# A Random-Parameter Negative Binomial Model for Assessing Freeway Crash Frequency by Injury Severity: Daytime versus Nighttime

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}(0.25) and predictive accuracy, along with a significantly positive $\alpha $ parameter. Moreover, transferability tests were conducted to confirm the rationality of separating the daytime and nighttime crashes. Based on the RPMNB models, several explanatory variables were observed to exhibit relatively stable effects whereas other variables presented obvious variations. This study can be of certain value in guiding highway design and policies and developing effective safety countermeasures.

## 1. Introduction

## 2. Literature Review

#### 2.1. Literature Review of Modeling Methodology for Crash Frequency

#### 2.2. Literature Review of Time-of-Day Variation and Temporal Instability

## 3. Methodology

#### 3.1. Poisson (Negative Binomial) Regression Model

#### 3.2. Negative Binomial Regression Model

#### 3.3. Random-Parameter Multivariate Model

#### 3.4. Elasticity Effect on Crash Frequency

#### 3.5. Model Estimation

## 4. Data Description

- (1)
- Light crash: a crash causing minor injuries to one to two persons, or causing property damage less than CNY 1000 (approximately USD 154.19);
- (2)
- Minor crash: a crash causing serious injuries to one to two persons, minor injuries to more than two, or property damage of more than CNY 1000 but less than CNY 30,000;
- (3)
- Severe crash: a crash causing one to two deaths, serious injuries to three to ten persons, or property damage of more than CNY 30,000 but less than CNY 60,000;
- (4)
- Very severe crash: a crash causing more than two deaths, serious injuries to more than 10 persons, or property damage of more than CNY 60,000. No very severe crashes were identified in this dataset. The crash frequency and outcomes regarding the three severity levels (light injury, minor injury, and severe injury) were calibrated and analyzed based on multivariate models.

^{2}; $i$ is the longitudinal grade; and $f$ is the longitudinal friction coefficient between the truck tires and road surface, and generally takes a value of 0.17.

## 5. Results

#### 5.1. Model Specification and Overall Measure of Fit

^{2}) (0.25) indicated that the RPMNB model outperforms the other models. Therefore, models accommodating unobserved effects performed better than their corresponding independent models (in both traditional and proposed regimes), highlighting the importance of accommodating the unobserved heterogeneity in examining crash counts by the different crash types.

#### 5.2. Model Estimation Result

#### 5.3. Transferability Tests

## 6. Discussions

#### 6.1. Traffic Characteristics

#### 6.2. Speed Characteristics

#### 6.3. Geometric Characteristics

#### 6.4. Sight Characteristics

#### 6.5. Elasticity Effects

## 7. Conclusions and Future Direction

^{2}(0.25) and predictive accuracy, along with a significantly positive $\alpha $ parameter. Based on the RPMNB models, several explanatory variables including were observed to exhibit relatively stable effects such as $\Delta {V}_{O-truck}$ and ${S}_{truck}$, whereas other variables were found to produce variations including AADT, ${R}_{present}$, ${L}_{present}$, $L{s}_{min}$, ${S}_{car}$, and ${S}_{car}$.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**A 95% confidence interval of elasticity effects of contributing factors on crash frequency. (

**a**) Average annual daily traffic (AADT). (

**b**) Speed difference of trucks with adjacent segment ($\Delta {V}_{O-truck}$) (10

^{−3}). (

**c**) Radius of the horizontal curve (${R}_{present}$). (

**d**) Length of the horizontal curve (${L}_{present}$). (

**e**) Length of the longitudinal slope corresponding to the minimum grade ($L{s}_{min}$). (

**f**) Stopping sight distance of cars (${S}_{car}$). (

**g**) Stopping sight distance of trucks (${S}_{truck}$). (

**h**) Horizontal clearance of trucks (${H}_{car}$).

Methodological Approach | Significant Variables | Previous Research |
---|---|---|

Random parameters count models | Average international roughness index (IRI); pavement condition; annual average daily travel (AADT); imposed speed limit; shoulder width, horizontal and vertical curves; weather condition | Anastasopoulos and Mannering [17], Buddhavarapu et al. [24] |

Random parameters Tobit model | IRI; pavement condition; horizontal curves; vertical grade; median barrier | Anastasopoulos et al. [25] |

Latent-class (finite mixture) models | Driver age; usage of alcohol/drugs; seat belt usage; lighting condition; speed; pavement condition; median barrier | Xie et al. [26], Behnood and Mannering [27] |

