A Study of Count Regression Models for Mortality Rate


CAUCHY – Jurnal Matematika Murni dan Aplikasi 
Volume 7(1) (2021), Pages 142-151 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: October 16, 2021 Reviewed: November 04, 2021 Accepted: November 05, 2021 
DOI: https://doi.org/10.18860/ca.v7i1.13642    

A Study of Count Regression Models for Mortality Rate 

Anwar Fitrianto 

 

Department of Statistics, Faculty of Mathematics and Natural Sciences, IPB University 
 

Email: anwarstat@gmail.com 

ABSTRACT 

In this study, Poisson regression model, Negative Binomial 1 regression model (NEGBIN 1) and 
Negative Binomial regression 2 (NEGBIN 2) model were proposed to fit mortality rate data. The 
method used is comparing the values of Akaike Information Criterion (AIC) and Bayesian 
Information Criterion (BIC) to find out which method suits the data the most. The results show 
that the data indeed display higher variability. Among the three models, the model preferred is 
NEGBIN 1 model. 
 
Keywords: Mortality; Poisson; Regression; Binomial; Overdispersion 

 

INTRODUCTION 

Count data contain variables that count how many times something has happened, 
such as the number of cases with a particular disease in epidemiology [1]. Linear 
regression models have often been applied to handle this kind of data, but the results are 
inefficient, inconsistent, and biased. This type of data is considered as count data with 
variable offset. Mortality data is considered as the amount of data that contains the offset 
variable.  

A study of mortality for middle-aged men on ischemic heart disease (IHD) that affects 
mortality has been conducted by [2]. The results showed that there were 46 of 109 deaths 
around 11.4 years of follow-up due to IHD. In addition to studies on causes of death other 
than IHD, [3] has researched the global impact of HIV/AIDS. Another study on mortality 
was conducted by [4] about the diarrheal disease. It has been found that diarrhea causes 
1 in 9 child deaths worldwide, the second leading cause among children under 5 years of 
age. In addition, [5] examined the global causes of death due to disease in children under 
5 years. In their study, diarrhea remained the second leading cause of death in children 
from infection in the last 30 years.  

In addition, malnutrition is said to be one of the world's worrisome problems. It affects 
about 6 million child deaths every year. [6] studied that poor nutrition during fetal 
development can cause severe physical damage, and malnutrition always increases 
susceptibility to disease. A study conducted by [7] stated that malnutrition (measured as 
poor anthropometric status) accounted for nearly 50% of childhood deaths. 

Regarding the problem of mortality due to disease, [8] stated that the trend of injuries 
and deaths from road traffic accidents (RTA) is becoming severe in countries such as 
India. Not a day goes by without an RTA in India; many people die or become disabled. In 

https://doi.org/10.18860/ca.v7i1.13642
mailto:anwarstat@gmail.com


A Study of Count Regression Models for Mortality Rate 

Anwar Fitrianto 143 

addition, suicide is one of the factors that contribute to the death rate. In a study by [9], 
suicidal behavior has always been a major health problem in many countries, both 
developed and developing countries.  

Poisson regression model is one of the general linear models for data with offset 
variables. It is also the standard model for calculating data and contingency tables. In this 
model, the response variable is assumed to have a Poisson distribution.  In addition to 
Poisson regression, Negative Binomial regression is also a generalized linear model where 
the dependent variable is the number of events. The Negative Binomial distribution is a 
two-parameter distribution that is generally more flexible than the Poisson model [2]. 
This model can also model scattered quantities, which the Poisson model cannot. The 
Negative Binomial model can be derived from the Poisson distribution and the 
generalized Poisson distribution. 

 [10] has discussed several other specific mortality measures, such as age-specific 
crude death rates, cause-specific mortality rates, and infant and maternal mortality rates. 
In the data collection process, there may be biased and inaccurate data measurements. 
The inaccuracy of this data collection will cause overdispersion. This study aims to 
identify the most suitable method when dealing with mortality data which usually has 
overdispersion. 

 

METHODS 

Data 

The data used in this study is mortality rate data which is available in [11]. The data 
consists of 163 observations (countries) with seven independent variables, which are the 
number of people dying per 100,000 live births due to IHD (𝑥1), diarrheal disease (𝑥2), 
HIV/AIDS (𝑥3), malaria (𝑥4), malnutrition (𝑥5), road accidents (𝑥6), and suicides (𝑥7).  

