Estimation Parameters and Modelling Zero Inflated Negative Binomial


CAUCHY – JURNAL MATEMATIKA MURNI DAN APLIKASI 
Volume 4(3) (2016), Pages 115-119 
 

Submitted: August, 20 2016 Reviewed: October, 31 2016 Accepted: November, 29 2016 
 
DOI: http://dx.doi.org/10.18860/ca.v4i3.3656 

Estimation Parameters and Modelling Zero Inflated Negative Binomial 

Cindy Cahyaning Astuti1, Angga Dwi Mulyanto2 

1 Muhammadiyah University of Sidoarjo, Sidoarjo, Indonesia 
2 Alpha Research, Malang, Indonesia 

 
Email: cindy.cahyaning@umsida.ac.id, angga.dwi.m@gmail.com 

ABSTRACT 

Regression model between predictor variables and the Poisson distributed response variable is called 
Poisson Regression Model. Since, Poisson Regression requires an equality between mean and variance, it is 
not appropriate to apply this model on overdispersion. Poisson regression can be used to analyze count data 
but it has not been able to solve problem of excess zero value on the response variable. An alternative model 
which is more suitable for overdispersion data and can solve the problem of excess zero value on the 
response variable is Zero Inflated Negative Binomial (ZINB). In this research, ZINB is applied on the case of 
Tetanus Neonatorum in East Java. The aim of this research is to examine the likelihood function and to form 
an algorithm to estimate the parameter of ZINB and also applying ZINB model in the case of Tetanus 
Neonatorum in East Java. Maximum Likelihood Estimation (MLE) method is used to estimate the parameter 
on ZINB and the likelihood function is maximized using Expectation Maximization (EM) algorithm. Test 
results of ZINB regression model showed that the predictor variable have a partial significant effect at 
negative binomial model is the percentage of pregnant women visits and the percentage of maternal health 
personnel assisted, while the predictor variables that have a partial significant effect at zero inflation model is 
the percentage of neonatus visits. 

Keywords: Overdispersion, Tetanus Neonatorum, Zero Inflation, Zero Inflated Negative Binomial (ZINB) 

INTRODUCTION 

Regression analysis is used to determine relationship between one or several response 
variable (Y) with one or several predictor variables (X). In the classical linear model assumptions 
are response variables follow a normal distribution, but in fact often found the response variable 
did not follow the normal distribution. To overcome this there is development in the classical linear 
model, namely the Generalized Linear Model (GLM) [1]. GLM assuming the response variable 
follows the exponential family distribution, which has a more general characteristic. In some 
research, there are often data with response variable that follows a Poisson distribution, regression 
analysis is used to this kind of data is the Poisson regression analysis. Poisson regression model is 
commonly used to analyze the data count (data count). Poisson regression there is an assumption 
on Y ~ Poisson (μ). A key assumption in the Poisson regression analysis is the variance should be 
equal to the average, the condition is called equidispersion. On the type of count data often 
encountered zero value is more than 50 percent on the response variable (zero inflation) [2]. Data 
proportion that has exaggeration zero value can lead to the accuracy of inference. Poisson 
regression can be used to analyze the data count but still cannot resolve the problem of excessive 
zero value. In modelling count data if there ar many zero observations on response variable it can 
be overcome by using Zero inflated Poisson regression (ZIP) model [3]. However, if there are many 

http://dx.doi.org/10.18860/ca.v4i3.3656
mailto:cindy.cahyaning@umsida.ac.id
mailto:angga.dwi.m@gmail.com


Estimation Parameters and Modelling Zero Inflated Negative Binomial 

Cindy Cahyaning Astuti 116 

zero observations and occurs overdispersion then Zero inflated Poisson regression (ZIP) 
inappropriately used. Overdispersion can be defined as a condition in which the Poisson 
distribution variance is greater than average. If in modelling count data (data count) there are 
many zero observations on response variable (zero inflation) and occurs overdispersion then the 
regression model can be used is Zero Inflated Generalized Poisson [2]. 

In progress there are other alternatives to modelling many zero observations and occurs 
overdispersion besides using Zero Inflated Generalized Poisson (ZIGP), the regression model is 
Zero Inflated Negative Binomial (ZINB). Zero Inflated Negative Binomial (ZINB) model is formed of 
Poisson Gamma mixture distribution [4]. Zero Inflated Negative Binomial (ZINB) can be used as an 
alternative to modelling many zero observations and occurs overdispersion because this model 
does not require the variance should be equal with average, in addition Zero inflated Negative 
Binomial (ZINB) model also has a dispersion parameter that useful to describe the variation of the 
data, which is commonly denoted by κ (kappa). The purpose of this research is examine the 
likelihood form, estimation parameters of Zero inflated Negative Binomial (ZINB) model and 
modelling Zero inflated Negative Binomial (ZINB) on Neonatorum Tetanus cases. 
 

