NETWORK ANALYSIS OF ROAD CRASH FREQUENCY USING SPATIAL MODELS

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NETWORK ANALYSIS OF ROAD CRASH FREQUENCY USING SPATIAL MODELS

ABSTRACT

Despite the evident spatial character of road crashes, limited research has been conducted in road safety analysis to account for spatial correlation; further, the practical consequences of this omission are largely unknown. The purpose of this research is to explore the effect of spatial correlation in models of road crash frequency at the segment level. Different segment neighboring structures are tested to establish the most promising one in the context of modeling crash frequency in road networks.

A Full Bayes hierarchical approach is used with conditional autoregressive effects for the spatial correlation terms. Analysis of crash, traffic and roadway inventory data from rural engineering districts in Pennsylvania and Washington indicate the importance of including spatial correlation in road crash models. Six different road classes were analyzed in a single network model to facilitate the inclusion of the spatial correlation structures in the whole state-maintained network simultaneously rather than in separate models by road type.

The inclusion of spatial correlation has an important impact on the estimation of crash frequency models by explaining additional extra-Poisson variability present in the data ; hence, producing better fitting models and improving the precision of the crash frequency and excess crash frequency estimates compared with models with heterogeneity-only random effects.

Pure distance-based neighboring models (i.e. exponential decay) performed poorly in comparison to adjacency-based or distance-order models. The results also suggest that spatial correlation is more important in distances of one mile or less. The inclusion of spatially correlated random effects significantly improves the precision of the estimates of the expected crash frequency for each segment by ‘pooling’ strength from

 

their neighbors; thus, reducing their standard deviation. This is a significant advantage of spatial models since poor estimates due to small sample sizes and low sample means is a frequent issue in highway safety analysis.

 

 

TABLE OF CONTENTS

LIST OF FIGURES…………………………………………………………………………………….. vii

LIST OF TABLES………………………………………………………………………………………. viii

ACKNOWLEDGEMENTS……………………………………………………………………………. x

Chapter 1  Introduction……………………………………………………………………………….. 1

Background………………………………………………………………………………………… 1

Literature Review…………………………………………………………………………………. 4

Spatial Statistics and Crashes …………………………………………………………. 4

Bayesian Modeling of Crashes ………………………………………………………… 8

Research Summary……………………………………………………………………………… 9

Chapter 2  Methodology ……………………………………………………………………………… 11

Functional Form Selection…………………………………………………………………….. 11Poisson-Gamma……………………………………………………………………………. 11

Poisson Log-Normal………………………………………………………………………. 13

Zero-Inflated Poisson……………………………………………………………………… 14

Zero-Inflated Poisson Log-Normal……………………………………………………. 15

Spatial Correlation……………………………………………………………………………….. 15

Prior Distribution Sensitivity Analysis………………………………………………………. 17

Multivariate Normal Prior ………………………………………………………………… 18

Alternative Priors…………………………………………………………………………… 18

Neighboring Structures…………………………………………………………………………. 19

Adjacency-Based Models ……………………………………………………………….. 19

Distance-Order Models…………………………………………………………………… 21

Distance-Exponential Decay Models ………………………………………………… 21

Adjacency-Route Information Models ……………………………………………….. 23

Adjacency and Network Distance Calculations…………………………………………. 24

Model Comparison ………………………………………………………………………………. 26

Chapter 3  Data Description ………………………………………………………………………… 28

PennDOT Data……………………………………………………………………………………. 28Crash Data…………………………………………………………………………………… 28

Road Inventory……………………………………………………………………………… 29

WSDOT data………………………………………………………………………………………. 34Accident File…………………………………………………………………………………. 35

Roadlog File …………………………………………………………………………………. 39

Curve File and Grade File……………………………………………………………….. 40

Ramp File…………………………………………………………………………………….. 40

Chapter 4  Results……………………………………………………………………………………… 41Functional Form Selection…………………………………………………………………….. 41Spatial Correlation……………………………………………………………………………….. 44

Sensitivity to Prior Distributions ……………………………………………………………… 51

Neighboring Structure Analysis………………………………………………………………. 57

Adjacency-Based Models ……………………………………………………………….. 59

Distance-Order Models…………………………………………………………………… 65

Distance-Exponential Decay Models ………………………………………………… 69

Adjacency-Route Information Models ……………………………………………….. 72

Effects of Spatial Correlation in Ranking of Sites for Engineering

Improvement ………………………………………………………………………………… 75

Chapter 5  Conclusions ………………………………………………………………………………. 80

Chapter 6  Recommendations for Future Research…………………………………………. 84

References……………………………………………………………………………………………….. 88

Appendix  Model Tables……………………………………………………………………………… 93

Chapter 1

 

Introduction

Despite the evident spatial character of road crashes, very little has been done in highway safety analysis to account for spatial correlation and the practical consequences of this omission are largely unknown. In recent years, spatial statistics have gained popularity due to the rediscovery by statisticians of Bayesian methods. Bayesian theory provides a flexible framework to handle spatial correlation structures that are too complex for traditional frequentist approaches. The purpose of this research is to analyze the impact of spatial correlation in highway safety models using Full Bayes hierarchical models.

