# Amoud Class for Hazard-Based and Odds-Based Regression Models: Application to Oncology Studies

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## Abstract

**:**

## 1. Introduction

## 2. Recent Literature Review and State of Art

#### 2.1. Hazard-Based Regression Models

#### 2.1.1. PH Model

#### 2.1.2. AFT Model

#### 2.1.3. AH Model

#### 2.1.4. GH Model

#### 2.1.5. Special Cases of the GH Model

**Proposition**

**1.**

**Proof**

**of**

**Proposition**

**1.**

#### 2.2. Odds-Based Regression Models

#### 2.2.1. PO Model

#### 2.2.2. Accelerated Forms

- i.
- Accelerated failure time (AFT) model; and
- ii.
- Accelerated odds model.

#### 2.2.3. Accelerated Odds Model

#### 2.2.4. General Odds Model

#### 2.2.5. Special Cases of the GO Model

**Proposition**

**2.**

**Proof**

**of**

**Proposition**

**2.**

## 3. The Proposed Class

#### 3.1. Why AM Class of Hazard-Based and Odds-Based Regression Models?

#### 3.2. Model Formulation

#### 3.3. Probabilistic Functions for the Amoud Class Model

#### 3.4. Special Sub-Models of the Proposed Class

**Proposition**

**3.**

**Proof**

**of**

**Proposition**

**3.**

## 4. Baseline Distributions

#### 4.1. Weibull Baseline for Hazard-Based Regression Models

#### 4.2. Log-Logistic Baseline for Odds-Based Regression Models

#### 4.3. Generalized Log-Logistic Baseline for All Models

## 5. Estimation Based on Frequentist and Bayesian Approaches

#### 5.1. MLE for Right-Censored Data

#### 5.2. The Log-Likelihood Functions

#### 5.3. Bayesian Inference

## 6. Model Comparison

#### 6.1. Classical Model Comparison

#### 6.1.1. Nested Models

- i.
- ${H}_{0}$: ${\beta}_{2}$ = ${\beta}_{1}$, that is the sample is from the GH model.${H}_{1}:$ if ${H}_{0}$ is false, then the sample is from the AM model.
- ii.
- ${H}_{0}$: ${\beta}_{1}$ = ${\beta}_{3}$, that is the sample is from the GO model.${H}_{1}:$ if ${H}_{0}$ is false, then the sample is from the AM model.
- iii.
- ${H}_{0}$: ${\beta}_{1}$ = ${\beta}_{2}$ = ${\beta}_{3}$ = 0, that is the sample is from the AFT model.${H}_{1}:$ if ${H}_{0}$ is false, then the sample is from the AM model.
- iv.
- ${H}_{0}$: ${\beta}_{1}$ = ${\beta}_{3}$ = 0, that is the sample is from the PO model.${H}_{1}:$ if ${H}_{0}$ is false, then the sample is from the AM model.
- v.
- ${H}_{0}$: ${\beta}_{1}$ = ${\beta}_{2}$ = 0, that is the sample is from the PH model.${H}_{1}:$ if ${H}_{0}$ is false, then the sample is from the AM model.
- vi.
- ${H}_{0}$: ${\beta}_{3}$–${\beta}_{1}$ = 0, ${\beta}_{2}$ = 0, that is the sample is from the AO model.${H}_{1}:$ if ${H}_{0}$ is false, then the sample is from the AM model.
- vii.
- ${H}_{0}$: ${\beta}_{2}$–${\beta}_{1}$ = 0, ${\beta}_{3}$ = 0, that is the sample is from the AH model.${H}_{1}:$ if ${H}_{0}$ is false, then the sample is from the AL model.

#### 6.1.2. Non-Nested Model

#### 6.2. Bayesian Model Comparison

## 7. Practical Illustrations

#### 7.1. IPASS Clinical Trial Data Set

#### 7.2. Bayesian Analysis

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Visual graph illustrating the relationship between the proposed Amoud Class (AM) and its sub-models including the proportional hazard (PH), general odds (GO), general hazard (GH), accelerated failure time (AFT), accelerated odds (AO), accelerated hazard (AH), and proportional odds (PO) models.

**Figure 2.**Illustrating the total time on test (TTT) plot, and the crossing survival curves for the two types of drugs determined using the Kaplan-Meier method for the IPASS dataset.

${\mathbf{\Delta}}_{\mathit{M}}$ | Level of Support of Model M |
---|---|

0–2 | Substantial |

4–7 | Considerably less |

>10 | Essentially none |

**Table 2.**Results from the fitted proposed fully-parametric odds-based and hazard-based regression models with W baseline distribution to IPASS dataset.

