category: literaturenote citekey: lordinvestigatingeffectsfixedvarying2008 title: Investigating the effects of the fixed and varying dispersion parameters of Poisson-gamma models on empirical Bayes estimates authors: "Lord, Dominique; Park, Peter Young-Jin" year: 2008 date: 2008-00-00 2008 doi: 10.1016/j.aap.2008.03.014 publication: "Accident Analysis & Prevention" url: "https://doi.org/10.1016/j.aap.2008.03.014" zotero_key: Z3Z8RX3U zotero_storage: D7R3WUP5 collections: transpordi_planeerimine folder: 001_artiklid firstAuthor: "Lord, Dominique"

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Accident Analysis and Prevention

journal homepage: www.elsevier.com/locate/aap

Investigating the effects of the fixed and varying dispersion parameters of Poisson-gamma models on empirical Bayes estimates

Dominique Lord^a,^∗, Peter Young-Jin Park ^b,¹

^a Zachry Department of Civil Engineering, Texas A&M University, 3136 TAMU, College Station, TX 77843-3136, United States
^b iTRANS Consulting Inc., 100 York Boulevard, Suite 300, Richmond Hill, ON, Canada L4B 1J8

article info

Article history: Received 18 September 2006 Received in revised form 10 March 2008 Accepted 21 March 2008

Keywords: Crash prediction models Dispersion parameter Empirical Bayes estimates Negative binomial Rural intersections

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 hotspots 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 overdispersion 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, especially for traffic flow-only models, as well as varying as a function of different time-periods, there is a need to determine how these models 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 traffic flow-only 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. Several statistical methods were used to compare the fitting performance of the various functional forms.

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, weight factors, and dispersion parameters. Timespecific models with a varying dispersion parameter provide better statistical performance in terms of goodness-of-fit (GOF) than aggregated multi-year models. Furthermore, the identification of hazardous sites, using the EB method, can be significantly affected when a GNB model with a time-varying dispersion parameter is used. Thus, erroneously selecting a functional form may lead to select the wrong sites for treatment. The study concludes that transportation safety analysts should not automatically use an existing functional form for modeling motor vehicle crashes without conducting rigorous analyses to estimate the most appropriate functional form linking crashes with traffic flow.

1. 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; Poch 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 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). [Note: more recently, Lord et al. (in press) have proposed the Conway–Maxwell–Poisson model as a substitute to the NB model.] 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) com-

^∗ Corresponding author. Tel.: +1 979 458 3949; fax: +1 979 845 6481. E-mail addresses: d-lord@tamu.edu (D. Lord), ppark@itransconsulting.com (P.Y.-J. Park).

¹ Tel.: +1 905 882 4100x5264; fax: +1 905 882 1557.

monly exceeds the mean value of the variable, which violates the main assumption associated with the Poisson model (i.e., "overdispersion" 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). Second, 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, 2007).

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 upon 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). 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 Saccomanno et al., 2001) or "sites with promise" (see Hauer, 1996) by ranking crash-prone locations by order 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:

$$\hat{\mu}{it} = (1 - \gamma{it})y{it} + \gamma{it}\hat{\mu}_{it} \tag{1}$$

where ˆ -ˆ it: EB estimate in crashes per year for given site i and year t; it: weight factor for given site i and year t; yit: observed number of crashes for given site i and year t; ˆit: 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 is given as follows:

$$\gamma{it} = \frac{1}{(1 + \alpha \hat{\mu}{it})} \tag{2}$$

where˛, the dispersion parameter for the given dataset [Note: in the safety literature, analysts sometimes report the inverse dispersion parameter = 1/˛].

Up until very recently, researchers did not estimate timevarying EB estimates (as currently defined in Eqs. (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 the sites' covariates, such as minor and major volumes (AADT) at intersections, vertical and horizontal curvatures 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 the 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. (2005b) for modeling the safety performance of freeways as a function of traffic flow characteristics. An exception is Lyon et al. (2005) 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.

There are many different functional forms that have been proposed to link crashes to the explanatory variables of traditional regressionmodels for segments (Martin, 2002; Abbas, 2004; Lord et al., 2005b; Fitzpatrick et al., in press) and intersections (Nicholson and Turner, 1996; Turner and Nicholson, 1998; Mountain et al., 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 in Eq. (1)], then it may also influence the precision of estimated dispersion parameters [i.e., ˛ in Eq. (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 traffic flow-only 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

Table 1 Summary Statistics for the Study Dataset

Variable (per site)	Mean	Standard deviation	Minimum	Maximum	Number of intersections
Total number of crashesa over 5 years	1.073	2.806	0	64	5588
Average crashesa per year	0.215	0.561	0	12.8
Average AADT of major road over 5 years	6953	7407	103	79,800
Average AADT of minor road over 5 years	268	683	1	23,111

^a All crash severities.

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 5 year (1997–2001) study period, a total of 5752 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 5588 three-legged rural intersections for the same 5-year period with a total of 5996 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 and Park and Lord, in press, about this assumption). Table 1 contains a brief summary of study dataset.

2. Model development

This section describes the characteristics of the traditional NB and GNB models.

2.1. 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

$$P(Y_{it} = yi | \mu{it}, \alpha)$$

$$= \frac{\Gamma(y{it} + (1/\alpha))}{y{it}! \cdot \Gamma(1/\alpha)} \cdot \left(\frac{1/\alpha}{(1/\alpha) + \mu{it}}\right)^{1/\alpha} \cdot \left(\frac{\mu{it}}{(1/\alpha) + \mu{it}}\right)^{y{it}}$$ (3)

In contrast to the Poisson distribution, the NB distribution allows for over-dispersion, and thus the mean (i.e., it = E{Yit} = exp(Xit-); Xit: a vector of covariates, and -: regression coefficients corresponding to the covariates) can be smaller than the variance (i.e., Var{Yit} = it + ˛ · -2 it) (Note: the model's output can also show signs of under-dispersion). When the dispersion parameter ˛= 1/ is equal to zero (that is →∞), the NB distribution reverts back to the Poisson distribution. Larger values of ˛ signifies a greater amount of over-dispersion. An important characteristic of the traditional NB model is that the dispersion parameter ˛ (or its inverse ) 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 Eq. (3) and estimates the number of crashes of each site, similar to the traditional NB model. However, instead of estimating a fixed dispersion parameter, the model estimates the varying dispersion parameters by using the following expression (Hardin and Hilbe, 2001):

$$\alpha{it} = \exp(\mathbf{Z}{it}\mathbf{\delta}_t)$$ (4)

where Zit: a vector of secondary covariates (are not necessarily the same as the covariates used for estimating the mean function ˆit); t: a vector of regression coefficients corresponding to covariates Zit.

