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Markov switching multinomial logit model: an application to accident injury severities
Nataliya V. Malyshkina, Fred L. Mannering
TL;DR
The paper addresses accident-injury severity modeling with standard single-state approaches that do not represent time-varying, unobserved roadway-safety heterogeneity. It proposes two-state Markov switching multinomial logit models estimated with Bayesian inference and MCMC, and finds superior statistical fit relative to standard multinomial logit models. The less frequent state is correlated with adverse weather conditions, although the two states are approximately equally dangerous in many cases.
Problem
Accident-severity modeling needs to account for possible heterogeneity from unobserved roadway-safety risk factors that can change over time.
Method
The study estimates two-state Markov switching multinomial logit models for Indiana accident-severity data using Bayesian inference and MCMC, alongside standard multinomial logit comparison models.
Results
The MSML models are strongly favored over corresponding ML models, improving the logarithm of marginal likelihood by 40.5 to 61.4.
Takeaways & Limitations
The less frequent roadway-safety state is correlated with adverse weather, while the two states are approximately equally dangerous for accident severity in many cases.
Takeaways & Limitations
MSML states were found for 1-vehicle accidents on high-speed roads but not for 2-vehicle accidents on high-speed roads.
Abstract
from arXiv · showhide
In this study, two-state Markov switching multinomial logit models are proposed for statistical modeling of accident injury severities. These models assume Markov switching in time between two unobserved states of roadway safety. The states are distinct, in the sense that in different states accident severity outcomes are generated by separate multinomial logit processes. To demonstrate the applicability of the approach presented herein, two-state Markov switching multinomial logit models are estimated for severity outcomes of accidents occurring on Indiana roads over a four-year time interval. Bayesian inference methods and Markov Chain Monte Carlo (MCMC) simulations are used for model estimation. The estimated Markov switching models result in a superior statistical fit relative to the standard (single-state) multinomial logit models. It is found that the more frequent state of roadway safety is correlated with better weather conditions. The less frequent state is found to be correlated with adverse weather conditions.
1 Introduction
The study extends accident-severity modeling with two-state Markov switching multinomial logit models that represent time-varying, unobserved roadway-safety conditions.
- 1 Introduction: Accident-severity research models injury outcomes using accident, roadway, driver, and environmental characteristics.Common approaches include multinomial logit, nested logit, mixed logit, and ordered probit models.
- 1 Introduction: The proposed models allow roadway entities to switch over time between two unobserved roadway-safety states.The switching process is Markovian and is intended to account for heterogeneity from unpredictable, unidentified, and unobservable risk factors.
- 1 Introduction: Each roadway-safety state has a separate multinomial logit process for generating accident-severity outcomes.This structure allows researchers to study heterogeneity effects in roadway safety.
2 Model specification
The model combines multinomial-logit severity processes with a stationary two-state Markov chain governing latent roadway-safety conditions over time.
- 2 Model specification: Accident observations are represented by severity-outcome indicators across accidents and time periods.The model specification includes accident characteristics such as weather, environment, vehicle and driver attributes, and roadway properties.
- 2 Model specification: A latent state variable assigns all roadway entities in each time period to one of two roadway-safety states.State 0 and state 1 are modeled as a stationary two-state Markov chain.
- 2 Model specification: The transition probabilities P(s_t+1 = 1|s_t = 0) = p_0→1 and P(s_t+1 = 0|s_t = 1) = p_1→0 govern switching between states.These transition probabilities are unknown parameters estimated from accident-severity data.
- 2 Model specification: The stationary unconditional state probabilities are p̄_0 = p_1→0/(p_0→1 + p_1→0) and p̄_1 = p_0→1/(p_0→1 + p_1→0).The model assumes p̄_0 ≥ p̄_1, labeling state 0 the more frequent state and state 1 the less frequent state.
- 2 Model specification: Each state uses a standard multinomial logit probability mass function with its own estimable coefficient vector.The first coefficient is an intercept, and coefficients for the last severity outcome are set to zero.
- 2 Model specification: The full parameter vector includes the state values, multinomial-logit coefficients, and transition probabilities.Together, the model equations define the two-state Markov switching multinomial logit model.
3 Model estimation methods
Because latent state variables complicate maximum-likelihood estimation, the study uses Bayesian inference and MCMC, with Bayes factors and goodness-of-fit tests for model comparison and evaluation.
