# A Class of Exponentiated Regression Model for Non Negative Censored Data with an Application to Antibody Response to Vaccine

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## Abstract

**:**

## 1. Introduction

## 2. Models for Asymmetric Data

#### The Centred Parametrization of the Skew-Normal Model

## 3. The Centred Exponentiated Log-Skew-Normal Family of Distribution for Censored Data

#### 3.1. The ELSN${}_{c}$ Regression Models

#### 3.2. The Censored ELSN${}_{c}$ Distribution

## 4. The Bernoulli/ Centred Log-Skew-Normal Alpha-Power Mixture Model

#### 4.1. The Logit/Centred Log-Skew-Normal Alpha-Power Model

- If ${I}_{i}=0$, i.e., the non-censored part, the covariates vector will be denoted by ${\mathit{x}}_{\left(2\right)i}={(1,{X}_{2i1},{X}_{2i2},\dots ,{X}_{2i{q}_{0}})}^{\top}$ with the parameter vector given by ${\mathit{\beta}}_{\left(2\right)}={({\beta}_{20},{\beta}_{21},\dots ,{\beta}_{2{q}_{0}})}^{\top}$.
- If ${I}_{i}=1$ and ${Y}_{i}=T$, i.e., the censored part, the covariates vector will be denoted by ${\mathit{x}}_{\left(1\right)i}={(1,{X}_{1i1},{X}_{1i2},\dots ,{X}_{1i{q}_{1}})}^{\top}$ with the parameter vector given by ${\mathit{\beta}}_{\left(1\right)}={({\beta}_{10},\phantom{\rule{4pt}{0ex}}{\beta}_{11},\dots ,\phantom{\rule{4pt}{0ex}}{\beta}_{1{q}_{1}})}^{\top}$.

#### 4.2. Fitting Model

## 5. An Application to Antibody Response to Vaccine

## 6. Final Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**(

**a**) ${\mathrm{ELSN}}_{c}(0,1,0.75,\alpha $) for $\alpha =0.35$ (solid line), $\alpha =0.5$ (dashed line), $\alpha =0.75$ (dotted line), $\alpha =2$ (dotted–dashed line), (

**b**) ${\mathrm{ELSN}}_{c}(0,1$,${\gamma}_{1},2$) with ${\gamma}_{1}=-0.50$ (solid line), ${\gamma}_{1}=-0.25$ (dashed line), ${\gamma}_{1}=0.50$ (dotted line), ${\gamma}_{1}=0.90$ (dotted–dashed line).

**Figure 2.**Log-likelihood profiled for ${\gamma}_{1}$ assuming ${\mathrm{ELSN}}_{c}$ distribution with samples sizes (

**a**) 50, (

**b**) 100 and (

**c**) 150 from a simulated $\mathrm{LN}(0,1)\equiv {\mathrm{ELSN}}_{c}(0,1)$ distribution.

**Figure 3.**Log-likelihood profiled for $\alpha $ assuming ${\mathrm{ELSN}}_{c}$ distribution with samples sizes (

**a**) 50, (

**b**) 100 and (

**c**) 150 from a simulated $\mathrm{LN}(0,1)\equiv {\mathrm{ELSN}}_{c}(0,1)$ distribution.

**Figure 4.**Histogram of the scaled residuals ${e}_{i}$ for (

**a**) LN model, (

**b**) LPN model, (

**c**) ELSN${}_{c}$ model.

**Figure 5.**QQ-plots of the scaled residuals ${e}_{i}$ for (

**a**) LN model, (

**b**) LPN model, (

**c**) ELSN${}_{c}$ model.

**Table 1.**Simulation study with 1000 iterations for $\alpha =0.75,1.5$, $\gamma =0.25,0.5,0.75$, ${\beta}_{0}=1.5$, and ${\beta}_{1}=2.5$, with sample sizes of $n=60,70,80$ and 500.

