# A Crash Data Analysis through a Comparative Application of Regression and Neural Network Models

^{1}

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## Abstract

**:**

## 1. Introduction

- To study crash data collected between 2014 and 2017 through a comparison of modelling methodologies, in terms of their performance and results, using four paradigms, namely, artificial neural networks (ANNs), generalized linear mixed-effects (GLME), multinomial regression (MNR), and general nonlinear regression (NLM);
- To find the analytical formulation that better describes the relationship between input and output;
- To analyze common variables of the models.

## 2. Methodology

- Analysis of data;
- Pre-processing and normalization of data;
- Model building;
- Check of model performance (if not satisfactory, we went back to step 3 for model building);
- Analysis of results and discussion.

## 3. The Data Set

#### 3.1. Database Information

- Crash type, with three categories, including between circulating vehicles, pedestrian hit, and isolated vehicle crash;
- Crash effects, with two categories, including injuries and fatalities.

#### 3.2. Data Set Variables

- Variables referring to the road conditions;
- Variable referring to the infrastructure;
- Variables referring to the crash characteristics;
- Variables for vehicle characteristics;
- Variables for driver’s description.

- 6.1.
- Crash effects, indicating the severity of the crash;
- 6.2.
- Type of crash, indicating the dynamic of the crash.

#### 3.3. Data Oversampling and Normalization

_{norm}is the normalized value of variable X and X

_{max}and X

_{min}are the maximum and minimum values of X, respectively. Binomial variables do not require normalization, while categorical variables are coded in a binary base (e.g., 6 is coded as “110” by using three independent inputs).

## 4. Models and Their Performance

#### 4.1. Back Propagation Artificial Neural Network

_{i}= observed values, and Y’

_{i}= predicted values:

#### 4.2. Generalized Linear Mixed Effects (GLME)

_{i}is the outcome variable, Distr is a specified conditional distribution of y given b, μ is the conditional mean of y

_{i}given b

_{i}and μ

_{i}is its ith element, σ

^{2}is the dispersion parameter, w is the effective observation weight vector, g(μ) is a link function, X is a matrix of the predictor variables, β is a column vector of the fixed-effect regression coefficients, Z plays the role of design matrix for the random effects, b is a vector of the random effects, and ε is a column vector of the residuals.

^{−1}is the inverse of the link function g(μ) and η is the linear predictor of the fixed and random effects of the GLME:

#### 4.3. Multinomial Regression (MNR)

_{n}is an input variable, θ

_{j}= P(y = j) is the probability of an outcome being in category j, k is the number of response categories, and n is the number of predictor variables. The last category was used as a reference variable, written as the kth category. Further, β

_{jn}are the coefficients in the model that realize the effects of the predictor variables on the log odds of being in category j versus the reference category k. The most important assumption is to set the kth category coefficients. Therefore, the probability of being in each category j is

#### 4.4. General Nonlinear Regression (NLM)

- Errors are independent;
- Errors have mean zero and constant variance;
- Errors are normally distributed.

_{0}is required before iteratively modifying the vector β to a vector with a minimum MSE.

+(β711 ∗ C7 ∗ C11) + (β913 ∗ C9 ∗ C13) + (β12 ∗ C12) + (β13 ∗ C13)

^{2}) + (β8 ∗ C8) + exp (β913 ∗ C9 ∗ C13) +

+ (exp (β99 ∗ C9) + exp (β11 ∗ C11))/(1 + β90 ∗ C9)

## 5. Analyses and Results

#### 5.1. Database Information Content

#### 5.2. Sensitivity Analysis

#### 5.3. Marginal Effects Analysis

#### 5.4. Model Comparison

_{ij}represents the number of predicted cases i and n is the matrix dimension.

