# Parameter Estimation and Hypothesis Testing of Multivariate Poisson Inverse Gaussian Regression

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Multivariate Poisson Inverse Gaussian Distribution (MPIGD)

_{1}, Y

_{2}, …, Y

_{m}as the response variables with the assumption Y

_{j}~ Poisson (μ

_{j}), j = 1, 2, …, m, with its mean and variance are the same or they can be written in the following regression equation:

_{j},

_{j}~ Mixed Poisson (μ

_{j}v

_{j}) [8,13]. The mixed Poisson distribution depends on the specific distribution of random variable V. In this study, variable V is an inverse Gaussian distributed. Hence, Y

_{1}, Y

_{2}, …, Y

_{m}are Poisson inverse Gaussian distribution (PIGD), with the probability mass function (pmf) being given by [3],

#### 2.2. Multivariate Poisson Inverse Gaussian Regression (MPIGR)

_{i1}, Y

_{i2}, …, Y

_{im}) ~ MPIG (vμ

_{ij}), where $i=1,2,\dots ,n\text{}$ $\mathrm{and}\text{}j=1,2,\dots .m$ then the MPIGR model can be stated as follows:

_{i}is an exposure variable which is defined as the weight of the i-th unit observation, ${x}_{i}^{T}=\left[\begin{array}{cccc}\begin{array}{cc}1& {x}_{1i}\end{array}& {x}_{2i}& \begin{array}{ccc}\cdots & {x}_{ki}& \cdots \end{array}& {x}_{pi}\end{array}\right]$ is the vector of predictor variables with (p + 1) dimension on the i-th observation (i = 1, 2, …, n), ${\mathsf{\beta}}_{j}^{T}=\left[\begin{array}{cccc}{\beta}_{j0}& {\beta}_{j1}& {\beta}_{j2}& \begin{array}{cc}\begin{array}{ccc}\dots & {\beta}_{jk}& \dots \end{array}& {\beta}_{jp}\end{array}\end{array}\right]$ is a (p + 1) × 1 vector of regression coefficient associated with the j-th response variable (j = 1, 2, …, m).

## 3. Results

#### 3.1. Parameter Estimation of MPIGR Model

**β**

_{j}and τ. The first step of the MLE method is by taking n random sample, $\left({Y}_{i1},{Y}_{i2},\dots ,{Y}_{ij},\dots ,{Y}_{im},{X}_{1i},{X}_{2i},\dots ,{X}_{ki},\dots {X}_{pi}\right)$ with j = 1, 2, …, m, k = 1, 2, …, p and i = 1, 2, …, n. The joint probability density function of ${Y}_{i1},{Y}_{i2},\dots ,{Y}_{im}$ is:

- ▪
- Step 1. Determine the initial value for parameter ${\widehat{\mathsf{\theta}}}^{\left(0\right)}={\left[\begin{array}{cccc}{\widehat{\mathsf{\beta}}}_{1}^{T\left(0\right)}& {\widehat{\mathsf{\beta}}}_{2}^{T\left(0\right)}& \begin{array}{cc}\begin{array}{ccc}\cdots & {\widehat{\mathsf{\beta}}}_{j}^{T\left(0\right)}& \cdots \end{array}& {\widehat{\mathsf{\beta}}}_{m}^{T\left(0\right)}\end{array}& {\widehat{\tau}}^{\left(0\right)}\end{array}\right]}^{T}$. The initial value of parameter ${\widehat{\mathsf{\theta}}}^{\left(0\right)}$ is obtained while using the separate univariate Poisson regression. The initial value for overdispersion parameter τ used the average of the observed overdispersion based on the variance of PIGD [9].
- ▪
- Step 2. Determine the gradient vector $g\left({\widehat{\mathsf{\theta}}}^{\left(r\right)}\right)$, which is the elements consist of the first derivative of the log-likelihood function, ${g}^{T}\left({\widehat{\mathsf{\theta}}}^{\left(r\right)}\right)={\left[\begin{array}{ccc}\frac{\partial l\left(\widehat{\mathsf{\theta}}\right)}{\partial {\widehat{\mathsf{\beta}}}_{1}}& \begin{array}{cc}\frac{\partial l\left(\widehat{\mathsf{\theta}}\right)}{\partial {\widehat{\mathsf{\beta}}}_{2}}& \begin{array}{cc}\begin{array}{ccc}\cdots & \frac{\partial l\left(\widehat{\mathsf{\theta}}\right)}{\partial {\widehat{\mathsf{\beta}}}_{j}}& \cdots \end{array}& \frac{\partial l\left(\widehat{\mathsf{\theta}}\right)}{\partial {\widehat{\mathsf{\beta}}}_{m}}\end{array}\end{array}& \frac{\partial l\left(\widehat{\mathsf{\theta}}\right)}{\partial \widehat{\tau}}\end{array}\right]}_{\widehat{\mathsf{\theta}}={\widehat{\mathsf{\theta}}}_{(t)}}$.
- ▪
- Step 3. Determine the Hessian matrix $H\left({\widehat{\mathsf{\theta}}}^{\left(r\right)}\right)$ where the elements consist of the second derivative of the log-likelihood function, as follows$$H\left({\widehat{\mathsf{\theta}}}^{\left(r\right)}\right)=\left[\begin{array}{cccccc}\frac{{\partial}^{2}l\left(\widehat{\mathsf{\theta}}\right)}{\partial {\widehat{\mathsf{\beta}}}_{1}\partial {\widehat{\mathsf{\beta}}}_{1}^{T}}& \cdots & \frac{{\partial}^{2}l\left(\widehat{\mathsf{\theta}}\right)}{\partial {\widehat{\mathsf{\beta}}}_{1}\partial {\widehat{\mathsf{\beta}}}_{j}^{T}}& \cdots & \frac{{\partial}^{2}l\left(\widehat{\mathsf{\theta}}\right)}{\partial {\widehat{\mathsf{\beta}}}_{1}\partial {\widehat{\mathsf{\beta}}}_{m}^{T}}& \frac{{\partial}^{2}l\left(\widehat{\mathsf{\theta}}\right)}{\partial {\widehat{\mathsf{\beta}}}_{1}\partial \widehat{\tau}}\\ & \ddots & \vdots & \vdots & \vdots & \vdots \\ & & \frac{{\partial}^{2}l\left(\widehat{\mathsf{\theta}}\right)}{\partial {\widehat{\mathsf{\beta}}}_{j}\partial {\widehat{\mathsf{\beta}}}_{j}^{T}}& \cdots & \frac{{\partial}^{2}l\left(\widehat{\mathsf{\theta}}\right)}{\partial {\widehat{\mathsf{\beta}}}_{j}\partial {\widehat{\mathsf{\beta}}}_{m}^{T}}& \frac{{\partial}^{2}l\left(\widehat{\mathsf{\theta}}\right)}{\partial {\widehat{\mathsf{\beta}}}_{j}\partial \widehat{\tau}}\\ & & & \ddots & & \\ & & & & \frac{{\partial}^{2}l\left(\widehat{\mathsf{\theta}}\right)}{\partial {\widehat{\mathsf{\beta}}}_{m}\partial {\widehat{\mathsf{\beta}}}_{m}^{T}}& \frac{{\partial}^{2}l\left(\widehat{\mathsf{\theta}}\right)}{\partial {\widehat{\mathsf{\beta}}}_{m}\partial \widehat{\tau}}\\ symmetric& & & & & \frac{{\partial}^{2}l\left(\widehat{\mathsf{\theta}}\right)}{\partial {\widehat{\tau}}^{2}}\end{array}\right]$$
- ▪
- Step 4. Start the Newton–Raphson iteration using the following formula,$${\widehat{\mathsf{\theta}}}^{(r+1)}={\widehat{\mathsf{\theta}}}^{(r)}-{H}^{-1}\left({\widehat{\mathsf{\theta}}}^{(r)}\right)g\left({\widehat{\mathsf{\theta}}}^{(r)}\right)$$
- ▪
- Step 5. The iteration will stop if $\Vert {\widehat{\mathsf{\theta}}}^{\left(r+1\right)}-{\widehat{\mathsf{\theta}}}^{\left(r\right)}\Vert \le \epsilon $, with ε is a very small value and it will produce the estimator value for each parameter.

#### 3.2. Factorial Simplification in the Third Modification of BESSEL Function

#### 3.3. Hypothesis Testing of MPIGR Model

_{j}

_{1}= β

_{j}

_{2}= … = β

_{j}

_{k}= … = β

_{jp}= 0 and τ = 0 and the alternative hypothesis is at least one β

_{j}

_{k}≠ 0 and τ ≠ 0, where j = 1, 2, …, m; and k = 1, 2,…, p.