Background

There are several reasons to consider spatial correlation in crash models. One of the most important is that by using spatial correlation, site estimates ‘pool strength’ from neighboring sites thus improving model estimation. This is especially true in circumstances with high random variability in the data, as is the case with most crash data. This issue is referred to as the “small area estimation” problem in statistics (Rao, 2003). There have been a series of papers concerning this issue in recent highway safety literature (Lord, Washington, and Ivan, 2005; Lord, 2006; and Lord, Washington, and Ivan, 2007). After conducting a simulation study, one group of researchers (Lord, Washington, and Ivan, 2005) concluded that excess zeros are likely under fairly common conditions and that the excess zeros arise because of low exposure and/or inappropriate selection of time/space scales. Crash reporting thresholds also influence road crashes reported to law enforcement, thus reducing the number of crashes that appear in agency databases compared to actual crash occurrence and what might be theoretically expected from a Poisson process. Crash sample size can also be reduced due to underreporting of low severity, property damage only crashes.

Recent enhancements in spatial modeling techniques have enabled researchers to investigate important issues related to risk estimation, unmeasured confounding variables, and spatial dependence in lattice systems such as road networks (Richardson, 1992). Around 1990, the “Markov Chain Monte Carlo (MCMC) revolution” took place, and methods like the Gibbs sampler and the Metropolis algorithm were coupled with faster computing to enable the evaluation of complicated integrals that are usually found in Bayesian methods (Banerjee, Carling, and Gelfand, 2004). Since then, Bayesian methods have been gaining popularity as the approach of choice when modelling multiple levels and incorporating random effects or complicated dependence structures.

Another very important advantage of spatial models is that spatial dependence can be a surrogate for unknown and relevant covariates and can adjust for them. By using spatial correlation as a surrogate for unmeasured covariates, model misspecification of the mean structure can be reduced by accounting for a variable that is spatially varying, improving model estimation (Dubin, 1988 and Cressie, 1993). In highway safety analysis, potential covariates that show variability in space include weather effects, driver population, and land use; these variables are rarely measured or accounted for in road safety models.

As in the case of temporal correlation, ignoring spatial dependence can lead to underestimation of variability (Congdon, 2001). This is particularly important in road safety models where random variability and small sample sizes are common issues. Here a more precise estimation of the variability in the parameters of interest is clearly advantageous for model interpretation.

Spatial and spatio-temporal models are better suited for Bayesian analyses where complex correlation structures can be more easily implemented. Full Bayes (FB) hierarchical models are the more flexible approach and offer several advantages in the particular case of crash data analysis.  One important characteristic of FB models is the explicit use of prior information to improve parameter estimates. The use of prior knowledge is a central part of the scientific method and priors have a natural implementation in Bayesian analysis.

Full Bayes models also take full account of the uncertainty associated with parameter estimates and provide exact measures of uncertainty on the posterior distributions of these parameters. Frequentist and Empirical Bayes (EB) methods traditionally ignore uncertainty in the correlation structures which is translated into overestimation of the precision of parameters associated with the covariates, which are usually the most important quantities to estimate in the model (Goldstein, 1995).

Another important advantage frequently cited by Bayesians is that Bayes methods provide confidence (credible) intervals that are more in line with commonsense interpretations. For instance, a Bayesian credible interval for an unknown quantity of interest can be directly regarded as having a high probability of containing the unknown quantity, while a frequentist confidence interval may be strictly interpreted only in relation to a sequence of similar inferences obtained by repeated sampling (Gelman et al, 2003).

Empirical Bayes methods have been used in highway safety to reduce regression to the mean bias (also known as selection bias) when selecting sites for engineering improvement. Typically, the prior in EB is obtained from the data (hence the empirical name), which is criticized by some as using the data twice (Carlin and Louis, 2000). FB models provide correction methods for regression-to-the-mean bias in a different way: spatially unstructured or/and structured random effects can be included to smooth the estimates and reduce regression-to-the-mean bias.

Finally, learning about spatial dependence of crashes is of interest in its own right. One might be interested in knowing how close intersections or segments should be to consider them correlated; or if the spatial correlation is different for different types of road; or how spatial relationships operate between segments at intersections.

Literature Review

This section is divided in two sub-sections, one reviewing the published studies that incorporate spatial statistics, frequentist or Bayesian, in highway safety analysis and the second one on Bayesian modeling of crashes.

