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Investigating the effects of the fixed and varying dispersion parameters of
Poisson-gamma models on empirical Bayes estimates
Dominique Lord, Ph.D., P.Eng.*
Assistant Professor
Department of Civil Engineering
Texas A&M University
3136 TAMU
College Station, TX 77843-3136
Tel. (979) 458-3949
Fax. (979) 845-6481
Email : d-lord@tamu.edu
Peter Young-Jin Park, Ph.D., P.Eng.
Transportation Engineer
iTRANS Consulting Inc.
100 York Boulevard, Suite 300
Richmond Hill, ON, Canada L4B 1J8
Tel: (905) 882-4100 ext.5264
Fax: (905) 882-1557
Email: ppark@itransconsulting.com
Paper submitted for publication
May 5th, 2007
* Corresponding author
* Manuscript
ABSTRACT
Traditionally, transportation safety analysts have used the empirical Bayes (EB) method to
improve the estimate of the long-term mean of individual sites; to correct for the regression-
to-the-mean (RTM) bias in before-after studies; and to identify hotspot or high risk locations.
The EB method combines two different sources of information: 1) the expected number of
crashes estimated via crash prediction models, and 2) the observed number of crashes at
individual sites. Crash prediction models have traditionally been estimated using a negative
binomial (NB) (or Poisson-gamma) modeling framework due to the over-dispersion
commonly found in crash data. A weight factor is used to assign the relative influence of each
source of information on the EB estimate. This factor is estimated using the mean and
variance functions of the NB model. With recent trends that illustrated the dispersion
parameter to be dependent upon the covariates of NB models as well as varying as a function
different time periods, there is a need to determine how this may affect EB estimates.
The objectives of this study are to examine how commonly used functional forms as
well as fixed and time-varying dispersion parameters affect the EB estimates. To accomplish
the study objectives, several crash prediction models were estimated using a sample of rural
three-legged intersections located in California. Two types of aggregated and time-specific
models were produced: 1) the traditional NB model with a fixed dispersion parameter and 2)
the generalized NB model (GNB) with a time-varying dispersion parameter, which is also
dependent upon the covariates of the model.
The results of the study show that the selection of the functional form of NB models
has an important effect on EB estimates both in terms of estimated values and dispersion
parameter. Time-specific models with a varying dispersion parameter provide better statistical
performance in terms of goodness-of-fit (GOF) than aggregated multi-year models. The
performance is even better when GNB are used both for time-specific and aggregated models.
Similar to past study findings, there might be no apparent benefits of introducing varying
dispersion parameters for identifying hotspots using the EB method. The study concludes that
transportation safety analysts should not automatically assume that existing functional forms
are adequate for modeling motor vehicle crashes and rigorous analyses should be used to
estimate the most appropriate functional form for linking crashes to explanatory variables,
including traffic flow.
Keywords: crash prediction models, dispersion parameter, empirical Bayes estimates,
negative binomial, rural intersections
2
INTRODUCTION
Statistical models or crash prediction models have been a very popular method for estimating
the safety performance of various transportation elements. The most common statistical
models used by transportation safety analysts are the Poisson and Negative Binomial (NB)
(or Poisson-gamma) regression models (Miaou, 1994; Pock and Mannering, 1996; Lord et al.,
2005a). NB models are usually the model of choice and have been applied extensively in
various types of highway safety studies, from the identification of hotspots or hazardous sites,
the prediction of motor vehicle collisions, to the development of accident modification factors
via the coefficients of the model (Harwood et al., 2000; Miaou, 1996; Vogt, 1999; Lord and
Bonneson, 2006). There are two main reasons why NB models are favored over Poisson
models for modeling motor vehicle collisions. First, the variance of the response variable (i.e.
crashes per unit of time) commonly exceeds the mean value of the variable, which violates
the main assumption associated with the Poisson model (i.e., “over-dispersion” phenomenon).
As a result, if a Poisson distribution is assumed in estimating the expected number of crashes,
larger discrepancies between the observed and the predicted crashes may be observed (Hauer,
2001). In addition, a mis-specified Poisson model may lead to the inclusion of covariates that
have been erroneously identified as being significant when, in fact, they are not (Park and
Lord, 2006). It has been reported that the over-dispersion is caused by some unmeasured
uncertainties associated with the unobserved or unobservable variables, resulting in the
omitted variable problem. However, although the latter problem can contribute to the over-
dispersion, it is mainly attributed to the nature of the crash process, namely the fact that
crashes are the product of Bernoulli trials with unequal probability of events (this is also
known as Poisson trials). Lord et al. (2005a) have reported that as the number of trials
increases and becomes very large, the distribution may be approximated by a Poisson process
(hence the use of Poisson-based or mixed-Poisson models), where the magnitude of the over-
dispersion is dependent on the characteristics of the Poisson trials. (Note: the over-dispersion
can be minimized using appropriate mean structures of statistical models, as discussed in
Miaou and Song, 2005, Mitra and Washington, 2007, and in the conclusions of this paper). In
short, the NB model can efficiently reduce these unmeasured uncertainties by allowing an
error term to capture the unmeasured heterogeneity in a study dataset (Miaou and Lord, 2003).
Therefore, in order to take into account the over-dispersion problem in a given study dataset,
transportation safety analysts normally adapt the NB modeling framework for developing
crash prediction models.
In highway safety, the dispersion parameter of NB models (note: some researchers
use the term over-dispersion parameter instead of the dispersion parameter) takes a central
role for calculating empirical Bayes (EB) estimates. These estimates are used to smooth the
random fluctuation of crash counts and generate a more accurate estimate of the long-term
mean at a given site. Inasmuch as the EB estimates are one of the main inputs for a sound EB
before-after study, the accuracy of EB estimates will definitely affect the precision of the
analysis output. The EB estimates can also be used to identify hotspots (see Saccommano et
al., 2001) or “sites with promise” (see Hauer, 1996) by ranking crash-prone locations by order
3
of magnitude or by computing the difference between the output of predictive models and the
EB estimate. As a result, rigorous statistical models based on an appropriate NB modeling
framework must be developed to obtain reliable EB estimates and to maximize the safety
benefit per dollar spent.
As discussed by Hauer (1997), the long-term mean for a site i over a period t can be
estimated using the EB method:
()
ˆ
ˆˆ
1
it it it it it
yµγγµ=−+ (1)
where,
it
γ= weight factor for given site i and year t;
it
y= observed number of crashes for given site i and year t;
ˆit
µ= the estimated number of crashes by crash prediction models for given site i and
year t (usually estimated using a NB model).
The weight factor it
γ
is given as follows:
()
ˆ
11
it it
γαµ=+
(2)
where,
α= the dispersion parameter for the given dataset [note: in the safety literature,
analysts have also used the inverse dispersion parameter 1φα=].
Up until very recently, researchers did not estimate time-varying EB estimates (as
currently defined in equations (1) and (2)); instead transportation safety analysts produced an
average EB estimates for the sites under study for the entire period by relying on a traditional
NB model (Harwood et al. 2000; Persaud et al., 2001; Vogt, 1999). The traditional NB model
uses a fixed dispersion parameter that is applied to the entire dataset in the study (Miaou,
1996). However, as pointed out by Hauer (2001), there is no tangible rationale that all sites in
the dataset should have a constant dispersion parameter over a given study period. Several
other researchers have also questioned the hypothesis that the dispersion parameter has a
fixed value over different sites and time-periods (Heydecker and Wu, 2001; Miaou and Lord,
2003; Lord et al., 2005b; Miranda-Moreno et al., 2005; El-Basyouny and Sayed, 2006).
Heydecker and Wu (2001) attempted to estimate varying dispersion parameters as a function
of sites’ covariates, such as AADT, lane and shoulder widths among others. They asserted that
the NB model with a varying dispersion parameter (henceforth defined as generalized NB
model or GNB) can better represent the nature of crash dataset than the traditional NB model
with a fixed dispersion parameter. The approach proposed by Heydecker and Wu (2001) was
also used by Lord et al. (2005) for modeling the safety performance of freeways as a function
of traffic flow characteristics. An exception is Lyon et al. (2005b) who introduced a time-
varying dispersion parameter using the traditional NB model. The dispersion parameter for
each year was estimated outside the model estimating process using the maximum likelihood
method.
