ZERO-INFLATED POISSON REGRESSION TO DETERMINE THE FACTORS THAT INFLUENCE THE MATERNAL MORTALITY RATE IN BANDUNG 2016

In Indonesia, especially Bandung, there are still many cases of maternal deaths during pregnancy, birth, and postpartum which can increase the Maternal Mortality Rate (MMR). Cases of maternal deaths are mostly due to exsanguination, infection, anemia, and the other causes that related to pregnancy. To reduce the MMR of Bandung, we can analyze to determine the factors that influence the MMR in order to maximize the maternal health care programs so as to prevent the possibility of death. The analysis is using Zero-Inflated Poisson (ZIP) regression because maternal mortality data is the result of counting that allows overdispersion due to excess zeros. Regression parameter estimation use expectation maximization algorithm followed by the NewtonRaphson iteration method. The analysis result showed that of the five suspected factors to affect the MMR -such as the first tetanus toxoid immunization (TT1), the provision of 90 Fe tablets (Fe3), postpartum care, pregnancy complications, and the first antenatal care (K1)only TT1 and postpartum care are significantly affect the MMR.


INTRODUCTION
The Maternal Mortality Rate (MMR) is the number of maternal deaths due to pregnancy, childbirth, and within 42 days postpartum every 100,000 live births.
The MMR is a measurement of the woman's death risk that related to pregnancy, as well as being an indicator of the quality of health care and health status of people in a region. Other than that, the MMR useful to describe the level awareness of healthy living behavior, nutritional status, maternal health, and level of health service. The MMR of Bandung in 2016 reached 62 every 100,000 life of birth, with 27 cases of maternal deaths (Dinas Kesehatan Kota Bandung).
In this paper, analysis of the factors that influence MMR is done through the data of number of maternal deaths in Bandung 2016 obtained from the official website of Bandung's health service. The data of number of maternal deaths in Bandung 2016 shown that the data are counting data that classified into discrete data. The results of this analysis are suppossed to help the government programs in resolving the MMR problems that are concerned to increase.
Analysis for discrete data can use Poisson regression which assumes that the average and the variance of the dependent variable with Poisson distribution has the same value (equdispersion) (Winkelmann, 2003). The data about death is a rare event data that contains many zero values (excess zeros) which has important meaning for the research. This can cause one of the assumption violations called overdispersion. Data which has overdispersion due to excess zeros can be analyzed by Zero-inflated Poisson (ZIP) regression.
ZIP regression model formed to determine the factors that influence the MMR in Bandung 2016. Regression parameter estimation in this paper using the Expectation Maximization (EM) algorithm and then followed by the Newton-Raphson iteration method.

Poisson Regression
Poisson regression assumes that the dependent variable (Y) which has a Poisson distribution with the following probability functions (Kutner et al, 2005): (2)

Parameters Estimation of Poisson Regression
Poisson regression parameter estimation using the Maximum Likelihood Estimation (MLE) method, followed by the Newton-Raphson iteration with these steps: 1. Forming the Poisson regression likelihood function using the equation (1) then transformed it into a log-likelihood function (Winkelmann, 2003): 2. Determining the value of ̂( ) .
3. Multiplying X by ̂( ) to form gradient vector which contains the first partial derivative value of equation (3) againts ̂. Gradient vector can be obtained with this formula: where is the transpose matrix of X, y is the gradient vector of dependent variable, and p is s vector which contains ( ̂( ) ) with .
4. Forming the inverse of the Hessian matrix [ ( ̂( ) )] which contains the second partial derivative value of equation (3) against ̂ or by calculating the following equation: where ( ).

Score Test
Excess zeros is a situation of many zeros in a research data. Excess zeros can cause violation of equidispersion assumption, for example condition where the value of the dependent variable variance exceeds the average value or called overdispersion (McCullagh & Nelder, 1989). Overdispersion in excess zeros data can be detected using a score test with the following test statistics (Rahayu, 2014): The test hypothesis used is : against the alternative : . estimated value ̂ and the independent variable matrix X has the following estimation model form (Mouatassim & Ezzahid, 2012): The logistic model ( ) has a regression parameter with an estimated value ̂ and the independent variable matrix X*. By using the logit link function, the estimation of the logistic model is (Mouatassim & Ezzahid, 2012): and The independent variables matrix of Poisson model can be different from logistic model with different number of regression parameters.