${\mathit{\beta}}_{0}=1.5$ | ${\mathit{\beta}}_{1}=2.5$ | $\mathit{\alpha}=0.75$ | ${\mathit{\gamma}}_{1}=0.25$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

n | Bias | RMSE | Bias | RMSE | Bias | RMSE | Bias | RMSE | |||

60 | −0.0307 | 0.5331 | 0.0019 | 0.1405 | −0.0168 | 0.3787 | 0.1614 | 0.5477 | |||

70 | −0.0296 | 0.5073 | 0.0015 | 0.1305 | −0.0130 | 0.3581 | 0.1410 | 0.5616 | |||

80 | −0.0170 | 0.4640 | −0.0012 | 0.1255 | −0.0204 | 0.3342 | 0.1083 | 0.4157 | |||

500 | 0.0080 | 0.1700 | 0.0011 | 0.0462 | −0.0046 | 0.1199 | 0.0068 | 0.1224 | |||

${\mathit{\beta}}_{\mathbf{0}}=\mathbf{1}.\mathbf{5}$ | ${\mathit{\beta}}_{\mathbf{1}}=\mathbf{2}.\mathbf{5}$ | $\mathbf{\alpha}=\mathbf{0}.\mathbf{75}$ | ${\mathbf{\gamma}}_{\mathbf{1}}=\mathbf{0}.\mathbf{50}$ | ||||||||

n | Bias | RMSE | Bias | RMSE | Bias | RMSE | Bias | RMSE | |||

60 | −0.0568 | 0.6625 | 0.0060 | 0.1384 | −0.0048 | 0.3726 | 0.2956 | 1.0945 | |||

70 | −0.0536 | 0.6290 | −0.0015 | 0.1276 | −0.0205 | 0.3309 | 0.2515 | 0.8024 | |||

80 | −0.0301 | 0.6118 | 0.0031 | 0.1189 | −0.0091 | 0.3135 | 0.2018 | 0.6585 | |||

500 | 0.0209 | 0.2970 | −0.0006 | 0.0448 | 0.0023 | 0.1156 | 0.0197 | 0.2412 | |||

${\mathit{\beta}}_{\mathbf{0}}=\mathbf{1}.\mathbf{5}$ | ${\mathit{\beta}}_{\mathbf{1}}=\mathbf{2}.\mathbf{5}$ | $\mathbf{\alpha}=\mathbf{0}.\mathbf{75}$ | ${\mathbf{\gamma}}_{\mathbf{1}}=\mathbf{0}.\mathbf{75}$ | ||||||||

n | Bias | RMSE | Bias | RMSE | Bias | RMSE | Bias | RMSE | |||

60 | −0.0508 | 0.6767 | 0.0072 | 0.1211 | 0.1848 | 1.9833 | 0.3443 | 1.2177 | |||

70 | −0.0481 | 0.67 | 0.0045 | 0.1174 | 0.184 | 2.0431 | 0.3243 | 1.1201 | |||

80 | −0.0368 | 0.6483 | 0.003 | 0.1047 | 0.1284 | 1.0011 | 0.2918 | 1.0173 | |||

500 | −0.0287 | 0.3889 | −0.0015 | 0.0387 | 0.0201 | 0.1639 | 0.1173 | 0.4708 | |||

n | Bias | RMSE | Bias | RMSE | Bias | RMSE | Bias | RMSE | |||

60 | −0.0973 | 0.5842 | 0.0500 | 0.1251 | 0.0519 | 0.4253 | 0.8683 | 3.2124 | |||

70 | −0.0815 | 0.5433 | 0.0102 | 0.1165 | 0.0360 | 0.4137 | 0.7459 | 3.0357 | |||

80 | −0.0716 | 0.4592 | −0.0068 | 0.1073 | 0.0081 | 0.3963 | 0.5457 | 2.2340 | |||

500 | −0.0151 | 0.1480 | 0.0013 | 0.0408 | 0.0001 | 0.1253 | 0.0564 | 0.2874 | |||

${\mathit{\beta}}_{\mathbf{0}}=\mathbf{1}.\mathbf{5}$ | ${\mathit{\beta}}_{\mathbf{1}}=\mathbf{2}.\mathbf{5}$ | $\mathbf{\alpha}=\mathbf{1}.\mathbf{5}$ | ${\mathbf{\gamma}}_{\mathbf{1}}=\mathbf{0}.\mathbf{5}$ | ||||||||

n | Bias | RMSE | Bias | RMSE | Bias | RMSE | Bias | RMSE | |||

60 | −0.0179 | 0.6981 | −0.0092 | 0.1203 | 0.0774 | 0.6987 | 1.0013 | 5.0588 | |||

70 | 0.0173 | 0.6716 | 0.0085 | 0.1153 | 0.0441 | 0.4326 | 0.6727 | 3.2734 | |||

80 | 0.0154 | 0.6311 | 0.0065 | 0.1057 | 0.0427 | 0.3569 | 0.6311 | 3.1535 | |||

500 | −0.0404 | 0.2858 | 0.0005 | 0.0395 | 0.0190 | 0.1496 | 0.1793 | 0.6559 | |||