4
There are many different functional forms that have been proposed to link crashes to
the explanatory variables of traditional regression models for segments (Martin, 2002; Abbas,
2004; Lord et al., 2005b) and intersections (Nicholson and Turner, 1996; Turner and
Nicholson, 1998; Mountain and Fawaz, 1998; Miaou and Lord, 2003). In the past, these
functional forms were adopted to determine the model that provided the best statistical fit
without considering the relationship between different functional forms and the dispersion
parameter (with the exception of Miaou and Lord, 2003). However, if the selection of a
functional form can influence the precision of the model estimates [i.e. the estimated number
of crashes; ˆit
µin equation (1)], then it may also influence the precision of estimated
dispersion parameters [i.e. α in equation (2)]. In evaluating the safety effects of treatments
(e.g., Persaud et al., 2001; Powers and Carson, 2004) and hotspot identification (e.g.,
Miranda-Moreno et al., 2005), the impact of the dispersion parameters on the EB estimates
calculated using the output of different functional forms merit further investigation. To
answer this question, this study has been motivated to address the following issues:
1) Develop a series of crash prediction models which can estimate a site and time-specific
number of crashes using both traditional NB and GNB models; estimate a fixed and
varying dispersion parameters across site i and time t; and, investigate the
characteristics of the dispersion in the data as a function of the selected covariates (i.e.,
traffic volumes).
2) Evaluate the statistical performance of crash prediction models using commonly used
goodness-of-fit (GOF) statistics as well as Cumulative Residual (CURE) plots.
3) Examine the relationship between the functional forms of crash prediction models and
the estimated dispersion parameters (i.e., both fixed and varying dispersion parameters).
4) Examine the differences on the EB estimates between the NB model with a fixed
dispersion parameter and GNB model with a varying dispersion parameter, and
compare both estimates in hotspot identification.
To accomplish the objectives of this study, several crash prediction models were
produced using a sample of three-legged rural intersections located in California. Crash,
traffic flow and geometric design data (to confirm the intersection geometry) were obtained
from the Highway Safety Information System (HSIS) managed by the University of North
Carolina in Chapel Hill, NC. For a given five-year (1997-2001) study period, a total of 5,752
three-legged rural intersections were included in the database. Intersections that contained
missing and questionable values were discarded from the dataset, resulting in a sample of
5,588 three-legged rural intersections for the same five-year period with a total of 5,996
reported crashes (all crash severities or the total number of crashes). Given the large sample
size, the inverse dispersion parameters in this study are assumed to be properly estimated (see
Lord, 2006 about this assumption). Table 1 contains a brief summary of study dataset.
MODEL DEVELOPMENT
This section describes the characteristics of the traditional NB and GNB models.
5
TRADITIONAL AND GENERALIZED NEGATIVE BINOMIAL MODELS
Properties of the traditional NB model have been illustrated by Cameron and Trivedi (1998).
The probability density function (pdf) of the NB distribution can be defined as
1
11
()
(|,) 11 1
!( )
ti
y
it it
it i it
it it it
y
PY y y
α
µ
αα
µα
µµ
αα α
⎛⎞⎛⎞
Γ+⎟⎟
⎜⎜
⎟⎟
⎜⎜
⎟⎟
== ⋅⋅
⎜⎜
⎟⎟
⎜⎜
⎟⎟
⋅Γ ++
⎜⎜
⎟⎟
⎝⎠⎝⎠
(3)
In contrast to the Poisson distribution, the NB distribution allows for over-dispersion,
and thus the mean (i.e.
{}
exp( )
it it it
EYµ==Xβ; i=Xa vector of covariates, and
=βregression coefficients corresponding to the covariates) can be smaller than the variance
(i.e.
{}
2
ititit
YVar
µαµ
⋅+= ) (Note: the data can also show signs of under-dispersion). When the
dispersion parameter 1αφ= is equal to zero, the NB distribution resorts back to the
Poisson distribution. Larger values of
α
signifies a greater amount of over-dispersion. An
important characteristic of the traditional NB model is that the dispersion parameter
α
(or its
inverse φ) would not vary from site to site. This type of model has only a single fixed value
over all observations without considering potential dependency on the covariates.
The GNB uses the same pdf shown in equation (3) and estimates the number of
crashes of each site, like the traditional NB model. However, instead of estimating a fixed
dispersion parameter, the model estimates varying dispersion parameters by using the
following expression (Hardin and Hilbe, 2001):
exp( )
it it t
Zδα=⋅ (4)
where,
Zit = a vector of secondary covariates (do not necessarily the same as the covariates
in estimating the mean function ˆit
µ),
t
δ = regression coefficients corresponding to covariates Zit.
With equation (4), the GNB model can be used for estimating a different over-
dispersion parameter according to the sites’ attributes (i.e., covariates). If there are no
significant secondary covariates for explaining the systematic dispersion structure, the
dispersion parameters will only contain a fixed value (i.e., constant term), resulting in a
traditional NB regression model.
6
In this study, STATA V.8.0 program (Stata, 2003) was used to estimate all the
coefficients of the traditional NB and GNB models, including the fixed and varying
dispersion parameters. In order to simplify the analysis, the serial correlation associated with
time-trend models was not included in this study. It should be pointed out that since the data
did not contain any missing values and because the model type is defined as a marginal model,
the coefficients of generalized linear models (GLM) are the same (or very similar) as the
values produced by the Generalized Estimating Equations (GEE), no matter which working
correlation matrix is used (or whether the correlation matrix is mis-specified). The only
difference is related to the standard errors of the coefficients. The standard errors are usually
underestimated when temporal effects are not included in the modeling process (see Lord and
Persaud, 2000 and Hardin and Hilbe, 2003 for additional information).
CRASH PREDICTION MODELS
Given the specific objectives of this study, instead of developing the models with the best
statistical fit considering every possible combination of covariates, only entering traffic
volumes (from major and minor intersecting roads) were used as covariates. Miaou and Lord
(2003) listed the most popular functional forms (referred in the study as Models 1 to 5) from
previously published studies:
1)
Model 1:
(
)
iitit FF 21ln
ˆ
ln 10
+
+=
β
β
µ
2)
Model 2: iitit FF 2ln1ln
ˆ
ln 210
β
β
β
µ
+
+=
3)
Model 3:
(
)
iitit FF 21ln
ˆ
ln 10
×
+=
β
β
µ
4)
Model 4:
(
)
(
)
iiiitit FFFF 12ln21ln
ˆ
ln 210
β
β
β
µ
+
+
+=
5)
Model 5:
iiitit FFF 22ln1ln
ˆ
ln 3210
β
β
β
β
µ
+
+
+=
where,
µit = the expected number of crashes at intersection i in year t;
F1i = average AADT over a given 5 years entering from major road at
intersection i; and,
F2i = average AADT over a given 5 years entering from minor road at
intersection i.
As reported by Miaou and Lord (2003), the functional forms described above are not
the most adequate for describing the relationship between crashes and exposure since the
forms do not appropriately fit the data near the boundary conditions. Nonetheless, they are
still relevant for this study, as they are considered established functional forms in the highway
7
safety literature. In addition, the most adequate functional form proposed by Miaou and Lord
(2003), a model with two distinct mean functions, cannot be estimated via a generalized
linear modeling (GLM) framework, as it was done in this study.
In this analysis, the approach proposed by Lyon et al.’s (2005) in which only the
intercept term ( 0t
β
) varies by year for both the traditional NB and GNB crash prediction
models was adopted. Their approach also assumes the same exposure for the entire study
period. Furthermore, the mean and dispersion functions have the same covariates (see Lord et
al., 2005b). However, in the final model selection, only the coefficients that passed the
significant test at a 95% confidence level were selected as covariates for the dispersion
parameters.
Tables 2 and 3 summarize the modeling results for the traditional NB and GNB
models, respectively. Five different functional forms are used for each model, and six time-
specific (1997, 1998, 1999, 2000, 2001, and All Year) crash prediction models are developed
for each functional form. To illustrate the application of the models, suppose the expected
number of crashes over a given five-year period is estimated using the GNB Model 1. If we
assumes for example that the average “Major AADT” and “Minor AADT” over the given
period is 3,000 vpd and 300 vpd, respectively, by employing the disaggregate time-specific
“GNB Model 1” in Table 3, one obtains the expected number of crashes per year as 0.095 [i.e.,
exp(-10.561+1.0136×ln(3000+300))], 0.075, 0.102, 0.104, and 0.109, respectively. By
summing up all these values, the expected number of crashes over the five-year period is
estimated to be 0.485. On the other hand, by employing the aggregate “All-Year Model” in
Table 3, the estimate of the crash frequency at the same intersection over the same 5-year
period equals 0.484 [i.e. exp(-8.9388+1.0136×ln(3000+300)) ≈ 0.484].
Using the same example intersection above, the time-specific inverse dispersion
parameters are estimated as 3.474 [i.e., exp(3.0961-0.2285×ln(3000+300))], 6.202, 2.722,
2.271, and 2.453, respectively, for each corresponding year. As opposed to the mean crashes,
these values cannot be aggregated; note that the average value equals 2.852. Using the “All-
Year Model”, the overall dispersion parameter over the five years equals 2.359.