With Eq. (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.

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 timetrend 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; Hardin and Hilbe, 2003 for additional information).

2.2. 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. Traffic flow-only models are the most common type of model used by transportation safety analysts (Mountain and Fawaz, 1996; Hauer, 1997; Hughes et al., 2005). In addition, this kind of model has been shown to exhibit a structured variance function (Miaou and Lord, 2003; Mitra and Washington, 2007; Geedipally and Lord, 2008). In other words, the variance function of the model (i.e., a function of the dispersion parameter) can be dependent upon the specific characteristics of each site (Miaou and Lord, 2003). Thus, investigating the effects of a varying dispersion parameter on the EB estimates for traffic flow-only models is important in the context described above. Miaou and Lord (2003) listed the most popular functional forms (referred in the study as Models 1–5) from previously published studies:

(1) Model 1 : ln it = ˇ0^t + ˇ¹ ln (F1it + F2it) (2) Model 2 : ln it = ˇ0^t + ˇ¹ ln F1it + ˇ² ln F2it (3) Model 3 : ln it = ˇ0^t + ˇ¹ ln (F1it × F2it) (4) Model 4 : ln it = ˇ0^t + ˇ¹ ln(F1it + F2it) + ˇ² ln (F2it/F1it) (5) Model 5 : ln it = ˇ0^t + ˇ¹ ln F1it + ˇ² ln F2it + ˇ³ F2it

where it: the expected number of crashes at intersection i in year t (Note: E{Y} = -); *F*1it = AADT entering from major road at intersection i in year t; and, *F*2it = AADT entering from minor road at intersection i in year t.

As reported by Miaou and Lord (2003), the functional forms described above are not the most adequate ones for describing the relationship between crashes and exposure since the forms do not

