Zero-state Markov switching count-data models: an empirical assessment.

Abstract

In this study, a two-state Markov switching count-data model is proposed as an alternative to zero-inflated models to account for the preponderance of zeros sometimes observed in transportation count data, such as the number of accidents occurring on a roadway segment over some period of time. For this accident-frequency case, zero-inflated models assume the existence of two states: one of the states is a zero-accident count state, which has accident probabilities that are so low that they cannot be statistically distinguished from zero, and the other state is a normal-count state, in which counts can be non-negative integers that are generated by some counting process, for example, a Poisson or negative binomial. While zero-inflated models have come under some criticism with regard to accident-frequency applications - one fact is undeniable - in many applications they provide a statistically superior fit to the data. The Markov switching approach we propose seeks to overcome some of the criticism associated with the zero-accident state of the zero-inflated model by allowing individual roadway segments to switch between zero and normal-count states over time. An important advantage of this Markov switching approach is that it allows for the direct statistical estimation of the specific roadway-segment state (i.e., zero-accident or normal-count state) whereas traditional zero-inflated models do not. To demonstrate the applicability of this approach, a two-state Markov switching negative binomial model (estimated with Bayesian inference) and standard zero-inflated negative binomial models are estimated using five-year accident frequencies on Indiana interstate highway segments. It is shown that the Markov switching model is a viable alternative and results in a superior statistical fit relative to the zero-inflated models.

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arXiv:0811.3639v2 [stat.AP] 2 Aug 2009
Zero-state Markov switching count-data
models: an empirical assessment
Nataliya V. Malyshkina∗and Fred L. Mannering
School of Civil Engineering, 550 Stadium Mall Drive, Purdue University, West
Lafayette, IN 47907, United States
Abstract
In this study, a two-state Markov switching count-data model is proposed as an alter-
native to zero-inflated models to account for the preponderance of zeros sometimes
observed in transportation count data, such as the number of accidents occurring on
a roadway segment over some period of time. For this accident-frequency case, zero-
inflated models assume the existence of two states: one of the states is a zero-accident
count state, in which accident probabilities are so low that they cannot be statisti-
cally distinguished from zero, and the other state is a normal count state, in which
counts can be non-negative integers that are generated by some counting process,
for example, a Poisson or negative binomial. In contrast to zero-inflated models,
Markov switching models allow specific roadway segments to switch between the
two states over time. An important advantage of this Markov switching approach
is that it allows for the direct statistical estimation of the specific roadway-segment
state (i.e., zero or count state) whereas traditional zero-inflated models do not. To
demonstrate the applicability of this approach, a two-state Markov switching nega-
tive binomial model (estimated with Bayesian inference) and standard zero-inflated
negative binomial models are estimated using five-year accident frequencies on Indi-
ana interstate highway segments. It is shown that the Markov switching model is a
viable alternative and results in a superior statistical fit relative to the zero-inflated
models.
Key words: Accident frequency count data models; zero-inflated models; negative
binomial; Markov switching; Bayesian; MCMC
∗Corresponding author.
Email addresses: nmalyshk@purdue.edu (Nataliya V. Malyshkina),
flm@ecn.purdue.edu (Fred L. Mannering).
Preprint submitted to Accident Analysis and Prevention2 August 2009

