# Binomial Regression Models with a Flexible Generalized Logit Link Function

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Binomial Regression Model

**X**denote $n\times \left(k+1\right)$ design matrix with rows ${\mathit{x}}_{\mathit{i}}^{\prime}$

**,**where ${\mathit{x}}_{\mathit{i}}=\left(1,{x}_{i1},{x}_{i2},\dots ,{x}_{ik}\right)\prime $ is a $\left(k+1\right)\times 1$ vector of the predictor variable, where value 1 corresponds to an intercept. Furthermore, we consider $\mathit{\beta}=\left({\beta}_{0},{\beta}_{1},\dots ,{\beta}_{k}\right)\prime $ as a $\left(k+1\right)\times 1$ corresponding the intercept and regression coefficient.

## 3. Flexible Generalized Logit Link

#### 3.1. Exponentiated-Exponential Logistic Distribution

**X**is an EEL distribution, with parameter $\theta =\left(\mu ,s,\alpha ,\lambda \right)$, where $\mu \in \Re $ and $s>0$ are the locations and the scale parameter of logistic distribution, $\alpha >0$ and $\lambda >0$ are the shapes and the scale parameter of exponentiated-exponential distribution. The probability density function (p.d.f.) of

**X**is given by:

**X**in (4) is reduced to the p.d.f. of the logistic distribution. In the special case of $\lambda =1$, the p.d.f. in (4) is the Type I GLD. Moreover, when $\alpha =1$, the p.d.f. reduces to the Type II GLD. The illustrations of some typical standard EEL p.d.f.s for different values of $\alpha $ and $\lambda $ in $\mu =0$ and $s=1$ are provided in Figure 1. Figure 1 shows that the density function of the EEL distribution can take different shapes. It has negative skewness for $\alpha <\lambda $, positive skewness for $\alpha >\lambda $, and symmetric for $\alpha =\lambda $. The greater skewness can be achieved when the difference between $\alpha $ and $\lambda $ is higher. Furthermore, the standard EEL p.d.f. has lighter tails compared to standard logistic p.d.f. when $\alpha >1$ and $\lambda >1$, otherwise if $\alpha <1$ and $\lambda <1$ then heavier tails are achieved.

#### 3.2. Binomial Regression Model with Glogit Link Function

model { |

for (i in 1:n) { |

y[i]~dbin(p[i], 1) |

omega[i] < -beta0 + beta1*x1[i] + beta2*x2[i] |

p[i] < -pow((1-pow((1+exp(omega[i])),-lambda)),alpha) |

} |

#Prior |

beta0~dnorm(0, 0.001) |

beta1~dnorm(0, 0.001) |

beta2~dnorm(0, 0.001) |

alpha~dlnorm(0, 1) |

lambda~dlnorm(0, 1) |

} |

## 4. Simulation

- Step 1: Specify a binomial regression model with one predictor variable and set a vector of the regression coefficients $\mathit{\beta}=\left({\beta}_{0},{\beta}_{1}\right)=\left(0.1,1\right)$;
- Step 2: Generate ${x}_{i1}~\mathrm{Normal}\left(0,1\right)$, then we create ${x}_{i}={\left(1,{x}_{i1}\right)}^{\prime}$;
- Step 3: Compute ${p}_{i}$ from a given link, so that ${p}_{i}=F({x}_{i}^{\prime}\mathit{\beta}$);
- Step 4: Generate ${n}_{i}~\mathrm{Poisson}\left(\delta \right)$ with $\delta $ = 100, and ${Y}_{i}~\mathrm{Binomial}\left({n}_{i},{p}_{i}\right)$, which is the binomial distribution with ${p}_{i}$ as the probability of success and ${n}_{i}$ is the sample size for the repeated Bernoulli trials.