NB del mo		ˇ0, t	ˇ1	ˇ2	ˇ3	log (˛ )	log like liho od	No . of ete par am rs	De via nce	AIC	BIC	G2- tist ics sta	R2- like MO F
Fun ctio nal For	199 7	−10 .53 71	1.01			0.9 483	−2 723 .68 47	2	0.9 76	544 7.3 69	−4 275 0.7 43	425 8.3 94	0.2 68
m 1 (NB Mo del 1)	199 8	(0.0 395 ) −10 .82 90	53 (0.0 ) 295			(0.0 865 ) 1.15 49	−2 251 .95 69	2	0.8 07	450 3.9 14	−43 694 .19 9	378 3.1 81	0.3 75
		(0.0 ) 441				(0.0 ) 929
	199 9	−10 .50 96 (0.0 367 )				0.6 553 (0.0 915 )	−2 847 .75 89	2	1.0 20	569 5.5 18	−4 250 2.5 95	426 7.8 75	0.2 67
	200 0	−10 .49 90 (0.0 ) 377				0.8 166 (0.0 ) 857	−2 883 .05 06	2	1.0 33	576 6.1 01	−4 243 2.0 11	440 0.6 01	0.2 80
	200 1	−10 .47 59				0.8 938	−2 959 .08 64	2	1.0 60	591 8.1 73	−4 227 9.9 40	459 5.4 27	0.2 85
	All -Ye ar	(0.0 379 ) −8 .95 41 (0.2 ) 609				(0.0 815 ) 0.6 871 (0.0 ) 399	−6 763 .09 79	3	2.4 22	135 26. 196	−34 663 .28 8	130 34. 299	0.4 75
nal Fun ctio For	199 7	−10 .00 62	730 (0.0 0.6	0.4 880 (0.0		0.3 490	−2 561 .01 24	2	0.9 17	512 2.0 25	−43 076 .08 8	371 9.8 03	0.3 79
m 2 (NB del 2) Mo	199 8	(0.0 371 ) −10 .27 09 424	266 )	162 )		(0.1 092 ) 0.7 101 081	−2 150 .46 60	2	0.7 70	430 0.9 32	−43 897 .18 0	339 0.9 54	0.4 31
	199 9	(0.0 ) −9 .96 27				(0.1 ) −0 .05 78	−2 664 .80 08	2	0.9 54	532 9.6 02	−4 286 8.5 11	370 1.5 63	0.3 77
	200 0	(0.0 341 ) −9 .94 85				(0.1 242 ) 0.2 597	−2 723 .34 76	2	0.9 75	544 6.6 95	−4 275 1.4 17	389 4.8 10	0.3 78
	200 1	(0.0 356 ) −9 .91 91				(0.1 055 ) 0.3 762	−2 806 .43 33	2	1.0 05	561 2.8 67	−4 258 5.2 46	408 1.5 85	0.3 71
	All -Ye ar	(0.0 ) 357 −8 .411 1				(0.0 ) 983 0.2 610	−6 420 .93 49	4	2.3 00	128 41. 870	−3 533 8.9 86	105 75. 656	0.6 14
		(0.2 325 )				(0.0 471 )
ctio nal Fun m 3 (NB For Mo del 3)	199 7	−9 .199 7 (0.0 ) 370	0.5 465 (0.0 ) 124			0.3 255 (0.1 ) 108	−2 559 .64 80	2	0.9 17	511 9.2 96	−43 078 .81 6	371 1.2 14	0.3 78
	199 8	−9 .45 43 (0.0 425 )				0.7 362 (0.1 076 )	−2 161 .77 96	2	0.7 74	432 3.5 59	−43 874 .55 3	341 3.7 11	0.4 08	D. Lord, P.YJ. Park / Accident Analysis and Prevention 40 (2008) 1441–1457
	199 9	−9 .149 3 (0.0 ) 341				−0 .06 35 (0.1 ) 248	−2 666 .83 49	2	0.9 55	533 3.6 70	−4 286 4.4 43	370 3.9 58	0.3 82
	200 0	−9 .134 2 (0.0 355 )				0.2 518 (0.1 062 )	−2 725 .12 03	2	0.9 76	545 0.2 41	−4 274 7.8 72	389 5.6 31	0.3 73
	200 1	−9 .102 8 (0.0 ) 357				0.3 626 (0.0 ) 991	−2 808 .07 82	2	1.0 06	561 6.1 56	−4 258 1.9 56	408 2.1 60	0.3 62
	All - Yea	−7. 597 6				0.2 689	−6 435 .98 94	3	2.3 05	128 71. 979	−3 531 7.5 05	105 90. 943	0.6 05
	r	(0.1 746 )				(0.0 472 )
Fun m 4 (NB ctio nal For Mo del 4)	199 7	−10 .31 25 (0.0 371 )	20 (0.0 1.16 ) 279	0.4 279 (0.0 ) 158		0.3 340 (0.1 099 )	−2 557 .61 53	2	0.9 16	511 5.2 31	−43 082 .88 2	369 8.5 55	0.3 84
	199 8	−10 .57 58 (0.0 ) 424				0.7 027 (0.1 ) 083	−2 147 .88 95	2	0.7 69	429 5.7 79	−43 902 .33 3	337 6.2 33	0.4 36
	199 9	−10 .26 90 (0.0 341 )				−0 .08 51 (0.1 258 )	−2 659 .53 72	2	0.9 53	531 9.0 74	−4 287 9.0 38	367 8.5 61	0.3 77
	200 0	−10 .25 55				0.2 339	−2 718 .59 47	2	0.9 74	543 7.1 89	−4 276 0.9 23	387 0.4 73	0.3 85
	200 1	(0.0 ) 355 −10 .22 69				(0.1 ) 070 0.3 471	−2 798 .35 13	2	1.0 02	559 6.7 03	−4 260 1.4 10	405 3.8 25	0.3 83
	All -Ye ar	(0.0 356 ) −8 .71 68 (0.2 ) 379				(0.0 995 ) 0.2 470 (0.0 ) 474	−6 409 .61 62	4	2.2 95	128 19.2 32	−3 536 1.6 23	104 58. 744	0.6 21
Fun ctio nal For m 5	199 7	−9 .713 9	0.6 706	0.4 196	0.0 002 (0.0 000 )	0.3 634	−2 560 .32 16	2	0.9 17	512 0.6 43	−43 077 .46 9	360 6.9 12	0.3 68
(NB del 5) Mo	199 8	(0.0 375 ) −9 .97 76	(0.0 264 )	(0.0 239 )		(0.1 067 ) 0.7 201	−2 149 .23 50	2	0.7 70	429 8.4 70	−43 899 .64 2	330 5.4 72	0.4 23
		(0.0 427 )				(0.1 064 )
	199 9	−9 .67 30 (0.0 346 )				−0 .02 39 (0.1 179 )	−2 663 .58 27	2	0.9 54	532 7.1 65	−4 287 0.9 47	359 1.3 59	0.3 58
	200 0	−9 .65 50				0.2 804	−2 722 .19 63	2	0.9 75	544 4.3 93	−4 275 3.7 20	379 4.2 44	0.3 69
	200 1	(0.0 359 ) −9 .62 40 361				(0.1 023 ) 0.3 941 957	−2 803 .63 78	2	1.0 04	560 7.2 76	−4 259 0.8 37	397 4.1 01	0.3 72
	All	(0.0 ) −8 .115 0				(0.0 ) 0.2 548	−6 413 .06 36	5	2.2 97	128 26. 127	−3 534 6.1 00	100 20. 232	0.5 93
	Yea r	(0.2 424 )				(0.0 471 )