Spatial Statistics and Crashes

One of the first crash analyses that included any type of spatial component was published by Levine, Kim, and Nitz  (1995a). Crashes were geocoded to the nearest intersection or ramp, then different ‘spatial’ statistics were calculated including mean center, standard distance deviation based on “great circle” distance, the standard deviational ellipse (1st and 2nd principal component), and the nearest neighbor index; based on the x and y coordinate of the accidents. The work concentrated on developing spatial probability ellipses for different categories of crashes, i.e. all crashes or alcohol related, one, two or three or more vehicles involved, etc. The analysis was descriptive rather than predictive in nature. In addition, the statistical assumptions upon which the work is tacitly based are commonly violated. For example, the assumption of normal spatial distribution of points in the x and y coordinate, which implies the absence of clustering that is clearly violated by crash data.

The work by Jones, Langford, and Bentham (1996) is another example of the use of spatial point pattern analysis in traffic accidents. The authors performed a classical K-function analysis on the residuals of a logit model where the log-odds were selected to be fatalities as opposed to seriously injured.  The variables of the model were: age, type of user (pedestrian, bicyclist, Motor Vehicle driver) and number of casualties. With this ad hoc approach, the authors found that, once the trend was removed from the data, the residuals presented clustering. Although this study included the analysis of certain contributing factors (as opposed to the work by Levine et al (1995a)), it still failed to directly include the spatial correlation into the coefficient estimation.

A theoretical analysis of spatial point-pattern distributions of accidents was performed by Nicholson (1999). One of the hypotheses of the study is that autocorrelation between underlying true accident rates at neighboring sites is likely to be found in crash data. Non-random distributions of accidents apart from the Complete Spatial Randomness (CSR) case were analyzed: stationary and isotropic (accidents not clustered but arranged regularly), nonstationary and isotropic (accidents clustered at randomly distributed points), and non-stationary and anisotropic (accident clustered along lines). Different statistical tests for spatial randomness such as quadrat methods, nearest neighbor methods, and K-function were analyzed. The author concluded that nearest neighbor methods appear more powerful and robust for detecting the kind of accident patterns that can be observed in practice. Nicholson also concluded that the Kfunction method enabled patterns at different spatial scales to be detected.

Levine, Kim, and Nitz (1995b) also estimate a spatial model at the census block level. They estimated what they called a “spatial lag” model. The spatial lag model is equivalent to the time series Autoregressive Lag 1 model (AR(1)), where the previous time is replaced by the weighted average of the neighbors.  The explanatory variables included in the model were: freeway crossing the block (dummy), miles of arterials or highways, miles of minor roads, miles of freeways, population, and employment. While the model takes into account the spatial correlation of the data, its weakness is the reliance on an assumed normal distribution for the number of crashes rather than a discrete count probability distribution such as Poisson or

Negative Binomial.

Black and Thomas (1998) studied spatial correlation of crash rates at the segment level for Belgium’s highway network, using data from 1996. Spatial correlation (or network autocorrelation as named by the authors) was quantified using the Moran’s Index (I). Moran’s I is a standard statistic used to measure the strength of spatial association among area units and it is analogous to the lagged autocorrelation coefficient in time series. The study concluded that there was a significant level of positive spatial correlation in the data, although, again, the results were descriptive.

A more advanced work in terms of spatial modeling of traffic crashes was developed by Miaou, Song, and Mallick (2003).  The authors estimated a series of spatial models of crashes at the county level for data from the state of Texas. Poisson-based Full Bayes hierarchical models of Fatal (K), incapacitating (A), and non-incapacitating (B) injuries were estimated using both frequency and rate values (using VMT as an offset term). A conditional Auto-Regressive model (CAR) was used to model spatial correlation and Markov Chain Monte Carlo (MCMC) was used to sample the posterior probability distribution. The main drawback of this work is the use of the surrogate variables: percent of time that the road is wet, sharp horizontal curves, and road side hazards.

Another analysis of crashes using spatial Full Bayes hierarchical models was performed by MacNab in 2004. Using hospitalization data for 83 local health areas in British Columbia (BC), Canada, between 1990 and 1999, determinants of motor vehicle accident injury were examined. Socioeconomic variables like marriage and immigration were used along with medical variables like life expectancy, health care providers, and hospital beds. In addition, the age effects were modeled using a spline regression. Other variables such as miles of roads and seatbelt violations were also used for the model. Random spatial effects were included in the model and assumed to have a CAR distribution. Considerable spatial correlation was found in the data.

Aguero-Valverde and Jovanis (2006) estimated Full Bayes hierarchical models (with spatial and temporal effects and space-time interactions) using injury and fatality data for Pennsylvania at the county level. Covariates included socio-demographics, weather conditions, transportation infrastructure and amount of travel. A CAR model was used for modelling spatial correlation and a time trend coefficient was included to model temporal effects. Space-time interactions were modeled using CAR random effects for each county times the time trend. Significant spatial correlation was found in the data even after including several spatially distributed covariates such as population and weather conditions. Given the evidence of spatial correlation at such an aggregated level, it is expected that spatial correlation would be more important at smaller spatial scales such as segment and intersection level.