Four different GOF statistical tests were employed for comparing the series of crash
prediction models. The tests, described in Hardin and Hilbe (2001) and Washington et al.
(2003), are as follows:
1) Akaike’s Information Criterion (AIC) = N
PML k2)(ln2
+
−
(5)
where,
ln L(Mk) = log likelihood of model k;
P = the number of parameters; and,
N = the number of observations (in our exercise = 5,588).
8
2) Bayesian Information Criteria (BIC) = D(Mk) – d.f.·lnN (6)
where,
D(Mk) = Deviance of model k; and,
d.f.= degrees of freedom.
3) Sum of Model Deviances (G2) =
()
∑
=
n
i
iii yy
1
ˆ
ln2
µ
(7)
where,
yi = the observed number of crashes at site i;
i
µ
ˆ= the expected number of crashes at site i.
4) R2 like measure of fit (MOF) based on Standardized Residuals (R2) =
()
()
()
()
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−− ∑∑ ==
2
1
2
1
1ˆˆ
1
n
i
i
n
i
iyyyy
µµ
(8)
where,
y= average number of observed number of crashes.
The model with the lowest value in AIC, BIC, and G2 is considered the model with
the best statistical fit. On the other hand, the model with the largest R2 like MOF value
indicates a superior fitted model. As shown in Table 4, in general Model 4 provides the best
fit for both the traditional NB and GNB models according to the three different test statistics
(i.e. AIC, BIC, and R2 like MOF). Model 5 is selected as the best fitted model based on the
G2-statistics, but is selected as the second worst model based on the R2 like MOF. On the
other hand, all the test statistics selected Model 1 as the worst statistical fit model regardless
of the model type (i.e., NB or GNB).
As pointed out by Miranda-Moreno et al. (2005), in general, GNB models fit the
data better than the traditional NB models on the basis of the three different test statistics (i.e.
AIC, BIC, and G2-statistics) with an exception the “Model 1” result in terms of the G2-
statistics. Karim and Sayed (2006) reported the same conclusion in their study. [Note: Using
the Deviance statistics, Miaou and Lord (2003) did not find a significant difference between
NB and GNB models in terms of GOF.] However, the R2 like MOF does not produce
consistent test results compared to the results based on the other test statistics. Examining the
analysis results of the four different test statistics, the following findings are worthwhile to be
noted (refer to the Table 4):
1) Model 4 can be considered as the best fitted model amongst the five alternate
Models regardless of the model type (i.e., NB and GNB model) based on the test
results of AIC, BIC, and R2 like MOF.
9
2) Overall, “GNB” models show a better fit than traditional “NB” models
regardless of the functional forms based on the three different test statistics (i.e.
AIC, BIC, and G2-statistics). However, since the R2 like MOF produced
inconsistent test results, this test statistics may not be suitable to determine the
best fitted model as well as the most suitable functional form at least for this
study dataset.
3) As a result, determining the best fitted model as well as the most suitable
functional form using a single (or a couple of) test statistics may potentially be
unreliable, and thus should be avoided.
GOODNESS OF FIT EVALUATION BASED ON RESIDUAL ANALYSIS
In the previous section, four different test statistics were utilized for measuring the GOF of
competitive functional forms or models. Another evaluation method initially proposed by
Hauer and Bamfo (1997) can also be used to evaluate the model fit adequacy. This method is
known as the CURE method and has since been used extensively by many transportation
safety analysts (e.g., Lord and Persaud, 2000; Washington et al., 2005; Wang and Abdel-Aty,
2007), to determine the most suitable model for their study data. The method requires
scrutinizing a graph (i.e., CURE plot), in which the cumulative residuals are plotted in
increasing order for each explanatory variable (i.e., major and minor road AADT in our case)
separately. The residuals (eit) represent the difference between the observed (yit) and the
expected number of crashes ( i
µ
ˆ) at a given study site i in a year t. The closer the curve
oscillates around zero-residual line, the better the model fits the data. The curious reader is
referred to the references listed above for additional details about the CURE method.
Figures 1 through 5 show a total of ten different CURE plots using five different
functional forms with two different model types based upon the “all year” crash data. As
discussed by Hauer and Bamfo (1997), the CURE plot will reveal how well the functional
forms fit the data with respect to each individual explanatory variable (in our case, F1 and F2)
and show systematic deviations of the cumulative residuals from the zero-residual line. Two
different CURE plots (i.e. Figure (a) and (b)) are generated for each Model to compare the
model adequacy between the two model types (i.e., NB and GNB models). To shorten the
illustration, the major road AADT (F1) has been used as a representative explanatory variable
for this analysis. Looking at Figures 1 to 5, several characteristics involving CURE plots can
be noticed:
1) Model 1 underestimates the expected number of crashes (i.e., yit > i
µ
ˆ) in the
range of between 0 and 65,000 major road AADT and slightly overestimates the
number (i.e., yit < i
µ
ˆ) where the major road AADT (F1) is higher than 65,000
(refer to Figure 1) regardless of the model type (NB or GNB). Moreover, in the
range of major road AADT (F1) between 5,000 and 12,000, Model 1 produces
significantly lower values in the expected number of crashes compared to the
observed number of crashes, resulting substantially greater values in the
cumulative residuals than the +2.0σ confidence interval. No practical difference
10
in the CURE plots between the two model types (i.e. NB and GNB model) is
found and the final cumulative residual line is reasonably close to 0.
2) NB Model 2 (refer to the Figure 2 (a)) and NB Model 4 (refer to the Figure 4 (a))
are similar in that both models underestimate the expected number of crashes
over the entire range of variable (i.e., F1) with greater than +2.0σ cumulative
residuals where the major road AADT is higher than about 50,000. In addition,
these two NB models do not have a feature that a good fitted model reasonably
has (i.e., the zero final cumulative residuals). In case of the NB Model 3, while
the expected number of crashes shows the overestimated results with the major
road AADT lower than about 20,000, the other features are similar to the NB
Model 2 and 4 (i.e., cumulative residuals greater than the +2.0σ confidence limit,
no zero cumulative residual ending).
3) Among the different CURE plots from Figure 1 through 5, the Model 5 CURE
plots (i.e., Figure 5-(a), (b)) showed the most unexpected results, including a
catastrophic drop in the cumulative residuals at the major road AADT around
42,000, regardless of the model type. In the previous section, this model was
selected as the best fitted model according to the G2-statistics and chosen as the
second best fitted model based on the AIC as well as BIC. A closer look at the
CURE plots as well as the raw dataset revealed that this sudden drop is caused
by the unusually higher number of minor road AADT (i.e., F2 = 23,111) at a
specific intersection. The value is at least twice higher than the other minor
roads’ AADT, and contributed to produce dramatically greater amount of
overestimation values (i.e., eit = -1165.6 in NB Model 5, eit = -1170.7 in GNB
Model 5). In fact, this sudden drop reveals that Model 5 is very sensitive in the
estimates by the higher number of minor road AADT (F2) by employing the third
variable in the functional form (i.e., 32i
F
β
). It should be recognized that the third
variable in Model 5 made the actual difference between Models 2 and 5, and
Model 2 does not show the sudden change in estimates. In truth, this site should
be investigated further to determine whether this observation is in fact an outlier
(e.g., error in reported flows, etc.) or an influence point. Statistical tests (not used
here), such as R-Student, DFFITS and Cooks’ D, can be used to identify
potential outliers and influence points (Myers, 2000).
4) GNB Model 2, 3, and 4 yielded improved CURE plots compared to the CURE
plots of the corresponding NB Models in that the amount of bias in the estimates
has been much reduced. The final cumulative residuals of these three GNB
models end close enough to zero value. Although there are just slight differences
in the CURE plots among these three GNB models, GNB Model 4 shows better
fitting result (it was already selected as the best fitted model based on the test
statistics analysis in previous section) since the cumulative residual lines are
oscillating reasonably across the zero cumulative residual line over the entire
explanatory values.
11
Table 5 contains the summary of the cumulative residuals for a total of 60 different
models [6 years (including all year) × 5 functional forms × 2 model types] to show the
difference between the time-specific model (yearly based model) and all-year model as well
as the difference between the NB (fixed dispersion) and GNB (varying dispersion) model.
