# Quantile Regression with Telematics Information to Assess the Risk of Driving above the Posted Speed Limit

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Objective

## 2. Background

## 3. Methods

#### 3.1. Quantile Regression

^{2}). When we model the conditional mean response, the Gaussian likelihood function is given by the following:

^{2}statistic in linear regression, which we have also implemented here. The criterion is calculated as 1 − $\widehat{{L}_{\alpha}\left(\beta \right)}/\tilde{{L}_{\alpha}\left(\beta \right)},$ where $\widehat{{L}_{\alpha}\left(\beta \right)}$ is the value of objective Function (1) where all covariates are included in the model specification (unrestricted model), whereas $\tilde{{L}_{\alpha}\left(\beta \right)}$ is the value of the objective function when only an intercept is considered (restricted model).

#### 3.2. The Data

## 4. Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Table A1.**Parameter estimates of the linear regression model. In the model, Tolerkm/1000 is the dependent variable and km_1000 (km/1000) and km_1000

^{2}are introduced in the model instead of lnkm as independent variables.

Parameter Estimate (p-Value) | |
---|---|

Intercept | 0.6397 (<0.0001) |

Km_1000 | 0.0292 (<0.0001) |

Km_1000^{2} | 0.0035 (<0.0001) |

Pkdr_vurba | −0.0137 (<0.0001) |

Pkdr_nocturn | 0.0018 (0.485) |

Age | −0.0079 (0.149) |

Gender | 0.2295 (<0.0001) |

R^{2} | 43.20% |

**Table A2.**Parameter estimates of the quantile regression model for different percentiles. In that case, Tolerkm/1000 is the dependent variable and km_1000 (km/1000) and km_1000

^{2}are introduced in the model, instead of lnkm, as independent variables.

50th Percentile (p-Value) | 75th Percentile (p-Value) | 90th Percentile (p-Value) | 95th Percentile (p-Value) | 97.5th Percentile (p-Value) | 99th Percentile (p-Value) | |
---|---|---|---|---|---|---|

Intercept | 0.1812 (0.0003) | 0.3845 (<0.0001) | 0.3681 (0.0805) | 0.4940 (0.0643) | 0.1147 (0.7889) | 0.8439 (0.1271) |

Km_1000 | 0.0113 (0.0399) | 0.0257 (0.0010) | 0.0595 (0.0001) | 0.0839 (<0.0001) | 0.0887 (0.0084) | 0.0632 (0.0243) |

Km_1000^{2} | 0.0035 (<0.0001) | 0.0056 (<0.0001) | 0.0079 (<0.0001) | 0.0087 (<0.0001) | 0.0107 (<0.0001) | 0.0138 (<0.0001) |

Pkdr_vurba | −0.0031 (<0.0001) | −0.0082 (<0.0001) | −0.0136 (<0.0001) | −0.0177 (<0.0001) | −0.0200 (<0.0001) | −0.0248 (<0.0001) |

Pkdr_nocturn | 0.0023 (0.0164) | 0.0010 (0.4777) | −0.0022 (0.6037) | 0.0028 (0.6140) | 0.0055 (0.5539) | 0.0095 (0.4052) |

Age | −0.0027 (0.1316) | −0.0001 (0.9749) | 0.0143 (0.0623) | 0.0216 (0.0266) | 0.0480 (0.0019) | 0.0416 (0.0725) |

Gender | 0.1132 (<0.0001) | 0.1734 (<0.0001) | 0.1975 (<0.0001) | 0.1510 (0.0082) | 0.1428 (0.1198) | 0.2360 (0.1646) |

Goodness-of-fit criterion | 23.62% | 33.45% | 43.70% | 49.62% | 54.10% | 59.67% |

**Table A3.**Estimates of the conditional percentiles for drivers with different characteristics, each of whom has driven 600 km above the posted speed limit. The models used in the calculations consider Tolerkm/1000 as the dependent variable and km_1000 (km/1000) and km_1000

^{2}are introduced in the model, instead of lnkm, as independent variables.

Driver 1 | Driver 2 | Driver 3 | |
---|---|---|---|

Km | 12,000 | 8000 | 5500 |

Pkdr_vurba | 80 | 75 | 80 |

Pkdr_noctur | 14 | 11 | 10.5 |

Age | 25 | 25 | 25 |

Gender | 1 | 1 | 1 |

Estimated conditional percentile ^{1} | 45th | 78th | 96th |

^{1}The estimated conditional percentile is found by locating the quantile level that produces a response equal to 600 km, given the exogenous characteristics (total kilometers driven, percent urban driving, percent nighttime driving, age and gender) in the three example columns.