Count Regression Models 

According to [12], the count regression model has been suggested to be used to model 
over-dispersed and zero-inflated count response variables. Poisson regression is the 
standard model for modeling count data, while the Negative Binomial regression model 
is often introduced to solve count data with overdispersion. Meanwhile, the zero-inflated 
Poisson model (ZIP) and the zero-inflated Negative Binomial model (ZINB) are introduced 
to solve a zero-inflated variable in which the data contains many zeros. Moreover, [13] 
found that ZIP and ZINB can be obtained by mixing a distribution degenerate at zero with 
a Poisson regression and Negative Binomial regression, respectively.  

The probability mass function of the ZIP is, 

𝑃(𝑌 = 𝑦𝑖) = {
𝜔𝑖 + (1 − 𝜔𝑖) 𝑒𝑥𝑝( 𝜆𝑖),      𝑦𝑖 = 0

(1 − 𝜔𝑖)
𝜆𝑖

𝑦𝑖!
𝑒𝑥𝑝( 𝜆𝑖),            𝑦𝑖 > 0

        (1) 

Meanwhile, the ZINB's probability mass function can be formulated as: 

𝑃(𝑌 = 𝑦𝑖) = {
𝜔𝑖 + (1 − 𝜔𝑖) (

𝜃

𝜃+𝜆𝑖
)
𝜃

,                      𝑦𝑖 = 0

(1 − 𝜔𝑖)
𝛤(𝑦𝑖+𝜃)

𝑦𝑖!𝛤(𝜃)
(

𝜃

𝜃+𝜆𝑖
) (

𝜆𝑖

𝜃+𝜆𝑖
)

𝑦𝑖
,     𝑦𝑖 > 0

       (2) 

with 𝜆𝑖 = 𝑒𝑥𝑝( 𝑥𝑖𝛽). The 0's arise with probability 𝜔 from a second process. The function 
F that relates to the product 𝑥𝑖𝛾 to the probability 𝜔𝑖 is named as the zero-inflated link 
function, 𝜔𝑖 = 𝐹(𝑥𝑖𝛾).         



A Study of Count Regression Models for Mortality Rate 

Anwar Fitrianto 144 

Poisson Regression Model 

[14] studied about Poisson regression model as the standard model for count data. A 
variable Y is a count of events of Poisson regression, and the marginal probability of 
Poisson regression is written as: 

𝑃(𝑌 = 𝑦𝑖) =
𝑒𝑥𝑝(−𝜆𝑖)𝜆𝑖

𝑦𝑖

𝛤(1+𝑦𝑖)
;            (3) 

with 𝜆𝑖 = 𝑒𝑥𝑝( 𝛼 + 𝑥𝑖𝛽); 𝑦𝑖 = 0,1, . . . 𝑁. The rate parameter of Poisson regression is 𝜆𝑖 and 
it is also known as its expected count is formulated as: 

𝜆𝑖 = 𝑒
𝛽0+𝛽1𝑥1+...+𝛽𝑝𝑥𝑝 .           (4) 

Based on Equation (4), the Log-linear model for mean rate is written as: 

𝑙𝑜𝑔(𝜆𝑖) = 𝛽0 + 𝛽1𝑥1+. . . +𝛽𝑝𝑥𝑝,          (5) 

with p is the number of predictors or covariates in the model, 𝛽0 is the intercept of the 
regression, 𝛽𝑝 are the regression coefficients, and 𝑥𝑖  is the independent variable. 

[14] formulated Maximum Likelihood Estimation (MLE) of Poisson regression. Let Y 
be a random variable with Poisson distribution and with an unknown parameter value 𝜃.  
The probability mass function of Y is obtained, which is 𝑃𝑦(𝑦; 𝜃) to emphasize the 

parameter 𝜃 and n is the independent trials in order to get the data 𝑦1, 𝑦2, 𝑦3, . . . , 𝑦𝑛. The 
joint probability mass function is as follows: 

𝑃𝑌...𝑌𝑛(𝑦1, . . . , 𝑦𝑛; 𝜃) = 𝑃𝑦(𝑦1; 𝜃). . . . 𝑃𝑦(𝑦𝑛; 𝜃).        (6) 

The likelihood of 𝜃 given data 𝑦1, . . . , 𝑦𝑛 can be obtained from Equation (6) by applying the 
logarithm as follows: 