Notable characteristics are:
1) The sums of residuals of time-specific models show a great amount of up-and-
down fluctuation in each year regardless of the functional forms employed. For
Models 2 though 4, the absolute values of the total residuals from time-specific
models (i.e., Sum 97-01) show lower values than those of the aggregated all-year
models. It indicates that the time-specific models produce more accurate results
than the aggregated all year models regardless of the model type (i.e. NB or
GNB) especially for Model 2, 3 and 4. The result is even more interesting if
recognized that we only used the constant entering traffic volumes (i.e. average
F1 and F2 over study period) as model inputs [Note: It is important to point out
that using the varying entering volumes for every year (i.e., time-varying F1t and
F2t) might show even better results than using a constant exposure (Mountain et
al., 1998; Lord and Persaud, 2000; Lord et al., 2005b).] On the other hand,
Model 1 and 5 do not show this characteristic.
2) In general, GNB model shows better fitting results with smaller residuals than
the traditional NB model, with the exception of Model 5. As mentioned
previously, Model 5 produced very different results compared to the results from
the other models because of the abnormally higher number of minor road AADT
(F2) at the previously identified intersection. This could be a unique
characteristic of the Model 5 at least for this study dataset. Note that the model is
usually adapted to estimate the crash frequencies at intersections in large urban
areas (Lord and Persaud, 2004; Miaou and Lord, 2003; Lyon et al. 2005),
therefore Model 5 may not be suitable to estimate the crashes at intersections in
rural areas. It is of interest to note that Model 5 was originally selected as the
best functional form in terms of the G2-statistics and the second best functional
form in terms of the AIC and BIC (refer to Table 4). The fact that the statistical
tests in Equations (5) through (8) are frequently used to determine and justify the
best model performance by transportation safety analysts without further looking
into raw data set using a model diagnosis tool (e.g., CURE plot) is a cause for
concern.
DISPERSION PARAMETERS AMONG DIFFERENT CRASH PREDICTION MODELS
Table 6 and Figure 6 summarize the estimated dispersion parameters obtained from different
models. A few notable features include:
1) The estimated values from this study are quite different from the reported values
in the previous study by Miaou and Lord (2003) since the study used a
12
completely different dataset from the City of Toronto. However, the same
patterns have been noticed, such that the estimated fixed dispersion parameters
among the traditional NB models 2, 3, 4, and 5 are fairly similar.
2) 3) In general, traditional NB models underestimate the inverse dispersion
parameters compared to GNB models, with the exception of Model 3. In terms of
the varying dispersion parameters, comparing the minimum/maximum values as
well as the average values, Model 5 again shows the biggest discrepancy amongst
all the models tested (as seen in Table 6). It may be caused by the observation
with abnormally higher number of minor road AADT or it is just a peculiar
characteristic of Model 5 since the similar discrepancy has been experienced by
Miaou and Lord (2003). No matter the reason since
{
}
2
ititit
YVar
µαµ
⋅+= , Model
5 estimates will produce larger variance than that of other models, implying
higher uncertainties associated with this model. As a result, compared to the other
models, Model 5 put more emphasis on the observed number of crashes than the
model estimates in obtaining EB estimates.
3) Inasmuch as a traditional NB model produces a fixed (i.e., constant) dispersion
parameter for each model, the weight factor is inversely related to model
estimates (i.e., tt
w
µ
1∝, refer to Figure 7). The higher the model estimates, the
smaller the weight factors, and vise versa.
4) GNB Models 2, 4, and 5 [refer to Figure 8-(b), 8-(d), and 8-(e), respectively]
allow different weight factors for intersections with the same model estimate. The
tendency is stronger for the intersections with higher model estimates. On the
other hand, GNB Models 1 and 3 [refer to Figure 8-(a) and 8-(c)] show the same
patterns with the corresponding NB models, and do not allow different dispersion
parameters unless the intersections have different model estimates. For Models 1
and 3, traffic volumes entering from the major and minor roads (i.e. F1 and F2,
respectively) are not treated as distinct traffic volumes, but rather as a single
traffic volume unit (i.e., F1+F2, F1·F2). Even with the exact same traffic volume
(e.g., F1 + F2 = 3,000/day), intersections could have different entering traffic
volumes from the major and minor roads (e.g., F1 = 1,500 and F2 = 1,500, F1 =
2,000 and F2 = 1,000, etc.). Since the same functional form is used to explain the
dispersion and the mean values in all GNB Models, GNB Models 1 and 3 should
have the same value for the weight factor if the model estimates are the same. On
the other hand, GNB Models 2, 4, and 5 could have different dispersion
parameters and model estimates even for intersections with the exactly same
aggregate traffic volumes. Even though the estimated dispersion parameters can
be different between the traditional NB and GNB models and if one only uses the
aggregated traffic volumes (i.e., F1+F2, F1·F2) in explaining the varying
dispersion parameters, there is no practical merit of using the GNB Model over
the traditional NB model for this case. Figures 9-(a) and 9-(c) clearly show that
the GNB Models 1 and 3 produce virtually the same weight factors as calculated
from the traditional NB model. It should be noted that GNB models produce a
13
slightly lower value for the weight factor (i.e., the coefficients in Figure 9 are less
than 1.0) than those of traditional NB models. Hence, GNB Models will give
slightly more weight to the observed number of crashes than the model estimates
when calculating the EB estimates compared to the traditional NB Models with
the same estimated value.
Figure 10 shows which covariates are more heavily associated with the degree of
dispersion amongst different GNB Models. In this illustration, GNB Models 1 and 3 were
disregarded since the models only contain a single aggregated traffic volume. GNB Models 2,
3, and 5 show that the major road AADT contributes more significantly to the variation in the
dispersion parameters than that of the minor road AADT. Obviously, for a given major road
AADT, a number of different dispersion parameters can be estimated. Since the weight
factors are a function of dispersion parameters, the variation in the weight factor in GNB
Models 2, 3, and 5 is mainly caused by the heterogeneity associated with the major road
covariate. Since the AADT has been used as a surrogate measure for explaining the possible
structure of un-modeled heterogeneities, as documented in Miaou and Lord (2003) and Mitra
and Washington (2007), the inclusion of other covariates describing characteristics associated
with the major approaches may help reduce the heterogeneity observed in the models.
The final objective of this study consisted of investigating the impact of varying
dispersions on identifying hotspots. Figure 11 illustrates the relationship between the hotspot
identification lists ranked by the traditional NB models and by the GNB models. Smaller
values in the ranking imply more hazardous intersections in terms of the EB estimates. The
association between the NB ranking and the GNB ranking is very strong and shows almost a
perfect correlation regardless of the functional forms used. However, a notable point to
observe is that the hotspot ranking for Models 1 and 3 is constantly lower for the traditional
NB model than that of the GNB model. The other functional forms do not show this pattern.
It is unclear at this time the cause of such pattern, but this characteristic should be
investigated further.
To evaluate the association between the two rankings, a Spearman rank-order
correlation coefficient (
ρ
s), which is a non-parametric statistical test, was applied to the data.
Similar to the findings documented in El-Basyouny and Sayed (2006), the association in
ranking between the traditional NB and GNB models was found to be very strong (i.e.,
ρ
s >
0.99, all cases). Therefore, it can be concluded that there is no substantial benefit of using
GNB model if the study purpose is solely in identifying the hotspots based on the EB
estimates (again if the same exposure level is used for model types).
CONCLUSIONS
In this paper, a number of important issues regarding the impact of the dispersion parameters
on EB estimates were presented. Several important conclusions can be reported:
1) Developing time-specific models is favored over developing aggregated models.
This supports the findings of Lord and Persaud (2000) who noted that models
with trend performed better in terms of GOF than models without trend (or
14
aggregated models). In the same line, using GNB models provide an even better
performance than NB models with a fixed dispersion parameter both for time-
specific and aggregated modeling frameworks in most cases.
2) If separate years and specific sites are analyzed individually, time-specific models
with a varying dispersion parameter will have a significant impact on the EB
estimate, as documented in Table 6 and Figure 6. This result concurs with those
of Miaou and Lord (2003). The varying dispersion parameters have shown a
significant level of variability, which will affect the weight factor associated with
the EB estimate as well as the computation of confidence intervals associated this
estimate.
3) Model 5 produced very sensitive result by the minor road AADT (F2) due to the
last component ( 32i
F
β
) in the functional form. Based on the empirical findings
of this study, including CURE plots, Model 5 (i.e., Functional Form 5) may not
seem to be an appropriate functional form to estimate the crashes at three-leg
intersections in rural areas, at least with the database used in this study. A similar
argument can be made about Model 1.
4) Developing reliable crash prediction models can never be overemphasized to
obtain rigorous EB estimates. Automatically adapting a functional form from
previous studies should be avoided, and a rigorous analysis not solely based on
GOF statistics to determine the appropriate functional form for a given study
dataset must be carried out. The residual diagnosis analysis using CURE plots
can provide much more detailed information beyond that from GOF statistics.