METHODS 

 In this research used secondary data sourced from East Java Health Profile 2012 [5]. Unit of 
observation in this research was 38 districts/cities in East Java province which covers 29 districts 
and 9 Cities. The response variable (Y) used in this research is number of cases of Tetanus 
Neonatorum in each district/city in East Java province, while the predictor variable (X) is used as 
much as 4 variables. Operational definition of each variable response and predictor variables will 
be described as 
a. The response variable (Y): Number of cases of Tetanus Neonatorum 
b. Predictor variable (X) 

1. The percentage of pregnant mothers visit K4 (X1) 
2. Percentage of immunization Tetanus Toxoid (TT) in pregnant women (X2) 
3. Percentage of maternal mothers assisted by health workers (X3) 
4. The percentage of neonates visits (X4) 

The method of analysis in this study is. 
a. Knowing the probability function of  Zero inflated Negative Binomial (ZINB) model. 

b. Determining the likelihood function of  Zero inflated Negative Binomial (ZINB) model based on 

probability function that are already known. 

c. Develop algorithms for estimation parameter process based on the likelihood function that is 

already known. Parameter estimation of Zero inflated Negative Binomial (ZINB) model. was 

performed using MLE method and solved using EM algorithm. 

d. Modelling Zero inflated Negative Binomial (ZINB) model  on Neonatorum Tetanus cases in East 

Java Province 

e. Significance test of parameters model carried out simultan and partial test. Statistical tests are 

used for simultan test is the test statistic G and to  partial test used test statistics t. 

RESULTS AND DISCUSSION 

Estimation parameter Zero inflated Negative Binomial (ZINB) was conducted using 
Maximum Likelihood Estimation (MLE) and to maximize the function is used EM (Expectation 
Maximization) algorithm. Probability Function of Zero inflated Negative Binomial (ZINB) model can be 
defined as : 

𝑃(𝑌𝑖 = 𝑦𝑖) =

{
 
 

 
 𝜋𝑖 + (1 − 𝜋𝑖)(

1

1+𝜿𝜇𝑖
)

1

𝜿
,𝑓𝑜𝑟  𝑦

𝑖
= 0

(1 − 𝜋𝑖)
 (𝑦𝑖+

1

𝜿
)

 (1
𝜿
)𝑦𝑖!

(
1

1+𝜿𝜇𝑖
)

1

𝜿
(
𝜿𝜇𝑖
1+𝜿𝜇𝑖

)
𝑦𝑖
,𝑓𝑜𝑟 𝑦

𝑖
> 0

                          



Estimation Parameters and Modelling Zero Inflated Negative Binomial 

Cindy Cahyaning Astuti 117 

 
 

EM algorithm consists of two stage, expectation and maximization stage. Expectation stage 
is expectation calculation of ln likelihood the function, the next stage  maximization is calculation to 
look for estimation parameter which maximizes the likelihood function. Probability function of 
ZINB model consist of  two conditions, yi = 0 and yi > 0. Response variable is also composed of two 
conditions, namely zero state and negative binomial state. To describe in detail the condition yi, 
then it will be redefined variables yi with latent variable Zi. 
 

𝑍𝑖 = {
1,𝑖𝑓  𝑦𝑖 𝑓𝑟𝑜𝑚 𝑧𝑒𝑟𝑜 𝑠𝑡𝑎𝑡𝑒

0,𝑖𝑓  𝑦𝑖 > 0𝑓𝑟𝑜𝑚 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑏𝑖𝑛𝑜𝑚𝑖𝑎𝑙 𝑠𝑡𝑎𝑡𝑒
 

 

Zero inflated Negative Binomial regression (ZINB) model can be defined as two models that are : 

Model for negative binomial �̂�𝑖   

𝑙𝑛�̂�𝑖 = �̂�0 + ∑�̂�0𝑥𝑖𝑗

𝑝

𝑗=1

, 𝑖 = 1,2,…,𝑛 𝑎𝑛𝑑𝑗 = 1,2, . . ,𝑝 

Model for  zero inflation �̂�𝑖 

𝑙𝑜𝑔𝑖𝑡�̂�𝑖 = 𝛾0 + ∑𝛾0𝑥𝑖𝑗

𝑝

𝑗=1

, 𝑖 = 1,2,…,𝑛 𝑎𝑛𝑑𝑗 = 1,2, . . ,𝑝 

  
 EM algorithm is alternative methods to maximize likelihood function on the data containing 
latent variables defining new variables such as variable Zi. EM algorithm consists of two stage: the 
expectation stage and maximization stage. Expectation stage is  calculation of the ln likelihood 
function, the next stage is maximization calculation stage to look for parameter estimation which 
maximizes the likelihood function ln results from stage earlier expectations. Estimation parameter 
and parameter test of Zero inflated Negative Binomial (ZINB) on Neonatorum Tetanus cases in East 
Java Province using SAS software, the result can see at table 1. 
 