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Predicted Values | ||||||
---|---|---|---|---|---|---|

Class | −1 | 0 | 1 | Total | PR | |

Real values | 0 | 230 | 33,325 | 1432 | 34,987 | 4.8% |

<1% | 95.2% | 4.8% | 100% | |||

1 | 0 | 0 | 35,100 | 35,100 | 0.0% | |

0.0% | 0.0% | 100% | 100% | |||

TotalPO | 230 | 33,325 | 36,532 | 70,087 | Accuracy97.6% | |

100% | 0.0% | 3.9% |

Predicted Values | |||||
---|---|---|---|---|---|

Class | 0 | 1 | Total | PR | |

Real values | 0 | 24,533 | 10,454 | 34,987 | 29.9% |

70.1% | 29.9% | 100% | |||

1 | 12,420 | 22,680 | 35,100 | 35.4% | |

35.4% | 64.6% | 100% | |||

TotalPO | 36,953 | 33,134 | 70,087 | Accuracy67.3% | |

33.6% | 31.6% |

Predicted Values | |||||
---|---|---|---|---|---|

Class | 0 | 1 | Total | PR | |

Real values | 0 | 23,592 | 11,395 | 34,987 | 32.6% |

67.4% | 32.6% | 100% | |||

1 | 12,960 | 22,140 | 35,100 | 36.9% | |

36.9% | 63.1% | 100% | |||

Total | 36,552 | 33,535 | 70,087 | Accuracy65.2% | |

PO | 35.5% | 34.0% |

Predicted Values | |||||
---|---|---|---|---|---|

Class | 0 | 1 | Total | PR | |

Real values | 0 | 22,695 | 12,292 | 34,987 | 35.1% |

64.9% | 35.1% | 100% | |||

1 | 10,080 | 25,020 | 35,100 | 28.7% | |

28.7% | 71.3% | 100% | |||

Total | 32,775 | 37,312 | 70,087 | Accuracy68.0% | |

PO | 30.8% | 32.9% |

Predicted Values | PR | |||||
---|---|---|---|---|---|---|

Class | 1 | 2 | 3 | Total | ||

Real values | 1 | 23,398 | 0 | 0 | 23,398 | 0.0% |

100% | 0.0% | 0.0% | 100% | |||

2 | 0 | 3898 | 1477 | 5375 | 27.5% | |

0.0% | 72.5% | 27.5% | 100% | |||

3 | 0 | 837 | 5572 | 6409 | 13.1% | |

0.0% | 13.1% | 86.9% | 100% | |||

Total | 23,398 | 4735 | 7049 | 35,182 | Accuracy93.4% | |

PO | 0% | 17.7% | 21.0% |

Predicted Values | ||||||
---|---|---|---|---|---|---|

Class | 1 | 2 | 3 | Total | PR | |

Real values | 1 | 23,398 | 0 | 0 | 23,398 | 0.0% |

100% | 0.0% | 0.0% | 100% | |||

2 | 0 | 3797 | 1578 | 5375 | 29.4% | |

0.0% | 70.6% | 29.4% | 100% | |||

3 | 0 | 901 | 5508 | 6409 | 14.1% | |

0.0% | 14.1% | 85.9% | 100% | |||

Total | 23,398 | 4698 | 7086 | 35,182 | Accuracy93.0% | |

PO | 0.0% | 19.2% | 22.3% |

Predicted Values | ||||||
---|---|---|---|---|---|---|

Class | 1 | 2 | 3 | Total | PR | |

Real values | 1 | 23,398 | 0 | 0 | 23,398 | 0.0% |

100% | 0.0% | 0.0% | 100% | |||

2 | 0 | 3768 | 1607 | 5375 | 29.9% | |

0.0% | 70.1% | 29.9% | 100% | |||

3 | 0 | 1322 | 5087 | 6409 | 20.6% | |

0.0% | 20.6% | 79.4% | 100% | |||

Total | 23,398 | 5090 | 6694 | 35,182 | Accuracy | |

PO | 0.0% | 26.0% | 24.0% | 91.7% |

Predicted Values | PR | |||||
---|---|---|---|---|---|---|

Class | 1 | 2 | 3 | Total | ||

Real values | 1 | 23,398 | 0 | 0 | 23,398 | 0.0% |

100% | 0.0% | 0.0% | 100% | |||

2 | 0 | 3640 | 1735 | 5375 | 32.3% | |

0.0% | 67.7% | 32.3% | 100% | |||

3 | 0 | 1180 | 5229 | 6409 | 18.4% | |

0.0% | 18.4% | 81.6% | 100% | |||

Total | 23,398 | 5090 | 6694 | 35,182 | Accuracy91.7% | |

PO | 0.0% | 28.5% | 21.9% |

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**Figure 1.**Distribution of crashes subdivided by (

**a**) crash severity (C15) and (

**b**) crash type (C16) on Milan metropolitan area.

**Figure 3.**PCA for C15 data set; the colors identify the subset of data according to the classification made by the GLME model.