Table 3Estimated coefficients and statistics for GNB crash prediction models

ode l GN B m		ˇ0, t	ˇ1	ˇ2	ˇ3	log (˛ )				log like liho od	. of No	ian Dev ce	AIC	BIC	G2- tist ics sta	R2- like
						0, t	1,t	2, t	3, t		ete par am rs					MO F
Fun ctio nal For m 1	199 7	−10 .52 40	1.01 36			3.1 238	−0 .23 14			−2 721 .63 94	3	0.9 75	544 3.2 79	−4 274 6.2 05	426 4.4 84	0.2 67
(GN ode l 1) B M	199 8	(0.0 397 ) −10 .79 70	(0.0 293 )			(1.0 521 ) 5.5 554	(0.1 121 ) −0 .46 18			−2 245 .69 74	3	0.8 05	449 1.3 95	−43 698 .08 9	375 2.7 62	0.3 85
		(0.0 ) 451				(1.1 ) 800	(0.1 ) 244
	199 9	−10 .49 30				3.0 740	−0 .25 47			−2 845 .55 06	3	1.0 20	569 1.1 01	−4 249 8.3 83	426 6.1 05	0.2 69
		(0.0 368 )				(1.1 049 )	(0.1 168 )
	200 0	−10 .48 64 378				2.5 738 505	−0 .186 5 116			−2 881 .66 36	3	1.0 32	576 3.3 27	−4 242 6.1 57	440 8.7 59	0.2 78
	200 1	(0.0 ) −10 .46 41				(1.0 ) 2.6 595	(0.1 ) −0 .187 5			−2 957 .43 37	3	1.0 60	591 4.8 67	−4 227 4.6 17	460 6.1 42	0.2 83
		(0.0 380 )				(0.9 712 )	(0.1 032 )
	All -Ye ar	−8 .93 88				2.1 562	−0 .160 2			−6 758 .33 75	4	2.4 20	135 16. 675	−34 664 .18 1	130 40. 488	0.4 75
		(0.2 ) 612				(0.4 ) 667	(0.0 ) 508
Fun ctio nal For m 2	199 7	−10 .00 43	0.6 594	0.5 114		3.6 978	−0 .134 2	−0 .33 78		−2 545 .67 97	4	0.9 13	509 1.3 59	−43 089 .49 6	367 7.3 20	0.3 80
(GN ode l 2) B M		(0.0 376 )	(0.0 269 )	(0.0 159 )		(1.1 761 )	(0.1 284 )	(0.0 652 )
	199 8	−10 .25 62 (0.0 434 )				5.7 060 (1.2 479 )	−0 .30 45 (0.1 324 )	−0 .34 40 (0.0 637 )		−2 133 .54 76	4	0.7 65	426 7.0 95	−43 913 .76 0	333 6.6 53	0.4 37
	199 9	−9 .96 00				3.7 594	−0 .178 9	−0 .33 95		−2 652 .43 10	4	0.9 51	530 4.8 62	−4 287 5.9 94	365 6.2 31	0.3 80
		(0.0 344 )				(1.3 150 )	(0.1 394 )	(0.0 726 )
	200 0	−9 .95 12				2.3 295	−0 .03 77	−0 .274 7		−2 714 .61 22	4	0.9 73	542 9.2 24	−4 275 1.6 31	386 6.9 42	0.3 68
	200 1	(0.0 359 ) −9 .92 26				(1.1 703 ) 2.1 886	(0.1 251 ) −0 .074 1	(0.0 650 ) −0 .179 4		−2 802 .10 68	4	1.0 04	560 4.2 14	−4 257 6.6 42	405 5.9 48	0.3 52
		(0.0 ) 361				(1.0 ) 892	(0.1 ) 128	(0.0 ) 603
nal Fun ctio For m 3 (GN ode l 3) B M	All -Ye ar	−8 .40 72				2.4 956	−0 .08 94	−0 .25 49		−63 83. 006 1	6	2.2 87	127 66. 012	−3 539 7.5 87	103 56. 610	0.6 13
		(0.2 317 )				(0.4 959 )	(0.0 549 )	(0.0 298 )
	199 7	−9 .30 48	0.5 541	(0.0 118 )		4.9 526	−0 .29 79			−2 545 .45 12	3	0.9 12	509 0.9 02	−43 098 .58 2	368 6.8 36	0.3 75
		(0.0 ) 373				(0.8 ) 032	(0.0 ) 529
	199 8	−9 .54 07 (0.0 435 )				6.1 682 −0 .34 82 (0.8 222 ) (0.0 537 ) 4.7 526 −0 .30 67			−2 142 .25 38	3	0.7 68	428 4.5 08	−43 904 .97 6	335 4.4 79	0.4 18
	199 9	−9 .25 09								−2 655 .23 37	3	0.9 51	531 0.4 67	−4 287 9.0 17	366 6.4 24	0.3 82
		(0.0 ) 343				(0.9 ) 146	(0.0 ) 602
	200 0	−9 .24 41				3.7 283	−0 .22 24			−2 717 .55 00	3	0.9 74	543 5.1 00	−4 275 4.3 84	388 1.4 48	0.3 67
	200 1	(0.0 358 ) −9 .213 0				(0.8 136 ) 2.9 010	(0.0 529 ) −0 .162 5			−2 804 .03 16	3	1.0 05	560 8.0 63	−4 258 1.4 21	406 8.3 60	0.3 54
		(0.0 ) 360				(0.7 ) 957	(0.0 ) 514
	All	−7. 699 5				3.5 021	−0 .22 03			−63 96. 836 7	4	2.2 91	127 93. 673	−3 538 7.1 82	104 23. 853	0.6 06
		(0.1 708 )				(0.3 422 )	(0.0 234 )
ctio nal Fun For m 4	199 7	−10 .35 13	1.17 24	0.4 418		3.5 748	−0 .44 36 ) (0.1 ) 320 −0 .63 13 845 381	−0 .30 57		−2 543 .56 30	4	0.9 12	508 7.1 26	−43 093 .73 0	366 7.2 46	0.3 82
(NB Mo del 4)		(0.0 ) 375	(0.0 271 )	(0.0 152 )		(1.2 182		(0.0 ) 640
	199 8	−10 .60 30 434			5.7	663		−0 .29 33 625		−2 131 .49 95	4	0.7 64	426 2.9 99	−43 917 .85 7	332 9.7 69	0.4 41
	199 9	(0.0 ) −10 .30 68				(1.2 ) 3.7 844	(0.1 ) −0 .50 71	(0.0 ) −0 .310 0		−2 647 .76 21	4	0.9 49	529 5.5 24	−4 288 5.3 31	364 2.7 07	0.3 75
		(0.0 ) 343				(1.3 ) 720	(0.1 ) 498	(0.0 ) 708
	200 0	−10 .29 81				2.1 518	−0 .28 66	−0 .25 74		−2 710 .29 36	4	0.9 71	542 0.5 87	−4 276 0.2 68	384 9.7 03	0.3 80
		(0.0 358 )				(1.2 163 )	(0.1 310 )	(0.0 636 )
	200 1	−10 .26 99 (0.0 ) 360				2.1 474 (1.1 ) 345	−0 .24 38 (0.1 ) 241	−0 .164 7		−2 794 .14 53	4	1.0 01	558 8.2 91	−4 259 2.5 65	403 3.6 09	0.3 77
	All -Ye ar	−8 .75 36				2.4 856	−0 .33 35	(0.0 ) 577 −0 .23 34	−63 72. 410 7	6	2.2 83	127 44. 821	−3 541 8.7 77	102 83. 859	0.6 21
		(0.2 378 )				(0.5 114 )	(0.0 570 )	(0.0 287 )