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1Introduction
The preponderance of zeros observed in many count-data applications has lead
researchers to consider the possibility that two states exist; one state that is a
“zero” state (where all counts are zero) and the other that is a normal count
state that includes zeros and positive integers. This two-state assumption has
led to the development of zero-inflated Poisson models and zero-inflated neg-
ative binomial models to account for possible overdispersion in the normal-
count state. These zero-inflated models have been applied to a number of fields
of study. For example, Lambert (1992) used a zero-inflated Poisson model to
study manufacturing defects. Lambert argued that unobserved changes in the
process caused manufacturing defects to move randomly between a state that
was near perfect (the zero state where defects were extremely rare) and an im-
perfect state where defects were possible but not inevitable (the normal count
state). Lamberts empirical assessment demonstrated that the zero-inflated
modeling approach fit the data much better than the standard Poisson. In
other work, van den Broek (1995) provided an application of the zero-inflated
Poisson to the frequency of urinary tract infections in men diagnosed with the
human immunodeficiency virus (HIV). In this case, it was postulated that a
zero-infection state existed for a portion of the patient population and that
this state generated a large number of zeros in the frequency data, which was
supported by the statistical findings. Also, Bohning et al. (1999) successfully
applied the zero-inflated Poisson to study the frequency of dental decay in
Portugal.
The frequency of vehicle accidents on a section of highway or at an intersection
(over some time period) often exhibit excess zeros. Similar to the literature
discussed above, the excess of zeros observed in the data could potentially be
explained by the existence of a two-state process for accident data genera-
tion (Shankar et al., 1997; Carson and Mannering, 2001; Lee and Mannering,
2002). In this case, roadway segments can belong to one of two states: a
zero-accident state (where zero accidents are expected) and a normal-count
state, in which accidents can happen and accident frequencies are generated
by some given counting process (Poisson or negative binomial). To account for
the two-state phenomena, zero-inflated Poisson (ZIP) and zero-inflated nega-
tive binomial (ZINB) models have been used in a number of roadway safety
studies (Miaou, 1994; Shankar et al., 1997; Washington et al., 2003). These
models explicitly account for an existence of the two states for accident data
generation and allow modeling of the probabilities of being in these states.
An application of ZIP and ZINB models was an empirical advance in statisti-
cal modeling of accident frequencies. However, although zero-inflated models
have become popular in a number of fields, they suffer from two important
drawbacks. First, these models do not deal directly with the states of road-
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way segments, instead they consider probabilities of being in these states. As
a result, zero-inflated models do not allow a direct statistical estimation of
whether individual roadway segments are in the zero or normal count state.
For example, suppose a given roadway segment has zero accidents observed
over a given time interval. This segment could truly be in the zero-accident
count state, or it may be in the normal-count state and just happened to have
zero accidents over the considered time interval (Shankar et al., 1997). Distin-
guishing between these two possibilities is not straightforward in zero-inflated
models. The second drawback of zero-inflated models is that, although they
allow roadway segments to be in different states during different observation
periods, zero-inflated models do not explicitly consider switching by the road-
way segments between the states over time. This switching is important from
the theoretical point of view because it is unreasonable to expect any roadway
segment to be in the zero-accident all the time and to have the long-term
mean accident frequency equal to zero (Lord et al., 2005).
In this study, we propose two-state Markov switching count-data models that
consider the zero-accident state and the normal-count state of roadway safety.
Similar to zero-inflated models, Markov switching models are intended to ex-
plain the preponderance of zeros observed in accident count data. However,
in contrast to zero-inflated models, Markov switching models allow a direct
statistical estimation of the states roadway segments are in at specific points
in time and explicitly consider changes in these states over time.
2Model specification
Two-state Markov switching count-data models of accident frequencies were
first presented in Malyshkina et al. (2009). Following that paper, we note that,
although there are several major differences between Malyshkina et al. (2009)
and this study, many ideas and statistical estimation methods developed in
Malyshkina et al. (2009) apply in this study as well. In that paper, two states
were assumed to exist but both were true count states (i.e., a zero-count
state did not exist). In the current paper, we take a different approach and
consider the case where one of the states is a zero state and the other is a
true count state and that individual roadway segments move between these
two states over time. This differs from Malyshkina et al. (2009) in that their
model assumes two true-count states and that all roadway segments are in the
same state at the same time.
To show this model, we note that Markov switching models are parametric
and can be fully specified by a likelihood function f(Y|Θ,M), which is the
conditional probability distribution of the vector of all observations Y, given
the vector of all parameters Θ of model M. In our study, we observe the
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number of accidents At,nthat occur on the nthroadway segment during time
period t. Thus Y = {At,n} includes all accidents observed on all roadway
segments over all time periods. Here n = 1,2,...,N and t = 1,2,...,T,
where N is the total number of roadway segments observed (it is assumed to
be constant over time) and T is the total number of time periods. Model M =
{M,Xt,n} includes the model’s name M (for example, M = “ZIP” or “ZINB”)
and the vector Xt,nof all roadway segment characteristic variables (segment
length, curve characteristics, grades, pavement properties, and so on).
To define the likelihood function, we introduce an unobserved (latent) state
variable st,n, which determines the state of the nthroadway segment during
time period t. Without loss of generality, it is assumed assume that the state
variable st,n can take on the following two values: st,n = 0 corresponds to
the zero-accident state, and st,n = 1 corresponds to the normal-count state
(n = 1,2,...,N and t = 1,2,...,T). It is further assumed that, for each road-
way segment n, the state variable st,nfollows a stationary two-state Markov
chain process in time,1which can be specified by time-independent transition
probabilities as
P(st+1,n= 1|st,n= 0) = p(n)
0→1,P(st+1,n= 0|st,n= 1) = p(n)
1→0.(1)
Here, for example, P(st+1,n = 1|st,n = 0) is the conditional probability of
st+1,n= 1 at time t + 1, given that st,n= 0 at time t. Transition probabilities
p(n)
(n = 1,2,...,N). The stationary unconditional probabilities of states st,n= 0
and st,n= 1 are ¯ p(n)
0
= p(n)
respectively.2If p(n)
0
segment n state st,n= 0 occurs more frequently than state st,n= 1. If p(n)
p(n)
0→1and p(n)
1→0are unknown parameters to be estimated from accident data
1→0/(p(n)
1→0, then ¯ p(n)
0→1+ p(n)
1→0) and ¯ p(n)
> ¯ p(n)
1
1
= p(n)
0→1/(p(n)
0→1+ p(n)
1→0)
0→1< p(n)
and, on average, for roadway
0→1>
1→0, then state st,n= 1 occurs more frequently for segment n.3
Next, consider a two-state Markov switching negative binomial (MSNB) model
that assumes a negative binomial (NB) data-generating process in the normal-
count state st,n= 1. With this, the probability of At,naccidents occurring on
roadway segment n during time period t is
1Markov property means that the probability distribution of st+1,ndepends only
on the value st,nat time t, but not on the previous history st−1,st−2,.... Stationarity
of {st,n} is in the statistical sense.
2These can be found from stationarity conditions ¯ p(n)
¯ p(n)
1010
3Here, Eq. (1) is a significant departure from Malyshkina et al. (2009) in that in-
dividual roadway segments can be in different states at the same time (i.e., the state
variable is subscripted by roadway segment n). Also, in contrast to Malyshkina et al.
(2009), here we do not restrict state st,n= 0 to be more frequent than state st,n= 1.
0
= [1−p(n)
0→1]¯ p(n)
0
+p(n)
1→0¯ p(n)
1,
= p(n)
0→1¯ p(n)
+ [1 − p(n)
1→0]¯ p(n)
and ¯ p(n)
+ ¯ p(n)
1
= 1.
4