𝐿(𝜃; 𝑦1, … , 𝑦1) = 𝑃𝑌...𝑌𝑛1
(𝑦1, . . . , 𝑦𝑛; 𝜃) = 𝑃𝑦(𝑦1; 𝜃). . . . 𝑃𝑦(𝑦𝑛; 𝜃).      (7) 

The estimated value maximizes the Maximum Likelihood Estimates 𝜃 = 𝜃. Y follows a 
Poisson distribution with unknown parameters, and the data is collected from the 
independent trials are of the form 𝑌1 = 𝑦1, 𝑌2 = 𝑦2, . . . , 𝑌𝑛 = 𝑦𝑛. On the other hand, the 
likelihood function of the Poisson regression is written as: 

𝐿 = ∏
𝑒−𝜆𝑖𝜆

𝑖

−𝑦𝑖

𝑦𝑖!

𝑁
𝑖=1 .            (8) 

The log-likelihood function of Poisson regression is obtained by applying the logarithm of 

Equation (8), l ∑ 𝑙𝑜𝑔 (
𝑒−𝜆𝑖𝜆

𝑖

−𝑦𝑖

𝑦𝑖!
)𝑁𝑖=1  . 

 

The Standard Negative Binomial Regression Model  

According to [1], in most applications, the mean of the data is usually greater than the 
variance. If otherwise, it is called overdispersion in the particular data. But, based on the 
study of [15], the Poisson regression model is inefficient when dealing with overdispersed 
data. While in a study by [16], the Negative Binomial distribution is more flexible than 
Poisson distribution as it is a two-parameter when modeling the data with 
overdispersion. Particularly, Negative Binomial regression can model overdispersed 
counts. The Negative Binomial model can be derived as a mixture of the Gamma-Poisson 
model. Starting from the conditional mean of the Poisson model, 

𝐸(𝑦𝑖|𝑥𝑖. 𝜀𝑖) = 𝑒𝑥𝑝( 𝛼 + 𝑥𝑖𝛽 + 𝜀𝑖) = ℎ𝑖𝜆𝑖,         (9) 



A Study of Count Regression Models for Mortality Rate 

Anwar Fitrianto 145 

where ℎ𝑖 = 𝑒𝑥𝑝(𝜀𝑖). In the case of the Poisson-Gamma distribution, 𝑔(𝜃, 𝜃) is the Poisson 
distribution while ℎ𝑖 = 𝑒𝑥𝑝(𝜀𝑖) follows Gamma distribution. The ℎ𝑖  is assumed to follow 
a two-parameter Gamma distribution, 

𝑓(ℎ𝑖) =
𝜃𝜃 𝑒𝑥𝑝(−𝜃ℎ𝑖)ℎ𝑖

𝜃−1

𝛤(𝜃)
.                     (10) 

Once ℎ𝑖  has been integrated out from the joint distribution, then the marginal 
probability of Negative Binomial distribution is obtained as follows: 

𝑃(𝑌 = 𝑦𝑖|𝑥𝑖) =
𝛤(𝜃+𝑦𝑖)

𝑦𝑖!𝛤(𝜃)
(

𝜃

𝜃+𝜆𝑖
)
𝜃

(
𝜆𝑖

𝜃+𝜆𝑖
)

𝑦𝑖
.                   (11) 

The mean of Negative Binomial is the same as Poisson regression, which is written as 
𝐸(𝑦𝑖|𝑥𝑖) = 𝜆𝑖 = 𝑒

𝑥𝑖𝛽 and the variance of a Negative Binomial is written as: 

𝑉𝑎𝑟(𝑦𝑖|𝑥𝑖) = 𝜆𝑖 [1 + (
1

𝜃
) 𝜆𝑖] = 𝜆𝑖(1 + 𝑘𝜆𝑖),                  (12) 

where 𝑘 = 𝑉𝑎𝑟(ℎ𝑖). Moreover, the rate parameter of negative regression 𝜆𝑖, which is also 
known as its expected counts, is written as: 

𝜆𝑖 = 𝑒
𝛽0+𝛽1𝑥1+...+𝛽𝑝𝑥𝑝 .                     (13) 

The Log-linear model for the mean rate of Negative Binomial regression can be 
obtained by applying the logarithm of Equation (13): 

𝑙𝑜𝑔(𝜆𝑖) = 𝛽0 + 𝛽1𝑥1+. . . +𝛽𝑝𝑥𝑝,                     (14) 

where p is the number of predictors or covariates in the model, 𝛽0 is the intercept of the 
regression, 𝛽𝑝 are the regression coefficients, and x's are the independent variables. 

