category: literaturenote citekey: naderanaggregatecrashpredictionmodels2010 title: "Aggregate crash prediction models: Introducing crash generation concept" authors: "Naderan, Ali; Shahi, Jalil" year: 2010 date: 2010-00-00 2010 doi: 10.1016/j.aap.2009.08.020 publication: "Accident Analysis & Prevention" url: "https://doi.org/10.1016/j.aap.2009.08.020" zotero_key: TPA6ZJQJ zotero_storage: KUZ9R2AK collections: transpordi_planeerimine folder: 001_artiklid/Road Safety Indicators firstAuthor: "Naderan, Ali"

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Accident Analysis and Prevention

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

Aggregate crash prediction models: Introducing crash generation concept

Ali Naderan∗, Jalil Shahi

Iran University of Science and Technology, Narmak, Tehran, Iran

article info

Article history: Received 15 March 2009 Received in revised form 31 July 2009 Accepted 17 August 2009

Keywords: Safety planning Trip generation Crash generation Negative binomial

abstract

Safety conscious planning is a new proactive approach towards understanding crashes. It requires a planning-level decision-support tool to facilitate proactive approach to assessing safety effects of alternative urban planning scenarios. The objective of this research study is to develop a series of aggregate crash prediction models (ACPM) that are consistent with the trip generation step of the conventional four-step demand models. The concept of crash generation models (CGMs) is introduced utilizing trip generation data in a generalized linear regression with the assumption of a negative binomial error structure. The relationship of crash frequencies in traffic analysis zones (TAZ) and number of trips generated by purpose is investigated. This translates into immediate checking of the impact of future trip generations on crash frequencies in comprehensive transportation-planning studies (i.e. ability to forecast crashes at each time-step trips are being forecasted). A good relation was seen between crash frequency and number of trips produced/attracted by purpose per TAZ.

1. Introduction

Aggregate crash prediction models (ACPMs) may still be called state of the art in urban safety planning. Although there are many references on the use of aggregate models for predicting transportation demand in urban areas, little attention has been paid to aggregate safety planning issues. It has been common practice to consider traffic safety as a short-term 3–5 years engineering problem, rather than a long-term 15–20 years planning issue. In other words, safety considerations are traditionally omitted from the long-range transportation studies and relatively little thought is given to how safety could be considered in the planning process (Washington et al., 2006; MTTO, 2007). The reason was that authorities blamed human factors and poor road conditions as major causes of crashes. The former needing publicity and education, the latter lots of funds for what was called black spot remedial measures. Nevertheless, proactive safety conscious planning is now considered a priority in many transportation-planning efforts. That is number of crashes need to be predicted/forecasted in the next 20 years or so.

ACPM may be used to develop four-step safety planning models similar to the process followed in four-step demand modeling. Fig. 1 depicts the idea of inter-relating transportation and safety planning. The concept behind this figure is that each trip generated

E-mail addresses: Ali@Naderan.Com (A. Naderan), Jalil@iust.ac.ir (J. Shahi).

from a TAZ will be subject to a probable risk of crash. In fact, the fundamental cause of a crash is the trip making itself: no trips results no crashes! The concept of four-step safety planning is still underdeveloped and needs to be defined properly, but as starting point, the following conceptualization may apply.

Crash generation models (CGMs) are developed based on results of trip generation models that provide number of trips made from/to a TAZ by different purposes. This includes an inherent measure of exposure: number of trips. It also provides a set of policy-related variables: number of trips by purpose. As will be demonstrated later herein, this part has proved appropriate and practical. The CGMs predict number of crashes in each TAZ based on number of trips by different purposes produced or attracted to that TAZ. Definition of other steps is in progress and needs further investigation.

This research paper proceeds with examining how to relate crash frequencies per urban TAZ with trip generation (i.e. production/attraction) data. This would provide a framework for predicting/forecasting future number of crashes based on changes in the number of trips produced/attracted to a TAZ.

2. Methodology for developing ACPM

The process for developing ACPM consists of four basic steps (Cameron and Trivedi, 1998) as described in the following subsections. Each step is performed considering its interaction with other steps. That is for example, there may be a feedback loop from step 2 to step 1, if a certain variable suggested in step 1 could not be made available.

^∗ Corresponding author at: Iran University of Science and Technology, Civil Engineering, Narmak, Tehran, Iran. Tel.: +98 9123803508.

Fig. 1. Four-step safety modeling.

