D
‐
1
Appendix D:
Negative Binomial Regression Models and Estimation Methods
By
Dominique Lord
Texas A&M University
ByungJung Park
Korea Transport Institute
This appendix presents the characteristics of Negative Binomial regression models and discusses
their estimating methods.
Probability Density and Likelihood Functions
The properties of the negative binomial models with and without spatial intersection are
described in the next two sections.
PoissonGamma Model
The PoissonGamma model has properties that are very similar to the Poisson model discussed in
Appendix C, in which the dependent variable
i
y
is modeled as a Poisson variable with a mean
i
where
the model error is assumed to follow a Gamma distribution. As it names implies, the PoissonGamma is a
mixture of two distributions and was first derived by Greenwood and Yule (1920). This mixture
distribution was developed to account for overdispersion that is commonly observed in discrete or count
data (Lord et al., 2005).
It became very popular because the conjugate distribution (same family of
functions) has a closed form and leads to the negative binomial distribution. As discussed by Cook
(2009), “the name of this distribution comes from applying the binomial theorem with a negative
exponent.” There are two major parameterizations that have been proposed and they are known as the
NB1 and NB2, the latter one being the most commonly known and utilized.
NB2 is therefore described
first. Other parameterizations exist, but are not discussed here (see Maher and Summersgill, 1996; Hilbe,
2007).
NB2 Model
Suppose that we have a series of random counts that follows the Poisson distribution:
;
!
i
i
ii
i
e
gy
y
(
D

1
)
where
i
y
is the observed number of counts for
1, 2,
i
n ; and
i
is the mean of the Poisson
distribution.
If the Poisson mean is assumed to have a random intercept term and this term enters the
conditional mean function in a multiplicative manner, we get the following relationship (Cameron and
Trivedi, 1998):
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1
0
0
1
0
1
exp
K
ij
j
j
i
K
ij
j
j
i
K
ii
j
j
i
j
x
i
x
i
i
x
ee
(
D

2
)
where,
0
exp
i
is defined as a random intercept;
0
1
exp
K
j
j
j
x
is the loglink
between the Poisson mean and the covariates or independent variables
xs
; and
s
are the regression
coefficients.
As discussed in Appendix C, the relationship can also be formulated using vectors, such
that
)
exp(
β
x
'
i
i
.
The marginal distribution of
i
y
can be obtained by integrating the error term,
i
,
;;
,
,
iii
i
i
o
f
yg
y
h
d
fy
E gy
(D3)
where
i
h
is a mixing distribution.
In the case of the PoissonGamma mixture,
;,
gy
is the
Poisson distribution and
i
h
is the Gamma distribution.
This distribution has a closed form and leads
to the NB distribution.
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 Summer '08
 Staff
 Normal Distribution, Regression Analysis, Probability theory, Maximum likelihood, Yi, yi ln yi

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