Markov switching count model | Pavement condition; AADT; percentage of single-unit trucks; season; the number of bridges per mile; shoulder width | Malyshkina et al. [28], Malyshkina and Mannering [29] |

Random-parameters multivariate models | AADT; unsignalized controlled intersection densities; business land use; lane number; lane width; shoulder width; posted speed | Barua et al. [21], Dong et al. [23] |

Code | Variables Name | Day | Night | Total | ||
---|---|---|---|---|---|---|

Frequency | Percentage | Frequency | Percentage | |||

1 | Light injury | 2083 | 65.94% | 869 | 27.52% | 2952 |

2 | Minor injury | 124 | 3.93% | 57 | 1.80% | 181 |

3 | Severe injury | 17 | 0.53% | 9 | 0.28% | 26 |

Total | 2224 | 70.40% | 935 | 29.60% | 3159 |

Variables Names | Definition | Min. | Max. | Mean | SD |
---|---|---|---|---|---|

Crash characteristic | |||||

Weather | 1, rainy or snowy day (6.1%); 0, otherwise (93.9%) | 0 | 1 | 0.06 | 0.2 |

Pavement condition | 1, ice pavement (2.1%); 0, otherwise (97.9%) | 0 | 1 | 0.02 | 0.1 |

Season | 1, occurred from February to April; 2, occurred from May to July; 3, occurred from August to October; 4, occurred in November, December, or January. | 1 | 4 | 2.5 | 1.4 |

Traffic characteristic | |||||

Interchange | 1, occurred near an interchange (25.8%); 0, otherwise (74.2%) | 0 | 1 | 0.1 | 0.3 |

Bridge | 1, occurred on bridge (12.4%); 0, otherwise (87.6%) | 0 | 1 | 0.3 | 0.5 |

AADT | Average annual daily traffic volume | 31,158 | 68,836 | 52,850.9 | 10,581.5 |

Speed characteristic | |||||

${V}_{O-car}$ (km/h) | Operating speed of cars | 95.1 | 193.8 | 119.6 | 21.5 |

$\Delta {V}_{O-car}$ (km/h) | Speed difference of cars with adjacent segment | −78.0 | 85.9 | −0.2 | 34.9 |

${V}_{O-truck}$ (km/h) | Operating speed of trucks | 61.3 | 104.8 | 79.3 | 12.1 |

$\Delta {V}_{O-truck}$ (km/h) | Speed difference of trucks with adjacent segment | −25.7 | 34.0 | −1.4 | 21.3 |

$\Delta {V}_{O}$ (km/h) | Speed difference between cars and trucks | 12.6 | 104.7 | 40.3 | 16.0 |

Geometric characteristics | |||||

${R}_{front}$ (m) | Radius of the plane curve of front section | 5597 | 1,000,000 | 429,300.6 | 490,879.2 |

${L}_{front}$ (m) | Length of the plane curve of front section | 450 | 3267 | 1224.2 | 711.3 |

${R}_{present}$ (m) | Radius of the horizontal curve (plane curve of present section) | 5597 | 1,000,000 | 380,669.6 | 481,087.0 |

${L}_{present}$ (m) | Length of the horizontal curve (plane curve of present section) | 680 | 3676 | 1638.4 | 639.0 |

${R}_{back}$ (m) | Radius of the plane curve of back section | 5597 | 1,000,000 | 438,578.3 | 492,254.3 |

${L}_{back}$ (m) | Length of the plane curve of back section | 450 | 3676 | 1233.8 | 759.3 |

${i}_{min}$ (%) | Minimum longitudinal grade of current section | −1.6 | 1.6 | 0.0 | 0.4 |

$L{s}_{min}$ (m) | Length of the longitudinal slope corresponding to the minimum grade | 240.0 | 1740.0 | 773.3 | 296.0 |

${i}_{max}$ (%) | Maximum longitudinal grade of current section | −2.50 | 2.50 | 0.00 | 0.97 |

$L{s}_{max}$ (m) | Length of the longitudinal slope corresponding to the maximum grade | 362.0 | 1740.0 | 652.6 | 248.0 |

Sight characteristics | |||||

${S}_{car}$ (m) | Stopping sight distance of cars | 244.0 | 1004.8 | 423.6 | 139.6 |

${S}_{truck}$ (m) | Stopping sight distance of trucks | 52.0 | 279.4 | 82.7 | 24.3 |

