# Flexible Parametric Accelerated Hazard Model: Simulation and Application to Censored Lifetime Data with Crossing Survival Curves

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## Abstract

**:**

## 1. Introduction

- i.
- It possesses the adaptability of parametric survival regression models.
- ii.
- It offers a continuous SF that makes it simple to find where two survival curves overlap.
- iii.
- It allows different shapes for the HRF and has the tractability of a parametric survival regression model.

## 2. AH Model Formulation

## 3. Baseline Hazard

## 4. The Proposed Model

#### 4.1. Submodels

#### 4.1.1. Submodel I: $\eta =1$

#### 4.1.2. Submodel II: $\eta =k$

#### 4.1.3. Submodel III: ${\eta}^{\alpha}\to 0$

## 5. Inferential Procedures

#### 5.1. Classical Approach

#### 5.2. Bayesian Approach

#### 5.2.1. Prior Distribution

#### 5.2.2. Likelihood Function

#### 5.2.3. Posterior Distribution

## 6. Simulation Study

- I.
- Procedure I: An MLE estimate technique.
- II.
- Procedure II: A Bayesian estimation technique with independent gamma priors for the baseline distribution parameters and a normal prior for the regression coefficients, as well as non-informative priors.

## 7. Applications

#### 7.1. Gastric Cancer Dataset

#### 7.2. Classical Analysis

#### 7.3. Likelihood Ratio Test

#### 7.4. Bayesian Analysis

#### 7.4.1. Numerical Summary

#### 7.4.2. Visual Summary

#### 7.4.3. Posterior Predictive Checks

#### 7.4.4. McMC Convergence Diagnostics

#### 7.4.5. Bayesian Model Selection

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Illustrating the overall survival curve and the crossing survival curves for the two types of treatment.

**Figure 10.**The empirical CDF, the dotted line and the CDF of the fitted model, the smooth curve, show that the fitted GLL–AH model predicts the future observations that are consistent with the current data.

**Figure 13.**Kaplan–Meier and fitted survival curve for the GLL–AH model of the gastric cancer dataset.

**Figure 14.**Kaplan–Meier and estimated survival plots for the competitive regression models with the GLL baseline distribution of the gastric cancer dataset.

**Figure 15.**Kaplan–Meier and estimated survival plots for the competitive AH models of the gastric cancer dataset.

**Table 1.**Simulation study for GLL–AH regression model. True values (True), Estimates (Est.), standard error (SE), average bias (AB), mean square error (MSE), and coverage probability (CP 95%) are presented for the parameters.

True | Est. | SE | AB | MSE | CP | Est. | SE | AB | MSE | $\widehat{\mathit{R}}$ | |
---|---|---|---|---|---|---|---|---|---|---|---|

Set I $\mathit{n}=50$ | |||||||||||

${\mathit{M}}_{\mathbf{2}}$ | MLE Approach | Bayesian | |||||||||

${\beta}_{1}$ | 0.75 | 0.800 | 0.100 | 0.050 | 0.037 | 93.85 | 0.790 | 0.002 | 0.040 | 0.036 | 1.002 |

${\beta}_{2}$ | 0.5 | 0.558 | 0.042 | 0.058 | 0.024 | 94.50 | 0.512 | 0.003 | 0.012 | 0.011 | 1.002 |

$\alpha $ | 1.50 | 1.590 | 0.010 | 0.090 | 0.008 | 95.20 | 1.505 | 0.001 | 0.005 | 0.003 | 1.000 |

k | 0.75 | 0.900 | 0.435 | 0.150 | 0.063 | 92.05 | 0.850 | 0.005 | 0.100 | 0.045 | 1.002 |

$\eta $ | 1.20 | 1.265 | 0.011 | 0.065 | 0.046 | 94.25 | 1.212 | 0.000 | 0.012 | 0.004 | 1.003 |

True | Est. | SE | AB | MSE | CP | Est. | SE | AB | MSE | $\widehat{\mathit{R}}$ | |

Set II $\mathit{n}=\mathbf{100}$ | |||||||||||

${\mathit{M}}_{\mathbf{2}}$ | MLE approach | Bayesian | |||||||||

${\beta}_{1}$ | 0.75 | 0.790 | 0.100 | 0.040 | 0.036 | 94.10 | 0.770 | 0.001 | 0.020 | 0.018 | 1.000 |

${\beta}_{2}$ | 0.5 | 0.530 | 0.030 | 0.030 | 0.024 | 94.80 | 0.510 | 0.002 | 0.010 | 0.010 | 1.001 |

