# A Comparative Study of Logistic Models Using an Asymmetric Link: Modelling the Away Victories in Football

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## Abstract

**:**

## 1. Introduction

## 2. Logit Specifications

#### 2.1. Frequentist Estimation

#### 2.2. Bayesian Estimation

#### 2.3. Bayesian Asymmetric Estimation

## 3. Description of Database

## 4. Empirical Results

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Away victories in four of the most important European football leagues from 1993–1994 to 2015–2016 seasons.

**Figure 2.**Home victories and draws versus away victories in four of the most important European football leagues from the 2012–2013 to 2015–2016 seasons.

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

Game statistics | |

HS | Home team shots. |

AS | Away team shots. |

AF | Fouls committed by the away team. |

HC | Corners in favour of the home team. |

AC | Corners in favour of the away team. |

HY | Yellow cards shown to the home team. |

AY | Yellow cards shown to the away team. |

HR | Red cards shown to the home team. |

AR | Red cards shown to the away team. |

Game variable | |

DERBY | Match played between teams from the same city or region or between the strongest teams in the league. |

Extra games | |

BUDH | Home team budget |

BUDA | Away team budget |

Referee | |

INTERNATIONAL | International experience |

ACIENT | Years of experience in the first division |

Variables | Frequentist | Bayesian | Asymmetric Bayesian | ||||||
---|---|---|---|---|---|---|---|---|---|

$\widehat{\mathit{\beta}}$ | Robust Sd | p -Value | $\widehat{\mathit{\beta}}$ | Sd | MC Error | $\widehat{\mathit{\beta}}$ | Sd | MC Error | |

Intercept | −2.417 *** | 0.929 | 0.009 | −1.313 *** | 0.504 | 0.000 | 12.58 *** | 1.343 | 0.009 |

HS | 0.006 | 0.031 | 0.836 | 0.006 | 0.031 | 0.000 | 0.020 | 1.343 | 0.0009 |

AS | 0.051 * | 0.030 | 0.100 | 0.052 * | 0.030 | 0.000 | 0.592 *** | 1.532 | 0.014 |

AF | 0.025 | 0.033 | 0.450 | 0.026 | 0.033 | 0.000 | 0.256 | 1.187 | 0.008 |

HC | 0.055 | 0.054 | 0.306 | 0.058 | 0.056 | 0.000 | 0.284 | 1.075 | 0.007 |

AC | −0.047 | 0.052 | 0.364 | -0.050 | 0.055 | 0.000 | −0.200 | 1.215 | 0.009 |

HY | 0.034 | 0.098 | 0.730 | 0.034 | 0.098 | 0.000 | 0.417 | 1.135 | 0.007 |

AY | −0.032 | 0.097 | 0.738 | −0.034 | 0.103 | 0.000 | 0.306 | 1.054 | 0.007 |

HR | 1.390 *** | 0.326 | 0.000 | 1.460 *** | 0.342 | 0.000 | 15.417 *** | 1.765 | 0.020 |

AR | −0.418 | 0.439 | 0.341 | −0.459 | 0.482 | 0.000 | −0.981 | 0.912 | 0.005 |

DERBY | −0.026 | 0.324 | 0.936 | −0.035 | 0.354 | 0.000 | −0.206 | 3.246 | 0.024 |

BUDH | −0.004 ** | 0.001 | 0.012 | −0.004 ** | 0.001 | 0.000 | −0.024 *** | 1.353 | 0.012 |

BUDA | 0.003 *** | 0.0009 | 0.001 | 0.003 *** | 0.0009 | 0.000 | 0.035 *** | 1.897 | 0.020 |

INTERNATIONAL | 0.369 | 0.276 | 0.182 | 0.389 * | 0.282 | 0.000 | 3.139 * | 2.345 | 0.024 |

ACIENT | 0.001 | 0.031 | 0.968 | 0.001 | 0.031 | 0.000 | 0.042 | 1.294 | 0.009 |

$\delta $ | −35.03 *** | 6.488 | 0.1034 | ||||||

AIC | 433.553 | 449.000 | 82.56 | ||||||

DIC | 403.553 | 434.096 | 99.95 | ||||||

% Correct Fitting | 73.68 | 71.58 | 100 | ||||||

${}^{\ast \ast \ast}$ indicates 1% significance or relevance level | |||||||||

${}^{\ast \ast}$ indicates 5% significance or relevance level | |||||||||

${}^{\ast}$ indicates 10% significance or relevance level |

**Table 3.**Frequentist, Bayesian and asymmetric Bayesian logit estimation results in the restricted models.

Variables | Frequentist | Bayesian | Asymmetric Bayesian | ||||||
---|---|---|---|---|---|---|---|---|---|

$\widehat{\mathit{\beta}}$ | Robust Sd | p -Value | $\widehat{\mathit{\beta}}$ | Sd | MC Error | $\widehat{\mathit{\beta}}$ | Sd | MC Error | |

Intercept | −0.985 *** | 0.131 | 0.000 | −1.231 *** | 0.225 | 0.000 | 11.55 *** | 2.859 | 0.131 |

AS | 0.158 | 0.131 | 0.227 | 0.156 | 0.135 | 0.000 | 2.63 *** | 1.381 | 0.039 |

HR | 0.494 *** | 0.115 | 0.000 | 0.517 *** | 0.115 | 0.000 | 5.542 *** | 1.763 | 0.063 |

BUDH | −0.578 ** | 0.228 | 0.011 | −0.641 *** | 0.23 | 0.000 | −3.119 *** | 0.992 | 0.030 |

BUDA | 0.409 *** | 0.127 | 0.001 | 0.423 *** | 0.126 | 0.000 | 4.571 *** | 1.79 | 0.057 |

INTERNATIONAL | 0.327 | 0.262 | 0.000 | 2.715 * | 2.13 | 0.06 | |||

$\delta $ | −33.19 *** | 7.125 | 0.335 | ||||||

AIC | 420.119 | 426.7 | 67.43 | ||||||

DIC | 410.119 | 420.75 | 108.105 | ||||||

% Correct Fitting | 72.89 | 70 | 100 | ||||||

${}^{\ast \ast \ast}$ indicates 1% significance or relevance level | |||||||||

${}^{\ast \ast}$ indicates 5% significance or relevance level | |||||||||

${}^{\ast}$ indicates 10% significance or relevance level |

© 2018 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–Sánchez, J.M.; Gómez–Déniz, E.; Dávila–Cárdenes, N.
A Comparative Study of Logistic Models Using an Asymmetric Link: Modelling the Away Victories in Football. *Symmetry* **2018**, *10*, 224.
https://doi.org/10.3390/sym10060224

**AMA Style**

Pérez–Sánchez JM, Gómez–Déniz E, Dávila–Cárdenes N.
A Comparative Study of Logistic Models Using an Asymmetric Link: Modelling the Away Victories in Football. *Symmetry*. 2018; 10(6):224.
https://doi.org/10.3390/sym10060224

**Chicago/Turabian Style**

Pérez–Sánchez, José María, Emilio Gómez–Déniz, and Nancy Dávila–Cárdenes.
2018. "A Comparative Study of Logistic Models Using an Asymmetric Link: Modelling the Away Victories in Football" *Symmetry* 10, no. 6: 224.
https://doi.org/10.3390/sym10060224