2.1. Variable selection

Identifying preliminary variables based on the purpose of model development and theoretical issues is the first step. To date, ACPM have used different explanatory variables to associate with crashes, including land use (Hadayeghi et al., 2007; Kim et al., 2006), network characteristics (Noland, 2002; de Guevara et al., 2004), environmental conditions (Aguero-Valverde and Jovanis, 2006), day of year (Qadeer Memon, 2006), demographic (Washington et al., 2006) and traffic related variables (Hadayeghi et al., 2003). These variables have been aggregated at different levels such as: state (Noland, 2002), county (Aguero-Valverde and Jovanis, 2006), TAZ (de Guevara et al., 2004) and (Hadayeghi et al., 2007), ward (Quddus, 2008), census block (Levine et al., 1995) and even smaller levels (Kim et al., 2006). Nevertheless, the logic behind selection of variables is more important than the goodness-of-fit results. For example, if the idea is to use ACPM to develop safety countermeasures, they should consist of variables that may be affected by engineering (e.g. improvement of geometric characteristics) or policymaking (road classification/construction). On the other hand, if they are meant purely for forecasting number of crashes in future, demographic and socio-economic variables may be best to use.

The primary purpose of this research is to investigate how travel demand in urban areas contributes to crash frequencies at TAZlevel, in order to forecast number of crashes in the future. This implies the following characteristics for model variables:

(1) Indicate a metric for number of trips.
(2) Capture the daily pattern of trips.
(3) Capture the nature of trips (i.e. why it is made).
(4) Credible for future growth in number of trips.
(5) Be easy to forecast.
(6) Account for exposure.
(7) Comply with the process of travel demand management.
(8) Do not burden additional costs of data collection.
(9) Results easily incorporated into the planning process.

The variables selected to consider here are results of trip generation models. The number of trips produced or attracted to each TAZ is available by purpose categories such as that presented in Table 1. The selected variables pass all the above-mentioned characteristics. Most importantly, results of any TDM scenario directly affects number of trips performed by different purposes to and from a TAZ, and employing them in an ACPM provides a means for the evaluation of these changes. This is in fact where the concept of crash generation starts, parallel to trip generation. Nevertheless, the pro-

Table 1 Preliminary variables for ACPM.

Category	Variable	Description	N	Minimum	Maximum	Mean	Std. deviation
General	TAZ	Traffic analysis zone	380	1	380	–	–
	pop	Population per TAZ	380	0	26074	5942.861	4283.858
	area m2	TAZ area in km2	380	.072234	18.71179	.7236114	1.242645
Crashes	pdo	No. of property damage only crashes	380	0	405	56.58421	61.63267
	injury	No. of injury crashes	380	0	61	8.052632	7.538984
	fatal	No. of fatal crashes	380	0	2	.1236842	.3675019
	severe	No. of fatal + injury crashes	380	0	61	8.176316	7.64468
	total	Total no. of crashes	380	0	436	64.76053	66.64325
Trip productions	pw	No. of work trip productions	380	0	22374	1870.637	1640.819
	pedu	No. of educational trip productions	380	0	9941	833.7237	1013.701
	peduu	No. of educational university trip productions	380	0	493	82.75263	85.23744
	psh	No. of shopping trip productions	380	0	2937	432.8026	387.138
	pre	No. of recreational trip productions	380	0	2805	527.7684	411.0565
	pper	No. of personal trip productions	380	0	3246	367.5632	360.4802
	ppilg	No. of pilgrimage trip productions	380	0	319	84.93684	60.59434
	pnhb	No. of non-home-base trip productions	380	0	5856	529.7026	641.9504
	ptotal	Total no. of trip productions	380	108	22834	4729.824	3005.952
Trip attractions	aw	No. of work trip attractions	380	0	8813	1797.261	1551.9
	aedu	No. of educational trip attractions	380	0	8884	869.8658	927.8295
	ash	No. of shopping trip attractions	380	0	23609	655.7421	2188.884
	are	No. of recreational trip attractions	380	0	7211	631.0132	948.3777
	aper	No. of personal trip attractions	380	0	2763	272.5	412.0852
	anhb	No. of non-home-base trip attractions	380	0	7633	557.4447	785.1203
	atotal	Total no. of trip attractions	380	30	45430	4783.718	5170.675
Trip attractions per	awa	No. of work trip attractions per km2	380	0	80972.95	4794.547	6469.753
unit of area	aedua	No. of educational trip attractions per km2	380	0	28670.71	2066.45	2702.631
	asha	No. of shopping trip attractions per km2	380	0	24956.26	1532.204	3673.977
	area	No. of recreational trip attractions per km2	380	0	14021.96	1341.469	1851.679
	apera	No. of personal trip attractions per km2	380	0	1098.409	662.5069	1098.409
	anhba	No. of non-home-base trip attractions per km2	380	0	24624.75	1405.424	2185.383

Fig. 2. Spatial distribution of total trips per TAZ.

posed variables are later selected analytically based on significance and goodness-of-fit tests in the regression model.