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

W-AM | ${\beta}_{1}$ | 0.167 | 0.008 | 5552.888 | 5583.409 | 5578.409 | 5552.849 | 5562.495 |

${\beta}_{2}$ | −1.346 | 0.109 | ||||||

${\beta}_{3}$ | 2.815 | 0.129 | ||||||

$\alpha $ | 1.845 | 0.002 | ||||||

$\kappa $ | 7.003 | 0.009 | ||||||

W-GH | ${\beta}_{1}$ | 1.398 | 0.007 | 5686.968 | 5711.385 | 5707.385 | 5686.943 | 5694.653 |

${\beta}_{2}$ | −0.862 | 0.129 | ||||||

$\alpha $ | 1.362 | 0.129 | ||||||

$\kappa $ | 6.613 | 0.129 | ||||||

W-GO | ${\beta}_{1}$ | −2.574 | 0.008 | 5609.716 | 5634.133 | 5630.133 | 5609.691 | 5617.402 |

${\beta}_{2}$ | 1.921 | 0.011 | ||||||

$\alpha $ | 1.596 | 0.003 | ||||||

$\kappa $ | 6.870 | 0.005 | ||||||

W-AH | $\beta $ | −0.875 | 0.005 | 5684.474 | 5702.787 | 5699.787 | 5684.460 | 5690.239 |

$\alpha $ | 1.365 | 0.004 | ||||||

$\kappa $ | 6.762 | 0.003 | ||||||

W-PH | $\beta $ | −0.320 | 0.007 | 5684.474 | 5702.787 | 5699.787 | 5684.460 | 5690.239 |

$\alpha $ | 1.365 | 0.004 | ||||||

$\kappa $ | 6.762 | 0.003 | ||||||

W-AO | $\beta $ | −0.567 | 0.001 | 5652.746 | 5671.058 | 5668.058 | 5652.731 | 5658.510 |

$\alpha $ | 1.419 | 0.003 | ||||||

$\kappa $ | 6.515 | 0.002 | ||||||

W-PO | $\beta $ | −0.003 | 0.017 | 5708.640 | 5726.952 | 5723.952 | 5708.625 | 5714.404 |

$\alpha $ | 1.341 | 0.010 | ||||||

$\kappa $ | 7.594 | 0.006 | ||||||

W-AFT | $\beta $ | −0.234 | 0.009 | 5684.474 | 5702.787 | 5699.787 | 5684.460 | 5690.239 |

$\alpha $ | 1.365 | 0.004 | ||||||

$\kappa $ | 6.762 | 0.003 |

**Table 3.**Results from the fitted proposed fully-parametric odds-based and hazard-based regression models with LL baseline distribution to IPASS dataset.

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

LL-AM | ${\beta}_{1}$ | 2.185 | 0.030 | 5690.051 | 5720.572 | 5715.572 | 5690.014 | 5699.658 |

${\beta}_{2}$ | −0.361 | 0.009 | ||||||

${\beta}_{3}$ | −0.171 | 0.011 | ||||||

$\alpha $ | 2.291 | 0.022 | ||||||

$\kappa $ | 5.498 | 0.006 | ||||||

LL-GH | ${\beta}_{1}$ | 1.074 | 0.018 | 5688.051 | 5712.468 | 5708.468 | 5688.027 | 5695.737 |

${\beta}_{2}$ | −0.088 | 0.019 | ||||||

$\alpha $ | 2.291 | 0.029 | ||||||

$\kappa $ | 5.498 | 0.109 | ||||||

LL-GO | ${\beta}_{1}$ | −0.845 | 0.038 | 5757.552 | 5781.968 | 5777.968 | 5757.527 | 5765.237 |

${\beta}_{2}$ | 0.726 | 0.129 | ||||||

$\alpha $ | 1.801 | 0.129 | ||||||

$\kappa $ | 5.478 | 0.129 | ||||||

LL-AH | $\beta $ | 1.119 | 0.014 | 5687.976 | 5706.289 | 5703.289 | 5687.961 | 5693.740 |

$\alpha $ | 2.259 | 0.008 | ||||||

$\kappa $ | 5.614 | 0.009 | ||||||

LL-PH | $\beta $ | −0.128 | 0.013 | 5751.598 | 5769.911 | 5766.911 | 5751.584 | 5757.362 |

$\alpha $ | 1.832 | 0.010 | ||||||

$\kappa $ | 5.156 | 0.009 | ||||||

LL-AO | $\beta $ | 0.061 | 0.007 | 5755.552 | 5773.864 | 5770.864 | 5755.537 | 5761.316 |