Table 3 (Continued )
GNB model		ˇ0,t	ˇ2 ˇ1	ˇ3	log(˛)				log likelihood	No. of	Deviance	AIC	BIC	G2-statistics	R2-like
					0,t	1,t	2,t	3,t		parameters					MOF
Functional Form 5	1997	−9.6933	0.6591	0.0002	4.1627	−0.1505	−0.4127	0.0001	−2545.6929	5	0.913	5091.386	−43080.841	3593.716	0.366
(GNB Model 5)		(0.0377)	(0.0254) (0.0267)0.4355	(0.0000)	(1.2342)	(0.1308)	(0.0773)	(0.0000)
	1998	−9.9432			6.1230	−0.2967	−0.4562	0.0002	−2130.9706	5	0.764	4261.941	−43910.286	3272.342	0.427
		(0.0435)			(1.3011)	(0.1352)	(0.0775)	(0.0000)
	1999	−9.6474			4.3724	−0.1788	−0.4812	0.0002	−2647.2761	5	0.949	5294.552	−42877.675	3566.666	0.360
		(0.0344)			(1.3838)	(0.1446)	(0.0851)	(0.0001)
	2000	−9.6352			2.5971	−0.0155	−0.3818	0.0001	−2711.3334	5	0.972	5422.667	−42749.560	3785.836	0.365
		(0.0360)			(1.2279)	(0.1281)	(0.0783)	(0.0000)
	2001	−9.6035			2.4494	−0.0710	−0.2491	0.0001	−2798.1022	5	1.003	5596.204	−42576.023	3964.343	0.368
		(0.0361)			(1.1481)	(0.1150)	(0.0739)	(0.0000)
	All-Year	−8.0915			2.7410	−0.0918)	−0.3113	0.0001	−6373.0739	8	2.284	12746.148	−35400.194	9932.313	0.594
		(0.2438)			(0.5148)	(0.0556)	(0.0360)	(0.0000)

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 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, a two-step calibration approach that was introduced by Lord and Persaud (2000) and applied by Lyon et al. (2005) was utilized to estimate the model parameters ( ˇ 0 t , ˇ 1 ˇ 2, and ˇ 3). They developed a series of traditional NB models assuming the varying intercept term ( ˇ 0 t,) by year, but constant model parameters for the covariates (ˇ1, ˇ2, and ˇ3).

Step (1) Develop the All-Year model using total crashes for 5 years and average AADT over the same 5 years to estimate the coefficients of themodel ( ˇ , ˇ 2, and ˇ 3) (themodel output is the number of crashes per 5 years).
Step (2) Calibrate the time-specific intercept terms ( ˇ ^t) using each year AADT and each year crash data with a constraint of holding the model parameters for covariates ( ˇ , ˇ 2, and ˇ 3) that were estimated in the first step.

In this study, only the intercept term ( ˇ ^t) varies by year for both the traditional NB and GNB crash prediction models. The output of All-Year models represents the total number of crashes over the 5-year study period. Instead, we can also obtain the total number of crashes for the same 5 years by aggregating the number of crashes for each year that was estimated using time-specific crash prediction models.

Similar to the study by Lord et al. (2005b), the same covariates were used for both the mean and dispersion functions of each functional form. Furthermore, in order to compare the impact of different functional forms on dispersion parameters, we kept every coefficient in the model. Consequently, several coefficients that failed to pass the significant test at a 5% confidence level were still reported as the dispersion parameters of GNB models.

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, the expected number of crashes per year is estimated using the time-specific GNB Model 1. If we assume, for example, that the major and minor AADTs for a particular year are 3000 vpd and 300 vpd, respectively, by employing the time-specific GNB Model 1 in Table 3, one obtains the expected number of crashes for 1997 as 0.099 [i.e., exp ( −10.5240 + 1.0136 × ln (3000 + 300))]. Similarly, the expected number of crashes for 1998, 1999, 2000, and 2001 is estimated to be 0.075, 0.102, 0.103, and 0.105, respectively. By summing up all these estimates, the expected number of crashes over the 5-year period by this disaggregate time-specific GNB Model 1 is estimated to be 0.485. On the other hand, by employing the aggregate All-Year GNB Model 1 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 traffic volumes above, the timespecific inverse dispersion parameters are equal to 3.488 [i.e., exp (3.1238 − 0.2314 × ln (3000 + 300))], 6.137, 2.747, 2.894, and 3.129, respectively, for each corresponding year. As opposed to the estimated number of crashes, these values cannot be aggregated (Note: the average value equals 3.679). Using the All-Year model, the overall inverse dispersion parameter over the 5 years equals 2.359. We found that the aggregate All-Year model always produces a smaller inverse dispersion parameter than the average of inverse dispersion parameters based on disaggregate time-specific models. It appears that the aggregate All-Year model using the average AADT does not adequately take into account the time variation in the dispersion.

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) = ⁻2 ln ^L(Mk)+2^P ^N (5) where ln L(Mk) = log likelihood of model k; P = the number of parameters; and, N = the number of observations (in our exercise = 5588).
(2) Bayesian Information Criteria (BIC) = D(Mk) − d.f. · ln N (6) where D(Mk) = Deviance of model k; and, d.f. = degrees of freedom.
(3) Sum of Model Deviances (G2) = 2 n yi ln yi ˆ i (7)

where yi: the observed number of crashes at site i; ˆi: the expected number of crashes at site i.

(4) R2-like measure-of-fit (MOF) based on Standardized Residuals

$$(R^{2}) = 1 - \left[ \frac{\left( \sum{i=1}^{n} \left( (y{i} - \hat{\mu}{1}) / \sqrt{\hat{\mu}} \right)^{2} \right)}{\left( \sum{i=1}^{n} \left( (y_{i} - \bar{y}) / \sqrt{\hat{y}} \right)^{2} \right)} \right]$$ (8)

where y¯: average number of observed number of crashes.

The model with the lowest value in AIC, BIC, andG² is considered the model with the best statistical fit. On the other hand, the model with the largest *R*2-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 *R*2-like MOF). Model 5 is selected as the best fitted model based on the *G*2-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 *G*2 statistics) with the exception of Model 1 for the *G*2-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 *R*2-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 *R*2-like MOF.
(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*G*2-statistics). However, since the *R*2-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.
(3) As a result, determining the model with the best statistical fit 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.

2.3. Goodness-of-fit evaluation based on residual analysis

In the previous section, four different statistical tests 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 for evaluating the fit of models. This method is known as the CURE method and has been used extensively by many transportation safety analysts (e.g., Lord and Persaud, 2000; Washington et al., 2005; Wang and Abdel-Aty, 2007). 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 estimated number of crashes ( ˆit) at a given study site i in 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.

