Collision Severity Model

• Collision Severity Model is a quantified framework that estimates crash outcomes by integrating physical crash mechanics, injury probability mappings, and contextual factors.
• It employs multilevel mixed-effects, ordered logit, and machine learning methods to provide statistically robust predictions and transparent risk assessments.
• Applications span safety engineering, ADAS/ADS validation, and policy development through targeted countermeasures and data-driven risk management.

A collision severity model is a quantified framework that estimates the severity of loss—injury, fatality, or property damage—resulting from on-road crash events. Such models serve as the analytical backbone for safety engineering, ADAS/ADS validation, and predictive crash analytics, integrating physical crash mechanics, injury probability mappings, traffic exposure, and behavioral heterogeneity. Both actuarial and mechanistic paradigms exist, spanning multilevel mixed-effects statistical models, damage-based energy integrals, machine learning classifiers, counterfactual simulation frameworks, and ethical cost-based planners. The rigorous mathematical and statistical foundations of collision severity models enable data-driven safety management, targeted countermeasure selection, and transparent explanation of risk at both system and case levels.

1. Mathematical Formulations and Severity Metrics

Collision severity quantification hinges upon mapping crash circumstances to probabilistic or deterministic outcomes. Foundational approaches include:

• Binary and Multinomial Regression Frameworks:

Logistic or logit models estimate $P_i = \Pr(\text{severity}_i > \text{PDO})$ via

$\logit(P_i) = \beta_0 + \sum_{h=1}^H \beta_h\,X_{hi},$

where predictors $X_{hi}$ cover driver, vehicle, and environment dimensions (Azhdari et al., 13 Aug 2025).

• Multilevel Mixed-Effects Models:

Severity is nested by crash and road, introducing random intercepts $u_{0j} \sim N(0, \sigma_u^2)$ and optionally random slopes $v_{kj} \sim N(0, \sigma_{v_k}^2)$ for context-specific sensitivity (Azhdari et al., 13 Aug 2025). The intraclass correlation coefficient (ICC),

$$\text{ICC} = \frac{\sigma_u^2}{\sigma_u^2 + \pi^2/3},$$

quantifies road-level variance in log-odds.

• Ordered Logit Models:

Ordinal models structure severity into ordered classes (PDO, Injury, Fatal), modeling latent severity with cutpoints and cumulative logistic functions:

$$P(Y_i \leq j) = \Lambda(C_j - X_i^\top\beta), \quad \Lambda(z) = \frac{1}{1 + e^{-z}}.$$

(Asgharpour et al., 2021)

• Damage-Based Energy Integrals:

The generalized CRASH3 algorithm models vehicle deformation as absorbed energy:

$$E_{3D} = \iint \frac{1}{2} B \left[c_p(w,h) + \frac{A}{B}\right]^2 dw\,dh,$$

with velocity change $\Delta V$ determined via energy conservation:

$\Delta V = \sqrt{\frac{2E_{3D}}{m}}.$

(Scurlock, 2014)

• Δv and Impact Kinematics:

Severity is empirically mapped from velocity differentials at impact:

$\Delta v = |v_{\text{after}} - v_{\text{before}}|,$

tied to injury and fatality risk via established epidemiological models (Porav et al., 2019, Roy-Singh et al., 9 Jun 2025).

• Machine Learning and Explainability:

Severe crash probability is learned as a classifier output (e.g., Ridge Logistic, Random Forest, XGBoost), with feature contributions explained via SHAP values $\phi_i$ and domain-specific importance ranking (Castellani et al., 15 Aug 2025, Adefabi et al., 2023, Chakraborty et al., 15 Sep 2025).

2. Covariates, Contextual Structure, and Hierarchical Effects

Models for collision severity incorporate multidimensional predictors:

• Crash-Level:

Demographics (age, gender, education), behavioral (seat belt, distraction), vehicular (model year, type), environmental (lighting, pavement, weather), and temporal (time-of-day, season) predictors.

• Road-Level:

AADT (log-transformed), truck share, terrain slope, access density, geometric features.

• Interaction and Heterogeneity:
  • Pavement status displays high between-road slope heterogeneity ($\sigma^2_{v,\text{pavement}}=0.259$).
  • Lighting, education, and age also have notable random variances, supporting tailored countermeasures.
• Cluster-Based Pattern Discovery:

Dimensionality-reduction and clustering (CCA) reveal patterns such as entry/yield, improper maneuver, fixed-object, and rear-end roundabout risks, with SHAP explanations exposing the drivers within each cluster (Chakraborty et al., 15 Sep 2025).

3. Estimation, Model Comparison, and Statistical Diagnostics

Collision severity models are calibrated and compared via:

• Likelihood and Information Criteria:

Deviance ($D = -2\ell(\theta)$), AIC ($-2\ell + 2p$), and BIC ($-2\ell + p\ln N$) underpin model selection, favoring more flexible frameworks (random-coefficient model accuracy 0.71 vs. GLM 0.62; AUC 0.775 vs. 0.570 (Azhdari et al., 13 Aug 2025)).

