Elements of Statistics and Crash Count Distributions

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Elements of Statistics and Crash Count
Distributions

• Spring 2007

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Theoretical Process of Motor Vehicle Crashes
Each time a vehicle enters an intersection, a
highway segment, or any other type of entity (a
trial) on a given transportation network, it will
either crash or not crash. For purposes of
consistency a crash is termed a success while
failure to crash is a failure. For the
Bernoulli trial, a random variable, defined as X,
can be generated with the following probability
model if the outcome w is a particular event
outcome (e.g. a crash), then X (?) 1 whereas if
the outcome is a failure then X (?) 0. Thus,
the probability model becomes
where p is the probability of success (a crash)
and q(1-p) is the probability of failure (no
crash).
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Binomial distribution
It can be shown that if there are N independent
trials (vehicle passing through an intersection,
road segment, etc.), the count of successes over
the number of trials give rise to a Bernoulli
distribution. Well define the term Z as the
number of successes over the N trials. Under the
assumption that all trials are characterized by
the same failure process (this assumption is
revisited later), the appropriate probability
model that accounts for a series of Bernoulli
trials is known as the binomial distribution, and
is given as
Equation 1
Where, n 0, 1, 2, , N (the number of
successes or crashes)
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Poisson Approximation
For typical motor vehicle crashes where the event
has a very low probability of occurrence and a
large number of trials exist (e.g. million
entering vehicles, vehicle-miles-traveled, etc.),
it can be shown that the binomial distribution is
approximated by a Poisson distribution. Under the
Binomial distribution with parameters N and p,
let p?/N , so that a large sample size N will
be offset by the diminution of p to produce a
constant mean number of events ? for all values
of p. Then as N - 8, it can be shown that
Equation 2
Where, n 0, 1, 2, , N (the number of
successes or crashes) ? the mean of a Poisson
distribution
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Poisson Approximation
The approximation illustrated in Equation (2)
works well when the mean ? and p are assumed to
be constant. In practice however, it is not
reasonable to assume that crash probabilities
across drivers and across road segments
(intersections, etc.) are constant. Specifically,
each driver-vehicle combination is likely to have
a probability that is a function of driving
experience, attentiveness, mental workload, risk
adversity, vision, sobriety, reaction times,
vehicle characteristics, etc. Furthermore, crash
probabilities are likely to vary as a function of
the complexity and traffic conditions of the
transportation network (road segment,
intersection, etc.). All these factors and others
will affect to various degrees the individual
risk of a crash.
These and other characteristics affecting the
crash process create inconsistencies with the
approximation illustrated in Equation (2).
Outcome probabilities that vary from trial to
trial are known as Poisson trials (note Poisson
trials are not the summation of independent
Poisson distributions this term is used to
designate Bernoulli trials with unequal
probability of events).
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Poisson Approximation
The equation below is used for determining if the
unequal event of independent probabilities can be
approximated by a Poisson process.
Equation 3
Where, dTV total variance between the two
probabilities measured L(Z) and Po(?) L(Z)
count data generated by unequal probability of
events Po(?) count data generated by unequal
events of independent probabilities with ?E(Z).
See Barbour et al. (1992) Poisson Approximation.
Clarendon Press, New York, NY for additional
information.
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Poisson Approximation
The equation below is used for determining if the
unequal event of independent probabilities leads
to over-dispersion, VAR(Z) gt E(Z).
If
Then
For any r gt 2, where
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Crash Data as Poisson Model
Given the characteristics described in the
previous overheads, it is often assumed that
crash data on a given site (or entity) follow a
Poisson a distribution. In other words, if one
were to count data over time for one site, the
data are assumed to be Poisson distributed.
Example
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7
0
3
1
4
Crash Count
Time t
1
2
3
4
i
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Poisson assumption Where, ? Mean of the
Poisson distribution x Crash count (0, 1, 2, )
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Crash Data as Poisson Model
If we have counts 3, 7, 0, and 3 on an entity,
what is ??
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Crash Data as Poisson Model
We can plot P(3, 7, 0, and 3) as a function of ?
(the likelihood function)
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Crash Data as Poisson Model