[17] has discussed the MLE of Negative Binomial regression in which random samples 
of n subjects are given. In a standard Negative Binomial model, the dependent variables 
𝑦𝑖 and the predictor variables  𝑥1𝑖, 𝑥2𝑖, … , 𝑥𝑝𝑖   are included. Predictor variables are 

combined to form the following matrix, 

𝑿 =

[
 
 
 
1 𝑥11 … 𝑥1𝑝
1 𝑥12 ⋯ 𝑥2𝑝
⋮ ⋮ ⋱ ⋮
1 𝑥1𝑛 ⋯ 𝑥𝑛𝑝]

 
 
 

. 

The 𝑖𝑡ℎ row of X is designated to be 𝑥𝑖 , from Equation (11), the 𝜆𝑖 in which is replaced by 
 𝑒𝑥𝑖𝛽. The Equation can be rewritten as, 

𝑃(𝑌 = 𝑦𝑖|𝑥𝑖) =
𝛤(𝜃+𝑦𝑖)

𝑦𝑖!𝛤(𝜃)
 (

𝜃

𝜃+𝑒𝑥𝑖𝛽
)
𝜃

(
𝑒𝑥𝑖𝛽

𝜃+𝑒𝑥𝑖𝛽
)

𝑦𝑖

.                  (15) 

 
The likelihood function of Negative Binomial is stated as below, 

𝐿 = ∏
Γ(𝜃+𝑦𝑖)

𝑦𝑖!Γ(𝜃)
𝑁
𝑖=1 (

𝜃

𝜃+𝑒𝑥𝑖𝛽
)
𝜃

(
𝑒𝑥𝑖𝛽

𝜃+𝑒𝑥𝑖𝛽
)

𝑦𝑖

,                   (16) 

and the log-likelihood function of Negative Binomial regression is obtained by applying 
the logarithm to obtain the following equation: 
 

         







ln1lnln1ln)
1

ln(
1


















 


ii

i

iii

N

i

yy
x

yxyyl
e .              (17) 



A Study of Count Regression Models for Mortality Rate 

Anwar Fitrianto 146 

Negative Binomial 1 Model and Negative Binomial P Model 

[18] have shown the Equation (9) is considered as a Negative Binomial 2 (NEGBIN 2) 
model. They re-parameterized the NEGBIN 2 model, and it is labeled as specification 
Negative Binomial 1 (NEGBIN 1), which is written as, 

𝑉𝑎𝑟(𝑦𝑖|𝑥𝑖) = 𝜆𝑖 + 𝑘𝜆𝑖 = 𝑉𝑎𝑟(𝑦𝑖|𝑥𝑖) = 𝜆𝑖(1 + 𝑘).                  (18) 

The marginal probability of NEGBIN 1 is obtained by replacing 𝛳 with 𝛳𝜆𝑖 in Equation 
(11), 

𝑃(𝑌 = 𝑦𝑖|𝑥) =
Γ(𝜃𝜆𝑖+𝑦𝑖)

𝑦𝑖!Γ(𝜃𝜆𝑖)
(

𝜃𝜆𝑖

𝜃𝜆𝑖+𝜆𝑖
)
𝜃

(
𝜆𝑖

𝜃𝜆𝑖+𝜆𝑖
)

𝑦𝑖
 .                  (19) 

By replacing 𝜃 with 𝜃𝜆𝑖
2−𝑃 in Equation 19, the Negative Binomial P (NEGBIN P) model is 

written as: 

𝑃(𝑌 = 𝑦𝑖|𝑥𝑖) =
𝛤(𝜃𝜆𝑖

2−𝑃+𝑦𝑖)

𝑦𝑖!𝛤(𝜃𝜆𝑖
2−𝑃)

(
𝜃𝜆𝑖

2−𝑃

𝜃𝜆𝑖
2−𝑃+𝜆𝑖

)
𝜃𝜆𝑖

2−𝑃

(
𝜆𝑖

𝜃𝜆𝑖
2−𝑃+𝜆𝑖

)
𝑦𝑖

 .                 (20) 

 

Overdispersion  

[19] have proposed that in almost the statistical study for the count data, it is always 
assumed that the dependent variable follows the Poisson distribution. The mean is 
assumed to be equal to the variance. However, in real life, the variance is usually larger 
than the mean. [19] also stated that overdispersion indicates high variability around a 
model's fitted values in the Poisson formulation. This case will lead to a Negative Binomial 
model as a proposal to correct this problem. 