${H}_{car}$ (m) | Horizontal clearance of cars | 0.02 | 6.5 | 1.4 | 1.5 |

${H}_{truck}$ (m) | Horizontal clearance of trucks | 0 | 0.2 | 0.05 | 0.04 |

Model | Log-Likelihood | AIC | BIC | R^{2} |
---|---|---|---|---|

MNB model | −1095.79 | 2245.58 | 2354.99 | 0.21 |

MP model | −1159.81 | 2371.63 | 2476.99 | 0.22 |

RPMNB model | −1084.76 | 2221.53 | 2326.88 | 0.25 |

RPMP model | −1174.98 | 2397.97 | 2495.22 | 0.24 |

Type | RMSE | MAE | MAPE | |||
---|---|---|---|---|---|---|

RPMNB | RPMP | RPMNB | RPMP | RPMNB | RPMP | |

All-day | 6.65 | 6.68 | 5.57 | 5.58 | 0.64 | 0.63 * |

Daytime | 4.87 | 4.28 * | 4.11 | 4.76 | 0.65 | 0.66 |

Nighttime | 0.95 | 0.96 | 1.92 | 1.92 | 0.66 | 0.68 |

Variables ^{1} | Parameter Estimate | t-Stat | Elasticity Effects | ||
---|---|---|---|---|---|

LI | MI | SI | |||

$\left(\mathrm{MI}\right)\text{}\mathrm{Constant}$ | −0.0044 | −2.87 | |||

Traffic characteristics | |||||

$\left(\mathrm{LI}\right)\text{}\mathit{AADT}$ | 3.36 × 10^{−5} | 11.54 | 1.41 | −1.10 | −0.31 |

Speed characteristics | |||||

$\left(\mathrm{LI}\right)\text{}\Delta {V}_{O-car}$ | 0.0045 | 6.19 | 6.26 × 10^{−4} | −2.12 × 10^{−4} | −4.14 × 10^{−4} |

$\left(\mathrm{SI}\right)\text{}\Delta {V}_{O-truck}$ | −0.022 | −2.73 | −5.26 × 10^{−3} | 9.22 × 10^{−3} | −3.96 × 10^{−3} |

Geometric characteristics | |||||

$\left(\mathrm{LI}\right)\text{}{R}_{present}$ | 1.17 × 10^{−6} | 4.51 | 0.46 | −0.01 | −0.45 |

$\left(\mathrm{SI}\right)\text{}{L}_{present}$ | 5.43 × 10^{−4} | 10.34 | 0.62 | −0.77 | 0.15 |

$\left(\mathrm{MI}\right)\text{}{L}_{front}$ | 7.84 × 10^{−5} | 3.98 | 0.086 | 0.034 | −0.12 |

$\left(\mathrm{SI}\right)\text{}L{s}_{min}$ | −2.90 × 10^{−4} | −2.51 | −0.20 | 0.65 | −0.45 |

Sight characteristics | |||||

$\left(\mathrm{LI}\right)\text{}{S}_{car}$ | −1.18 | −3.05 | −383.33 | 61.11 | 322.22 |

$\left(\mathrm{LI}\right)\text{}{S}_{truck}$ | 1.17 | 5.13 | 75.61 | −11.31 | −64.30 |

$\left(\mathrm{MI}\right)\text{}{H}_{truck}$ | 2.958 | 4.16 | 0.15 | 0.34 | −0.49 |

Number of observations | 3159 | ||||

AIC | 2221.53 | ||||

BIC | 2326.88 | ||||

R^{2} | 0.25 | ||||

$\alpha $ | 2.43 |

^{1}Variable definition and unit can be seen in Table 3. Parameter defined for (LI) Light injury; (MI) Minor Injury; (SI) Severe Injury.

Variables | Parameter Estimate | t-Stat | Elasticity Effects | ||
---|---|---|---|---|---|

LI | MI | SI | |||

$\left(\mathrm{LI}\right)\text{}\mathrm{Constant}$ | 0.0082 | 3.10 | |||

Traffic characteristics | |||||

(LI) AADT | 4.89 × 10^{−5} | 15.44 | 1.94 | −2.96 | 1.02 |

Speed characteristics | |||||

$\left(\mathrm{LI}\right)\text{}\Delta {V}_{O-truck}$ | −0.023 | −2.34 | −2.68 × 10^{−3} | 5.30 × 10^{−3} | −2.62 × 10^{−3} |

Geometric characteristics | |||||

$\left(\mathrm{LI}\right)\text{}{R}_{present}$ | 1.02 × 10^{−6} | 6.42 | 0.37 | −0.08 | −0.29 |

$\left(\mathrm{LI}\right)\text{}{L}_{present}$ | 5.81 × 10^{−4} | 10.20 | 0.71 | −0.34 | −0.37 |