**Figure 4.**PCA for C16; the colors identify the subset of data according to the classification made by the GLME model.

Var. | Name | Type | Min | Median | Max | Label | Description | Frequency | Percentage |
---|---|---|---|---|---|---|---|---|---|

C1 | Day of week | C | 1 | 4 | 7 | 1 | Sunday | 3421 | 10% |

2 | Monday | 5153 | 14% | ||||||

3 | Tuesday | 5585 | 16% | ||||||

4 | Wednesday | 5642 | 16% | ||||||

5 | Thursday | 5537 | 16% | ||||||

6 | Friday | 5617 | 16% | ||||||

7 | Saturday | 4227 | 12% | ||||||

C2 | Hour (daytime/night-time) | B | 0 | 0 | 1 | 0 | Day time | 24,503 | 70% |

1 | Night-time | 10,679 | 30% | ||||||

C3 | Road typology | C | 1 | 2 | 4 | 1 | One-way carriag. | 6719 | 19% |

2 | Two-way carriag. | 14,117 | 40% | ||||||

3 | Two carriageways | 8297 | 24% | ||||||

4 | >two carriageways | 6049 | 17% | ||||||

C4 | Type of road infrastructure | B | 0 | 1 | 1 | 0 | Intersection | 16,906 | 48% |

1 | Section | 18,276 | 52% | ||||||

C5 | Road conditions | C | 1 | 1 | 3 | 1 | Dry | 28,690 | 81.50% |

2 | Wet | 6231 | 17.71% | ||||||

3 | Slippery/Icy/Frozen | 261 | 0.74% | ||||||

C6 | Meteorological conditions | C | 1 | 1 | 4 | 1 | Serene | 30,582 | 87.00% |

2 | Wind | 25 | 0.07% | ||||||

3 | Fog | 284 | 0.81% | ||||||

4 | Rain/Snow/Hail | 4291 | 12.00% | ||||||

C7 | Type of vehicle A | C | 1 | 2 | 3 | 1 | Two-wheeled | 14,355 | 41% |

2 | Passenger car | 18,375 | 52% | ||||||

3 | Other-heavy veh. | 2452 | 7% | ||||||

C8 | Age A [years] | N | 4 | 41 (mean = 42, std = 15) | 96 | 0 | Unknown/not present | 1359 | 4% |

[1–99] | years | 33,823 | 96% | ||||||

C9 | Gender A | C | 0 | 1 | 2 | 0 | Unknown | 916 | 2% |

1 | Male | 26,576 | 76% | ||||||

2 | Female | 7690 | 22% | ||||||

C10 | Years of driving license A | N | 0 | 4 (mean = 9, std = 10) | 58 | 0 | Unknown/not present | 7511 | 21% |

[1–99] | years | 27,793 | 79% | ||||||

C11 | Type of vehicle B | C | 0 | 1 | 3 | 0 | Unknown | 11,784 | 34% |

1 | Two-wheeled | 6844 | 19% | ||||||

2 | Passenger car | 14,998 | 43% | ||||||

3 | Other-heavy veh. | 1556 | 4% | ||||||

C12 | Age B [years] | N | 4 | 42 (mean = 42, std = 14) | 93 | 0 | Unknown/not present | 12,326 | 35% |

[1–99] | years | 22,356 | 65% | ||||||

C13 | Gender B | C | 0 | 1 | 2 | 0 | Unknown | 12,050 | 34% |

1 | Male | 17,154 | 49% | ||||||

2 | Female | 5978 | 17% | ||||||

C14 | Years of driving license B | N | 0 | 5 (mean = 9, std = 11) | 58 | 0 | Unknown/not present | 16,789 | 48% |

[1–99] | years | 18,393 | 52% | ||||||

C15 | Crash effects | B | 0 | 0 | 1 | 0 | Injuries | 34,987 | 99.5% |

1 | Fatalities | 195 | 0.5% | ||||||

C16 | Crash types | C | 1 | 2 | 3 | 1 | Between circulating vehicles | 23,398 | 67% |