When the data are over-dispersed, the variance is not the same as its mean, or 
𝑉𝑎𝑟(𝑥𝑖) = 𝜑𝜆, where 𝜆 is the mean.  If  𝜑 = 1, the Poisson model is ordinary; if 𝜑 > 1, it 
means that the model is overdispersed model. Consequently, [20] stated that a unique 
property of distributions in exponential families is the conditional variance equal the 
conditional mean. The dispersion parameter, 𝜑. In the Poisson model, the dispersion 
parameter is set to constant value 𝜑 = 1. 
 

Count Data 

According to [16], count data indicates how many times or how frequent something 
happens. Furthermore, [18] stated that an event outcome is the number of times an event 
occurs while an event count is a nonnegative random variable. The examples of count data 
included the number of patients hospitalized, the number of thieves arrested, and the 
number of natural disasters.  

In some cases of count, data have offset variables. [21] said that offset variable is 
always being analyzed by the generalized linear model (GLM) and count regression 
model. The analysis is usually used whenever the data is recorded over an observed 
period. Offset is used to denote the period observed in GLM. Other than that, offset is 
usually defined as a measure of exposure. The exposure can be the number of house years 
incurred, and the response will be the number of claims incurred.  

The log-linear mean rate for Poisson regression and Negative Binomial model is, 
𝑙𝑜𝑔(𝜆𝑖) = 𝛽0 + 𝛽1𝑥1+. . . +𝛽𝑝𝑥𝑝,                    (21) 

when applying Poisson regression or Negative Binomial regression, the offset variable, 
𝑙𝑜𝑔(𝑡) is added  



A Study of Count Regression Models for Mortality Rate 

Anwar Fitrianto 147 

𝑙𝑜𝑔(𝜆𝑖) = 𝛽0 + 𝛽1𝑥1 + ⋯ + 𝛽𝑝𝑥𝑝 + 𝑙𝑜𝑔( 𝑡)                         (22) 

𝑙𝑜𝑔(𝜆𝑖) − 𝑙𝑜𝑔( 𝑡) = 𝛽0 + 𝛽1𝑥1+. . . +𝛽𝑝𝑥𝑝 

𝑙𝑜𝑔 (
𝜆𝑖

𝑡
) = 𝛽0 + 𝛽1𝑥1+. . . +𝛽𝑝𝑥𝑝.        

𝑙𝑜𝑔(𝜆𝑖) − 𝑙𝑜𝑔( 𝑡) = 𝛽0 + 𝛽1𝑥1+. . . +𝛽𝑝𝑥𝑝 

𝑙𝑜𝑔 (
𝜆𝑖

𝑡
) = 𝛽0 + 𝛽1𝑥1+. . . +𝛽𝑝𝑥𝑝,                    

 
where  p is the number of predictors or covariates in the model, 𝛽0 is the intercept of the 
regression, 𝛽𝑝 are the coefficients of the regression, 𝑥 is the independent variable, t is the 

period observed (exposure), log (t) is the offset variable and  
𝜆𝑖

𝑡
 is the rate. 

In this study, our interest is in modeling for the mortality data, which is count data. 
Poisson regression and Negative Binomial regression are generally appropriate to deal 
with the count data. In this research, our interest is to find out which regression best fits 
the mortality data.  

 

Modelling the Mortality Rate Data 

Poisson regression and Negative Binomial regression are the main study in this 
research in modeling the data. The model for Poisson model and Negative Binomial model 
are written as Equation (22), where p is the number of predictors or covariates in the 
model, 𝛽0 is the intercept of the regression, 𝛽𝑝 is the covariate coefficients, and 𝑥 is the 

independent variable. The 𝑙𝑜𝑔 (
𝜆𝑖

𝑡
) represents the number of people dying per time unit 

and the function βx is the relationship of death rate changes as a function of subject 
covariates. The null hypothesis states the slope is equal to zero, whereas the alternative 
hypothesis indicates the slope is not equal to zero.  
 