Figs. 1–5 show a total of 10 different CURE plots using five different functional forms with the two model types based upon the "All-Year" crash data. As discussed by Hauer and Bamfo (1997), the CURE plot reveals how well the functional forms fit the data with respect to each individual explanatory variable and show systematic deviations of the cumulative residuals from the zeroresidual line. Two different CURE plots (i.e., figure (a) and (b)) are generated for each model to compare the model performance

Fig. 1. Cumulative residual plots for Model 1.

Table 4 Summary of statistical tests

Functional form (All-Year model)	AIC		BIC		G2-statistics		R2-like MOF
	NB model	GNB model	NB model	GNB model	NB model	GNB model	NB model	GNB model
Model 1	2.422	2.420a	−34663.288	−34664.181	13034.299	13040.488	0.475	0.475
Model 2	2.300	2.287	−35338.986	−35397.587	10575.656	10356.610	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	−35400.194	10020.232	9932.313	0.593	0.594
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

^a Bold values represent a better model between the NB and GNB model within the same functional form.

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 Figs. 1–5, several characteristics involving CURE plots can be noticed:

(1) Model 1 underestimates the expected number of crashes (i.e., yit > -ˆ it) for the range between 100 and 65,000 major road AADT and slightly overestimates the expected number (i.e., yit < -ˆ it) when themajor road AADT (*F*1it) is higher than 65,000 (refer to Fig. 1) regardless of the model type (NB or GNB). Moreover, for the range of major road AADT (*F*1) between 5000 and 12,000, Model 1 produces larger cumulative residual values than the +2.0 confidence interval boundary. No practical difference in the CURE plots between the two model types (i.e., NB and GNB model) is found and the final cumulative residual curve is relatively close to 0.

(2) NB Model 2 (refer to the Fig. 2(a)) and NB Model 4 (refer to the Fig. 4(a)) are similar in that both models underestimate the expected number of crashes over the entire range of the variable (i.e., *F*1it). The cumulative residual values are larger than the +2.0 boundary for AADT values above 50,000. In addition, the cumulative residual curves for these two NB models do not end near the zero-residual line.

Fig. 2. Cumulative residual plots for Model 2.

Fig. 3. Cumulative residual plots for Model 3.

Fig. 4. Cumulative residual plots for Model 4.

Fig. 5. Cumulative residual plots for Model 5.

(3) Among the different CURE plots, the Model 5 CURE plots (i.e., Fig. 5(a) and (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 one with the best statistical fit according to the G2-statistics and chosen as the second best based on the AIC as well as BIC criteria. 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., *F*2it = 23,111) at that specific intersection. The value is at least twice as high as 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 with respect to large values of minor road AADT (*F*2it), as captured by the third variable in the functional form (i.e., ˇ3*F*2it). It should be noted that the third variable in Model 5 actually influenced the fit of the model, since Model 2 does not show the sudden change for the cumulative residual plot. In truth, this site or observation should be investigated further to determine whether this observation is an outlier (e.g., error in reported flows) or an influence point.

Table 5 Cumulative residuals between NB and GNB models

	Residualsa (yit − ˆ it)	Yearly based model						All-Year model
		1997	1998	1999	2000	2001	Total 97–01
NB	Model 1	−17.77	41.17	3.53	−24.29	−32.76	−30.12	−16.58
Mod	Model 2	13.92	47.40	26.66	1.84	−3.37	86.45	124.99
els	Model 3	15.84	44.07	26.65	2.40	−1.93	87.03	123.52
	Model 4	7.86	42.68	22.15	−1.88	−5.54	65.27	100.41
	Model 5	−266.29	−176.82	−273.27	−311.03	−330.18	−1357.59	−1332.83
GNB	Model 1	−14.60	26.33	2.64	−20.03	−27.14	−32.82	−13.38
Mod	Model 2	−4.07	21.69	6.84	−11.19	−15.48	−2.21	24.69
els	Model 3	4.17	16.93	9.09	−5.00	−9.45	15.72	43.22
	Model 4	−6.59	20.61	5.56	−12.39	−15.92	−8.73	16.32
	Model 5	−262.74	−189.16	−276.38	−304.66	−324.34	−1357.29	−1332.38

^a yit represents the observe number of crashes per site per each year.