2 | Pedestrian hit | 5375 | 15% | ||||||

3 | Isolated vehicle crash | 6509 | 18% | ||||||

Total observations | 35,182 | 100% |

AIC | Likelihood | ||||
---|---|---|---|---|---|

89,319 | −44,652 | ||||

Name | Estimate | p Value | SE | Lower Limit | Upper Limit |

C8 | 0.29417 | <10^{−3} | 0.018389 | 0.25812 | 0.33021 |

C9 | 10.769 | <10^{−3} | 1.5483 | 7.7595 | 13.833 |

C42 | −0.09477 | <10^{−3} | 0.0067284 | −0.10797 | −0.08159 |

C92 | −8.0156 | <10^{−3} | 1.0326 | −10.041 | −5.9904 |

Group variables | Estimate | ||||

Intercept | 4.2363 | ||||

C2 (Intercept) | −0.92965 | ||||

C2 (Intercept) | 0.26753 |

AIC | Likelihood | ||||
---|---|---|---|---|---|

81,010 | −40,497 | ||||

Name | Estimate | p Value | SE | Lower Limit | Upper Limit |

C4 | −0.0445 | <10^{−3} | 0.0088324 | −0.0618 | −0.02729 |

C5 | 0.05081 | 0.01044 | 0.0198400 | 0.0119 | 0.08969 |

C7 | −0.1832 | <10^{−3} | 0.0150050 | −0.2126 | −0.15379 |

C9 | 0.5606 | <10^{−3} | 0.0866160 | 0.3908 | 0.73038 |

C92 | −0.3508 | <10^{−3} | 0.0621540 | −0.4726 | −0.22903 |

Group variables | Estimate | ||||

Intercept | 0.40461 | ||||

C2 (Intercept) | 1 | ||||

C2 (Intercept) | 0.021654 |

Name | Estimate | SE | p Value |
---|---|---|---|

α | −0.635 | 0.040 | <10^{−3} |

β11 | 0.518 | 0.026 | <10^{−3} |

β12 | −0.743 | 0.018 | 0 |

β13 | −0.457 | 0.025 | <10^{−3} |

β14 | 0.422 | 0.017 | <10^{−3} |

β15 | 1.614 | 0.065 | <10^{−3} |

β16 | −0.310 | 0.038 | <10^{−3} |

β17 | −0.680 | 0.026 | <10^{−3} |

β18 | −1.864 | 0.051 | <10^{−3} |

β19 | 1.921 | 0.046 | 0 |

β110 | 0.740 | 0.050 | <10^{−3} |

β111 | 0.045 | 0.044 | 0.306 |

β112 | −1.174 | 0.054 | <10^{−3} |

β113 | 1.944 | 0.046 | 0 |

β114 | 1.664 | 0.068 | <10^{−3} |

Name | Estimate | SE | p Value | Name | Estimate | SE | p Value |
---|---|---|---|---|---|---|---|

