# A Bayesian Mixture Cure Rate Model for Estimating Short-Term and Long-Term Recidivism

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## Abstract

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## 1. Introduction

## 2. Mixture Cure Rate Models

#### Likelihood Function for the Mixture Cure Rate Model

## 3. Bayesian Analysis

- Sample ${U}_{i}$, $i=1,\dots ,n$, from$$P({U}_{i}=0|{c}_{i}=0)=\frac{\pi \left({z}_{i}\right)}{1-\pi \left({z}_{i}\right)+\pi \left({z}_{i}\right){S}_{s}\left({t}_{i}\right|{U}_{i}=1,{x}_{i})}$$
- Sample $\beta $ from$$\pi \left(\beta \right|U)\propto {L}_{c}(\beta ,\alpha ,r|{d}_{n},U)\xb7\pi \left(\beta \right)\propto \prod _{i=1}^{n}{\left(\frac{{e}^{{z}_{i}^{\prime}\beta}}{1+{e}^{{z}_{i}^{\prime}\beta}}\right)}^{{u}_{i}}{\left(1-\frac{{e}^{{z}_{i}^{\prime}\beta}}{1+{e}^{{z}_{i}^{\prime}\beta}}\right)}^{(1-{u}_{i})(1-{c}_{i})}\xb7\pi \left(\beta \right)$$
- Sample $\alpha $ from$$\pi \left(\alpha \right|\beta ,U)\propto {L}_{c}(\beta ,\alpha ,r|{d}_{n},U)\xb7\pi \left(\alpha \right)\propto \prod _{i=1}^{n}{\left({e}^{{x}_{i}^{\prime}\alpha}\right)}^{{u}_{i}{c}_{i}}{\left({e}^{-{e}^{{x}_{i}^{\prime}\alpha}{t}^{r}}\right)}^{{u}_{i}}\xb7\pi \left(\alpha \right)$$
- Sample r from$$\pi \left(r\right|\alpha ,\beta ,U)\propto {L}_{c}(\beta ,\alpha ,r|{d}_{n},U)\xb7\pi \left(r\right)\propto \prod _{i=1}^{n}{\left(r{t}^{r}\right)}^{{u}_{i}{c}_{i}}{\left({e}^{-{e}^{{x}_{i}^{\prime}\alpha}{t}^{r}}\right)}^{{u}_{i}}\xb7\pi \left(r\right).$$

## 4. A Real Data Example

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Survival function estimates using the Kaplan–Meier estimator considered in the recidivism data example.

**Figure 3.**Dependence of survival function of the age of the victim for the recidivism example. Note that the time axis is divided into four-year periods.

Model | Without Covariates | With Covariates |
---|---|---|

Standard Weibull | 30,662.57 | 29,890.05 |

Weibull–SPM${}_{1}$ | 30,100.01 | 29,424.12 |

Weibull–SPM${}_{2}$ | 29,780.55 | |

Weibull–SPM${}_{3}$ | 29,338.96 |

Risk Factor | $\mathit{\beta}$ | 95% Credible Interval (CI) | ||
---|---|---|---|---|

Probability | Intercept | 0.573 | 0.357 | 0.794 |

of Recidivism | NP | 0.532 | 0.459 | 0.611 |

NPS | 0.167 | −0.064 | 0.416 | |

AGE | −0.405 | −0.466 | −0.344 | |

UND16 | 0.314 | 0.135 | 0.493 | |

Survival model | Intercept | 0.665 | −1.079 | 2.434 |

NP | 1.097 | 0.867 | 1.318 | |

NPS | −0.786 | −1.842 | 0.147 | |

AGE | −1.614 | −2.251 | −1.015 | |

UND16 | 0.061 | −1.189 | 1.345 |

**Table 3.**Estimate odds ratio (OR) and hazard ratio (HR) by Weibull mixture cure rate model (Weibull–SPM${}_{3}$).

Risk Factor | Long-Term Survival OR (95% CI) | Short-Term Survival HR (95% CI) |
---|---|---|

NP | 1.70 (1.58, 1.84) | 3.00 (2.38, 3.74) |

NPS | 1.18 (0.94, 1.52) | 0.46 (0.16, 1.16) |

AGE | 0.67 (0.63, 0.71) | 0.20 (0.11, 0.36) |

Age Victim | ||

≥16 | 1 | 1 |

<16 | 1.38 (1.15, 1.65) | 1.07 (0.31, 3.85) |

**Table 4.**Percentages of global and by-group recidivism for the logistic regression, Cox and mixture cure rate models at 3 and 10 years.

Real | Logistic | Cox | Mixture Cure Rate Model | |
---|---|---|---|---|

Regression | Regression | (Weibull) | ||

Recidivism general | 51.9 | 52.0 (22.1) | 47.8 (20.2) | 52.6 (21.8) |

Recidivism at 10 years | 47.5 | 47.3 (21.8) | 47.7 (19.7) | 48.1 (22.5) |

Recidivism at 3 years | 33.4 | 33.3 (18.5) | 33.3 (17.1) | 31.7 (18.3) |

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**MDPI and ACS Style**

de la Cruz, R.; Fuentes, C.; Padilla, O. A Bayesian Mixture Cure Rate Model for Estimating Short-Term and Long-Term Recidivism. *Entropy* **2023**, *25*, 56.
https://doi.org/10.3390/e25010056

**AMA Style**

de la Cruz R, Fuentes C, Padilla O. A Bayesian Mixture Cure Rate Model for Estimating Short-Term and Long-Term Recidivism. *Entropy*. 2023; 25(1):56.
https://doi.org/10.3390/e25010056

**Chicago/Turabian Style**

de la Cruz, Rolando, Claudio Fuentes, and Oslando Padilla. 2023. "A Bayesian Mixture Cure Rate Model for Estimating Short-Term and Long-Term Recidivism" *Entropy* 25, no. 1: 56.
https://doi.org/10.3390/e25010056