Wang and Abdel-Aty (2006) performed a temporal and spatial analysis of rear-end crashes at signalized intersections using the Generalized Estimating Equations (GEE) approach. In the analysis, intersections were grouped into clusters based on their spatial location, distances, and corridor location; resulting in clusters varying from 1 to 13 intersections per cluster (all intersections in a cluster belong to the same corridor). Intersections within a cluster were considered correlated while intersections from different clusters were considered independent. Note that while the correlation between intersections in the same corridor was estimated, spatial correlation between intersections in corridors that intersect each other was ignored. For the spatial models, the authors explored three different correlation structures: independent correlation, exchangeable correlation (constant correlations between any two intersections within a cluster), and Autoregressive (AR-1) correlation; where the correlation decreases as the gap between intersections increase. A fourth spatial correlation structure, unstructured correlation was tried but failed to converge. In this structure different correlations are estimated for each intersection pair within a cluster. The models showed high spatial correlations between intersections for rear-end crashes.  More important yet, when spatial correlation structures were introduced, several coefficient estimates changed noticeably, this appears to indicate model bias due to misspecification of the mean structure.

Bayesian Modeling of Crashes

Empirical Bayes methods for safety analysis were proposed as early as 1981 (Abess, Jarret, and Wright, 1981).  These methods are frequently used to correct for regression-to-themean bias. EB methods for ranking of sites by expected accident frequency as well as expected excess accident frequency have been used in several studies (e.g. Persaud, Lyon, and Nguyen,

1999; Heydecker, and  Wu, 2001; and Miranda-Moreno, Saccomanno, and Labbe, 2005). These methods have been also suggested for observational before-after studies in highway safety (Hauer, 1997).

Full Bayes hierarchical models have being used in highway safety only recently. In 1997, Schlüter, Deely, and Nicholson proposed the use of FB models for ranking of sites using three different criteria: posterior probability of selecting the worse site, predictive probability of future accident numbers and expected number of future accidents. This approach was also used by Tunaru to model crash frequency with respect to several covariates (Tunaru, 1997; Tunaru, 2002).

Miaou, Song, and Mallick (2003) used FB models for road traffic crash mapping and Miaou and Song proposed the use of these models for ranking of sites using two different ranking criteria: ranking by probability that the site is the worst and ranking by posterior distribution of ranks (Miaou and Song, 2005). They also suggested the additional concept of a decision parameter which is site-specific and can include traffic flow, covariates, space and time effects as well as random effects. Aguero-Valverde and Jovanis (2007) also used FB hierarchical (Poisson Log-normal) models for ranking of road segments for engineering improvement for roads in Pennsylvania.

Qin et al (2005) used FB hierarchical models to fit Zero-inflated Poisson models to four different crash types for highway segments from Michigan, California, Washington, and Illinois. Lord and Miranda-Moreno (2007) used the FB approach along with Monte Carlo simulation to estimate the effect of low sample mean and small sample size on the estimation of the dispersion parameter in Poisson-Gamma models. Miranda-Moreno and Fu (2007) also used FB models in a comparison with EB approaches.  Bayesian multivariate Poisson-log-normal models of crash counts were proposed by Ma, Kockelman and Damien (2007) as well as Park and Lord (2007).

Research Summary

The purpose of this research is to explore the effect of spatial dependence in models of road crash frequency at the segment level.  Different segment neighboring structures are explored to establish the most promising one in the context of modeling crash frequency in road networks. Multivariate coefficients for the covariates are estimated by road type in a single network model. This facilitates the analysis of the spatial correlation structures in the whole state-maintained network simultaneously rather than in separate models by road type (as is common in highway safety practice). Spatial dependency is modeled in terms of network distances and relationships such as in a directed graph.

Other issues related to the main goal of the research that are explored are:

  • Are spatially distributed omitted variables present in the models? In other words, is the bias due to these variables detectable and important in the models?
  • What is the extent of spatial correlation: is there correlation only between segments next to each other? How far does the spatial correlation go?
  • Is the spatial correlation stronger between segments that belong to the same route? Is spatial correlation only serial correlation? Or is the contribution of intersecting routes also important?

Figure 1-1 shows the adjacency-based relationships used in the analysis to establish the basic neighboring structure for the spatial analysis. More details about the neighboring structures as well as the general methodology are discussed in the following section. Next, the sources and nature of the data analyzed in the study are presented, followed by the presentation and discussion of results, and finally conclusions and recommendations for future research.

NETWORK ANALYSIS OF ROAD CRASH FREQUENCY USING SPATIAL MODELS

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