
• The conclusions regarding the importance of socioeconomic factors over longer time scales and the influence of meteorological conditions in short-term variations are insightful and well-supported by the data.

 

Weaknesses of the Manuscript

• While the authors mention using data from 2020 to 2023, it would be beneficial to provide a more detailed description of the data sources for each variable (temperature, humidity, precipitation, population density, passenger car registrations, road infrastructure).
• The article does not include a cognitive model of the human as a transport operator. This omission might limit the understanding of how human behavior influences accident rates.
• The manuscript does not specify the size of the transport vehicles involved, i.e., whether the data pertains to passenger cars or freight trucks. This distinction could significantly impact the analysis.
• The authors list the predictive variables, but it would be helpful to briefly justify why these specific variables were chosen and what the expected relationships with RTA frequency are.
• It would be valuable to add a section or subsection discussing the potential limitations of this study. For example, are there any unaccounted variables that might also influence RTA frequency (e.g., road surface conditions, traffic intensity during specific periods, police effectiveness, etc.)?
• Although visualizations are not typically included in an abstract, it would be extremely beneficial to include graphs in the full article illustrating the relationships between the predictive variables and RTA frequency, as well as model comparisons.
• The article lacks a section discussing the practical implications of the obtained results for government authorities responsible for road safety. This would strengthen the applied value of the research.

Recommendation

Overall, the manuscript represents a high-quality scientific study that warrants publication after addressing the minor additions and clarifications outlined above.

Decision

Accept with minor revisions.

 

Author Response

We sincerely thank the Reviewer for their time and thoughtful feedback on our manuscript. Their comments have been very helpful in improving the clarity and quality of the paper. Below, we provide detailed responses to each point along with a summary of the corresponding changes made in the revised version.

Regarding Weaknesses of the Manuscript

Comment 1. While the authors mention using data from 2020 to 2023, it would be beneficial to provide a more detailed description of the data sources for each variable (temperature, humidity, precipitation, population density, passenger car registrations, road infrastructure).

Response 1. We appreciate this suggestion. In response, we have expanded the Data section to provide extensive information about the data sources for each variable used in our analysis.

Revised Manuscript Section. [Section 2.1. Data, lines 164–218] This dataset was developed in our previous work [45], where traffic accident data was sourced from the Accident and Collision Recording System (SEWIK), the official police-maintained registry of traffic incidents, socioeconomic metrics from Statistics Poland (GUS), Poland's principal government agency for official statistics, and meteorological data from the Institute of Meteorology and Water Management (IMGW), the state service for meteorological and hydrological monitoring. [...] For our analysis, we examined the period from 2020 to 2023, utilizing weather parameters (temperature, humidity, and precipitation) based on data from IMGW and three socioeconomic indicators (population density, passenger car registrations, and road density) reported by GUS.

Comment 2. The article does not include a cognitive model of the human as a transport operator. This omission might limit the understanding of how human behavior influences accident rates.

Response 2. We agree with this point. While a cognitive model of the human as a transport operator was beyond the scope of the current study, we recognize its relevance to understanding accident dynamics. Accordingly, we have acknowledged this limitation and included a suggestion to incorporate such a model in the Conclusions section of the revised manuscript.

Revised Manuscript Section. [Section 4. Conclusions, lines 490–534] Future research should pursue several extensions. First, the Bayesian Poisson model adopted here could be expanded to include spatial and spatio-temporal components, enabling more refined estimation of regional and temporal patterns. Second, incorporating a cognitive model of the human as a transport operator would help capture behavioral dimensions of accident risk, particularly under adverse weather. Third, aggregating data by vehicle type could uncover mode-specific risk factors. Lastly, conducting cross-country comparisons would strengthen the generalizability and transferability of the results.

Comment 3. The manuscript does not specify the size of the transport vehicles involved, i.e., whether the data pertains to passenger cars or freight trucks. This distinction could significantly impact the analysis.

Response 3. We agree with this observation. In response, we have clarified in the Data section that our dataset encompasses all recorded traffic incidents in Poland involving various vehicle types.

Revised Manuscript Section. [Section 2.1. Data, lines 164–218] For our analysis, we examined the period from 2020 to 2023, utilizing [...] While the original dataset provides a detailed breakdown of vehicle registrations by category (e.g., passenger cars, trucks, buses), the analytical scope of this paper is focused on modeling the overall frequency of traffic accidents. Consequently, our models aggregate accidents across all vehicle types, allowing for a comprehensive assessment of the combined impact of the selected factors on total accident risk at the county level. Based on the original dataset, specific preprocessing was conducted for each analytical problem.

Comment 4. The authors list the predictive variables, but it would be helpful to briefly justify why these specific variables were chosen and what the expected relationships with RTA frequency are.

 Response 4. We agree with this point. In response, we have added a directed acyclic graph (DAG) to visualize the model structure and enhanced the variable selection rationale with detailed justifications based on established literature and our preliminary findings.

Revised Manuscript Section. [Section 2.2. Models, Figure 3] Directed acyclic graph (DAG) of the model structure; pc - passenger_cars, pd - population_density, rd - road_density, t - temperature, h - humidity, p - precipitation.