Goodness-of-fit Test 

Deviance and Person's Chi-Square will be carried out to check if the data has over-
dispersion or under-dispersion. The results of deviance and Pearson's Chi-Square that are 
divided by the degree of freedom (df) should be approximately equal to one. If the values 
are more than one, this indicates that the data is overdispersion. Goodness-of-fit is 
performed by using the PROC GENMOD statement in SAS.  Deviance for fitted Poisson 
regression and Negative Binomial regression is written as: 

 

𝐷 = 2 ∑ {𝑦𝑖𝑙𝑜𝑔 (
𝑥𝑖

𝑦𝑖
) − (𝑥𝑖 − 𝜆𝑖)}

𝑛
𝑖=1 .                                (23) 

And the Pearson’s Chi-Square is defined as,  

𝜒2 = ∑
(𝑥𝑖−𝜆𝑖)

2

𝑉𝑎𝑟(𝑥𝑖)
𝑛
𝑖=1 ,                      (24) 

where  𝜆𝑖 = 𝑒
𝛽0+𝛽1𝑥1+...+𝛽𝑝𝑥𝑝 . 



A Study of Count Regression Models for Mortality Rate 

Anwar Fitrianto 148 

RESULTS AND DISCUSSION  

Mortality Rate Data Models 

PROC GENMOD statement in SAS version 9.4 was used to run the Poisson regression 
analysis. At 5% level of significance, all independent variables contributed significantly to 
the mortality rate with the following estimated Poisson regression model (Table 1): 

(
𝑙𝑜𝑔(𝜆𝑖)

𝑡
)

̂
=6.5834 + 0.0008𝑥1+ 0.0039𝑥2+0.0010𝑥3+0.004𝑥4-0.003𝑥5– 0.0123𝑥6+0.0081𝑥7 

 
Table 1. Analysis of Maximum Likelihood Parameter Estimates for Poisson Regression 

Parameter 
Degree of 
Freedom 

Estimate 
Standard 

Error 
Chi-Square 

Pr > Chi-
Square 

Intercept 1 6.5834 0.0083 6255069.00 <.0001 
𝑥1 1 0.0008 0.0000 507.71 <.0001 
𝑥2 1 0.0039 0.0002 312.25 <.0001 
𝑥3 1 0.0010 0.0000 1787.60 <.0001 
𝑥4 1 0.0046 0.0002 529.74 <.0001 
𝑥5 1 -0.0032 0.0003 95.57 <.0001 
𝑥6 1 -0.0123 0.0004 1092.88 <.0001 
𝑥7 1 0.0081 0.0004 463.89 <.0001 

 

The estimated Poisson model, along with the standard error of each estimated 
coefficient and p values, indicated that the IHD, diarrheal disease, AIDS/HIV, malaria, 
malnutrition, road accidents and suicides were significant predictors contributing to the 
mortality rate.  As an alternative to the Poisson regression model, the data were also 
analyzed using the Negative Binomial model. Table 2 displays the result of the analysis 
based on maximum likelihood estimation for the Negative Binomial regression.  
 

Table 2. Analysis of Maximum Likelihood Parameter Estimates for Negative Binomial Regression 

Parameter 
Degree of 
Freedom 

Estimate 
Standard 

Error 
Chi-Square 

Pr > Chi -
Square 

Intercept 1 6.5602 0.00875 5616.57 <.0001 
𝑥1 1 0.0008 0.0004 4.02 0.0451 
𝑥2 1 0.0046 0.0026 3.16 0.0755 
𝑥3 1 0.0011 0.0003 13.13 0.0003 
𝑥4 1 0.0054 0.0022 5.95 0.0147 
𝑥5 1 -0.0041 0.0038 1.19 0.2750 
𝑥6 1 -0.0138 0.0037 13.88 0.0002 
𝑥7 1 0.0112 0.0045 6.13 0.0133 

 

Fitting the data using the Negative Binomial regression model found that all 
independent variables are except 𝑥2 (diarrheal disease) and 𝑥5(malnutrition) contribute 
significantly to the mortality rate. Both variables have a more considerable p value 
(0.0755 for diarrheal diseases and 0.2750 for malnutrition). Hence, diarrheal disease and 
malnutrition were not significant predictors, while the other variables IHD, AIDS/HIV, 
malaria, road accidents, and suicides, were the significant predictors. The predicted 
model using the Negative Binomial regression model for the mortality rate data is written 
as, 

(
𝑙𝑜𝑔(𝜆𝑖)

𝑡
)