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**Figure 2.**Parameter estimates at different levels of the quantile. Confidence intervals at a 5% level of significance. The horizontal red line represents the corresponding parameter estimate in a classical linear regression model. (

**a**) Intercept; (

**b**) lnkm; (

**c**) pkdr_vurba; (

**d**) pkdr_nocturn; (

**e**) age; and (

**f**) gender.

Variable | Description |
---|---|

Tolerkm | Number of kilometers driven at speeds above the posted limit during 2010. |

Km | Total number of kilometers driven during 2010. |

Lnkm | Logarithm of the total number of kilometers driven during 2010. |

Pkdr_vurba | % of kilometers driven on urban roads during 2010. |

Pkdr_nocturn | % of kilometers driven at night (between midnight and 6 am.) during 2010. |

Age | Age of the driver at the beginning of 2010. |

Gender | 1 = Male, 0 = Female |

Variable | Min | 1st Qu | Median | Mean | 3rd Qu | Max | St. Dev. | Skewness |
---|---|---|---|---|---|---|---|---|

Tolerkm | 0.00 | 282.40 | 689.20 | 1398.20 | 1701.60 | 23,500.20 | 1995.37 | 3.64 |

Km | 0.69 | 7530.56 | 11,697.82 | 13,063.71 | 17,337.00 | 57,756.98 | 7715.80 | 1.08 |

Lnkm | −0.37 | 8.93 | 9.37 | 9.27 | 9.76 | 10.96 | 0.75 | −1.87 |

Pkdr_vurba | 0.00 | 15.60 | 23.39 | 26.29 | 34.32 | 100.00 | 14.18 | 1.03 |

Pkdr_nocturn | 0.00 | 2.48 | 5.31 | 7.02 | 9.84 | 78.56 | 6.13 | 1.67 |

Age | 18.11 | 22.66 | 24.63 | 24.78 | 26.88 | 35.00 | 2.82 | 0.11 |

Parameter Estimate (p-Value) | |
---|---|

Intercept | −8082.506 (<0.0001) |

Lnkm | 1064.506 (<0.0001) |

Pkdr_vurba | −21.868 (<0.0001) |

Pkdr_nocturn | 7.536 (0.0101) |

Age | −1.131 (0.8565) |

Gender | 328.009 (<0.0001) |

R^{2} | 25.96% |

50th Percentile (p-Value) | 75th Percentile (p-Value) | 90th Percentile (p-Value) | 95th Percentile (p-Value) | 97.5th Percentile (p-Value) | 99th Percentile (p-Value) | |
---|---|---|---|---|---|---|

Intercept | −4496.53 (<0.0001) | −6250.34 (<0.0001) | −6418.11 (<0.0001) | −6009.63 (<0.001) | −5137.24 (<0.0001) | −2451.17 0.5780 |

Lnkm | 597.60 (<0.0001) | 892.80 (<0.0001) | 1074.66 (<0.0001) | 1094.57 (<0.0001) | 1119.94 (<0.0001) | 1180.21 (<0.001) |

Pkdr_vurba | −9.19 (<0.0001) | −22.26 (<0.0001) | −39.59 (<0.0001) | −53.44 (<0.0001) | −68.58 (<0.0001) | −87.12 (<0.0001) |

Pkdr_nocturn | 5.41 (<0.0001) | 6.71 (0.0363) | 21.76 (0.0226) | 37.49 (0.0086) | 20.01 (0.4266) | 43.86 (0.4014) |

Age | −2.56 (0.1632) | 1.84 (0.7298) | 5.16 (0.7419) | 40.29 (0.2086) | 71.28 (0.1094) | 36.87 (0.7009) |

Gender | 206.76 (<0.0001) | 377.94 (<0.0001) | 574.08 (<0.0001) | 755.87 (<0.0001) | 1070.06 (<0.0001) | 1091.38 (0.0624) |

Goodness-of-fit criterion | 14.19% | 18.26% | 20.23% | 20.27% | 20.56% | 20.06% |

**Table 5.**Estimates of the conditional percentiles for drivers with different characteristics, each of whom has driven 600 km above the posted speed limit.

Driver 1 | Driver 2 | Driver 3 | |
---|---|---|---|

Km | 12,000 | 8000 | 5500 |

Pkdr_vurba | 80 | 75 | 80 |

Pkdr_noctur | 14 | 11 | 10.5 |

Age | 25 | 25 | 25 |

Gender | 1 | 1 | 1 |

Estimated conditional percentile ^{1} | 50th | 75th | 90th |

^{1}The estimated conditional percentile is found by locating the quantile level that produces a response equal to 600 km, given the exogenous characteristics (total kilometers driven, percent urban driving, percent nighttime driving, age, and gender) in the three example columns.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Pérez-Marín, A.M.; Guillen, M.; Alcañiz, M.; Bermúdez, L.
Quantile Regression with Telematics Information to Assess the Risk of Driving above the Posted Speed Limit. *Risks* **2019**, *7*, 80.
https://doi.org/10.3390/risks7030080

**AMA Style**

Pérez-Marín AM, Guillen M, Alcañiz M, Bermúdez L.
Quantile Regression with Telematics Information to Assess the Risk of Driving above the Posted Speed Limit. *Risks*. 2019; 7(3):80.
https://doi.org/10.3390/risks7030080

**Chicago/Turabian Style**

Pérez-Marín, Ana M., Montserrat Guillen, Manuela Alcañiz, and Lluís Bermúdez.
2019. "Quantile Regression with Telematics Information to Assess the Risk of Driving above the Posted Speed Limit" *Risks* 7, no. 3: 80.
https://doi.org/10.3390/risks7030080