5) There is no substantial benefit in obtaining the EB estimates by applying varying
dispersion parameters (rather than a fixed dispersion parameter) if the study
purpose is to identify hotspots (e.g. also an aggregated-level analysis). However,
there might be a considerable impact on evaluating the treatment effects via EB
before-after studies and the impact should be worthy of further investigation. As
described above, time-varying exposure should also be investigated in this
context.
In conclusion, as pointed out by Miaou (2005) and to some degree in Miaou and Lord
(2003), statistics is just one of many sciences. The "science part" of a statistical model is
related to the mean function. What the transportation safety analyst needs the most is to better
understand the functional structure of the mean function. While this study focused on “the
structure of the variance function,” especially the one related to the dispersion parameter (or
its inverse), transportation safety analysts should never lose sight of the most important part
of a statistical model (i.e., the structure of the mean function) In theory, any modifications to
the structure of the mean function (via the inclusion or exclusion of covariates in crash
prediction models) will affect the structure of the variance function.
Ideally, with any types of model, a good structure and the proper selection of the
covariates for the mean function would make the structure of the variance function vanish or
at least significantly minimize the magnitude of the variance (e.g., see Miaou and Song, 2005
15
and Mitra and Washington, 2007). However, since it is practically unachievable to obtain a
perfect mean function (see Xie et al., 2006), transportation safety analysts will continue to
work with the variance function for the following three rationales: 1) looking for clues to
improve the deficiency of the mean function, 2) reducing the bias in the mean function, and
3) hopefully providing more accurate statistical inferences for decision-making purposes.
These are important issues that need to be addressed in future research projects.
16
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19
Table 1. Summary Statistics for the Study Dataset
Variable (per site) Mean Std. Dev. Min. Max. No. of
intersections
No. of crashes* /5 years 1.073 2.806 0 64
No. of average crashes (KABCO)
/year
0.215 0.561 0 12.8
Avg. AADT of major road over 5
years
6,953 7,407 103 79,800
Avg. AADT of minor road over 5
years
268 683 1 23,111
5,588
* = All crash severities
20
Table 2. Estimated Coefficients and Statistics for NB Crash Prediction Models
NB Model β0,t β1,t β2,t β3,t Log(α) Log Likelihood No. of Parameters Deviance AIC BIC G2 - Statistics R2 like MOF
-10.5741 0.9383 1997
(0.0395) (0.0867)
-2717.7926 2 5435.585
0.973
-42762.527 4241.982
0.275
-10.8353 1.1580 1998
(0.0441) (0.0928)
-2252.1741 2 4504.348
0.807
-43693.764 3788.404
0.375
-10.5141 0.6542 1999
(0.0367) (0.0915)
-2846.5973 2 5693.195
1.020
-42504.918 4271.889
0.275
-10.4896 0.8203 2000
(0.0377) (0.0856)
-2885.1331 2 5770.266
1.033
-42427.846 4408.752
0.276
-10.4433 0.8974 2001
(0.0379) (0.0815)
-2962.3282 2 5924.656
1.061
-42273.456 4603.335
0.283
-8.9541 0.6871
Functional
Form 1
(NB Model 1)
All Year
(0.2609)
1.0153
(0.0295)
- -
(0.0399)
-6763.0979
3
13526.196
2.422
-34663.288
13034.299
0.475
-10.0342 0.3407 1997
(0.0371) (0.1092)
-2556.6668 2 5113.334
0.916
-43084.779 3710.145
0.382
-10.2757 0.7072 1998
(0.0423) (0.1081)
-2148.9187 2 4297.837
0.770
-43900.275 3391.635
0.434
-9.9651 -0.0623 1999
(0.0341) (0.1244)
-2663.3955 2 5326.791
0.954
-42871.321 3702.554
0.382
-9.9418 0.2665 2000
(0.0356) (0.1053)
-2725.6695 2 5451.339
0.976
-42746.773 3900.767
0.376
-9.8969 0.3854 2001
(0.0358) (0.0981)
-2811.3501 2 5622.700
1.007
-42575.412 4089.771
0.363
-8.4111 0.2610
Functional
Form 2
(NB Model 2)
All Year
(0.2325)
0.6730
(0.0266)
0.4880
(0.0162)
-
(0.0471)
-6420.9349
4
12841.870
2.300
-35338.986
10575.656
0.614
-9.2226 0.3190 1997
(0.0370) (0.1108)
-2556.1857 2 5112.371
0.916
-43085.741 3703.764
0.381
-9.4585 0.7331 1998
(0.0425 (0.1076)
-2160.1023 2 4320.205
0.774
-43877.908 3413.441
0.411
-9.1513) -0.0660 1999
(0.0341) (0.1250)
-2665.9620 2 5331.924
0.955
-42866.188 3705.128
0.386
-9.1288 0.2582 2000
(0.0355) (0.1060)
-2727.2530 2 5454.506
0.977
-42743.606 3900.835
0.370
-9.0851 0.3707 2001
(0.0357) (0.0989)
-2812.5312 2 5625.062
1.007
-42573.050 4089.798
0.354
-7.5976 0.2689
Functional
Form 3
(NB Model 3)
All Year
(0.1746)
0.5465
(0.0124)
- -
(0.0472)
-6435.9894 3 12871.979
2.305
-35317.505 10590.943
0.605
-10.3397 0.3265 1997
(0.0371) (0.1099)
-2553.4246 2 5106.849
0.915
-43091.263 3690.981
0.388
-10.5805 0.6996 1998
(0.0423) (0.1083)
-2146.2885 2 4292.577
0.769
-43905.535 3376.622
0.439
-10.2712 -0.0882 1999
(0.0340) (0.1259)
-2658.3654 2 5316.731
0.952
-42881.382 3679.758
0.382
-10.2486 0.2405 2000
(0.0355) (0.1068)
-2720.8898 2 5441.780
0.975
-42756.333 3875.938
0.383
-10.2045 0.3563 2001
(0.0357) (0.0992)
-2803.4481 2 5606.896
1.004
-42591.216 4060.822
0.375
-8.7168 0.2470
Functional
Form 4
(NB Model 4)
All Year
(0.2379)
1.1620
(0.0279)
0.4279
(0.0158)
-
(0.0474)
-6409.6162
4
12819.232
2.295
-35361.623
10458.744
0.621
-9.7416 0.3552 1997
(0.0374) (0.1067)
-2555.7990 2 5111.598
0.915
-43086.514 3597.308
0.372
-9.9824 0.7168 1998
(0.0426) (0.1064)
-2147.7370 2 4295.474
0.769
-43902.638 3306.568
0.426
-9.6755 -0.0294 1999
(0.0345) (0.1182)
-2662.1116 2 5324.223
0.954
-42873.889 3592.798
0.364
-9.6482 0.2873 2000
(0.0359) (0.1021)
-2724.6199 2 5449.240
0.976
-42748.873 3800.043
0.366
-9.6017 0.4041 2001
(0.0361) (0.0955)
-2808.7386 2 5617.477
1.006
-42580.635 3981.894
0.365
-8.1150 0.2548
Functional
Form 5
(NB Model 5)
All Year
(0.2424)
0.6706
(0.0264)
0.4196
(0.0239)
0.0002
(0.0000)
(0.0471)
-6413.0636
5
12826.127 2.297 -35346.100
10020.232 0.593
21
Table 3. Estimated Coefficients and Statistics for GNB Crash Prediction Models
Log(α) GNB Model β0,t β1,t β2,t β3,t
γ0,t γ 1,t γ 2,t γ 3,t
Log Likelihood No. of Parameters Deviance AIC BIC G2 - Statistics R2 like MOF
