# Geographically Weighted Three-Parameters Bivariate Gamma Regression and Its Application

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Gamma Distribution

**Hypothesis**

**1**

**(H1).**

**Hypothesis**

**2**

**(H2).**

**Hypothesis**

**3**

**(H3).**

**Hypothesis**

**4**

**(H4).**

- Fixed function Gaussian kernel$${w}_{ii*}=\mathrm{exp}\left[-\frac{1}{2}{\left(\frac{{d}_{ii*}}{g}\right)}^{2}\right],$$${w}_{ii*}$ = the weighting of observations on the i-th location and the location of all ${i}^{*},$g = bandwidth,${d}_{ii*}$ = Euclidian distance.
- Adaptive kernel bisquare$${w}_{ii*}=\{\begin{array}{ll}{\left[-\frac{1}{2}{\left(\frac{{d}_{ii*}}{{g}_{i}}\right)}^{2}\right]}^{2}& \mathrm{when}{d}_{ii*}\le {g}_{i},\\ 0& \mathrm{when}{d}_{ii*}{g}_{i}.\end{array}$$
- Fixed bisquare kernel$${w}_{{}_{ii*}}=\{\begin{array}{ll}{\left[1-{\left(\frac{{d}_{ii*}}{g}\right)}^{2}\right]}^{2}& \mathrm{when}{d}_{ii*}\le g,\\ 0& \mathrm{when}{d}_{ii*}g.\end{array}$$
- Adaptive Gaussian kernel$${w}_{ii*}=\mathrm{exp}\left(-\frac{1}{2}{\left(\frac{{d}_{ii*}}{{g}_{i}}\right)}^{2}\right).$$

## 3. Data and Method

## 4. Discussion and Analysis

#### 4.1. Parameter Estimation of GWBGR with Three-Parameters

#### 4.2. The Hypothesis Testing for GWBGR Model

#### 4.2.1. The Similarity Model Test

**Hypothesis**

**5**

**(H5).**

**Hypothesis**

**6**

**(H6).**

_{1}and df

_{2}global model is a local model-free interval and with a certain significance level (α). The deviance for GWBGR models can be obtained by looking $\mathrm{ln}L({\widehat{\omega}}_{\mathrm{GWBGR}})$ and $\mathrm{ln}L({\widehat{\mathsf{\Omega}}}_{\mathrm{GWBGR}}).$ $\mathrm{ln}L({\widehat{\omega}}_{\mathrm{GWBGR}})$ can be obtained by maximizing the likelihood function under the null hypothesis. The parameters were set under the null hypothesis ${\omega}_{\mathrm{GWBGR}}=\{\theta ,{\gamma}_{1},{\gamma}_{2},$${\beta}_{10}({u}_{i},{v}_{i}),{\beta}_{20}({u}_{i},{v}_{i})$$,i=1,2,\cdots ,n\}$. The log-likelihood function under the null hypothesis can be defined as follows.

#### 4.2.2. The Simultaneous Test

^{2}. The hypothesis used for simultaneous testing is as follows.

**Hypothesis**

**7**

**(H7).**

**Hypothesis**

**8**

**(H8).**

^{2}> ${\chi}_{(\alpha ;v)}^{2}$ with significance level α and v is the difference $n(\mathsf{\Omega})-n(\omega ).$

#### 4.2.3. The Partial Test

**Hypothesis**

**9**

**(H9).**

**Hypothesis**

**10**

**(H10).**

#### 4.3. Descriptive Statistics and Testing Assumptions

_{table}, |t| = 3.2213 > t table (0.025; 32) = 2.0369, then the conclusion is to reject the null hypothesis means that there is a relationship between the response variables Y1 and Y2.

#### 4.4. Modeling RIM and RMM with BGR Three-Parameters

^{2}. Simultaneous test statistic values obtained 11,165.08 because the value is greater than ${\chi}_{(0.05,12)}^{2}=$ 3.94, then the null hypothesis is rejected it means at least one independent variable that significantly influences the response variables. Then followed by a partial test pass. The value of the test statistic Z is shown in Table 3 and Table 4. With a significance level of 5% of the obtained, all predictor variables significantly influence the RIM and RMM for a value $\left|Z\right|$ greater than ${Z}_{0.025}=1.96$.

#### 4.5. Modeling the RIM and RMM with GWBGR Three-Parameters Models

**Hypothesis**

**11**

**(H11).**

**Hypothesis**

**12**

**(H12).**

^{2}test statistic obtained by 213,372.4116 where the value is greater than ${\chi}_{(0.05,408)}^{2}=234.806$ then the decision is to reject the null hypothesis, which is to say, simultaneously predictor variables affect the response variables with a significance level of 5%. It is next necessary partial testing to determine any predictor variables that affect the response variables.

_{table}with a significance level of 5% so that the GWBGR model for three-parameters as follows.

#### 4.6. Selection of Best Model

_{C}value is a test statistic that can be used to determine which the best model among global models and local models in the modeling of RIM and RMM in the province of North Sulawesi, Gorontalo and Central Sulawesi.

_{C}value. Table 7 shows that the model GWBGR with a bisquare fixed kernel function is the best model to model RIM and RMM in the province of North Sulawesi, Gorontalo and Central Sulawesi in 2016.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Rate of infant mortality (RIM) histogram with gamma distribution of two parameters and three-parameters.