^{2}for the MPIGR model determined by substituting Equations (9) and (22) to Equation (23) and the result is as follows:

^{2}based on the likelihood ratio test method is described in Appendix A. The statistics G

^{2}follows the asymptotic of the Chi-square distribution, such that the significant level α reject the null hypothesis when G

^{2}value falls into the rejection region, i.e., when ${G}^{2}>{\chi}_{\alpha ,\left(p+1\right)k}^{2}$.

#### 3.4. Application

_{1}), under-five children mortality(Y

_{2}), and maternal mortality (Y

_{3}). There are eight predictor variables, such as the percentage of antenatal care visit by pregnant women (X

_{1}), the percentage of pregnant women who received Fe3 tablet (X

_{2}), the percentage of complete neonatal visits (X

_{3}), the percentage of Low Birth Weight (LBW) (X

_{4}), the percentage of healthy house (X

_{5}), the percentage of active integrated service post (X

_{6}), the percentage of infants received vitamin A (X

_{7}), and the percentage of births that are assisted by health workers (X

_{8}). Banten Province does not have several predictor variables selected, which is quite important for modelling the response variable. Thus, Banten Province was excluded from the study on the consideration of the selected predictor variables.

_{1}, Y

_{2}, and Y

_{3}differ greatly, we need to measure the spread of the data. The coefficient of variation (CoV) can be used to measure data distribution. The CoV of Y

_{1}, Y

_{2}, and Y

_{3}are 63.4, 425.6, and 89.1. The number of child mortality (Y

_{2}) has the highest CoV, which means that variable Y

_{2}is more heterogenous than the other two variables. This evidence is also supported by histogram for Y

_{1}, Y

_{2}, and Y

_{3}in Figure 1. It shows that the Y

_{2}curve is quite skewed to the right than Y

_{1}and Y

_{3}.

_{1}and Y

_{2}is 0.543 (p-value = 6.97 × 10

^{−10}). The coefficient of Pearson’s correlation between variable Y

_{1}and Y

_{3}is 0.587 (p-value = 1.29 × 10

^{−11}). Otherwise, the coefficient of Pearson’s correlation between variable Y

_{2}and Y

_{3}is 0.130 (p-value = 0.172). Even though there is one pair of the response variables that has significantly no correlation, we need to make sure whether there is dependency among the response variables in multivariate way. Therefore, we calculated the correlation using Bartlett’s test. The result shows that ${\chi}^{2}(92.238)>{\chi}_{3,0.05}^{2}(7.815)$ and p-value (7.20 × 10

^{−20}) < α (0.05). The decision is to reject the null hypothesis, stating that Pearson correlation matrix not equal to an identity matrix. Thus, the response variable can be used in multivariate analysis while using the MPIGR model.

^{2}= 39.86 × 10

^{8}is higher than ${\chi}_{0.05,25}^{2}=37.653$; hence, the decision to reject the null hypothesis. It means that there is at least one predictor variable that significantly influences the number of infants, child, and maternal mortality.

_{1}and Y

_{3}are better than those of Y

_{2}. This empirical results, of course, can be improved. These findings become the big concerns for the next research that are possibly related to the spatial dependencies among the responses that are discussed in the coming section.

## 4. Discussion

_{1}), the percentage of pregnant women who received Fe3 tablet (X

_{2}), the percentage of complete neonatal visits (X

_{3}), the percentage of Low Birth Weight (LBW) (X

_{4}), the percentage of healthy house (X

_{5}), the percentage of active integrated service post (X

_{6}), the percentage of infants received vitamin A (X

_{7}), and the percentage of births that are assisted by health workers (X

_{8}).

_{4}) gave the greatest effect to the number of infant mortality (Y

_{1}), the number of under-five children mortality (Y

_{2}), and the number of maternal mortality (Y

_{3}). However, it has inappropriate dependencies with Y

_{1}, Y

_{2}, and Y

_{3}. Based on these results, we need to look at the pattern of the relationship between the predictor and the response variables below.

_{1}) has a negative relationship with Y

_{1}and Y

_{2}, while it has positive or inappropriate relationship with Y

_{3}. The percentage of pregnant women who received Fe3 tablet (X

_{2}) has negative dependencies with Y

_{2}, while it has a positive or inappropriate relationship with Y

_{1}and Y

_{3}. On the other side, the percentage of complete neonatal visits (X

_{3}) has negative or appropriate dependencies with Y

_{1}and Y

_{3}. Meanwhile, the percentage of Low Birth Weight (LBW) (X

_{4}) has positive dependencies with all of the response variables. However, conflicting results were obtained for all of the response variables.