We can plot P(3, 7, 0, and 3) as a function of ?
(the likelihood function)
is maximum at
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Crash Data as Poisson Model
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Crash Data as Poisson Model
Accuracy of estimation (?)
Counts and Predicted Values
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Crash Data as Poisson Model
Crash counts of entities which have recorded x
crashes per unit of time
Finding the mean
Where
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Overdispersion (aka heterogeneity)
Crash data can rarely be exhibited as pure
Poisson distribution. Usually, the data display a
variance that is greater than the mean, VAR(?) gt
E(?). This is known as over-dispersion. The
principal cause of over-dispersion was explained
in the previous overheads (Bernoulli process with
unequal probability of events). Over-dispersion
can also caused by numerous factors. For other
types of processes (not based on a Bernoulli
trial) over-dispersion can be explained by the
clustering of data (neighborhood, regions, wiring
boards, etc.), unaccounted temporal correlation,
and model mis-specification. These factors also
influences the heterogeneity found in crash data.
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Overdispersion (aka heterogeneity)
In order to account for over-dispersion commonly
found in crash data, it has been hypothesized
that the mean (?) found in a population of sites
follows a two-parameter Gamma probability density
function. In other words, if we have a population
of entities (say 100 intersections) their mean ?s
(if everything else remain constant) would follow
a Gamma distribution. The Gamma probability
density function if defined by
for ? gt 0 Where, ? the mean of the
selected site F, d parameters of the Gamma
distribution Gamma function (?e-u
u(F-1) du)
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Overdispersion (aka heterogeneity)
There are three reasons why the Gamma probability
function has been a popular assumptions 1. The
mean ? is allowed only to take a positive
value 2. The Gamma PDF is very flexible it can
move and stretched to fit a variety of shapes
and 3. It makes the algebra simple and often
yields closed form results. Note Nobody has
proved so far that the mean varies according to a
Gamma probability model. People use it because it
is easy to manipulate. Some researchers have used
a lognormal distribution. You can also use more
complicated distributions.
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Overdispersion (aka heterogeneity)
The mean and variance of Gamma probability
density function can be estimated as
follows To estimate F and d from data,
you can use the following equations
and s2 are estimated using the equations shown
above.
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Overdispersion (aka heterogeneity)
Note for the Poisson-gamma assumption to hold, F
and d must be equal This is known as the
one parameter gamma distribution. The
relationships shown above will become useful for
describing the mean and variance of the negative
binomial distribution.
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Negative Binomial (or Poisson-gamma)
It can be shown that if the mean of a Poisson
distribution is Gamma distributed, the joint
mixed distribution gives rise to the Negative
Binomial distribution. The derivation is shown as
follows
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Negative Binomial (or Poisson-gamma)
ALTERNATIVE FORMS
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Negative Binomial (or Poisson-gamma)
The mean and variance of the Negative Binomial
distribution are estimated using the following
equations Note if F -gt 8, the second part
of the variance function tend towards 0. This
means that the Negative Binomial becomes a
Poisson distribution since the mean and variance
are now equal. Note For modeling purposes, the
term F is usually estimated directly from the
data. This will be addressed later in the course.
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Causal-Effect Relationship
Analysis method known as in-dept analysis or
clinical study. This method does not rely on the
statistical nature of the crash process. It seeks
to determine the deterministic mechanism that
lead to an accident or a crash. This method is
very common for extremely rare events, such
aircraft or space shuttle crashes. The data
needed to carry out such investigations and
reconstruction are very extensive. Often, it is
not available on typical computerized crash
records.
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Causal-Effect Relationship
Site Characteristics
Outcome
Speed Distribution
Speed Limit Policy
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Causal-Effect Relationship
An example analysis of de-icing roads and
traffic club for children using causal chain
approach. This study aims at estimating the
reduction in risk factors for each chain.
See Elvik (2003) in Accident Analysis
Prevention, Vol. 35, pp. 741-748
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De-icing Roads
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De-icing Roads
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Traffic Club