- 3 Model estimation methods: Unobserved state variables make traditional maximum-likelihood estimation of Markov switching models of limited use.The state variables create a large latent-state space; with T = 208 time periods, there are 2^208 possible state-vector combinations.
- 3 Model estimation methods: Bayesian inference combines the likelihood of the observed data with prior knowledge to obtain the posterior distribution of model parameters.The posterior is proportional to the product of the likelihood and prior distribution.
- 3 Model estimation methods: Markov Chain Monte Carlo simulations provide a practical way to sample from the posterior distribution when direct integration is infeasible.The difficulty arises because the parameter vector contains too many components for direct computation.
- 3 Model estimation methods: Bayes factors compare models through ratios of their marginal likelihoods and penalize models that include too many parameters.With equal prior model probabilities, the posterior-probability ratio equals the Bayes factor.
- 3 Model estimation methods: Model fit is evaluated using a Pearson χ2 goodness-of-fit test whose p-value is obtained from Monte Carlo simulations.The Pearson χ2 quantity measures discrepancy between observations and model predictions.
4 Empirical results
Using Indiana accident data from 2003–2006, the study finds that two-state MSML models exist for several roadway and accident-type combinations and are strongly favored over standard ML models when both states exist. The results associate roadway-safety states with road class, weather conditions, and accident characteristics, while often finding similar severity probabilities across states.
- Data and model estimation: 811720 Indiana accidents from 2003–2006 were analyzed in weekly periods across roadway-class and accident-type combinations.The analysis used 208 weeks and excluded accidents involving more than two vehicles.
- Data and model estimation: Two roadway-safety states were identified for 1-vehicle accidents on all roadway classes and for 2-vehicle accidents on streets.Two states were not found for 2-vehicle accidents on interstate highways, US routes, state routes, or county roads.
- Model comparison: MSML models improved the logarithm of marginal likelihood by 40.5–61.4 over corresponding ML models and were strongly favored whenever both states existed.Related unreported combinations also favored MSML models, including 1-vehicle accidents on county roads and streets and 2-vehicle accidents on streets.
- Roadway-safety-state patterns: The MSML models showed positive correlations among safety states for 1-vehicle accidents on high-speed roads, with coefficients ranging from 0.263 to 0.688.The authors interpret this pattern as suggesting common unobservable factors affecting switching between roadway-safety states.
- Roadway-safety-state patterns: The less frequent state for 1-vehicle accidents on high-speed roads correlated positively with extreme temperatures, precipitation, snowfall, strong winds, fog, frost, and low visibility.The authors relate the two-state pattern to differences in roadway safety between bad and better weather.
- Roadway-safety-state patterns: Street-road safety states showed low correlation with other roads and weather, while averaged fatality and injury probabilities often differed little between states.Significant fatality-probability differences were found only for 1-vehicle accidents on US routes, county roads, and streets.
5 Conclusions
The study finds that two-state MSML models apply to one-vehicle accidents on high-speed roads but not two-vehicle accidents there. Across cases with two roadway-safety states, MSML models fit accident severity substantially better than standard ML models, while the states are often similarly dangerous despite adverse-weather associations.
- Model applicability: Two-state MSML models exist for one-vehicle accident severity on high-speed roads, but not for two-vehicle accident severity on those roads.The authors suggest that one- and two-vehicle accidents may differ in their underlying nature, and call for further study.
- Severity interpretation: States s_t = 0 and s_t = 1 are approximately equally dangerous for accident severity in many cases, despite state s_t = 1 being correlated with adverse weather.The paper explains that bad weather may increase both serious and minor accidents, leaving their relative fraction approximately steady.
Variable
The model uses indicators and quantitative variables describing accident timing, precipitation, roadway conditions, vehicle and driver characteristics, traffic control, location, and season.
- Timing and season: Accident timing and seasonal conditions include late-night hours, summer, winter, Sunday, and Thursday indicators.
- Weather and roadway: Weather and roadway-environment variables include precipitation type, snow or slush coverage, snowing weather, road geometry, construction, road junctions, traffic control, road type, and roadway median.
- Location and context: Location and traffic-context variables include private drives, non-roadway crashes, four-way intersections, urban versus rural road type, and the license state of the vehicle at fault.
- Vehicles and drivers: Vehicle and driver characteristics include vehicle and driver ages, occupant count, speed limit, driver gender, vehicle type, fire involvement, and primary accident cause.
- State-weather relationships: Table 5 reports correlations between posterior state probabilities P(s_t = 1|Y) for MSML models and weather-condition variables.