α1 | −31.400 | 0.482 | 0 | α2 | −0.348 | 0.0912 | 0.0001 |

β11 | −0.127 | 0.362 | 0.723 | β21 | 0.145 | 0.0700 | 0.0370 |

β12 | −1.739 | 0.244 | <10^{−3} | β22 | −0.724 | 0.0487 | <10^{−3} |

β13 | 0.809 | 0.339 | 0.017 | β23 | 0.163 | 0.0690 | 0.0170 |

β14 | −0.455 | 0.224 | 0.042 | β24 | 1.045 | 0.0453 | <10^{−3} |

β15 | 1.348 | 0.797 | 0.091 | β25 | −1.641 | 0.1480 | <10^{−3} |

β16 | −0.027 | 0.494 | 0.955 | β26 | 0.630 | 0.0880 | <10^{−3} |

β17 | −14.595 | 0.424 | <10^{−3} | β27 | 3.897 | 0.0890 | 0.0000 |

β18 | 0.118 | 0.662 | 0.857 | β28 | −0.129 | 0.1310 | 0.3240 |

β19 | −0.866 | 0.468 | 0.064 | β29 | −1.977 | 0.0980 | <10^{−3} |

β110 | 0.010 | 0.644 | 0.987 | β210 | 0.776 | 0.1280 | <10^{−3} |

β111 | 226.185 | 1.115 | 0 | β211 | −2.699 | 1.3720 | 0.0490 |

β112 | 31.872 | 1.625 | <10^{−3} | β212 | 2.290 | 2.4240 | 0.3440 |

β113 | 2.755 | 1.053 | 0.009 | β213 | 1.837 | 1.5880 | 0.2470 |

β114 | 2.303 | 1.447 | 0.111 | β214 | 0.022 | 2.1290 | 0.9910 |

Name | Estimate | SE | p Value |
---|---|---|---|

β4 | 0.81083 | 0.0061205 | 0 |

β5 | 0.17332 | 0.0056368 | <10^{−3} |

β6 | −0.29411 | 0.010835 | <10^{−3} |

β7 | −0.19772 | 0.016101 | <10^{−3} |

β9 | −0.14945 | 0.019974 | <10^{−3} |

β12 | −0.40295 | 0.0094004 | 0 |

β13 | −0.23681 | 0.016157 | <10^{−3} |

β44 | 0.13937 | 0.010201 | <10^{−3} |

β411 | −0.14607 | 0.016511 | <10^{−3} |

β711 | −0.25411 | 0.026614 | <10^{−3} |

β913 | 0.54439 | 0.040193 | <10^{−3} |

Name | Estimate | SE | p Value |
---|---|---|---|

β2 | 0.16018 | 0.0097000 | <10^{−3} |

β4 | −0.31900 | 0.0049883 | 0 |

β5 | −0.67276 | 0.0075298 | 0 |

β7 | 0.15286 | 0.0070069 | <10^{−3} |

β8 | 0.14588 | 0.0328520 | <10^{−3} |

β11 | −0.20741 | 0.0056216 | <10^{−3} |

β44 | −0.43761 | 0.0222610 | <10^{−3} |

β90 | 7.13780 | 0.0892520 | 0 |

β99 | 2.00000 | 0.0146530 | 0 |

β913 | 0.64146 | 0.0622290 | <<10^{−3} |

**Table 8.**Synoptic table for performance model comparison (PRi and POi are reported for each class by row and column).

Model | Evaluation Method | Output C15 (Crash Severity) | Output C16 (Crash Type) | ||
---|---|---|---|---|---|

Relevant Variables | Accuracy [PR1,2] [PO1,2] | Relevant Variables | Accuracy [PR1,2,3 [PO1,2,3] | ||

ANN | Sensitivity analysis | C8: Driver A age C3: Road typology C2: Hour of crash C13: Gender B | 97.6% [4.8, 0.0]% [0.0, 3.9]% | C8: Driver A age C7: Type of vehicle A C4: Type of road infrastructure C11: Type of vehicle B | 93.4% [0.0,27.5,13.1]% [0.0,17.7,21.0]% |

GLME | Marginal Effects | C8: Driver A age C9: Gender A C4: Type of road infrastructure | 67.3% [29.9, 35.4]% [33.6, 31.6]% | C5: Road conditions C9: Gender A C4: Type of road infrastructure C7: Type of vehicle A | 93.0% [0.0,29.4,14.1]% [0.0,19.2,22.3]% |

NLM | Modelcoefficients | C4: Type of road infrastructure C6: Meteorological conditions C12: Drive B age C13: Gender B | 65.2% [32.6, 36.9]% [35.5, 34.0]% | C5: Road Conditions C4: Type of road infrastructure C11: Type of vehicle B C9: Gender A | 91.7% [0.0,29.9,20.6]% [0.0,26.0,24.0]% |

MNR | Modelcoefficients | C13: Gender B C9: Gender A C14: Years of driving license B C8: Driver A age | 68.0% [35.1, 28.7]% [30.8, 32.9]% | C12: Driver B age C11: Type of vehicle B C7: Type of vehicle A C13: Gender B | 91.7% [0.0,32.3,18.4]% [0.0,28.5,21.9]% |

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## Share and Cite

**MDPI and ACS Style**

Mussone, L.; Alizadeh Meinagh, M.
A Crash Data Analysis through a Comparative Application of Regression and Neural Network Models. *Safety* **2023**, *9*, 20.
https://doi.org/10.3390/safety9020020

**AMA Style**

Mussone L, Alizadeh Meinagh M.
A Crash Data Analysis through a Comparative Application of Regression and Neural Network Models. *Safety*. 2023; 9(2):20.
https://doi.org/10.3390/safety9020020

**Chicago/Turabian Style**

Mussone, Lorenzo, and Mohammadamin Alizadeh Meinagh.
2023. "A Crash Data Analysis through a Comparative Application of Regression and Neural Network Models" *Safety* 9, no. 2: 20.
https://doi.org/10.3390/safety9020020