$\alpha $ | 1.801 | 0.009 | ||||||

$\kappa $ | 5.478 | 0.010 | ||||||

LL-PO | $\beta $ | 0.049 | 0.004 | 5755.552 | 5773.864 | 5770.864 | 5755.537 | 5761.316 |

$\alpha $ | 1.801 | 0.009 | ||||||

$\kappa $ | 5.478 | 0.010 | ||||||

LL-AFT | $\beta $ | 0.027 | 0.009 | 5755.552 | 5773.864 | 5770.864 | 5755.537 | 5761.316 |

$\alpha $ | 1.801 | 0.009 | ||||||

$\kappa $ | 5.478 | 0.010 |

**Table 4.**Results from the fitted proposed fully-parametric odds-based and hazard-based regression models with GLL baseline distribution to IPASS dataset.

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

GLL-AM | ${\beta}_{1}$ | 0.179 | 0.003 | 5554.864 | 5591.489 | 5585.489 | 5554.810 | 5566.392 |

${\beta}_{2}$ | −1.344 | 0.001 | ||||||

${\beta}_{3}$ | 2.824 | 0.002 | ||||||

$\alpha $ | 1.853 | 0.129 | ||||||

$\kappa $ | 0.143 | 0.129 | ||||||

$\eta $ | 0.010 | 0.129 | ||||||

GLL-GH | ${\beta}_{1}$ | 2.049 | 0.001 | 5605.079 | 5635.600 | 5630.600 | 5605.041 | 5614.686 |

${\beta}_{2}$ | −0.835 | 0.002 | ||||||

$\alpha $ | 1.818 | 0.129 | ||||||

$\kappa $ | 0.151 | 0.129 | ||||||

$\eta $ | 0.045 | 0.129 | ||||||

GLL-GO | ${\beta}_{1}$ | −0.635 | 0.002 | 5645.507 | 5676.028 | 5671.028 | 5645.470 | 5655.114 |

${\beta}_{2}$ | 0.339 | 0.001 | ||||||

$\alpha $ | 1.364 | 0.129 | ||||||

$\kappa $ | 0.145 | 0.129 | ||||||

$\eta $ | 0.000 | 0.129 | ||||||

GLL-AH | $\beta $ | 1.545 | 0.001 | 5656.576 | 5680.993 | 5676.993 | 5656.551 | 5664.262 |

$\alpha $ | 1.886 | 0.129 | ||||||

$\kappa $ | 0.154 | 0.129 | ||||||

$\eta $ | 0.101 | 0.129 | ||||||

GLL-PH | $\beta $ | −0.064 | 0.014 | 5686.187 | 5710.604 | 5706.604 | 5686.162 | 5693.872 |

$\alpha $ | −0.171 | 0.129 | ||||||

$\kappa $ | −0.171 | 0.129 | ||||||

$\eta $ | −0.171 | 0.129 | ||||||

GLL-AO | $\beta $ | −0.708 | 0.002 | 5661.339 | 5685.755 | 5681.755 | 5661.314 | 5669.024 |

$\alpha $ | 1.482 | 0.129 | ||||||

$\kappa $ | 0.158 | 0.129 | ||||||

$\eta $ | 0.000 | 0.129 | ||||||

GLL-PO | $\beta $ | −0.008 | 0.021 | 5708.832 | 5733.249 | 5729.249 | 5708.808 | 5716.518 |

$\alpha $ | 1.406 | 0.129 | ||||||

$\kappa $ | 0.139 | 0.129 | ||||||

$\eta $ | 0.031 | 0.129 | ||||||

GLL-AFT | $\beta $ | −0.166 | 0.014 | 5688.582 | 5712.999 | 5708.999 | 5688.557 | 5696.267 |

$\alpha $ | 1.359 | 0.129 | ||||||

$\kappa $ | 0.143 | 0.129 | ||||||

$\eta $ | 0.000 | 0.129 |

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

GH | ${H}_{0}$: ${\beta}_{2}$ = ${\beta}_{1}$, ${H}_{1}$: ${H}_{0}$ is false, | 52.214 | <0.0001 |

GO | ${H}_{0}$: ${\beta}_{3}$ = ${\beta}_{1}$, ${H}_{1}$: ${H}_{0}$ is false, | 92.644 | <0.0001 |

AH | ${H}_{0}$: ${\beta}_{2}$–${\beta}_{1}$ = 0,${\beta}_{3}$ = 0, ${H}_{1}$: ${H}_{0}$ is false, | 105.712 | <0.0001 |