$\alpha $ | 1.50 | 1.610 | 0.040 | 0.110 | 0.087 | 93.40 | 1.553 | 0.001 | 0.053 | 0.041 | 1.003 |

k | 0.75 | 0.850 | 0.250 | 0.100 | 0.056 | 93.20 | 0.800 | 0.004 | 0.050 | 0.037 | 1.002 |

$\eta $ | 1.20 | 1.250 | 0.008 | 0.050 | 0.034 | 94.80 | 1.205 | 0.000 | 0.005 | 0.003 | 1.001 |

Set III $\mathit{n}=\mathbf{300}$ | |||||||||||

True | Est. | SE | AB | MSE | CP | Est. | SE | AB | MSE | $\widehat{\mathit{R}}$ | |

${\mathit{M}}_{\mathbf{2}}$ | MLE approach | Bayesian | |||||||||

${\beta}_{1}$ | 0.75 | 0.78 | 0.092 | 0.030 | 0.032 | 94.40 | 0.768 | 0.001 | 0.018 | 0.016 | 1.000 |

${\beta}_{2}$ | 0.5 | 0.525 | 0.013 | 0.025 | 0.021 | 93.90 | 0.503 | 0.001 | 0.003 | 0.002 | 1.000 |

$\alpha $ | 1.50 | 1.592 | 0.021 | 0.042 | 0.030 | 93.85 | 1.506 | 0.001 | 0.006 | 0.006 | 1.001 |

k | 0.75 | 0.844 | 0.212 | 0.094 | 0.049 | 93.46 | 0.798 | 0.003 | 0.048 | 0.036 | 1.000 |

$\eta $ | 1.20 | 1.252 | 0.008 | 0.052 | 0.034 | 94.60 | 1.205 | 0.000 | 0.005 | 0.003 | 1.001 |

True | Est. | SE | AB | MSE | CP | Est. | SE | AB | MSE | $\widehat{\mathit{R}}$ | |

Set IV $\mathit{n}=\mathbf{500}$ | |||||||||||

${\mathit{M}}_{\mathbf{2}}$ | MLE approach | Bayesian | |||||||||

${\beta}_{1}$ | 0.75 | 0.775 | 0.065 | 0.025 | 0.017 | 95.10 | 0.752 | 0.000 | 0.002 | 0.002 | 1.000 |

${\beta}_{2}$ | 0.5 | 0.526 | 0.013 | 0.026 | 0.021 | 94.00 | 0.503 | 0.001 | 0.003 | 0.002 | 1.000 |

$\alpha $ | 1.50 | 1.550 | 0.040 | 0.050 | 0.037 | 94.70 | 1.503 | 0.001 | 0.003 | 0.001 | 1.000 |

k | 0.75 | 0.825 | 0.110 | 0.075 | 0.048 | 94.07 | 0.780 | 0.003 | 0.030 | 0.027 | 1.001 |

$\eta $ | 1.20 | 1.205 | 0.005 | 0.005 | 0.003 | 95.04 | 1.203 | 0.000 | 0.003 | 0.001 | 1.001 |

**Table 2.**Results from the fitted proposed fully parametric AH regression model and other survival regression models with the GLL baseline distribution to gastric cancer dataset.

Models | Parameter(s) | Estimate | SE | AIC | CAIC | HQIC |
---|---|---|---|---|---|---|

GLL-AH | $\beta $ | 2.690 | 0.021 | 244.318 | 242.845 | 248.351 |

$\alpha $ | 1.505 | 0.040 | ||||

k | 0.542 | 0.036 | ||||

$\eta $ | 0.133 | 0.022 | ||||

GLL-PO | $\beta $ | 0.750 | 0.101 | 251.816 | 250.522 | 255.848 |

$\alpha $ | 1.382 | 0.100 | ||||

k | 0.650 | 0.074 | ||||

$\eta $ | 0.500 | 0.042 | ||||

GLL-PH | $\beta $ | 0.130 | 0.241 | 255.565 | 254.345 | 259.598 |

$\alpha $ | 1.302 | 0.140 | ||||

k | 0.759 | 0.136 | ||||

$\eta $ | 0.580 | 0.222 | ||||

GLL-AFT | $\beta $ | 0.540 | 0.135 | 252.139 | 250.851 | 256.171 |

$\alpha $ | 1.545 | 0.127 | ||||

k | 0.557 | 0.106 | ||||

$\eta $ | 0.728 | 0.231 |

**Table 3.**Results from the fitted proposed fully parametric AH regression model with different baseline distributions to gastric cancer dataset.