$\left(\mathrm{LI}\right)\text{}L{s}_{min}$ | −2.82 × 10^{−4} | −2.28 | −0.21 | 0.13 | 0.08 |

Sight characteristics | |||||

$\left(\mathrm{SI}\right)\text{}{S}_{car}$ | −0.73 | −7.27 | −319.39 | 360.13 | −40.74 |

$\left(\mathrm{LI}\right)\text{}{S}_{truck}$ | 0.72 | 4.46 | 63.06 | −23.02 | −40.04 |

$\left(\mathrm{MI}\right)\text{}{H}_{truck}$ | 3.12 | 4.64 | 0.18 | 0.11 | −0.29 |

Number of observations | 2224 | ||||

AIC | 1986.82 | ||||

BIC | 2092.17 | ||||

R^{2} | 0.27 | ||||

α | 2.41 |

Variables | Parameter estimate | t-Stat | Elasticity effects | ||
---|---|---|---|---|---|

LI | MI | SI | |||

(LI) Constant | −0.03 | −2.11 | |||

Traffic characteristics | |||||

(MI) AADT | 3.20 × 10^{−5} | 10.54 | 1.23 | 3.23 | −4.46 |

Speed characteristics | |||||

$\left(\mathrm{LI}\right)\text{}\Delta {V}_{O-car}$ | 0.0045 | 6.41 | 1.38 × 10^{−3} | 1.64 × 10^{−3} | −3.02 × 10^{−3} |

$\left(\mathrm{SI}\right)\text{}\Delta {V}_{O-truck}$ | −0.024 | −3.08 | −3.57 × 10^{−3} | 7.28 × 10^{−3} | −3.71 × 10^{−3} |

Geometric characteristics | |||||

$\left(\mathrm{LI}\right)\text{}{R}_{present}$ | 2.02 × 10^{−6} | 4.13 | 0.58 | −2.60 | 2.02 |

$\left(\mathrm{SI}\right)\text{}{L}_{present}$ | 3.07 × 10^{−4} | 10.34 | 0.33 | −0.60 | 0.27 |

$\left(\mathrm{LI}\right)\text{}{L}_{front}$ | 7.84 × 10^{−5} | 3.89 | 0.28 | −0.82 | 0.54 |

$\left(\mathrm{MI}\right)\text{}L{s}_{min}$ | −3.11 × 10^{−4} | −6.97 | −0.27 | −1.15 | 1.42 |

Sight characteristics | |||||

$\left(\mathrm{MI}\right)\text{}{S}_{car}$ | −2.75 | −2.74 | −887.39 | −366.50 | 520.89 |

$\left(\mathrm{LI}\right)\text{}{S}_{truck}$ | 2.64 | 3.24 | 176.75 | −73.02 | −103.73 |

$\left(\mathrm{LI}\right)\text{}{H}_{truck}$ | 4.53 | 2.55 | 0.09 | −0.59 | 0.50 |

Number of observations | 935 | ||||

Log-likelihood | −796.78 | ||||

AIC | 1645.56 | ||||

BIC | 1750.91 | ||||

R^{2} | 0.29 | ||||

$\alpha $ | 2.51 |

${\mathit{t}}_{\mathbf{1}}$ | ${\mathit{t}}_{\mathbf{2}}$ | |
---|---|---|

Daytime | Nighttime | |

Daytime | - | 123.562 (10) (>99.99%) |

Nighttime | 105.628 (9) (>99.99%) | - |

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## Share and Cite

**MDPI and ACS Style**

Zhang, P.; Wang, C.; Chen, F.; Cui, S.; Cheng, J.; Bo, W.
A Random-Parameter Negative Binomial Model for Assessing Freeway Crash Frequency by Injury Severity: Daytime versus Nighttime. *Sustainability* **2022**, *14*, 9061.
https://doi.org/10.3390/su14159061

**AMA Style**

Zhang P, Wang C, Chen F, Cui S, Cheng J, Bo W.
A Random-Parameter Negative Binomial Model for Assessing Freeway Crash Frequency by Injury Severity: Daytime versus Nighttime. *Sustainability*. 2022; 14(15):9061.
https://doi.org/10.3390/su14159061

**Chicago/Turabian Style**

Zhang, Ping, Chenzhu Wang, Fei Chen, Suping Cui, Jianchuan Cheng, and Wu Bo.
2022. "A Random-Parameter Negative Binomial Model for Assessing Freeway Crash Frequency by Injury Severity: Daytime versus Nighttime" *Sustainability* 14, no. 15: 9061.
https://doi.org/10.3390/su14159061