**Figure 2.**Rate of maternal mortality (RMM) histogram with gamma distribution of two parameters and three-parameters.

Variables | Mean | Std. Dev | Min | Med | Max |
---|---|---|---|---|---|

Rate of infant mortality (Y1) | 10.463 | 5.558 | 1.139 | 10.682 | 28.571 |

Rate of maternal mortality (Y2) | 209.6 | 125.8 | 66.3 | 182.8 | 655.2 |

The percentage of poor people (X1) | 15.95 | 12.96 | 5.24 | 14.09 | 78.36 |

The percentage of obstetric complications treated (X2) | 76.55 | 28.92 | 7.24 | 79.28 | 154.85 |

The percentage of pregnant mothers who received Fe3 (X3) | 71.23 | 18.45 | 21.78 | 71.57 | 106.83 |

The percentage of first-time pregnant mothers under seventeen years of age (X4) | 11.851 | 4.239 | 4.770 | 12.645 | 20.850 |

The percentage of use of health facilities (X5) | 65.52 | 25.62 | 0.00 | 71.93 | 98.15 |

The percentage of households with clean and healthy lifestyle (X6) | 57.06 | 17.29 | 27.12 | 57.36 | 93.90 |

X1 | X2 | X3 | X4 | X5 | X6 | X7 | |
---|---|---|---|---|---|---|---|

VIF | 1.26 | 1.61 | 1.38 | 1.66 | 1.77 | 1.29 | 1.22 |

Parameters | Rate of Infant Mortality (Y1) | |||
---|---|---|---|---|

Estimate | Std. Error | Z | p-Value | |

β10 | −3.3738 | −3.74 × 10^{−7} | 9.02 × 10^{6} | 0.000 |

β11 | 0.0238 | −6.94 × 10^{−5} | −3.43 × 10^{2} | 0.000 |

β12 | 0.0052 | −1.43 × 10^{−5} | −3.61 × 10^{2} | 0.000 |

β13 | 0.0251 | −4.01 × 10^{−5} | 5.06 × 10^{3} | 0.000 |

β14 | −0.0493 | −9.73 × 10^{−6} | 3.31 × 10^{2} | 0.000 |

β15 | −0.0122 | −3.69 × 10^{−5} | −2.34 × 10^{2} | 0.000 |

β16 | 0.0143 | −6.12 × 10^{−5} | 1.06 × 10^{7} | 0.000 |

Parameters | Rate of Maternal Mortality (Y2) | |||
---|---|---|---|---|

Estimate | Std. Error | Z | p-Value | |

β20 | −5.9847 | −5.65 × 10^{−7} | 1.06 × 10^{7} | 0.000 |

β21 | −0.0175 | −6.37 × 10^{−6} | 2750.48 | 0.000 |

β22 | −0.0055 | −4.90 × 10^{−5} | 112.034 | 0.000 |

β23 | 0.0089 | −4.23 × 10^{−5} | −212.293 | 0.000 |

β24 | 0.0302 | −5.70 × 10^{−6} | −5291.62 | 0.000 |

β25 | 0.0062 | −4.36 × 10^{−5} | −143.424 | 0.000 |

β26 | −0.0007 | −3.25 × 10^{−5} | 20.8328 | 0.000 |

**Table 5.**Values of RIM and geographically weighted bivariate gamma regression (GWBGR) parameter estimation models in the Bolaang Mongondow Regency.

Parameters | Rate of Infant Mortality (Y1) | |||
---|---|---|---|---|

Estimate | Std. Error | Z | p-Value | |

β10 | 3.3621 | 1.02 × 10^{−7} | 3.28 × 10^{7} | 0.000 |

β11 | 0.0022 | 2.81 × 10^{−6} | 774.9121 | 0.000 |

β12 | 0.0046 | 2.18 × 10^{−5} | 211.2963 | 0.000 |

β13 | −0.0192 | 9.85 × 10^{−6} | −1.950.324 | 0.000 |

β14 | −0.1095 | 1.43 × 10^{−6} | −7.65 × 10^{4} | 0.000 |

β15 | −0.0094 | 2.3 × 10^{−5} | −76.558.74 | 0.000 |

β16 | −0.0004 | 1.64 × 10^{−5} | −25.4718 | 0.000 |

Parameters | Rate of Maternal Mortality (Y2) | |||
---|---|---|---|---|

Estimate | Std. Error | Z | p−Value | |

β20 | 5.9086 | 2.11 × 10^{−8} | 2.79 × 10^{8} | 0.000 |

β21 | 0.0190 | 4.07 × 10^{−7} | 4.68 × 10^{4} | 0.000 |

β22 | 0.0076 | 1.26 × 10^{−6} | 6076.77 | 0.000 |

β23 | −0.0110 | 1.45 × 10^{−6} | −7579.55 | 0.000 |

β24 | −0.0264 | 2.94 × 10^{−7} | −8.98 × 10^{4} | 0.000 |

β25 | −0.0051 | 1.57 × 10^{−6} | −3251.85 | 0.000 |

β26 | −0.0010 | 1.33 × 10^{−6} | −746.945 | 0.000 |

Model | AIC_{C} |
---|---|

BGR | 991.9833 |

Fixed GWBGR bisquare | 973.3614 |

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**MDPI and ACS Style**

Purhadi; Rahayu, A.; Wenur, G.H.
Geographically Weighted Three-Parameters Bivariate Gamma Regression and Its Application. *Symmetry* **2021**, *13*, 197.
https://doi.org/10.3390/sym13020197

**AMA Style**

Purhadi, Rahayu A, Wenur GH.
Geographically Weighted Three-Parameters Bivariate Gamma Regression and Its Application. *Symmetry*. 2021; 13(2):197.
https://doi.org/10.3390/sym13020197

**Chicago/Turabian Style**

Purhadi, Anita Rahayu, and Gabriella Hillary Wenur.
2021. "Geographically Weighted Three-Parameters Bivariate Gamma Regression and Its Application" *Symmetry* 13, no. 2: 197.
https://doi.org/10.3390/sym13020197