${\mathit{\beta}}_{\mathbf{0}}=\mathbf{1}.\mathbf{5}$ | ${\mathit{\beta}}_{\mathbf{1}}=\mathbf{2}.\mathbf{5}$ | $\mathbf{\alpha}=\mathbf{1}.\mathbf{5}$ | ${\mathbf{\gamma}}_{\mathbf{1}}=\mathbf{0}.\mathbf{75}$ | ||||||||

n | Bias | RMSE | Bias | RMSE | Bias | RMSE | Bias | RMSE | |||

60 | 0.0789 | 0.6885 | 0.0053 | 0.1218 | 0.1973 | 1.4280 | 0.4340 | 3.1930 | |||

70 | 0.0784 | 0.6413 | 0.0025 | 0.1134 | 0.1630 | 1.0152 | 0.4212 | 2.4414 | |||

80 | 0.0623 | 0.6046 | 0.0012 | 0.1059 | 0.1366 | 0.6108 | 0.3358 | 1.8002 | |||

500 | 0.0316 | 0.3572 | −0.0005 | 0.0380 | 0.0723 | 0.3008 | 0.0890 | 0.6704 |

Density | AIC | ${\mathit{\beta}}_{10}$ | ${\mathit{\beta}}_{11}$ | ${\mathit{\beta}}_{12}$ | ${\mathit{\gamma}}_{1}/\mathit{\alpha}$ | ${\mathit{\beta}}_{20}$ | ${\mathit{\beta}}_{23}$ | |
---|---|---|---|---|---|---|---|---|

Logit/LN | 986.19 | 0.652 | 0.808 | 0.422 | −0.401 | 0.264 | ||

(0.220) | (0.304) | (0.288) | (0.112) | (0.155) | ||||

Logit/LSN${}_{c}$ | 944.15 | 0.503 | 0.648 | 0.974 | 0.899 | −0.284 | 0.108 | |

(0.203) | (0.277) | (0.303) | (0.537) | (0.059) | (0.079) | |||

Logit/LPN | 976.11 | 0.640 | 0.765 | 0.357 | 9.660 | −3.030 | 0.221 | |

(0.209) | (0.280) | (0.269) | (4.306) | (0.607) | (0.138) |

AIC | ${\mathit{\beta}}_{10}$ | ${\mathit{\beta}}_{11}$ | ${\mathit{\beta}}_{12}$ | ${\mathit{\beta}}_{20}$ | ${\mathit{\beta}}_{21}$ | ${\mathit{\beta}}_{22}$ | ${\mathit{\beta}}_{23}$ | ${\mathit{\gamma}}_{1}$ | $\mathit{\alpha}$ |
---|---|---|---|---|---|---|---|---|---|

985.21 | 1.106 | – | – | −1.786 | −0.166 | 0.115 | 0.176 | 0.291 | 4.281 |

(0.134) | – | – | (0.633) | (0.136) | (0.138) | (0.137) | (0.432) | (1.459) | |

938.32 | 0.349 | 0.975 | 0.689 | 0.045 | – | – | 0.156 | 0.923 | 0.613 |

(0.199) | (0.274) | (0.295) | (0.101) | – | – | (0.084) | (0.407) | (0.090) |

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**MDPI and ACS Style**

Martínez-Flórez, G.; Vergara-Cardozo, S.; Tovar-Falón, R.
A Class of Exponentiated Regression Model for Non Negative Censored Data with an Application to Antibody Response to Vaccine. *Symmetry* **2021**, *13*, 1419.
https://doi.org/10.3390/sym13081419

**AMA Style**

Martínez-Flórez G, Vergara-Cardozo S, Tovar-Falón R.
A Class of Exponentiated Regression Model for Non Negative Censored Data with an Application to Antibody Response to Vaccine. *Symmetry*. 2021; 13(8):1419.
https://doi.org/10.3390/sym13081419

**Chicago/Turabian Style**

Martínez-Flórez, Guillermo, Sandra Vergara-Cardozo, and Roger Tovar-Falón.
2021. "A Class of Exponentiated Regression Model for Non Negative Censored Data with an Application to Antibody Response to Vaccine" *Symmetry* 13, no. 8: 1419.
https://doi.org/10.3390/sym13081419