-10.5610 3.0961 -0.2285 - - 1997
(0.0397) (1.0610) (0.1125) - -
-2715.8124 3 5431.625
0.973
-42757.859 4248.340
0.274
-10.8026 5.6048 -0.4666 - - 1998
(0.0452) (1.1962) (0.1261) - -
-2245.9615 3 4491.923
0.805
-43697.561 3756.577
0.385
-10.4971 3.0128 -0.2483 - - 1999
(0.0368) (1.1197) (0.1183) - -
-2844.5574 3 5689.115
1.019
-42500.369 4268.989
0.277
-10.4742 0.8202 n.a. - - 2000
(0.0377) (0.0856) n.a. - -
-2885.0900 2 5770.180
1.033
-42427.932 4409.860
0.277
-10.4280 0.8972 n.a. - - 2001
(0.0378) (0.0815) n.a. - -
-2962.2666 2 5924.533
1.061
-42273.579 4604.552
0.284
-8.9388 2.1562 -0.1602 - -
Functional
Form 1
(NB Model 1)
All
(0.2612)
1.0136
(0.0293)
- -
(0.4667) (0.0508) -
-6758.3375
4
13516.675
2.420
-34664.181
13040.488
0.475
-10.0277 2.4832 n.a. -0.3441 - 1997
(0.0374) (0.3928) n.a. (0.0644) -
-2542.5005 3 5085.001
0.911
-43104.483 3663.926
0.380
-10.2571 5.7276 -0.3104 -0.3385 - 1998
(0.0435) (1.2646) (0.1340) (0.0641) -
-2132.6150 4 4265.230
0.765
-43915.626 3334.523
0.437
-9.9598 2.1456 n.a. -0.3507 - 1999
(0.0343) (0.4425) n.a. (0.0729) -
-2651.7545 3 5303.509
0.950
-42885.975 3656.838
0.380
-9.9414 2.0073 n.a. -0.2788 - 2000
(0.0359) (0.3947) n.a. (0.0641) -
-2716.8689 3 5433.738
0.973
-42755.746 3870.922
0.368
-9.8967 1.5581 n.a. -0.1879 - 2001
(0.0361) (0.3784) n.a. (0.0603) -
-2806.9713 3 5613.943
1.006
-42575.541 4059.650
0.352
-8.4040 1.7312 n.a. -0.2640 -
Functional
Form 2
(NB Model 2)
All
(0.2314)
0.6578
(0.0269)
0.5134
(0.0158)
-
(0.1669) n.a. (0.0295) -
-6384.3005
5
12768.601
2.287
-35403.626
10347.664
0.613
-9.3283 4.8790 -0.2924 - - 1997
(0.0373) (0.8095) (0.0531) - -
-2542.6011 3 5085.202
0.911
-43104.282 3679.690
0.378
-9.5443 6.1169 -0.3448 - - 1998
(0.0435) (0.8307) (0.0542) - -
-2141.2279 3 4282.456
0.767
-43907.028 3352.992
0.422
-9.2523 4.7512 -0.3067 - - 1999
(0.0343) (0.9192) (0.0604) - -
-2654.4034 3 5308.807
0.951
-42880.677 3665.911
0.386
-9.2387 3.7656 -0.2247 - - 2000
(0.0358) (0.8101) (0.0527) - -
-2719.4728 3 5438.946
0.974
-42750.538 3887.016
0.364
-9.1952 3.0049 -0.1691 - - 2001
(0.0360) (0.7839) (0.0508) - -
-2808.0660 3 5616.132
1.006
-42573.352 4076.810
0.346
-7.6995 3.5021 -0.2203 - -
Functional
Form 3
(NB Model 3)
All
(0.1708)
0.5541
(0.0118)
- -
(0.3422) (0.0234) - -
-6396.8367 4 12793.673
2.291
-35387.182 10423.853
0.606
-10.3783 3.4820 -0.4323 -0.3011 - 1997
(0.0375) (1.2291) (0.1327) (0.0640) -
-2540.0314 4 5080.063
0.911
-43100.793 3659.810
0.386
-10.6065 5.7966 -0.6326 -0.2878 - 1998
(0.0434) (1.3023) (0.1399) (0.0628) -
-2130.3818 4 4260.764
0.764
-43920.092 3327.764
0.445
-10.3085 3.7436 -0.5032 -0.3103 - 1999
(0.0343) (1.3887) (0.1514) (0.0710) -
-2646.6847 4 5293.369
0.949
-42887.486 3642.744
0.382
-10.2915 2.2491 -0.2956 -0.2555 - 2000
(0.0359) (1.2112) (0.1307) (0.0636) -
-2712.5502 4 5425.100
0.972
-42755.755 3855.607
0.378
-10.2480 2.3097 -0.2609 -0.1668 - 2001
(0.0361) (1.1175) (0.1225) (0.0576) -
-2798.9706 4 5597.941
1.003
-42582.914 4041.863
0.368
-8.7536 2.4856 -0.3335 -0.2334 -
Functional
Form 4
(NB Model 4)
All
(0.2378)
1.1724
(0.0271)
0.4418
(0.0152)
-
(0.5114) (0.0570) (0.0287) -
-6372.4107
6
12744.821
2.283
-35418.777
10283.859
0.621
-9.7060 2.7813 n.a. -0.4151 0.0001 1997
(0.0376) (0.4407) n.a. (0.0781) (0.0000)
-2542.4047 4 5084.809
0.911
-43096.046 3578.725 0.369
-9.9335 6.1316 -0.3005 -0.4513 0.0002 1998
(0.0436) (1.3185) (0.1369) (0.0780) (0.0000)
-2130.2489 5 4260.498
0.764
-43911.729 3269.585
0.429
-9.6373 2.7523 n.a. -0.4903 0.0002 1999
(0.0344) (0.4787) n.a. (0.0865) (0.0001)
-2646.5215 4 5293.043
0.949
-42887.813 3567.426
0.364
-9.6153 2.4632 n.a. -0.3823 0.0001 2000
(0.0360) (0.4373) n.a. (0.0780) (0.0000)
-2713.7821 4 5427.564
0.973
-42753.291 3789.491
0.362
-9.5673 1.8322 n.a. -0.2544 0.0001 2001
(0.0362) (1.8322) n.a. (0.0744) (0.0000)
-2803.1294 4 5606.259
1.005
-42574.597 3967.084
0.360
-8.0781 1.9449 n.a. -0.3182 0.0001
Functional
Form 5
(NB Model 5)
All
(0.2437)
0.6568
(0.0267)
0.4367
(0.0254)
0.0002
(0.0000)
(0.1883) n.a. (0.0360) (0.0000)
-6374.4074
7
12748.815 2.284 -35406.156
9920.566 0.592
22
Table 4. Summary of Statistical Tests
AIC BIC G2 - Statistics R2 like MOF*
Functional Form
(All Year Model) NB Model GNB
Model NB Model GNB
Model NB Model GNB
Model NB Model GNB
Model
Model 1 2.422 2.420 -34663.288 -34664.181 13034.299 13040.488 0.475 0.475
Model 2 2.300 2.287 -35338.986 -35403.626 10575.656 10347.664 0.614 0.613
Model 3 2.305 2.291 -35317.505 -35387.182 10590.943 10423.853 0.605 0.606
Model 4 2.295 2.283 -35361.623 -35418.777 10458.744 10283.859 0.621 0.621
Model 5 2.297 2.284 -35346.100 -35406.156 10020.232 9920.566 0.593 0.592
Best Model NB
Model 4
GNB
Model 4
NB
Model 4
GNB
Model 4
NB
Model 5
GNB
Model 5
NB
Model 4
GNB
Model 4
Second Best
Model
NB
Model 5
GNB
Model 5
NB
Model 5
GNB
Model 5
NB
Model 4
GNB
Model 4
NB
Model 2
GNB
Model 2
Worst Model NB
Model 1
GNB
Model 1
NB
Model 1
GNB
Model 1
NB
Model 1
GNB
Model 1
NB
Model 1
GNB
Model 1
* = Bold values represent a better model between the NB and GNB model within the same functional form
23
Table 5 Cumulative Residuals between NB and GNB Models
Yearly Based Model Residuals*
()
∑−itit
y
µ
ˆ 1997 1998 1999 2000 2001 Total
97-01
All Year
Model
Model 1 -17.88 42.62 6.58 -22.82 -33.18 -24.68 -16.58
Model 2 13.72 49.29 28.90 1.57 -5.72 87.76 124.99
Model 3 15.71 45.65 28.29 1.98 -3.80 87.85 123.52
Model 4 8.60 44.58 24.14 -2.34 -8.81 66.17 100.41
NB
Models
Model 5 -268.74 -173.43 -269.18 -309.85 -334.16 -1355.37
-1332.83
Model 1 -14.58 27.11 5.11 -22.16 -32.42 -36.94 -13.38
Model 2 -6.16 22.37 9.09 -12.40 -20.09 -7.19 20.62
Model 3 4.19 18.03 10.01 -5.26 -11.00 15.97 43.22
Model 4 -5.79 21.50 7.05 -12.65 -18.68 -8.58 16.32
GNB
Models
Model 5 -278.88 -196.69 -284.04 -316.64 -343.74 -1419.99
-1393.78
* = yit represents the observe number of crashes per site per each year
24
Table 6. Inverse Dispersion Parameters among Different Models
Model 1 Model 2 Model 3 Model 4 Model 5 Model 1 ~ Model 5
Avg. Min. Max. Avg. Min. Max. Avg. Min. Max. Avg. Min. Max. Avg. Min. Max. Avg. Min. Max.
