Abstract
We propose a multivariate regression model called Multivariate Zero Inflated Generalized Poisson Regression (MZIGPR) type II. This model further develops the Bivariate Zero Inflated Generalized Poisson Regression (BZIGPR) type II. This study aims to develop parameter estimation, test statistics, and hypothesis testing, both simultaneously and partially, for significant parameters of the MZIGPR model. The steps of the EM algorithm for obtaining the parameter estimator are also described in this article. We use Berndt–Hall–Hall–Hausman (BHHH) numerical iteration to optimize the EM algorithm. Simultaneous testing is carried out using the maximum likelihood ratio test (MLRT) and the Wald test to partially assess the hypothesis. The proposed MZIGPR model is then used to model the three response variables: the number of maternal childbirth deaths, the number of postpartum maternal deaths, and the number of stillbirths with four predictors. The units of observation are the sub-districts of the Pekalongan Residency, Indonesia. The indicate overdispersion in the data on the number of maternal childbirth deaths and stillbirths, and underdispersion in the data on the number of postpartum maternal deaths. The empirical studies show that the three response variables are significantly affected by all the predictor variables.
Highlights
The Poisson regression model is commonly used to analyze data in which the response variable follows a Poisson distribution
Research on generalized Poisson distribution (GPD) has developed it into generalized Poisson regression (GPR) known as the GP-1 regression model, GP-2 regression, the bivariate GPR (BGPR) model [6], and the multivariate GPR (MGPR) model [7]
The log model shows that the variable coefficient X1 is positive. This means that every 1% increase in the number of childbirths assisted by medical personnel will increase the average number of maternal childbirth deaths (Y1) by 1.06 people and the average number of stillbirths (Y3) by 1.03 people, but decrease the average number of postpartum maternal deaths (Y2) by 0.98 people when the other predictor variables were held constant
Summary
The Poisson regression model is commonly used to analyze data in which the response variable follows a Poisson distribution. When the sample variance is greater (less) than the sample mean, it is called overdispersion (underdispersion). The use of Poisson regression on over- or underdispersed data tends to make the standard error and test statistics derived from the model inaccurate, resulting in invalid conclusions [1,2]. Several distribution models have been developed, such as negative binomial distribution (NBD) and log-normal Poisson distribution, which can be used to resolve overdispersion. Alternative models for count data that can overcome both overand underdispersion include double Poisson distribution, gamma count distribution, and generalized Poisson distribution (GPD). The first two distribution models are weak in their probability function, which is complex, and the variance and mean have no explicit form [3]. Research on GPD has developed it into generalized Poisson regression (GPR) known as the GP-1 regression model (or the classic GPR model [4]), GP-2 regression (or the restricted GPR model [5]), the bivariate GPR (BGPR) model [6], and the multivariate GPR (MGPR) model [7]
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