Models | Parameter(s) | Estimate | SE | AIC | CAIC | HQIC |
---|---|---|---|---|---|---|

GLL-AH | $\beta $ | 2.690 | 0.021 | 244.318 | 242.845 | 248.351 |

$\alpha $ | 1.505 | 0.040 | ||||

k | 0.542 | 0.036 | ||||

$\eta $ | 0.133 | 0.022 | ||||

PGW-AH | $\beta $ | 1.930 | 0.082 | 251.186 | 249.878 | 255.218 |

$\alpha $ | 1.687 | 0.142 | ||||

k | 0.821 | 0.066 | ||||

$\eta $ | 2.226 | 0.102 | ||||

GG-AH | $\beta $ | 2.688 | 0.130 | 252.645 | 251.368 | 256.677 |

$\alpha $ | 1.821 | 0.122 | ||||

k | 0.482 | 0.236 | ||||

$\eta $ | 0.737 | 0.042 | ||||

EW-AH | $\beta $ | 2.066 | 0.110 | 252.667 | 251.390 | 256.699 |

$\alpha $ | 0.789 | 0.212 | ||||

k | 0.911 | 0.086 | ||||

$\eta $ | 2.283 | 0.052 | ||||

LL-AH | $\beta $ | 1.097 | 0.020 | 247.492 | 246.686 | 250.517 |

$\alpha $ | 1.913 | 0.052 | ||||

k | 1.213 | 0.019 | ||||

LN-AH | $\beta $ | 0.261 | 0.120 | 263.830 | 263.197 | 266.854 |

$\alpha $ | 0.065 | 0.101 | ||||

k | 1.260 | 0.032 | ||||

BXII-AH | $\beta $ | 0.923 | 0.142 | 249.144 | 248.359 | 252.168 |

$\alpha $ | 0.880 | 0.119 | ||||

k | 1.890 | 0.120 | ||||

W-AH | $\beta $ | 2.581 | 0.214 | 256.776 | 256.078 | 259.800 |

$\alpha $ | 1.013 | 0.049 | ||||

k | 1.818 | 0.112 | ||||

G-AH | $\beta $ | 2.367 | 0.430 | 255.121 | 254.406 | 258.145 |

$\alpha $ | 1.495 | 0.039 | ||||

k | 1.252 | 0.123 |

Model | Hypothesis | LRT | p-Value |
---|---|---|---|

GLL-AH vs. BXII-AH | ${H}_{0}:\eta =1,{H}_{1}:{H}_{0}$ is false, | 6.999 | 0.008 |

GLL-AH vs. LL-AH | ${H}_{0}$: $\eta =k,{H}_{1}:{H}_{0}$ is false, | 5.347 | 0.021 |

GLL-AH vs. W-AH | ${H}_{0}$: ${\eta}^{\alpha}\to 0$, ${H}_{1}:{H}_{0}$ is false, | 14.533 | <0.001 |

Models | Par (s) | Estimate | SE | SD | 2.5% | Medium | 97.5% | ${\mathit{N}}_{\mathit{eff}}$ | $\widehat{\mathit{R}}$ |
---|---|---|---|---|---|---|---|---|---|

GLL–AH | $\beta $ | 1.016 | 0.009 | 0.476 | 0.030 | 1.027 | 1.909 | 2684 | 1.001 |

$\alpha $ | 0.836 | 0.002 | 0.106 | 0.648 | 0.829 | 1.064 | 3097 | 1.002 | |

k | 1.553 | 0.004 | 0.196 | 1.205 | 1.544 | 1.969 | 2714 | 1.001 | |

$\eta $ | 0.674 | 0.003 | 0.191 | 0.353 | 0.653 | 1.105 | 3023 | 1.001 | |

GLL–PO | $\beta $ | 0.565 | 0.006 | 0.353 | −0.135 | 0.562 | 1.268 | 3617 | 1.001 |

$\alpha $ | 1.414 | 0.003 | 0.156 | 1.136 | 1.405 | 1.741 | 3257 | 1.000 | |

k | 0.804 | 0.002 | 0.115 | 0.600 | 0.796 | 1.054 | 2951 | 1.001 | |

$\eta $ | 0.806 | 0.004 | 0.214 | 0.429 | 0.792 | 1.262 | 2918 | 1.000 | |

GLL–PH | $\beta $ | 0.106 | 0.004 | 0.224 | −0.330 | 0.107 | 0.540 | 3216 | 1.000 |