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Fig. 1. Graphical demonstration of a two-state Markov switching model.
P(A)
t,n =

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


I(At,n)if st,n= 0
NB(At,n) if st,n= 1
,(2)
I(At,n) = {1 if At,n= 0 and 0 if At,n> 0},(3)
NB(At,n) =Γ(At,n+ 1/α)
Γ(1/α)At,n!
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1
1 + αλt,n
?1/α?
αλt,n
1 + αλt,n
?At,n
,(4)
λt,n= exp(β′Xt,n),t = 1,2,...,T,n = 1,2,...,N.(5)
Here, Eq. (3) is the probability mass function that reflects the fact that acci-
dents never happen in the zero-accident state st,n= 0.4Eq. (4) is the standard
negative binomial probability mass function, Γ( ) is the gamma function, and
prime means transpose (so β′is the transpose of β). Parameter vector β and
the over-dispersion parameter α ≥ 0 are unknown estimable model parame-
ters.5Scalars λt,nare the accident rates in the normal-count state. We set
the first component of Xt,nto unity, and, therefore, the first component of β
is the intercept.
A two-state Markov switching model of accident frequencies is graphically
demonstrated in Figure 1. In the two states s = 0 and s = 1 shown in the
figure, the accident frequency data are generated by two different processes,
shown by the circles (for state s = 0) and the diamonds (for s = 1). In this
study, we assume that accident frequency is generated according to the zero-
accident distribution I(At,n) in state s = 0, and according to the standard
4Although Eq. (3) formally assumes st,n= 0 to be a zero-accident state, in which
accidents never happen, this state can be viewed as an approximation for a nearly
safe state, in which the average accident rate is negligible (λt,n≪ 1) and accidents
are extremely rare (over the considered time period).
5To ensure that α is non-negative, we estimate its logarithm instead of it.
5