Parameter Estimation of Zero-inflated Poisson Regression
The Expectation Maximization (EM) algorithm is used to estimate the Zero-inflated Poisson (ZIP) regression parameter with the following steps (Mouatassim & Ezzahid, 2012):  Determining the distribution of indicator variable (Z) Assume the dependent variable (Y) is related to the indicator variable (Z), such as: so has Bernoulli distribution.
 Determining the joint probability distribution function of and :  Forming the likelihood function of equation (13) and then transforming it into log-likelihood form, therefore obtained: and  The expectation stage aims to find conditional expectations of indicator variables ( ( | )) which will be used to determine the value of ( ) : Zero-Inflated Poisson Regression 99  The maximization stage aims to find the convergent values of ̂ and ̂ from the log-likelihood function in equations (14) and (15) using the Newton-Raphson iteration method.
Newton-Raphson iteration steps to estimate ZIP regression parameters are as follows: 1. Determining the value of ̂( ) and ̂( ) .
2. Forming a zik vector with an entry ( ) where .
3. Multiplying the matrix X by ̂( ) and the matrix X* by ̂( ) to form the gradient vector as follows:  Gradient vector for Poisson model containing the first partial derivative value of equation (14) against ̂.
where M is the diagonal matrix with the entry ( ) and u is the vector with the entry . ̂( ) /.
 Gradient vector for logistik model containing the first partial derivative value of equation (15) against ̂.
( ̂( ) ) ( ) where is the diagonal matrix with the entry ( ̂) and v is the vector with the entry ( ̂).

Forming the inverse of the Hessian matrix as follows:
 0 . ̂( ) /1 containing the second partial derivative value of equation (14) against ̂ which is inversed.

Likelihood Ratio Test
The method used to test the overall of ZIP regression parameter is the Likelihood Ratio Test (LRT) with the following test statistics (Agresti, 2002;Hosmer & Lemeshow, 2000): Test statistics (Winkelmann, 2003): The value of ( ̂) is obtained from the main diagonal entry ( ) of the inverse Hessian matrix in equation (19 Test statistics (Winkelmann, 2003): The value of ( ̂) is obtained from the main diagonal entry ( ) of the inverse Hessian matrix in equation (20). Test criteria: reject if test statistics fulfill ( ) . If is accepted then the independent variable of ̂ must be removed from the model.

RESULT AND DISCUSSION
The research is performed by taking the dependent variable (Y), which is the number of maternal death cases in Bandung 2016 that is 30 observations and the independent variable (X), which is the suspected factors to affect MMR, including TT1 or the first tetanus toxoid immunization ( ), Fe3 or provision 90 Fe tablets for pregnant women ( ), postpartum care ( ), pregnancy complication ( ), and K1 or first antenatal care ( ).

Model
Estimated parameter model of Poisson regression that convergent using Newton-Raphson is obtained in the 6th iteration where and ̂( ) , -, so that the Poisson regression model is as follows: Detection of overdispersion in excess zeros data use the score test method that calculated from equation (7). Based on the hypothesis : and : , the value of the test statistic is | | ( ) so is rejected at a significance level of 5%. The result shows that the dependent variable in the data has overdispersion caused by excess zeros. Poisson regression cannot be used to analyze or model the data.

Analysis Result Using the Zero-inflated Poisson Regression
Estimated parameter of Zero-Inflated Poisson (ZIP) regression model which is convergent using the Newton-Raphson iteration method is obtained in the 51st iteration where , ̂( ) ̂ , -, the matrix of X equals to the matrix of X* which include and , so the Poisson regression model is as follows:

CONCLUSION
After analyzing the data of maternal deaths in Bandung 2016 using Zero-Inflated Poisson (ZIP) regression it can be concluded that the factors that significantly affect the MMR in Bandung 2016 are the first tetanus toxoid immunization (TT1) and postpartum maternal care. The goverment of Bandung is expected to pay more attention to TT1 immunizaion for pregnant women which is useful to prevent maternal deaths due to infection. In addition, the postpartum care program needs to be improved and make sure that all postpartum women get health care.
In this paper overdispersion was caused only because of excess zeros. For the next research is expected to use a method that can handle all cause of overdispersion.