$\alpha $ | 1.341 | 0.002 | 0.146 | 1.077 | 1.332 | 1.646 | 3588 | 1.001 | |

k | 0.876 | 0.002 | 0.122 | 0.662 | 0.869 | 1.134 | 3068 | 1.001 | |

$\eta $ | 0.837 | 0.004 | 0.221 | 0.452 | 0.820 | 1.315 | 3239 | 1.001 | |

GLL–AFT | $\beta $ | 0.418 | 0.005 | 0.269 | −0.116 | 0.415 | 0.949 | 3396 | 1.000 |

$\alpha $ | 1.435 | 0.003 | 0.177 | 1.124 | 1.423 | 1.804 | 3373 | 1.000 | |

k | 0.809 | 0.002 | 0.114 | 0.609 | 0.801 | 1.060 | 2963 | 1.000 | |

$\eta $ | 0.850 | 0.004 | 0.210 | 0.479 | 0.836 | 1.311 | 2728 | 1.000 |

**Table 6.**Results for the posterior properties of the submodels of the GLL–AH model including LL–AH, W–AH, and BXII–AH models.

Models | Par (s) | Estimate | SE | SD | 2.5% | Medium | 97.5% | ${\mathit{N}}_{\mathit{eff}}$ | $\widehat{\mathit{R}}$ |
---|---|---|---|---|---|---|---|---|---|

LL–AH | $\beta $ | 0.764 | 0.007 | 0.385 | −0.073 | 0.800 | 1.421 | 3228 | 1.001 |

$\alpha $ | 1.636 | 0.004 | 0.197 | 1.261 | 1.629 | 2.039 | 2930 | 1.000 | |

k | 0.879 | 0.002 | 0.107 | 0.688 | 0.873 | 1.109 | 3681 | 1.001 | |

W–AH | $\beta $ | −0.007 | 0.014 | 0.949 | −1.850 | −0.019 | 1.860 | 4377 | 1.000 |

$\alpha $ | 0.984 | 0.001 | 0.085 | 0.821 | 0.982 | 1.152 | 3521 | 1.000 | |

k | 0.559 | 0.001 | 0.068 | 0.437 | 0.554 | 0.702 | 3875 | 1.001 | |

BXII–AH | $\beta $ | 0.678 | 0.007 | 0.378 | −0.135 | 0.697 | 1.345 | 3291 | 1.000 |

$\alpha $ | 1.627 | 0.004 | 0.209 | 1.247 | 1.620 | 2.062 | 3099 | 1.000 | |

k | 0.949 | 0.002 | 0.115 | 0.740 | 0.943 | 1.186 | 3932 | 1.000 |

Model | WAIC | LOOIC |
---|---|---|

GLL–AH | 243.20 | 243.20 |

GLL–PO | 251.40 | 251.42 |

GLL–AFT | 251.80 | 251.90 |

GLL–PH | 254.80 | 254.82 |

**Table 8.**Bayesian model comparison for the GLL–AH and its special cases including LL–AH, W–AH, and BXII–AH models.

Model | WAIC | LOOIC |
---|---|---|

GLL–AH | 243.20 | 243.20 |

LL–AH | 249.30 | 249.40 |

W–AH | 255.01 | 255.00 |

BXII–AH | 247.05 | 247.08 |

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**MDPI and ACS Style**

Muse, A.H.; Chesneau, C.; Ngesa, O.; Mwalili, S.
Flexible Parametric Accelerated Hazard Model: Simulation and Application to Censored Lifetime Data with Crossing Survival Curves. *Math. Comput. Appl.* **2022**, *27*, 104.
https://doi.org/10.3390/mca27060104

**AMA Style**

Muse AH, Chesneau C, Ngesa O, Mwalili S.
Flexible Parametric Accelerated Hazard Model: Simulation and Application to Censored Lifetime Data with Crossing Survival Curves. *Mathematical and Computational Applications*. 2022; 27(6):104.
https://doi.org/10.3390/mca27060104

**Chicago/Turabian Style**

Muse, Abdisalam Hassan, Christophe Chesneau, Oscar Ngesa, and Samuel Mwalili.
2022. "Flexible Parametric Accelerated Hazard Model: Simulation and Application to Censored Lifetime Data with Crossing Survival Curves" *Mathematical and Computational Applications* 27, no. 6: 104.
https://doi.org/10.3390/mca27060104