Revised Manuscript Section. [Section 2.1. Data, lines 164–218] The dataset employed in this study is a comprehensive aggregation of road traffic accidents recorded across Poland between 2015 and 2023, […], and road density). The selection of these variables is directly informed by established road safety literature and the preliminary findings presented in our foundational work [45]. Weather parameters such as temperature, humidity, and precipitation were chosen for their well-documented effects on driver behavior and road surface conditions. Our previous analysis confirmed these relationships, revealing a consistent positive correlation between temperature and accident frequency (correlation coefficient of 0.09) and a notable negative correlation for humidity (-0.10). The selected socioeconomic indicators serve as robust proxies for traffic exposure and infrastructure characteristics. Specifically, population density and passenger car registrations were included due to their strong positive correlation with total accident counts (0.45 and 0.82, respectively), while road density provides a crucial measure of infrastructure development, showing a meaningful correlation (0.31). Based on the original dataset, specific preprocessing was conducted for each analytical problem.

Comment 5. It would be valuable to add a section or subsection discussing the potential limitations of this study. For example, are there any unaccounted variables that might also influence RTA frequency (e.g., road surface conditions, traffic intensity during specific periods, police effectiveness, etc.)?

 Response 5. We appreciate this suggestion. In response, we have expanded on the study’s limitations in the Conclusions section.

Revised Manuscript Section. [Section 4. Conclusions, lines 490-534] Despite its contributions, this study has several limitations. Temporal aggregation at the weekly and yearly levels may obscure short-term dynamics and transient weather effects. The absence of key covariates—such as driver behavior and road surface conditions—may introduce unmeasured confounding. The aggregation of counties' socioeconomic data could hide differences between urbanized and rural areas. Additionally, the analysis does not distinguish between types of transport vehicles, which may exhibit different risk profiles. Finally, the reliance on historical data limits the model’s immediate applicability for real-time prediction or adaptive interventions.

Comment 6. Although visualizations are not typically included in an abstract, it would be extremely beneficial to include graphs in the full article illustrating the relationships between the predictive variables and RTA frequency, as well as model comparisons.

Response 6. We agree with this observation. In response, we added a directed acyclic graph of the model structure.

Revised Manuscript Section. [Section 2.2. Models, Figure 3] Directed acyclic graph (DAG) of the model structure; pc - passenger_cars, pd - population_density, rd - road_density, t - temperature, h - humidity, p - precipitation.

Comment 7. The article lacks a section discussing the practical implications of the obtained results for government authorities responsible for road safety. This would strengthen the applied value of the research.

Response 7. We agree with the reviewer that discussing practical implications is important. However, as this is our first study on the topic, we consider it primarily as an exploratory analysis rather than a foundation for direct policy recommendations. The findings offer initial insights that we hope will guide future, more comprehensive research. To address this point, we have expanded the paragraph about future research to outline the steps needed before drawing robust, actionable conclusions for government authorities.

Revised Manuscript Section. [Section 4. Conclusions, lines 490–534] Future research should pursue several extensions. First, the Bayesian Poisson model adopted here could be expanded to include spatial and spatio-temporal components, enabling more refined estimation of regional and temporal patterns. Second, incorporating a cognitive model of the human as a transport operator would help capture behavioral dimensions of accident risk, particularly under adverse weather. Third, aggregating data by vehicle type could uncover mode-specific risk factors. Lastly, conducting cross-country comparisons would strengthen the generalizability and transferability of the results.

Reviewer 5 Report

Comments and Suggestions for Authors

Several sections, especially Introduction (lines 19–156) and Discussion (410–438), shows signs of AI-generated writing, such as: excessively balanced sentence structures "...socioeconomic factors emerge as critical determinants... while meteorological conditions prove most significant...", generic phrases like “offers a data-driven approach” or “underscores the need for sophisticated modeling approaches.” Consider revising these sections with a more human voice: clearer transitions, more variation in sentence length, and a stronger narrative arc.
Model 2 (Lines 256–261): model structure is mathematically well-defined, but some descriptive explanations e.g., rationale for variable selection, justification for prior parameters could be improved for non-statistical readers. A conceptual diagram summarizing the four models is needed.
Figures 11–18: Some scatter plots and histograms lack clear axis labeling or are visually dense. Please provide consistent scales across models and highlighting key model differences visually.
Table 4 and 5: need summarizing paragraph to contextualize ELPD differences e.g., whether 83.61 or 78.99 is considered large in practice

Comments on the Quality of English Language

Overuse of technical jargon e.g., "prior predictive check," "hyperparameter tuning" without lay explanation. Overly long sentences in discussion and introduction e.g. Lines 419-424 - consider splitting into two sentences for better flow

Author Response

We sincerely thank the Reviewer for their time and thoughtful feedback on our manuscript. Their comments have been very helpful in improving the clarity and quality of the paper. Below, we provide detailed responses to each point along with a summary of the corresponding changes made in the revised version.