• Simulation and Cross-Validation:

Machine learning models undergo stratified cross-validation, hyperparameter tuning, and out-of-sample hold-out performance evaluation (e.g., Ridge LR AUC=84.9% (Castellani et al., 15 Aug 2025), XGBoost AUC=0.82 (Chakraborty et al., 15 Sep 2025), Random Forest AUC=0.80 (Adefabi et al., 2023)).

• Feature Selection and Explainability:

Statistically Equivalent Signature (SES), SHAP, and Granger causality identify and validate leading predictors, revealing surprising importance patterns (environmental/contextual variables often outperform behavioral predictors (Castellani et al., 15 Aug 2025, Chakraborty et al., 2021)).

4. Physical and Biomechanical Severity Mapping

Mechanistic approaches compute severity indices from crash physics:

• Delta-V and Injury Probability:

Δv models—derived from vehicle and pedestrian dynamics—calibrate fatality risk curves (e.g., $P_{\text{ped}}(v) = 1/(1 + e^{6.9 - 0.09|v|})$, $$P_{\text{occ}}(\Delta v) = \min\{1, (0.621\Delta v/71)^4\}$$ (Porav et al., 2019)) and are embedded in RL policy reward functions to guide active collision mitigation.

• Crush Energy Integrals:

The 3D CRASH3 algorithm calculates total crush energy as a surface integral, with severity translated to velocity change and thus potential for injury (Scurlock, 2014).

• Organ Trauma and Peak Virtual Power:

Detailed biomechanical models (OTM/PVP) compute the instantaneous rate of mechanical work in organ tissues, scaled by impact speed and location, and mapped to AIS levels via cubic risk escalation

$$\mathrm{PVP}_{\rm AIS=i} = \mathrm{PVP}_{\rm AIS=4} \left(\frac{i}{4}\right)^{3/4}.$$

Ageing, material degradation, and subdural hematoma corrections further refine forensic severity prediction (Bastien et al., 2020).

5. Simulation, Counterfactual Analysis, and Validation

Simulation techniques enable forward estimation and policy evaluation:

• Scenario Generation:

Synthetic crash scenario sets use multivariate kinematics, mixed driving behavior models, and weighted sampling (IPF, kNN) to reflect representative severity distributions for ADS/ADAS validation. Δv-based outcomes are validated against reference datasets via KS and t-SNE multivariate fit tests (Wu et al., 2024).

• Fractional Collision Risk:

Counterfactual simulation frameworks estimate the distribution of severity levels (L0, L1, L2) by sampling behavioral responses (reaction time, acceleration) and aggregating probabilities:

$P(L_x) = \sum_{j} p_j 1_{L_x^{(j)}}$

yielding fractional collisions in probabilistic terms suitable for ADS benchmarking (Roy-Singh et al., 9 Jun 2025).

• Optimal and Ethical Path Planning:

Severity maps with scalar weights $C_o$ (reflecting social/ethical cost) underpin two-level optimal control: first minimizing integrated severity over time and proximity, then minimizing steering effort among lowest-severity paths. This quantifies the explicit trade-off when collisions are unavoidable, and exposes the influence of severity ratings on AV routing (Wang et al., 2024, Pickering et al., 2022).

6. Contextual Insights, Applications, and Policy Implications

Collision severity modeling generates actionable knowledge:

• Targeted Countermeasures:

Multilevel slope estimates indicate where pavement or lighting interventions yield highest severity reduction (priority by $\sigma^2_{v,k}$ (Azhdari et al., 13 Aug 2025)). Rear-end models guide belt law enhancements, large-truck speed/following policies, and driver response programs (Yuan et al., 2023).

• Vision Zero and Systemic Risk Management:

Leading indicators (e.g., SHM hazard scores (Antonsson et al., 2022)) and interpretable dashboards inform systematic risk control strategies—speed management, road design improvements, targeted education—over traditional accident statistics.

• Explainability and Auditable Reporting:

SHAP-based models (Castellani et al., 15 Aug 2025, Chakraborty et al., 15 Sep 2025) and pattern-specific cluster frameworks provide transparent case-level attribution of severity, supporting audit-ready screening and deployment decisions in traffic safety analytics.

7. Limitations, Validation, and Future Directions

Recognized constraints and open research challenges include:

• Model Assumptions:

Homogeneous stiffness in damage models, independence in SHM scoring, absence of continuous injury risk curves in some Δv frameworks, and lack of secondary impact modeling in organ trauma scenarios (Scurlock, 2014, Bastien et al., 2020, Roy-Singh et al., 9 Jun 2025).

• Validation and Data Bias:

Synthetic scenario generation requires careful marginal and correlation matching to avoid bias toward high-severity or non-representative crash populations (Wu et al., 2024).

• Uncertainty Propagation and Counterfactual Robustness:

Monte-Carlo sampling, behavioral distributions, and sensor noise integration underpin uncertainty quantification, but published variance estimates remain limited (Roy-Singh et al., 9 Jun 2025).

Advances in real-time risk prediction, explainable AI, simulation-based ethics, and biomechanical severity mapping will further enhance the precision, interpretability, and policy utility of future collision severity models.