_{5}) has negative relationship with Y

_{1}and Y

_{3}, while it has positive or inappropriate sign with Y

_{2}. The percentage of active integrated service post (X

_{6}) has negative dependencies with Y

_{1}and Y

_{3}, while it has a positive effect on Y

_{2.}The percentage of infants received vitamin A (X

_{7}) has a positive relationship with Y

_{1}and Y

_{3}, while it has negative or appropriate dependencies with Y

_{2}. The latter one, the percentage of births assisted by health workers (X

_{8}), has negative dependencies with Y

_{3}, while it has an inappropriate relationship with Y

_{1}and Y

_{3}. This finding means that, even though all predictor variables are statistically significant, not all of them have an appropriate relationship with all of the response variables.

_{0}) is no spatial heterogeneity and the alternative hypothesis (H

_{1}) is that there is spatial heterogeneity. The results of the test is G

^{2}=1309.84, which is higher than ${\chi}_{0.05,24}^{2}=36.415$. Therefore, the decision is to reject H

_{0}, which means that the response variables have a spatial heterogeneity and can be modelled using a spatial model in future work. The local model, for example, geographically weighted regression, for MPIGR will be the big concern for our future research.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

PIG | Poisson Inverse Gaussian |

PIGD | Poisson Inverse Gaussian Distribution |

PIGR | Poisson Inverse Gaussian Regression |

MPIGD | Multivariate Poisson Inverse Gaussian Distribution |

MPIGR | Multivariate Poisson Inverse Gaussian Regression |

MLE | Maximum Likelihood Function |

MLRT | Maximum Likelihood Ratio Test |

CoV | Coefficient of Variation |

LBW | Low Birth Weight |

## Appendix A

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Variable | Mean | SD | Coefficient of Variation | Min | Max |
---|---|---|---|---|---|

The number of infant mortality (Y_{1}) | 118.08 | 72.885 | 63.4 | 7 | 403 |

The number of child mortality (Y_{2}) | 20.41 | 36.554 | 425.6 | 0 | 278 |

The number of maternal mortality (Y_{3}) | 16.4 | 12.233 | 89.1 | 0 | 59 |

**Table 2.**Descriptive statistics of predictor variables that are based on city or municipality in Java.

Variable | Province | ||||
---|---|---|---|---|---|

Jakarta | Yogyakarta | Central Java | West Java | East Java | |

The percentage of antenatal care visit by pregnant women | 99.26 (6.10) ^{a} | 90.92 (3.61) | 92.86 (3.62) | 97.44 (8.20) | 89.34 (5.47) |

The percentage of pregnant woman who received Fe3 tablet | 95.14 (4.35) | 88.05 (4.41) | 92.85 (4.01) | 95.88 (9.50) | 88.37 (5.67) |

The percentage of complete neonatal visits | 95.44 (2.13) | 77.32 (28.53) | 92.97 (10.29) | 94.30 (16.78) | 96.34 (3.82) |

The percentage of Low Birth Weight (LBW) | 1.07 (1.45) | 5.26 (1.14) | 4.54 (0.93) | 2.87 (1.66) | 4.20 (1.41) |

The percentage of healthy house | 66.33 (18.85) | 70.59 (17.97) | 85.27 (14.17) | 71.25 (15.84) | 70.53 (16.40) |

The percentage of active integrated service post | 100 (0.00) | 76.99 (9.36) | 66.98 (18.88) | 63.07 (20.58) | 78.14 (14.53) |

The percentage of infants received vitamin A | 92.52 (8.13) | 90.92 (16.40) | 97.25 (8.43) | 91.76 (16.01) | 98.30 (7.97) |

The percentage of births assisted by health workers | 98.00 (5.56) | 100.0 (0.00) | 99.14 (1.56) | 97.94 (8.39) | 94.04 (4.16) |

The number of live births ^{b} | 34649 (23137) | 8470 (4548) | 15424 (7310) | 33903 (25870) | 15144 (10125) |

Variable | Deviance | df | Deviance/df |
---|---|---|---|

Number of infant mortality (Y_{1}) | 4462.60 | 102 | 43.75 |

Number of child mortality (Y_{2}) | 2181.11 | 102 | 21.38 |

Number of maternal mortality (Y_{3}) | 670.05 | 102 | 6.57 |

Parameter | The Number of Infant Mortality | The Number of Under-Five Children Mortality | The Number of Maternal Mortality | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Est | Se | Z | P | Est | Se | Z | P | Est | Se | Z | P | |