̂
=6.5602+0.0008𝑥1+0.0046𝑥2+0.0011𝑥3+0.0054𝑥4-0.0041𝑥5-.0138𝑥6+0.0112𝑥7 

 



A Study of Count Regression Models for Mortality Rate 

Anwar Fitrianto 149 

Descriptive Statistics of the Variables for Checking Overdispersion 

When the variance of a particular variable is higher than its mean, it indicates that the 
data has overdispersion. In this study, the dependent variable's mean and variance were 
824.0061 and 105125.22, respectively, indicating overdispersion. Table 3 displays the 
means and variances of all the variables in the study. All the variables were over-
dispersed and more considerable variability was given around a model's fitted values in 
Poisson regression, 𝑉𝑎𝑟(𝑥𝑖) =𝜑𝜆, 𝜑 >1. As a consequence, the Negative Binomial 
regression was the better approach for modeling over-dispersed count data.  
 

Table 3. The Mean and the Variance for Each Variable 
Variable Mean Variance 

Mortality 824.0061 105125.22 
IHD 114.5747 6031.84 
Diarrhoel Disease 22.0791 1118.02 
AIDS/HIV 46.6652 11754.18 
Malaria 11.5688 481.2379 
Malnutrition 12.7489 477.6099 
Road Accidents 17.2767 91.6093 
Suicides 10.0120 52.6231 

 

Goodness-of-fit Test for Poisson Regression and Negative Binomial Regression 

The main purpose of the goodness-of-fit test is to determine a more appropriate 
model. Table 4 presents the deviance and Pearson's Chi-Square to observe whether the 
deviance and Pearson's Chi-Square obtained close to one.  
 

Table 4. Goodness-of-fit Test for Poisson Regression and Negative Binomial Regression 
Model Criterion df Value Value/df 

Poisson Regression Deviance 155 15081.6228 97.3008 

Pearson's Chi-Square 155 14196.5148 91.5904 

Negative Binomial Regression Deviance 155 167.1663 1.0785 

Pearson's Chi-Square 155 131.1002 0.8458 

 

The value/df column of Deviance and Pearson's Chi-Square for the Poisson model 
were 97.3008 and 91.5904, respectively, which were remarkably higher than one. The 
Poisson model did not correctly describe the data. There was more significant variability 
among counts than will be expected for Poisson distribution. This situation arises because 
repeated subjects may not be independent.  One of the possible reasons for the 
overdispersion is that experimental conditions are not under control, hence 𝜆𝑖  varies with 
uncontrolled factors. The table shows that the Negative Binomial regression was the 
better alternative to model the mortality rate. The value/df of the Deviance and Pearson's 
Chi-Square were 1.0785  and 0.8458, respectively. Both values were closer to one as 
compared to the corresponding values in the Poisson regression model.  

 

Comparison between Poisson Regression, Negative Binomial 1 and Negative 
Binomial 2.  

 

Comparisons between all the three proposed models for the mortality data were given in 
Table 5. The AIC for Poisson regression was larger compared to the other two. The AIC 
value for NEGBIN 1 was slightly smaller than the one for NEGBIN 2. It indicated that 
NEGBIN l was a better fit than Poisson regression and NEGBIN 2. On the other hand, the 
BIC values for the three regressions were 16501, 2345, and 2347, respectively, for 



A Study of Count Regression Models for Mortality Rate 

Anwar Fitrianto 150 

Poisson, NEGBIN 1, and NEGBIN 2. The BIC value for Poisson regression was much higher 
when compared to the Negative Binomial regressions. Thus, with lower AIC and BIC 
values,  the NEGBIN 1 was the better approach for the mortality rate data since it can 
explain more variation with the same number of independent variables..  
 

Table 5. AIC and BIC Values Between Fifferent Regression Models 
Regressions AIC BIC 

Poisson  16476 16501 

NEGBIN 1 2317 2315 

NEGBIN 2 2319 2347 

 

CONCLUSIONS 

The analysis was conducted to compare the performance of three models: Poisson 
regression, NEGBIN 1 and NEGBIN 2. The NEGBIN 1 has been proven that it is the most 
appropriate model for overdispersed data. The mean and the variance were calculated to 
ensure that data has overdispersion. Since the data were overdispersed, the results of 
deviance and Pearson's Chi-Square showed that Negative Binomial was a better model for 
the data. Then, the performance of AIC and BIC showed that NEGBIN 1 is a better model, 
followed by NEGBIN 2 and Poisson regression. 

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