	Mo del 1		Mo del 2			Mo del 3			Mo del 4			Mo del 5			Mo del	s 1 –5
		Avg	Mi n.	Ma x.	Avg	Mi n.	Ma x.	Avg	Mi n.	Ma x.	Avg	Mi n.	Ma x.	Avg	Mi n.	Ma x.	Avg	Mi n.	Ma x.
Fix ed dis sio per	E{ Di} 97	2.5 81	2.5 81	2.5 81	1.4 18	1.4 18	1.4 18	1.3 85	1.3 85	1.3 85	1.3 96	1.3 96	1.3 96	1.4 38	1.4 38	1.4 38	1.6 44	1.3 85	2.5 81
n ( NB del ) mo	Di} E{ 98	3.1 74	3.1 74	3.1 74	2.0 34	2.0 34	2.0 34	2.0 88	2.0 88	2.0 88	2.0 19	2.0 19	2.0 19	2.0 55	2.0 55	2.0 55	2.2 74	2.0 19	3.1 74
	E{ Di} 99	1.9 26	1.9 26	1.9 26	0.9 44	0.9 44	0.9 44	0.9 39	0.9 39	0.9 39	0.9 18	0.9 18	0.9 18	0.9 76	0.9 76	0.9 76	1.1 41	0.9 18	1.9 26
	E{ Di} 00	2.2 63	2.2 63	2.2 63	1.2 97	1.2 97	1.2 97	1.2 86	1.2 86	1.2 86	1.2 64	1.2 64	1.2 64	1.3 24	1.3 24	1.3 24	1.4 87	1.2 64	2.2 63
	Di} E{ 01	2.4 44	2.4 44	2.4 44	1.4 57	1.4 57	1.4 57	1.4 37	1.4 37	1.4 37	1.4 15	1.4 15	1.4 15	1.4 83	1.4 83	1.4 83	1.6 47	1.4 15	2.4 44
	E{ Di} All	1.9 88	1.9 88	1.9 88	1.2 98	1.2 98	1.2 98	1.3 09	1.3 09	1.3 09	1.2 80	1.2 80	1.2 80	1.2 90	1.2 90	1.2 90	1.4 33	1.2 80	1.9 88
ing dis sio Va ry per n	E{ Di} 97	3.3 33	1.6 92	7.0 15	3.3 20	0.3 25	19. 819	3.6 62	0.2 98	29 .20 9	3.1 66	0.3 14	17. 150	3.5 23	0.2 05	28 .94 8	3.4 01	0.2 05	29 .20 9
( GN B m ode l)	Di} E{ 98	5.8 86	1.4 22	26 .31 6	6.1 96	0.3 64	53 .77 4	6.8 78	0.3 48	66 .70 4	5.9 44	0.3 46	47. 169	6.8 17	0.7 33	85 .30 7	6.3 44	0.3 46	85 .30 7
	E{ Di} 99	2.6 09	1.2 26	6.1 32	2.4 43	0.2 09	15. 614	2.6 93	0.1 99	20 .46 7	2.3 72	0.1 92	14. 418	2.7 68	0.4 31	28 .84 1	2.5 77	0.1 92	28 .84 1
	E{ Di} 00	2.7 66	1.6 03	5.8 48	2.3 58	0.4 34	8.3 06	2.6 18	0.4 10	11. 885	2.2 43	0.4 21	7.2 89	2.4 99	0.8 17	12. 307	2.4 97	0.4 10	12. 307
	E{ Di} 01	2.9 81	1.6 82	6.3 45	2.2 24	0.6 64	5.8 77	2.3 57	0.6 22	7.2 83	2.1 54	0.6 34	5.4 80	2.2 76	1.0 34	7.7 63	2.3 98	0.6 22	7.7 63
	E{ Di} All	2.2 64	1.4 15	3.8 92	1.9 70	0.3 60	7.3 15	0.8 53	0.0 69	6.1 56	1.9 21	0.3 43	6.8 11	2.0 13	0.6 54	9.2 24	1.8 04	0.0 69	9.2 24
Fix ed- var y	E{ Di} 97	-0. 752	0.8 90	−4 .43 4	−1 .90 2	1.0 93	−1 8.4 02	−2 .27 7	1.0 87	−2 7.8 24	−1 .77 0	1.0 83	−1 5.7 54	−2 .08 5	1.2 33	−2 7.5 10
ing Dis sio per n	E{ Di} 98	−2 .71 2	1.7 52	−2 3.1 43	−4 .16 2	1.6 70	−5 1.7 39	−4 .79 0	1.7 40	−6 4.6 16	−3 .92 5	1.6 74	−4 5.1 50	−4 .76 2	1.3 22	−8 3.2 52
	Di} E{ 99	−0 .68 3	0.7 00	−4 .20 6	−1 .49 9	0.7 35	−14 .67 0	−1 .75 5	0.7 39	−1 9.5 28	−1 .45 4	0.7 27	−1 3.4 99	−1 .79 1	0.5 45	−2 7.8 65
	E{ Di} 00	−0 .50 4	0.6 59	−3 .58 5	−1 .06 1	0.8 63	−7 .00 9	−1 .33 2	0.8 76	−1 0.5 99	−0 .97 9	0.8 43	−6 .02 5	−1 .17 6	0.5 07	−1 0.9 83
	E{ Di} 01	−0 .53 6	0.7 62	−3 .90 0	−0 .76 7	0.7 93	−4 .42 1	−0 .91 9	0.8 16	−5 .84 6	−0 .73 9	0.7 81	−4 .06 5	−0 .79 3	0.4 49	−6 .28 0
	Di} All E{	−0 .27 6	0.5 73	−1 .90 4	−0 .67 2	0.9 38	−6 .01 7	0.4 55	1.2 40	−4 .84 7	−0 .64 1	0.9 37	−5 .53 0	−0 .72 3	0.6 36	−7 .93 4
are:
tional forms Table 5
	values.				points (

						Fig. 6.
×

					Myers, 2000).

Statistical tests (not used here), such as

Cooks' D, can be used to identify potential outliers and influence

(4) GNB Models 2, 3, and 4 yielded improved CURE plots com-

R-Student, DFFITS and

Dispersion Parameters based on All-Year Models.

values. the zero-residual line over the entire range of explanatory the cumulative residual curves are oscillating reasonably across based on the test statistics analysis in previous section) since 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 and GNB (varying dispersion) model. Notable characteristics as well as the difference between the NB (fixed dispersion) the time-specific model (yearly based model) and All-Year model

properties (it was already selected as the best fitted model these three GNB models, GNB Model 4 shows better fitting Although there are slight differences in the CURE plots among three GNB models end close enough to the zero-residual line. much reduced. The final cumulative residual curves for these that the amount of bias in the estimate values has been pared to the CURE plots produced from the NB Models in

(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 2–4, 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 Models 2, 3 and 4. On the other hand, Models 1 and 5 do not show this characteristic.
(2) In general, GNB model shows either better fitting performances with much smaller residuals than the traditional NB model (i.e., Models 2, 3, and 4) or almost same performance with negligible difference in residuals (i.e., Models 1 and 5).
(3) Model 5 produced very different results in cumulative residuals compared to those produced by the other four models because of the abnormally high volume for the minor approach at the previously identified intersection. This intersection should be investigated further to ensure it is not an outlier and removed, if necessary. However, it is of interest to note that Model 5 was originally selected as the best functional form in terms of the *G*2-statistics and the second best functional form according to the AIC and BIC output (refer to Table 4). The fact that the statistical tests in Eqs. (5)–(8) are frequently used to deter-

mine and justify the best model performance by transportation safety analysts without further looking into raw dataset using a model diagnosis tool (e.g., CURE plot) is a cause for concern.