Table 1. Estimation Parameter and Parameter Test of ZINB 
Parameter Estimation SE       t value (Pr >| t|) 
�̂�
0
 -5,847 3,602 -1,623 0,105 

�̂�
1
 -0,145 0,055 -2,644 0,008* 

�̂�
2
 -0,006 0,010 -0,599 0,549 

�̂�
3
 0,233 0,101 2,295 0,022* 

�̂�
4
 -0,023 0,067 -0,339 0,735 

�̂�
0
 11,325 13,409 0,845 0,398 

�̂�
1
 0,223 0,169 1,316 0,188 

�̂�
2
 -0,296 0,179 -1,653 0,098 

�̂�
3
 0,835 0,503 1,660 0,096 

�̂�
4
 -1,078 0,539 -2,000 0,045* 

Test Statistic G = 581,24 
   
 Results of  simultan parameter test  based on test statistic G. G test is 581.24. Value of G test 
is greater than 2

(0,05;8)
15, 507  . This shows that simultaneously the predictor variables X1, 

X2, X3 and X4 have significant effect on response variable. While results of  partial parameter test  
based on  test statistical t. According to Table 1, there are two predictor variables in negative 
binomial state model and one predictor variables in zero inflation state model that has t value 
greater than or equal to t (α / 2; 37 = 2.00) and has p-value less than α (0.05). This indicates that the 
predictor variables were partial significant effect in negative binomial state model are pregnant 
mothers visit K4 (X1) and maternal mothers assisted by health workers (X3), while the predictor 



Estimation Parameters and Modelling Zero Inflated Negative Binomial 

Cindy Cahyaning Astuti 118 

variables were partial significant effect in zero inflation state model is the percentage of neonates 
visits (X4). So that Zero inflated Negative Binomial (ZINB) model can be defined as : 
a. Negative binomial state model for �̂� 

�̂� = 𝑒𝑥𝑝(−5,847 − 0,145 𝑋1 − 0,006 𝑋2 + 0,233 𝑋3 − 0,023 𝑋4) 
 
 

b. Zero inflation state model for �̂� 

�̂� =
exp (11,325 + 0,223 𝑋1 − 0,296 𝑋2 + 0,835 𝑋3 − 1,078 𝑋4)

1 + exp (11,325 + 0,223 𝑋1 − 0,296 𝑋2 + 0,835 𝑋3 − 1,078 𝑋4)
 

 
All coefficient parameter which aren’t significant still is exist in Negative binomial state  Zero 

inflation state model because it is intended to determine the contribution of each predictor variable on the 

response variable can be defined as : 
Zero inflation  model for ˆ   

1. Each additional 1 percent of pregnant mothers visit K4 (X1) it will increase the chances of the number of 
Tetanus Neonatorum by exp (0.223) = 1.249 times the number of cases of Tetanus Neonatorum original, if 
the other variables constant value. 

2. Each additional 1 percent immunization Tetanus Toxoid (TT) in pregnant women (X2) will decrease 
the chances of the number of Tetanus Neonatorum by exp (0.296) = 1.344 times the number of cases of 
Tetanus Neonatorum original, if the other variables constant value. 

3. Each additional 1 percent of maternal mothers assisted by health workers (X3) then it will increase 
the chances of the number of Tetanus Neonatorum by exp (0.835) = 2.305 times the number of cases of 
Tetanus Neonatorum original, if the other variables constant value. 
4. Each additional 1 percent of neonates visits (X4) will decrease the chances of the number of Tetanus 
Neonatorum by exp (1.078) = 2.939 times the number of cases of Tetanus Neonatorum original, if the other 
variables constant value. 

Let’s discussion, based on the negative binomial state model and zero inflation state model, 
there are signs of regression coefficient as opposed to the theory are percentage of maternal 
mothers assisted by health workers (X3) to model negative binomial state model and the 
percentage of pregnant mothers visit K4 (X1) and the percentage of maternal mothers assisted by 
health workers (X3) for zero inflation state model . The existence of the regression coefficient has a 
sign contrary to the theory of probability caused by the effect of the multikolinieritas. Moreover 
sign contrary to the theory also caused by the shape of the data pattern of the predictor variables 
that have a positive correlation with the response variable. In a subsequent study if there are 
multikolinieritas the predictor variables can be addressed using Principal Component Analysis 
(PCA). 

CONCLUSION 

 Based on the results, estimation parameter of Zero inflated Negative Binomial (ZINB) model 
was conducted using Maximum Likelihood Estimation (MLE) and to maximize the likelihood 
function used the EM (Expectation Maximization) algorithm. For parameter test predictor variable 
that has significant effect on the number of cases of Tetanus Neonatorum are are pregnant mothers 
visit K4 (X1) and maternal mothers assisted by health workers (X3) for the negative binomial state 
models, while zero inflation state model predictor variable that has significant effect on the number 
of cases of Tetanus Neonatorum include the percentage of neonates visit (X4). 

 
 

  



Estimation Parameters and Modelling Zero Inflated Negative Binomial 

Cindy Cahyaning Astuti 119 

REFERENCES 

 

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[4]  J. M. Hilbe, Negative Binomial Regression, New York: Cambridge University Press, 2011.  

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