- Scenario 1: ${F}^{-1}\left({p}_{i}\right)=\mathrm{glogit}({p}_{i}|\alpha =1,\lambda =1)$, same with ${F}^{-1}\left({p}_{i}\right)=\mathrm{logit}\left({p}_{i}\right)$;
- Scenario 2: ${F}^{-1}\left({p}_{i}\right)=\mathrm{glogit}({p}_{i}|\alpha =4,\lambda =4)$, represent the lighter tails compared to the logit;
- Scenario 3: ${F}^{-1}\left({p}_{i}\right)=\mathrm{glogit}({p}_{i}|\alpha =0.5,\lambda =0.5)$, represent the heavier tails compared to the logit;
- Scenario 4: ${F}^{-1}\left({p}_{i}\right)=\mathrm{glogit}({p}_{i}|\alpha =3,\lambda =0.3)$, represent the positive skewness;
- Scenario 5: ${F}^{-1}\left({p}_{i}\right)=\mathrm{glogit}({p}_{i}|\alpha =0.3,\lambda =3)$, represent the negative skewness;
- Scenario 6: ${F}^{-1}\left({p}_{i}\right)=\mathrm{cloglog}\left({p}_{i}\right)$;
- Scenario 7: ${F}^{-1}\left({p}_{i}\right)=\mathrm{probit}\left({p}_{i}\right)$.

## 5. Application

#### 5.1. Beetle Mortality Dataset

^{th}dose. A comparison of the five models was considered by using different link functions ${F}^{-1}$.

#### 5.2. The Potency of Three Different Poisons Dataset

^{th}log(dose), rotenone dummy variable, and deguelin dummy variable. The mixture variable is considered as the reference poison (rotenone = 0 and deguelin = 0). A comparison to the models was considered by applying different ${F}^{-1}$ (link functions). They are the SP, AEP, SW, and glogit models.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Probability density function (p.d.f.) of the exponentiated-exponential logistic (EEL) distribution when $\mu =0$ and $s=1$ at various choices of $\alpha $ and $\lambda $. The black solid line shows the p.d.f. of EEL when $\alpha =1$ and $\lambda =1$ that are equal to standard logistic distribution. The red dashed line shows the p.d.f. of EEL when $\alpha =4$ and $\lambda =4$ that has lighter tails compared to the standard logistic. The blue dotted line shows the p.d.f. of EEL when $\alpha =0.5$ and $\lambda =0.5$ that has heavier tails compared to the standard logistic. The yellow dotted dashed line shows the p.d.f. of EEL when $\alpha =3$ and $\lambda =0.3$ that has positive skewness. The green long dashed line shows the p.d.f. of EEL when $\alpha =0.3$ and $\lambda =3$ that has negative skewness.

**Figure 2.**Cumulative distribution functions (c.d.f.) of the EEL distribution when $\mu =0$ and $s=1$ at various choices of $\alpha $ and $\lambda $, which correspond to Figure 1.

**Table 1.**Model comparison using average of deviance information criterion (DIC) and absolute errors (AE) corresponding to the link functions at several scenarios.

Scenarios | Glogit Link | Logit Link | Cloglog Link | Probit Link | ||||
---|---|---|---|---|---|---|---|---|

DIC | AE | DIC | AE | DIC | AE | DIC | AE | |

1 | 815.21 ** | 0.1903 ** | 813.74 * | 0.1589 * | 1116.61 | 1.7661 | 824.27 | 0.3413 |

2 | 697.04 * | 0.1409 * | 746.97 | 0.5093 | 1903.64 | 2.5617 | 712.61 ** | 0.2909 ** |

3 | 826.91 * | 0.1980 * | 827.74 ** | 0.2021 ** | 916.40 | 0.9986 | 833.12 | 0.2904 |

4 | 462.63 * | 0.0545 * | 539.94 | 0.2971 | 557.11 | 0.3286 | 477.06 ** | 0.1396 ** |

5 | 597.04 * | 0.0895 * | 822.00 | 0.8047 | 599.08 ** | 0.1038 ** | 655.23 | 0.4173 |

6 | 745.41 ** | 0.1934 ** | 1613.28 | 2.4169 | 742.98 * | 0.1369 * | 1378.13 | 2.0713 |

7 | 776.19 ** | 0.2139 ** | 825.32 | 0.6017 | 1611.83 | 2.4673 | 771.02 * | 0.1249 * |

log(dose) | n | ndead | p |
---|---|---|---|

49.1 | 59 | 6 | 0.102 |

53.0 | 60 | 13 | 0.217 |

56.9 | 62 | 18 | 0.290 |

60.8 | 56 | 28 | 0.500 |

64.8 | 63 | 52 | 0.825 |

68.7 | 59 | 53 | 0.898 |

72.6 | 62 | 61 | 0.984 |

76.5 | 60 | 60 | 1 |

**Table 3.**Summary of posterior estimations and model comparison for the several models fitted to the beetle mortality dataset.