2.4. Dispersion parameters among different crash prediction models

Table 6 and Fig. 6 summarize the estimated dispersion parameters obtained from different models. A few notable features include:

(1) 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 largest discrepancy amongst all the models evaluated (as seen in Table 6). This may be caused by the observation with an abnormally large minor road AADT or it may just be the peculiar characteristic of Model 5 since a similar discrepancy has been noted by Miaou and Lord (2003). Since Var{Yit} = it + ˛ · it, Model 5 estimates will produce a larger variance than that of the other models, implying a higher level of uncertainty associated with this model. As a result, compared to the other

Fig. 7. Relationships between the weight factor and NB model estimates.

Fig. 8. Relationships between the weight factor and the GNB model estimates.

four models, Model 5 puts more emphasis on the observed number of crashes than the model estimates in obtaining EB estimates.

(2) 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., wit ∝ 1/it, refer to Fig. 7 and Eq. (2)). The higher the model estimates, the smaller the weight factors, and vise versa.
(3) GNB Models 2, 4, and 5 [refer to Fig. 8(b), (d), and (e), respectively] allow different weight factors for intersections with the same model estimate ( ˆit). The tendency is stronger for the intersections with higher model estimates. On the other hand, GNB Models 1 and 3 [refer to Fig. 8(a) and (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., *F*2it and *F*2it, respectively) are not treated as distinct traffic volumes, but rather as a single traffic volume unit (i.e., *F*1it + *F*2it, *F*1it·*F*2it). Even with the exact same traffic volume unit

(e.g., *F*1it + *F*2it = 3000/day), intersections could have different entering traffic volumes from the major and minor approaches (e.g., *F*1t = 1500 and *F*2it = 1500, *F*1it = 2000 and *F*2it = 1000, 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 aggregated 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., *F*1it + *F*2it, *F*1it·*F*2it) in explaining the varying dispersion parameters, there is no practicalmerit of using the GNBModel over the traditional NB model for this case. Fig. 9(a) and (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 slightly lower value for the weight factor (i.e., the coefficients in Fig. 9 are less than 1.0) than those of traditional NB models. Hence, GNB Models will give

Fig. 9. Associations of weight factors between NB and GNB models.

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.

Fig. 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,

Table 7 Differences in ranking between GNB and NB models using the EB estimates

Model	Non-identical ranking percentagea	Ranking difference beyond 10 positions	Ranking difference beyond 50 positions	Ranking difference beyond 100 positions	Ranking difference beyond 200 positions
Model 1	97.39%	4665	2526	541	298
Model 2	98.89%	4945	3413	2330	1189
Model 3	98.69%	4898	3618	2688	632
Model 4	98.77%	4815	3196	2114	1015
Model 5	99.12%	4914	3300	2203	1176

There are 5588 intersections in the database.

^a Ranking difference by at least 1 position.

Fig. 10. Relationships between the varying dispersions and traffic volumes.

and 5 show that themajor road AADT contributesmore 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. Given that 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) andMitra andWashington (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. Fig. 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 EB estimates. A preliminary examination of the figure seems to indicate a positive association between the NB ranking and the GNB ranking. This is supported by the large R² values and the analysis results of the Spearman rank-order correlation test (s) as well as, to a lesser degree, the Kendall's ranking test (). Using some of these tests, El-Basyouny and Sayed's (2006) reported that using GNB models did not influence the identification of hazardous sites.

Looking more closely at the graphs however, one can observe a greater variation in the ranking below the 45-degree line (Note: more than 2/3 of all observations lie above the line), which means that, for the same observation, some sites ranked based on the GNB model tend to be ranked as more dangerous than those ranked based on the NB model; the sites below the line are usually associated with larger weight factors, which imply that the variation at time t is relatively small. Furthermore, a detailed analysis of the ranking, as seen in Table 7, shows that the ranking is actually poles apart. In fact, for some models, about a fifth of the observations had a ±200 position difference in ranking between the models with a fixed and varying dispersion parameters. This table clearly shows

Fig. 11. Relationships in ranking by varying dispersions and traffic volumes.

that, despite the outcome of traditional ranking tests, the functional form of GNB models strongly influences the identification of hazardous sites. The results also show that the tests may not be appropriate for this kind of analysis. Given this characteristic, further work is needed on this topic.

3. Summary and conclusions

In this paper, a number of important issues regarding the impact of the dispersion parameters on EB estimates were presented. Several key conclusions can be reported:

(1) Developing GNB models provide better statistical properties in terms of fitted values than NB models with a fixed dispersion parameter both for time-specific and aggregated modeling frameworks in most cases. For traffic flow-only models, GNB should be estimated over traditional NB models, since the variance function is most likely to be structured. The varying dispersion parameter can be used to characterize the structure of the variance.
(2) Time-specific models with a varying dispersion parameter will have a significant impact on the EB estimate, as documented in Table 6 and Fig. 6. In addition, the selection of the functional form will affect the value of weight factor used for estimating the EB output. This means that for the same observation, different weight factors can be estimated and they will be dependent upon the functional form and the variance function of themodel selected, as detailed in Figs. 8–10.
(3) Similar to point (2), the identification of hazardous sites, using the EB method, can be significantly affected when a GNB model is used. For the same observation, some sites ranked based on the GNB model tend to be ranked as more dangerous than those ranked based on the NB model. The weight factor seems to play a role for the observed differences in ranking. Even though the traditional Spearman and Kandall's ranking tests showed a positive correlation, a detailed analysis of the ranking showed opposite results, suggesting that these tests may not be appropriate for this kind of analysis.
(4) Thus, automatically adapting a functional form, especially for flow-only models, from previous studies should be avoided, as well as models that are solely based on a single GOF test statistics. Transportation safety analysts should evaluate different functional forms (when a varying dispersion parameter is employed) using a combination of GOF test statistics, including CURE plots, since the selection of the functional form could affect methods in which the EB output is used.

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., Miaou and Song, 2005; Mitra and Washington, 2007). However, since it is practically unachievable to obtain a perfect mean function (e.g., Xie et al., 2007), 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. Finally, although interesting results were found in this study, further research should be conducted using different datasets (urban intersections, rural and urban segments, etc.). This may tell us whether the issues raised in this paper are a common problem, an isolated problem, or specific to rural intersections or this dataset.

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