E{Di}_97 2.556 2.556 2.556 1.406 1.406 1.406 1.376 1.376 1.376 1.386 1.386 1.386 1.426 1.426 1.426 1.630 1.376 2.556
E{Di}_98 3.183 3.183 3.183 2.028 2.028 2.028 2.082 2.082 2.082 2.013 2.013 2.013 2.048 2.048 2.048 2.271 2.013 3.183
E{Di}_99 1.924 1.924 1.924 0.940 0.940 0.940 0.936 0.936 0.936 0.916 0.916 0.916 0.971 0.971 0.971 1.137 0.916 1.924
E{Di}_00 2.271 2.271 2.271 1.305 1.305 1.305 1.295 1.295 1.295 1.272 1.272 1.272 1.333 1.333 1.333 1.495 1.272 2.271
E{Di}_01 2.453 2.453 2.453 1.470 1.470 1.470 1.449 1.449 1.449 1.428 1.428 1.428 1.498 1.498 1.498 1.660 1.428 2.453
Fixed
Dispersion
(NB
Model)
E{Di}_All 1.988 1.988 1.988 1.298 1.298 1.298 1.309 1.309 1.309 1.280 1.280 1.280 1.290 1.290 1.290 1.433 1.280 1.988
E{Di}_97 3.302 1.676 7.093 2.867 0.377 11.980 3.605 0.305 25.158 3.093 0.324 15.466 2.965 0.250 16.140 3.166 0.250 25.158
E{Di}_98 5.933 1.401 26.653 6.156 0.370 53.100 12.431 1.051 86.754 5.914 0.350 46.729 6.681 0.199 84.114 7.423 0.199 86.754
E{Di}_99 2.582 1.233 5.913 1.994 0.252 8.547 3.172 0.268 22.140 2.350 0.192 14.164 2.202 0.114 15.679 2.460 0.114 22.140
E{Di}_00 2.271 2.271 2.271 2.294 0.452 7.443 1.184 0.100 8.263 2.276 0.418 7.546 2.435 0.252 11.742 2.092 0.100 11.742
E{Di}_01 2.453 2.453 2.453 2.111 0.719 4.750 0.553 0.047 3.862 2.220 0.617 5.907 2.121 0.485 6.248 1.892 0.047 6.248
Varying
Dispersion
(GNB
Model)
E{Di}_All 2.264 1.415 3.892 1.845 0.398 5.648 0.910 0.077 6.349 1.921 0.343 6.811 1.848 0.286 6.993 1.758 0.077 6.993
E{Di}_97 -0.746 0.879 -4.537 -1.461 1.028 -10.574 -2.229 1.071 -23.783 -1.707 1.063 -14.080 -1.539 1.177 -14.713
E{Di}_98 -2.749 1.783 -23.469 -4.128 1.658 -51.072 -10.349 1.030 -84.673 -3.901 1.663 -44.716 -4.634 1.849 -82.067
E{Di}_99 -0.659 0.691 -3.989 -1.055 0.688 -7.607 -2.236 0.668 -21.204 -1.434 0.723 -13.248 -1.231 0.857 -14.707
E{Di}_00 0.000 0.000 0.000 -0.988 0.853 -6.138 0.111 1.195 -6.968 -1.004 0.853 -6.275 -1.102 1.080 -10.409
E{Di}_01 0.000 0.000 0.000 -0.641 0.751 -3.280 0.896 1.402 -2.413 -0.792 0.811 -4.478 -0.624 1.013 -4.750
Fixed-
Varying
Dispersion
E{Di}_All -0.276 0.573 -1.904 -0.547 0.900 -4.349 0.399 1.232 -5.040 -0.641 0.937 -5.530 -0.558 1.004 -5.703
25
CURE Plot for the NB Model 1 (All Year)
-250
-200
-150
-100
-50
0
50
100
150
200
250
0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 90,000
Major AADT (F1)
Cumulative Residuals
-2 Std.Dev. Cumulaive Residual +2 Std.Dev
(a)
CURE Plot for the GNB Model 1 (All Year)
-250
-200
-150
-100
-50
0
50
100
150
200
250
0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 90,000
M ajor AADT (F 1)
Cumulative Residuals
-2 Std.Dev. Cumulaive Residual +2 Std.Dev
(b)
Figure 1. Cumulative Residual Plots for Model 1
26
CURE Plot for the NB Model 2 (All Year)
-250
-200
-150
-100
-50
0
50
100
150
200
250
0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 90,000
Major AADT (F1)
Cumulative Residuals
-2 Std.Dev. Cumulaive Residual +2 Std.Dev
(a)
CURE Plot for the GNB Model 2 (All Year)
-250
-200
-150
-100
-50
0
50
100
150
200
250
0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 90,000
Major AADT (F1)
Cumulative Residuals
-2 Std.Dev. Cumulaive Residual +2 Std.Dev
(b)
Figure 2. Cumulative Residual Plots for Model 2
27
CURE Plot for the NB Model 3 (All Year)
-250
-200
-150
-100
-50
0
50
100
150
200
250
0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 90,000
Major AADT (F1)
Cumulative Residuals
-2 Std.Dev. Cumulaive Residual +2 Std.Dev
(a)
CURE Plot for the GNB Model 3 (All Year)
-250
-200
-150
-100
-50
0
50
100
150
200
250
0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 90,000
Major AADT (F1)
Cumulative Residuals
-2 Std.Dev. Cumulaive Residual +2 Std.Dev
(b)
Figure 3. Cumulative Residual Plots for Model 3
28
CURE Plot for the NB Model 4 (All Year)
-250
-200
-150
-100
-50
0
50
100
150
200
250
0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 90,000
Major AADT (F1)
Cumulative Residuals
-2 Std.Dev. Cumulaive Residual +2 Std.Dev
(a)
CURE Plot for the GNB Model 4 (All Year)
-250
-200
-150
-100
-50
0
50
100
150
200
250
0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 90,000
Major AADT (F1)
Cumulative Residuals
-2 Std.Dev. Cumulaive Residual +2 Std.Dev
(b)
Figure 4. Cumulative Residual Plots for Model 4
29
CURE Plot for the NB Model 5 (All Year)
-1500
-1000
-500
0
500
1000
1500
0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 90,000
Major AADT (F1)
Cumulative Residuals
-2 Std.Dev. Cumulaive Residual +2 Std.Dev
(a)
CURE Plot for the GNB Model 5 (All Year)
-1500
-1000
-500
0
500
1000
1500
0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 90,000
Major AADT (F1)
Cumulative Residuals
-2 Std.Dev. Cumulaive Residual +2 Std.Dev
(b)
Figure 5. Cumulative Residual Plots for Model 5
30
0.0
0.5
1.0
1.5
2.0
2.5
Model 1 Model 2 Model 3 Model 4 Model 5
Inverse Dispersion (Average)
Fixed Dispersion
Varying Dispersion
0.0
0.5
1.0
1.5
2.0
2.5
Model 1 Model 2 Model 3 Model 4 Model 5
Inverse Dispersion (Minimum)
Fixed Dispersion
Varying Dispersion
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Model 1 Model 2 Model 3 Model 4 Model 5
Inverse Dispersion (Maximum)
Fixed Dispersion
Varying Dispersion
Figure 6. Dispersion Parameters based on All-Year Models
31
Model 1 using Fixed Dispersion (All Year)
0.0
3.0
6.0
9.0
12.0
15.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Weight Fa ctor
E{Yi}
(a)
Model 2 using Fixed D ispersion (All Year)
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Weight Factor
E{Yi}
(b)
Model 3 using Fixed Dispersion (All Year)
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Weight Factor
E{Yi}
(c)
Model 4 using Fix ed Dispersion (All Yea r)
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Weight Fa ctor
E{Yi}
(d)
Model 5 using Fixed Dispersion (All Year)
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Weig ht Fact or
E{Yi}
(e)
Figure 7. Relationships between the Weight Factor and NB Model Estimates
32
Mode l 1 using Varying Dispersion (All Year )
0.0
3.0
6.0
9.0
12.0
15.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Weight Fact or
E{Yi}
(a)
Model 2 using Varying Dispersion (All Year)
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Weight Factor
E{Yi}
(b)
Model 3 using Varying Dispersion (All Year)
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Weight Factor
E{Yi}
(c)
Model 4 using Fixed Dispersion (All Year)
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Weight Fa ctor
E{Yi}
(d)
Model 5 us ing Varying Disper sion (All Year)
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Weight Fac tor
E{Yi}
(e)
Figure 8. Relationships between the Weight Factor and the GNB Model Estimates
33
Y = 0. 9256X
R
2
= 0.993
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 0.1 0.2 0.3 0.4 0 .5 0.6 0.7 0.8 0.9 1
Model 1 Weight Factor by Fixed Dispersion (All Year)
Model 1 weight Fact or by Va rying Dis persio
n
(All Yea r)
(a)
Y = 0.8885 X
R
2
= 0.9309
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Model 2 Weight Factor by Fixed Dispersion (All Year)
Model 2 Weight Factor by Varying Dispersion
(All Year)
(b)
Y = 0.8318X
R
2
= 0.9355
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Model 3 We ight Factor by Fixed Dispersion (All Year)
Model 3 Weight Factor by Varying Dispersion
(All Year)
(c)
Y = 0.8678X
R2 = 0.9362
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Model 4 We ight Factor by Fixed Dispersion (All Year)
Model 4 Weight Factor by Varying Dispersion
(All Year)
(d)
Y = 0.8888X
R
2
= 0.818
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Model 5 Weight Factor by Fixed Disper sion (All Year)
Model 5 Weight Factor by Varying Dispersion
(All Year)
(e)
Figure 9. Associations of Weight Factors between NB and GNB Models
34
Model 2 Varying Dispersion (All Year)
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 90, 000
Majo r Road Avg. AADT
Dispersio n Parameter
(a)
Model 2 Varying Dispersion (All Year)
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
0 5000 10000 15000 20000 25000
Minor Road Avg. AADT
Dispersion Parameter
(b)
Model 4 Varying Dispersion (All Year)
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 90,000
Avera ge Ma jor AADT
Dispersion Paramter
(c)
Model 4 Varying Dispersion (All Year)
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
0 5,000 10,000 15,000 20,000 25,000
Mino r Road Avg. AADT