Parameters | Model [Posterior Mean (Standard Deviation)] | |||||
---|---|---|---|---|---|---|

Probit | T(8) | SP | SGT | AEP | Glogit | |

${\beta}_{0}$ | −33.88 (1.91) | −35.17 (2.03) | −34.45 (2.06) | −8.3 (1.73) | −35.16 (0.36) | −16.56 (5.08) |

${\beta}_{1}$ | 19.14 (1.04) | 19.86 (1.14) | 19.43 (1.14) | 4.69 (0.98) | 19.69 (0.21) | 0.25 (0.08) |

- | $\delta =0.08\left(0.45\right)$ | ${v}_{1}=9.4\left(14.8\right)$ | ${\theta}_{1}=1.04\left(0.15\right)$ | $\alpha =0.86\left(0.59\right)$ | ||

- | $\delta =0.22\left(0.1\right)$ | ${\theta}_{1}=0.47\left(0.07\right)$ | $\lambda =2.61\left(2.37\right)$ | |||

DIC | 39.26 | 41.22 | 40.22 | 36.72 | 36.43 ** | 35.09 * |

AE | 0.35 | 0.37 | 0.35 | 0.25 | 0.15 * | 0.18 ** |

**Table 4.**Relative potency of three different poisons (rotenone, deguelin, and a mixture of the two) to kill insects.

Rotenone | Deguelin | Mixture | ||||||
---|---|---|---|---|---|---|---|---|

log(dose) | dead | n | log(dose) | dead | n | log(dose) | dead | n |

1.01 | 44 | 50 | 1.7 | 48 | 48 | 1.4 | 48 | 50 |

0.89 | 42 | 49 | 1.61 | 47 | 50 | 1.31 | 43 | 46 |

0.71 | 24 | 46 | 1.48 | 47 | 49 | 1.18 | 38 | 48 |

0.58 | 16 | 48 | 1.31 | 34 | 48 | 1 | 27 | 46 |

0.41 | 6 | 50 | 1 | 18 | 48 | 0.71 | 22 | 46 |

0.71 | 16 | 49 | 0.4 | 7 | 47 |

**Table 5.**Summary of posterior estimations and model comparison for the several models fitted to the potency of three different poisons dataset.

Parameters | Model [Posterior Mean (Standard Deviation)] | |||
---|---|---|---|---|

SP | AEP | SW | glogit | |

${\beta}_{0}$ | −2.19 (0.40) | −4.98 (1.51) | 0.27 (0.20) | −10.78 (0.81) |

${\beta}_{1}$ | 2.65 (0.22) | 5.27 (1.48) | 0.78 (0.23) | 9.08 (0.74) |

${\beta}_{2}$ | 0.38 (0.13) | 1.23 (0.46) | 0.12 (0.04) | 1.89 (0.37) |

${\beta}_{3}$ | −0.50 (0.13) | −1.01 (0.38) | −0.15 (0.06) | −1.74 (0.40) |

- | $\delta =-0.04\left(0.55\right)$ | ${\theta}_{1}=0.45\left(0.12\right)$ | $\gamma =4.03\left(1.20\right)$ | $\alpha =0.26\left(0.03\right)$ |

- | ${\theta}_{1}=0.91\left(0.19\right)$ | $\lambda =1.17\left(0.72\right)$ | ||

DIC | 751.92 | 749.01 ** | 751.38 | 747.75 * |

MAE | 0.0662 | 0.0564 ** | 0.0644 | 0.0497 * |

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**MDPI and ACS Style**

Prasetyo, R.B.; Kuswanto, H.; Iriawan, N.; Ulama, B.S.S.
Binomial Regression Models with a Flexible Generalized Logit Link Function. *Symmetry* **2020**, *12*, 221.
https://doi.org/10.3390/sym12020221

**AMA Style**

Prasetyo RB, Kuswanto H, Iriawan N, Ulama BSS.
Binomial Regression Models with a Flexible Generalized Logit Link Function. *Symmetry*. 2020; 12(2):221.
https://doi.org/10.3390/sym12020221

**Chicago/Turabian Style**

Prasetyo, Rindang Bangun, Heri Kuswanto, Nur Iriawan, and Brodjol Sutijo Suprih Ulama.
2020. "Binomial Regression Models with a Flexible Generalized Logit Link Function" *Symmetry* 12, no. 2: 221.
https://doi.org/10.3390/sym12020221