Regarding Comments and Suggestions for Authors

Comment 1. Several sections, especially Introduction (lines 19–156) and Discussion (410–438), shows signs of AI-generated writing, such as: excessively balanced sentence structures "...socioeconomic factors emerge as critical determinants... while meteorological conditions prove most significant...", generic phrases like “offers a data-driven approach” or “underscores the need for sophisticated modeling approaches.” Consider revising these sections with a more human voice: clearer transitions, more variation in sentence length, and a stronger narrative arc.

 Response 1. We appreciate the reviewer’s insightful feedback. In response, we thoroughly revised the manuscript to improve clarity and narrative flow. We have reworded overly generic phrases, introduced more natural transitions, and varied sentence length to achieve a more engaging writing style throughout the manuscript.

Comment 2. Model 2 (Lines 256–261): model structure is mathematically well-defined, but some descriptive explanations e.g., rationale for variable selection, justification for prior parameters could be improved for non-statistical readers. A conceptual diagram summarizing the four models is needed.

 Response 2. We agree with this observation. In response, we have added a directed acyclic graph (DAG) to visualize the model structure and enhanced the variable selection rationale with detailed justifications based on established literature and our preliminary findings.

Revised Manuscript Section. [Section 2.2. Models, Figure 3] Directed acyclic graph (DAG) of the model structure; pc - passenger_cars, pd - population_density, rd - road_density, t - temperature, h - humidity, p - precipitation.

Revised Manuscript Section. [Section 2.1. Data, lines 164–218] The dataset employed in this study is a comprehensive aggregation of road traffic accidents recorded across Poland between 2015 and 2023, […], and road density). The selection of these variables is directly informed by established road safety literature and the preliminary findings presented in our foundational work [45]. Weather parameters such as temperature, humidity, and precipitation were chosen for their well-documented effects on driver behavior and road surface conditions. Our previous analysis confirmed these relationships, revealing a consistent positive correlation between temperature and accident frequency (correlation coefficient of 0.09) and a notable negative correlation for humidity (-0.10). The selected socioeconomic indicators serve as robust proxies for traffic exposure and infrastructure characteristics. Specifically, population density and passenger car registrations were included due to their strong positive correlation with total accident counts (0.45 and 0.82, respectively), while road density provides a crucial measure of infrastructure development, showing a meaningful correlation (0.31). Based on the original dataset, specific preprocessing was conducted for each analytical problem.

 

Comment 3. Figures 11–18: Some scatter plots and histograms lack clear axis labeling or are visually dense. Please provide consistent scales across models and highlighting key model differences visually.

Response 3. We appreciate this feedback. In response, we have improved Figures 11–18 to enhance their clarity and comparability. Specifically, we have added more descriptive axis labels, included informative titles to provide better context, removed some supplementary information from the figures such as the exact numerical values of the median and confidence intervals to reduce visual clutter, and standardized the axis scales within each group of models.

Comment 4. Table 4 and 5: need summarizing paragraph to contextualize ELPD differences e.g., whether 83.61 or 78.99 is considered large in practice.

Response 4. We agree with the reviewer that ELPD differences require contextual interpretation. As ELPD is a relative measure of predictive accuracy, absolute values (e.g., 83.61 or 78.99) cannot be meaningfully interpreted in isolation, as the scale can be different in each analysis. Apart from the general rule that higher ELPD indicates better predictive performance, no fixed threshold defines how large a value is in practical terms. Nevertheless, ELPD remains a widely accepted and robust tool for comparing models. In response to this comment, we have expanded second paragraph of the Model Comparison to clarify the interpretation of ELPD differences.

Revised Manuscript Section. [Section 3.2. Model Comparison, lines 438–489] The expected log pointwise predictive density (ELPD) serves as our primary comparison metric. While higher ELPD values indicate better posterior predictive accuracy when using the model to predict new data points, the metric itself has no interpretable unit and cannot be compared across different analyses or datasets. Even within a single analysis, absolute ELPD values lack contextual meaning. However, because ELPD reliably ranks models by predictive accuracy, it remains a robust and informative criterion for comparing models within the same domain. When comparing models, we also consider standard errors (SE) and the difference in standard errors (dSE) to assess the statistical significance of performance differences [47].

Regarding Comments on the Quality of English Language

Comment 5. Overuse of technical jargon e.g., "prior predictive check," "hyperparameter tuning" without lay explanation. Overly long sentences in discussion and introduction e.g. Lines 419-424 - consider splitting into two sentences for better flow.

Response 5. We appreciate this valuable suggestion. In response, we conducted a thorough revision of the manuscript to improve clarity and readability. Technical terms have been explained more clearly, and overly long sentences—particularly in the Introduction and Discussion sections—have been revised or split to enhance the flow and overall quality of the English language.

Revised Manuscript Section. [Section 2.3. Prior Predictive Checks, lines 320–357] Before fitting the models to observed data, prior predictive checks were performed to assess whether the specified priors lead to plausible predictions. This involves generating simulated data using only the prior distributions—without conditioning on any observed outcomes—to evaluate whether the model can produce realistic values given the assumed parameter ranges. Each model was executed with 500 warm-up iterations and 2,000 sampling iterations across four chains to generate these prior-based simulations.

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

My comments have been addressed.