AO | ${H}_{0}$: ${\beta}_{3}$–${\beta}_{1}$ = 0,${\beta}_{2}$ = 0, ${H}_{1}$: ${H}_{0}$ is false, | 135.322 | <0.0001 |

PH | ${H}_{0}$: ${\beta}_{1}$ = ${\beta}_{2}$ = 0, ${H}_{1}$: ${H}_{0}$ is false, | 110.474 | <0.0001 |

PO | ${H}_{0}$: ${\beta}_{1}$ = ${\beta}_{3}$ = 0, ${H}_{1}$: ${H}_{0}$ is false, | 157.968 | <0.0001 |

AFT | ${H}_{0}$: ${\beta}_{1}$ = ${\beta}_{2}$ = ${\beta}_{3}$, ${H}_{1}$: ${H}_{0}$ is false, | 137.718 | <0.0001 |

Models | Hypothesis | LRT Statistic | p-Value |
---|---|---|---|

GH vs. AH | ${H}_{0}$: ${\beta}_{2}$ = 0, ${H}_{1}$: ${H}_{0}$ is false, | 53.498 | <0.0001 |

GH vs. PH | ${H}_{0}$: ${\beta}_{1}$ = 0, ${H}_{1}$: ${H}_{0}$ is false, | 83.108 | <0.0001 |

GH vs. AFT | ${H}_{0}$${\beta}_{1}$ = ${\beta}_{2}$, ${H}_{1}$: ${H}_{0}$ is false, | 85.504 | <0.0001 |

Models | Hypothesis | LRT Statistic | p-Value |
---|---|---|---|

GO vs. AO | ${H}_{0}$: ${\beta}_{2}$ = 0, ${H}_{1}$: ${H}_{0}$ is false, | 17.830 | <0.0001 |

GO vs. PO | ${H}_{0}$: ${\beta}_{1}$ = 0, ${H}_{1}$: ${H}_{0}$ is false, | 65.324 | <0.0001 |

GO vs. AFT | ${H}_{0}$: ${\beta}_{1}$ = ${\beta}_{2}$, ${H}_{1}$: ${H}_{0}$ is false, | 45.074 | <0.0001 |

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

GLL-AM | ${\beta}_{1}$ | 0.293 | 0.003 | 0.166 | −0.032 | 0.291 | 0.618 | 4108 | 1.000 |

${\beta}_{2}$ | −1.287 | 0.002 | 0.096 | −1.473 | −1.288 | −1.096 | 3309 | 1.000 | |

${\beta}_{3}$ | 2.810 | 0.004 | 0.242 | 2.349 | 2.811 | 3.279 | 3431 | 1.000 | |

$\alpha $ | 1.953 | 0.001 | 0.072 | 1.818 | 1.952 | 2.095 | 3477 | 1.000 | |

k | 0.151 | 0.000 | 0.004 | 0.143 | 0.151 | 0.160 | 4309 | 1.001 | |

$\eta $ | 0.052 | 0.000 | 0.010 | 0.032 | 0.052 | 0.073 | 3613 | 1.001 | |

GLL-GH | ${\beta}_{1}$ | 1.839 | 0.005 | 0.219 | 1.444 | 1.828 | 2.296 | 2068 | 1.001 |

${\beta}_{2}$ | −0.672 | 0.003 | 0.139 | −0.968 | −0.663 | −0.425 | 2232 | 1.000 | |

$\alpha $ | 1.851 | 0.001 | 0.075 | 1.712 | 1.849 | 2.004 | 2672 | 1.002 | |

k | 0.155 | 0.000 | 0.005 | 0.145 | 0.155 | 0.165 | 2431 | 1.001 | |

$\eta $ | 0.059 | 0.000 | 0.013 | 0.036 | 0.059 | 0.087 | 1739 | 1.002 | |

GLL-GO | ${\beta}_{1}$ | −0.395 | 0.001 | 0.065 | −0.524 | −0.395 | −0.263 | 3169 | 1.001 |

${\beta}_{2}$ | 0.157 | 0.001 | 0.077 | 0.005 | 0.158 | 0.307 | 3303 | 1.001 | |

$\alpha $ | 1.450 | 0.001 | 0.041 | 1.371 | 1.450 | 1.533 | 3576 | 1.000 | |

k | 0.145 | 0.000 | 0.004 | 0.137 | 0.145 | 0.154 | 3523 | 1.001 | |

$\eta $ | 0.008 | 0.000 | 0.007 | 0.000 | 0.007 | 0.027 | 3664 | 1.000 | |

GLL-AH | $\beta $ | 1.448 | 0.003 | 0.143 | 1.163 | 1.445 | 1.728 | 3090 | 1.000 |