${\beta}_{0}$ | 4.101 | 5.27 × 10^{−4} | −7.78 × 10^{3} | p < 0.001 | −2.549 | 5.17 × 10^{−3} | 4.92 × 10^{2} | p < 0.001 | 3.613 | 3.59 × 10^{−2} | −1.00 × 10^{2} | p < 0.001 |

${\beta}_{1}$ | −0.032 | 7.40 × 10^{−8} | 4.31 × 10^{5} | p < 0.001 | −0.072 | 7.24 × 10^{−7} | 9.89 × 10^{4} | p < 0.001 | 0.014 | 1.06 × 10^{−5} | −1.34 × 10^{3} | p < 0.001 |

${\beta}_{2}$ | 0.004 | 6.23 × 10^{−8} | 6.16 × 10^{4} | p < 0.001 | −0.018 | 6.15 × 10^{−7} | −2.87 × 10^{4} | p < 0.001 | 0.007 | 3.05 × 10^{−6} | 2.31 × 10^{3} | p < 0.001 |

${\beta}_{3}$ | −0.003 | 5.16 × 10^{−9} | −5.33 × 10^{5} | p < 0.001 | 0.001 | 5.03 × 10^{−9} | −2.64 × 10^{5} | p < 0.001 | −0.003 | 4.43 × 10^{−8} | −6.09 × 10^{4} | p < 0.001 |

${\beta}_{4}$ | −0.076 | 6.83 × 10^{−7} | 1.11 × 10^{5} | p < 0.001 | −0.449 | 1.41 × 10^{−5} | 3.18 × 10^{4} | p < 0.001 | −0.119 | 3.54 × 10^{−5} | 3.36 × 10^{3} | p < 0.001 |

${\beta}_{5}$ | −0.002 | 3.88 × 10^{−9} | 4.88 × 10^{5} | p < 0.001 | 0.005 | 2.04 × 10^{−8} | 2.24 × 10^{5} | p < 0.001 | −0.005 | 3.99 × 10^{−7} | 1.28 × 10^{4} | p < 0.001 |

${\beta}_{6}$ | −0.005 | 3.67 × 10^{−9} | 1.37 × 10^{6} | p < 0.001 | 0.019 | 2.53 × 10^{−8} | −7.59 × 10^{5} | p < 0.001 | −0.013 | 2.60 × 10^{−7} | 5.34 × 10^{4} | p < 0.001 |

${\beta}_{7}$ | 0.004 | 6.67 × 10^{−9} | −6.03 × 10^{5} | p < 0.001 | −0.005 | 2.02 × 10^{−8} | −2.57 × 10^{5} | p < 0.001 | 0.006 | 8.32 × 10^{−7} | −7.68 × 10^{3} | p < 0.001 |

${\beta}_{8}$ | 0.041 | 3.26 × 10^{−8} | 1.24 × 10^{6} | p < 0.001 | 0.144 | 1.94 × 10^{−6} | 7.42 × 10^{4} | p < 0.001 | 3.613 | 3.59 × 10^{−2} | −1.00 × 10^{2} | p < 0.001 |

**Table 5.**The Mean Squared Error (MSE) and Root Mean Squared Error (RMSE) for each response variable.

Y_{1} | Y_{2} | Y_{3} | |
---|---|---|---|

MSE | 4821.05 | 707.31 | 104.17 |

RMSE | 69.43 | 26.59 | 10.21 |

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**MDPI and ACS Style**

Mardalena, S.; Purhadi, P.; Purnomo, J.D.T.; Prastyo, D.D.
Parameter Estimation and Hypothesis Testing of Multivariate Poisson Inverse Gaussian Regression. *Symmetry* **2020**, *12*, 1738.
https://doi.org/10.3390/sym12101738

**AMA Style**

Mardalena S, Purhadi P, Purnomo JDT, Prastyo DD.
Parameter Estimation and Hypothesis Testing of Multivariate Poisson Inverse Gaussian Regression. *Symmetry*. 2020; 12(10):1738.
https://doi.org/10.3390/sym12101738

**Chicago/Turabian Style**

Mardalena, Selvi, Purhadi Purhadi, Jerry Dwi Trijoyo Purnomo, and Dedy Dwi Prastyo.
2020. "Parameter Estimation and Hypothesis Testing of Multivariate Poisson Inverse Gaussian Regression" *Symmetry* 12, no. 10: 1738.
https://doi.org/10.3390/sym12101738