Dispersio n Paramter
(d)
Model 5 Varying Dispersion (All Year)
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 90,000
Maj or Road AADT
Dispersion Parameter
(e)
Model 5 Varying Disper sion (All Year)
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
0 5,000 10,000 15,000 20,000 25,000
Minor Road AADT
Dispersion Parameter
(f)
Figure 10. Relationships between the Varying Dispersions and Traffic Volumes
35
Y = 0.9996X
R2 = 0.9969
0
1,000
2,000
3,000
4,000
5,000
0 1,000 2,000 3,000 4,000 5,000
Model 1 Ranking (Fixed Dispers ion; Sum 97-01)
Model 1 Ranking (Varying Dispersion ; Sum 97-01)
(a) Spearman
ρ
s = 0.998
Y = 0.9983X
R
2
= 0.986
0
1,000
2,000
3,000
4,000
5,000
0 1,000 2,000 3,000 4,000 5,000
Model 2 Ranking (Fixed Dispersion; Sum 97-01)
Model 2 Ranking (Varying Dispersion; Sum 97-01)
(b) Spearman
ρ
s = 0.993
Y = 0.9978X
R
2
= 0.982
0
1,000
2,000
3,000
4,000
5,000
0 1,000 2,000 3,000 4,000 5,000
Model 3 Ranking (Fixed Dispersion; Sum 97-01)
Model 3 Ranking (Varying Dispersion; Sum 97-01)
(c) Spearman
ρ
s = 0.991
Y = 0.9982X
R
2
= 0.9857
0
1,000
2,000
3,000
4,000
5,000
0 1,000 2,000 3,000 4,000 5,000
Model 4 Ranking (Fixed Dispersion; Sum 97-01)
Model 4 Ran king (Varying Disp ersion; Sum 97- 01)
(d) Spearman
ρ
s = 0.993
Y = 0.9999X
R
2
= 0.9989
0
1,000
2,000
3,000
4,000
5,000
0 1,000 2,000 3,000 4,000 5,000
Model 5 Ranking (Fixed Dispers ion; Sum 97-01)
Model 5 Ranking (Varying Dispersion; Sum 97-01)
(e) Spearman
ρ
s = 0.999
Figure 11. Relationships in Ranking by Varying Dispersions and Traffic Volumes
Reviewer #1: The topic is an important one and the overall approach is reasonable, so I would like
to see the paper improved and salvaged. I am focusing on major issues only in this review because
the paper will, with the additional work, need another round of reviewing before it can be
considered further. At the very least, if my concerns turn out to be unfounded, I would be interested
to see the authors' response, before vetting the paper.
Response: We thank the reviewer for the thoughtful comments. We believe the comments
significantly improved the paper.
Here are my concerns:
1. Testing various models on the data from which they were calibrated is not considered
acceptable statistical practice, especially when assessing model performance by comparing the
sums of observations and predictions.
Response: The comparison of the observed and predicted values is only one of several tools that we
have used in this assessment. We have added discussions to this issue and included CURE plots as a
separate section of the paper. We have also added other goodness-of-fit statistics for evaluating the
models.
2. The authors use only one data set for their investigation, so it is difficult to make such
sweeping conclusions as have been made.
Response: Since we are not examining the predicting capabilities of statistical models (e.g., see Xie
at al., AA&P, in press), using one dataset is adequate for this kind of exercise. This is not different
than what other researchers have done for examining the performance of statistical models. Several
researchers have used one dataset for this kind of analysis, including previous work done by Mitra
and Washington (as noted by the reviewer), Hauer, Miaou, Mannering, Sayed, Abdel-Aty, Ivan, and
many others. This decision is also based on costs and resources available.
3. Theoretically, if the APM multiplier or AADT varies non-linearly over time, then it does make
sense to estimate time dependent models rather than use an average AADT and a single
multiplier - regardless of how we treat the dispersion parameter. So the authors conclusion on
this issue -- that time dependent models may be unnecessary — is a good example of the type
of conclusion that should not be drawn from such a limited empirical investigation.
Response: Given the additional analyses performed in this work, some of the conclusions have been
revised. Time-specific models are preferred over aggregated models as well as a varying dispersion
parameter is favored over a fixed dispersion, when it is warranted or when the number of covariates
is small. Many conclusions support previous work conducted on these topics. We have added some
references in the paper to support these conclusions.
4. The authors argue in favor of using simple models for their investigation. I wonder if
estimating well specified models will not resolve the issue of a varying dispersion parameter.
Washington and colleagues in Arizona have addressed this question I believe. If published, that
work should be discussed in the context of the authors' investigation.
* Response to Reviewers
Response: This research has been conducted before the publication of Mitra and Washington. Since
many models available in the literature are traffic-only models (including baseline models proposed
for the upcoming Highway Safety Manual), there is a need to examine how the varying dispersion
parameter affects the performance of such models. This and other papers have shown that the
dispersion parameter should not be assumed to be fixed when traffic-flow only models are used or
estimated. We have added the paper by Mitra and Washington and discussed its application to our
work.
5. The authors' model performance measures may not be the best ones, especially in the light of
the first comment. On this issue, I have two points.
a. The models are all fitted by taking logs to linearize them. Thus, it should not be surprising if
the sum of the observations does not match the sum of the predictions; so assessing model
performance by comparing observations to predictions seems inappropriate. To see why,
consider the worst possible model that says that every site has the same expected number of
accidents, equal to the mean if the population. Assessing this model by comparing the sum
of observations to the sum of predictions would suggest that this model is better than any of
those that the authors have estimated, since the sum of observations would exactly equal the
sum of predictions.
Response: We agree that the model is fitted in its logarithmic form and the sum should theoretically
be the same. However, the model will always be used in its non-logarithmic form. When a large
discrepancy exists, the model will be biased and provide inaccurate estimates. This has been shown
in previous work done by one of the authors. To confirm this point, we modified Table 5, and have
added Cumulative Residual (CURE) plots showing the differences between the different model
outputs. The CURE method has been used extensively by other transportation safety analysts to
evaluate the fit of models. Again, this is only one of the tools that safety analysts should use in
modeling development. Due to space constraints, we did not discuss the characteristic associated
with the fit in the logarithmic form of the equation.
b. It should not be surprising that the sums of the EB estimates are similar for the models and
quite close to the sums of the observations. I believe that this similarity should be
theoretically so. (That one model does not perform well with this test suggests to me that
there may be an error in the calculations.) Even if this were not theoretically so, the worst
possible model will again do best with this test. Consider one that is so bad that the model
prediction weight is zero for the EB estimate. Then the accident count has a weight of 1,
which would guarantee that the sum of the EB estimates would equal the sum of the
observations.
Response: We agree and have noted an error in the spreadsheet. We have removed this part of the
analysis from the paper. Thank you for noting this problem.
Given these concerns, I suggest that the authors consider more appropriate model performance
measures and use data other than the calibration set.
Response: To summarize, using one dataset is adequate for the objective of this study. We have
included additional measures to evaluate the performance of the models, including GOF measures
proposed by Mitra and Washington. We believe that all the different criteria presented in the paper
are helpful for determining the most suitable models whether they are developed using a fixed or a
varying dispersion parameter and how they affect the EB estimates.