$\alpha $ | 1.912 | 0.002 | 0.089 | 1.742 | 1.911 | 2.087 | 2446 | 1.000 | |

k | 0.159 | 0.000 | 0.006 | 0.148 | 0.159 | 0.170 | 2406 | 1.000 | |

$\eta $ | 0.112 | 0.000 | 0.013 | 0.088 | 0.112 | 0.137 | 1989 | 1.000 | |

GLL-PH | $\beta $ | −0.294 | 0.001 | 0.066 | −0.427 | −0.294 | −0.168 | 3606 | 1.000 |

$\alpha $ | 1.526 | 0.001 | 0.054 | 1.425 | 1.524 | 1.635 | 3214 | 1.001 | |

k | 0.167 | 0.000 | 0.007 | 0.155 | 0.167 | 0.180 | 3082 | 1.000 | |

$\eta $ | 0.071 | 0.000 | 0.016 | 0.042 | 0.071 | 0.104 | 2693 | 1.002 | |

GLL-AO | $\beta $ | −0.557 | 0.001 | 0.092 | −0.740 | −0.556 | −0.378 | 4181 | 1.000 |

$\alpha $ | 1.481 | 0.001 | 0.041 | 1.403 | 1.480 | 1.561 | 3743 | 1.001 | |

k | 0.163 | 0.000 | 0.005 | 0.153 | 0.163 | 0.174 | 3575 | 1.001 | |

$\eta $ | 0.042 | 0.000 | 0.011 | 0.022 | 0.042 | 0.067 | 3654 | 1.002 | |

GLL-PO | $\beta $ | −0.011 | 0.002 | 0.105 | −0.214 | −0.011 | 0.196 | 2226 | 1.001 |

$\alpha $ | 1.412 | 0.002 | 0.059 | 1.308 | 1.408 | 1.537 | 1028 | 1.003 | |

k | 0.140 | 0.000 | 0.007 | 0.128 | 0.140 | 0.156 | 837 | 1.003 | |

$\eta $ | 0.033 | 0.001 | 0.019 | 0.003 | 0.032 | 0.075 | 726 | 1.004 | |

GLL-AFT | $\beta $ | −0.188 | 0.001 | 0.054 | −0.291 | −0.190 | −0.078 | 3662 | 1.000 |

$\alpha $ | 1.481 | 0.001 | 0.051 | 1.388 | 1.479 | 1.584 | 3041 | 1.000 | |

k | 0.161 | 0.000 | 0.006 | 0.150 | 0.161 | 0.174 | 2976 | 1.000 | |

$\eta $ | 0.061 | 0.000 | 0.016 | 0.032 | 0.060 | 0.095 | 2601 | 1.000 |

**Table 9.**Bayesian model selection between the proposed AM class and its sub-models using the GLL baseline distribution.

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

GLL-AM | 5559.50 | 5559.54 |

GLL-GH | $5608.30$ | $5608.28$ |

GLL-GO | $5651.76$ | $5651.69$ |

GLL-AH | $5657.40$ | $5697.43$ |

GLL-PH | $5692.50$ | $5692.51$ |

GLL-AO | $5666.60$ | $5666.61$ |

GLL-PO | $5708.60$ | $5708.62$ |

GLL-AFT | $5698.00$ | $5698.04$ |

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**MDPI and ACS Style**

Muse, A.H.; Mwalili, S.; Ngesa, O.; Chesneau, C.; Alshanbari, H.M.; El-Bagoury, A.-A.H.
Amoud Class for Hazard-Based and Odds-Based Regression Models: Application to Oncology Studies. *Axioms* **2022**, *11*, 606.
https://doi.org/10.3390/axioms11110606

**AMA Style**

Muse AH, Mwalili S, Ngesa O, Chesneau C, Alshanbari HM, El-Bagoury A-AH.
Amoud Class for Hazard-Based and Odds-Based Regression Models: Application to Oncology Studies. *Axioms*. 2022; 11(11):606.
https://doi.org/10.3390/axioms11110606

**Chicago/Turabian Style**

Muse, Abdisalam Hassan, Samuel Mwalili, Oscar Ngesa, Christophe Chesneau, Huda M. Alshanbari, and Abdal-Aziz H. El-Bagoury.
2022. "Amoud Class for Hazard-Based and Odds-Based Regression Models: Application to Oncology Studies" *Axioms* 11, no. 11: 606.
https://doi.org/10.3390/axioms11110606