# Investigating Spatial Autocorrelation and Spillover Effects in Freeway Crash-Frequency Data

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Spatial Autocorrelation Effect

#### 2.2. Spatial Spillover Effect

## 3. Data Assembly and Preliminary Analysis

#### 3.1. Roadway Inventory

#### 3.2. Crash Data

#### 3.3. Traffic Data

## 4. Methodology

#### 4.1. Model Specification

#### 4.1.1. CAR Model

_{i}is the observed crash count at segment i, and λ

_{i}is the expected Poisson crash rate, which is modeled as a generalized linear function of the observed risk factors

**X**

_{i}:

_{0}and

**β**are the estimable coefficients corresponding to the crash exposure variable and risk factors

**X**

_{i}respectively. θ

_{i}is a residual term to account for unstructured heterogeneous effects which basically reflect unmeasured differences among freeway segments. It is assumed to follow an ordinary, exchangeable normal distribution [4]:

_{i}represents the spatial autocorrelation effect, and is modeled by using the intrinsic CAR prior, first proposed by Besag [33]:

#### 4.1.2. Spatial Spillover Model

_{i}:

#### 4.1.3. Hybrid Model

_{i}and the observed adjacent factors ${X}_{i}^{adj}$ are incorporated into the link function, that is,

#### 4.2. Model Assessment

#### 4.3. Model Estimation

## 5. Results Analysis and Discussion

#### 5.1. Model Comparison

_{c}(=1/τ

_{c}), which represent the spatial autocorrelation effect, is significantly positive at the 95% credibility level in the CAR and hybrid models. The spatial autocorrelation effect is expected and attributable to the missing variables that are spatially clustered, thus affecting many adjacent segments [16,17]. Examples of such missing variables include terrain features and weather conditions. The spatial spillover effect is confirmed by the significant effects of the exogenous variables observed at adjacent segments (i.e., Curvature_adj in the spatial spillover model, and Curvature_adj and Grade_adj in the hybrid model). While the spatial spillover effect has been found at the macro-level crash frequencies [6], this seems to be the first time that it is found at the micro-level crash frequencies.

#### 5.2. Interpretation of Parameter Estimates

## 6. Conclusions and Future Research

## Author Contributions

## Funding

## Conflicts of Interest

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Variable | Description | Mean | S.D. | Min. | Max. |
---|---|---|---|---|---|

Response variable | |||||

Crash | Crash count per road segment | 4.46 | 3.32 | 0 | 23 |

Crash exposure variable | |||||

DVKT | Daily vehicle kilometers traveled (10^{3} km·pcu ^{a}) | 44.3 | 16.9 | 9.98 | 129 |

Risk factors | |||||

Curvature | Horizontal curvature (0.1 km^{−1}) | 1.77 | 1.27 | 0 | 4.35 |

Grade | Vertical grade (%) | 0.741 | 0.568 | 0 | 2.91 |

Bridge | A part of bridge: yes = 1, no = 0 | 0.5 | 0.502 | 0 | 1 |

Ramp | Presence of ramp: yes = 1, no = 0 | 0.208 | 0.407 | 0 | 1 |

Curvature_adj | Curvature of adjacent segments | 1.79 | 0.951 | 0 | 4.35 |

Grade_adj | Grade of adjacent segments | 0.731 | 0.435 | 0.15 | 2.15 |

Bridge_adj | A part of bridge on adjacent segments: yes = 1, no = 0 | 0.747 | 0.436 | 0 | 1 |

Ramp_adj | Presence of ramp on adjacent segments: yes = 1, no = 0 | 0.383 | 0.488 | 0 | 1 |

^{a}pcu: passenger car unit.

Variable | CAR | Spatial Spillover | Hybrid Model |
---|---|---|---|

Constant | −9.37 (1.49) ^{a} ** | −9.26 (1.74) ** | −8.27 (1.34) ** |

Log(DVKT) | 0.987 (0.139) ** | 0.978 (0.162) ** | 0.886 (0.125) ** |

Curvature | 0.072 (0.040) * | 0.100 (0.050) ** | 0.106 (0.043) ** |

Grade | 0.171 (0.088) * | 0.109 (0.099) | 0.170 (0.088) * |

Curvature_adj | — | −0.150 (0.068) ** | −0.150 (0.062) ** |

Grade_adj | — | 0.061 (0.124) | 0.227 (0.129) * |

σ_{h} (=1/τ_{h}) | 0.025 (0.026) * | 0.138 (0.046) ** | 0.021 (0.024) |

σ_{c} (=1/τ_{c}) | 0.033 (0.013) ** | — | 0.032 (0.013) ** |

α | 0.747 (0.112) ** | — | 0.772 (0.109) ** |

DIC | 689 | 702 | 687 |

^{a}Posterior mean (standard deviation) for the parameter. ** Significant at the 95% credible level. * Significant at the 90% credible level. DIC: deviance information criterion.

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**MDPI and ACS Style**

Wen, H.; Zhang, X.; Zeng, Q.; Lee, J.; Yuan, Q.
Investigating Spatial Autocorrelation and Spillover Effects in Freeway Crash-Frequency Data. *Int. J. Environ. Res. Public Health* **2019**, *16*, 219.
https://doi.org/10.3390/ijerph16020219

**AMA Style**

Wen H, Zhang X, Zeng Q, Lee J, Yuan Q.
Investigating Spatial Autocorrelation and Spillover Effects in Freeway Crash-Frequency Data. *International Journal of Environmental Research and Public Health*. 2019; 16(2):219.
https://doi.org/10.3390/ijerph16020219

**Chicago/Turabian Style**

Wen, Huiying, Xuan Zhang, Qiang Zeng, Jaeyoung Lee, and Quan Yuan.
2019. "Investigating Spatial Autocorrelation and Spillover Effects in Freeway Crash-Frequency Data" *International Journal of Environmental Research and Public Health* 16, no. 2: 219.
https://doi.org/10.3390/ijerph16020219