It has also been suggested that factors affecting crashesmay vary by crash severity (Hadayeghi et al., 2007; Quddus, 2008). Therefore, TAZ-level crash data were stratified into Property Damage Only, injury and fatality crashes. Total crashes, defined as sum of all crashes reported to the police and severe crashes defined as sum of injury and fatal crashes were also considered in the modeling effort. The latter is the most reliable because all severe crashes are mandated by law to be reported to the police (MTTO, 2007). Besides, it compensates for the many zero observations of fatality crashes (Quddus, 2008).

2.2. Data collection/process

Development of a database that contains variables used for modeling is the second practical step for the modeling effort. This study is based on the transportation database of Mashhad city, Iran, because comprehensive transportation studies are completed and updated for this city (MTTO, 2007). The city is divided into 380 TAZs and trip generation model predictions are available for the year 2006. Crash data of the same year are also geo-coded into the GIS environment. The initial databases were in GIS format, Microsoft Access/Excel files. All these data were first entered into a single GIS file, and then were segregated between TAZs. As a result, a random error detested while creating the databases. The error relates to crash data assigned to points. Given the varying accuracy of their measurements, these data may produce seemingly random errors when GIS tools attempted to assign them to TAZs. This error is thought to be small (Hadayeghi et al., 2003; de Guevara et al., 2004). Descriptive analysis of the variables is presented in Table 1. Figs. 2 and 3 show the spatial distribution of total trips produced/attracted and total crash frequency per TAZ in the city of Mashhad.

2.3. Model postulation

Hypothesizing the mathematical form of the relationship between variables is the foundation of the statistical modeling. The purpose of study, nature of available data, statistical software packages and computational skills required to calibrate statistical models greatly influence the mathematical formulations in previous researches. Linear regression (Clark and Cushing, 2004); linear regression with a spatial lag parameter (Levine et al., 1995); Poisson models (Chatterjee et al., 2004); negative binomial (NB) models (Hadayeghi et al., 2003, 2007; Maher and Summersgill, 1996; Abdel-Aty and Radwan, 2000; Miaou and Lord, 2003; Qadeer Memon, 2006; Mitra and Washington, 2007; Noland and Quddus, 2004); zero-inflated Poisson and NB models (ZIP/ZINB) (Qin et al., 2004; Shankar et al., 1997); simultaneous estimation of NB models (de Guevara et al., 2004); full Bayes hierarchical models with spatial effects (Aguero-Valverde and Jovanis, 2006; Quddus, 2008), geographically weighted regression (GWR) (Hadayeghi et al., 2003); log-linear models using ordinary least squares (OLS) estimation technique (Washington et al., 2006; Wier et al., 2009) were used successfully by previous researchers in different approaches to crash studies. Hadayeghi et al. (2006) tested temporal transferability of the macro-models and concluded that model forecasts are within acceptable range in application context.

Most of the above-mentioned models were calibrated to explain the underlying process of crashes and the effect of each independent variable on crash frequency. But, some researchers have argued against validity of linear (Miaou and Lum, 1993) and ZIP/ZINB (Lord et al., 2005, 2007) models for explaining crashes. Despite improvements provided by accounting for spatial correlation in ACPMs, the results from the spatial Bayesian hierarchical models were found quite similar to the standard non-spatial NB models in many cases (Quddus, 2008). That is, the values of

Fig. 3. Spatial distribution of total crashes per TAZ.

coefficients in both spatial and non-spatial models were significantly similar. It was also concluded that spatial dependence among spatial units would decrease at higher levels of spatial aggregation.

On the other hand, similar to Washington et al. (2006), the focus of this paper is prediction/forecasting and not explanation of the cause-effect issues related to crashes. Therefore, this research paper assumes standard NB formulation for the following reasons:

(1) Increased simplicity and applicability, less computational efforts in the state of practice.
(2) Although spatial NB models produce better results for the explanation purposes, the model coefficients are reported to be close to standard NB models (Quddus, 2008), justifying use NB models in concept development for prediction purposes.
(3) Crash data are clustered into TAZs, accounting for some of the heterogeneity (Karlaftis and Tarko, 1998).
(4) Trip generation models are calibrated in the linear form to predict number of trips that themselves are non-negative integer with spatial correlation, but have been accepted worldwide. So the NB formulation with log-link function was thought reasonable for the starting step in the concept development for CGMs.
(5) Focus is prediction, not explanation. Models for prediction have fewer restrictions than models for explanation: inference, or effects of isolated variables (estimated coefficients) not too important, multi-colinearity tolerated; goodness of fit and predictive-ability most important (Washington et al., 2006).

One requirement of the Poisson distribution is that the mean of the count data equals its variance. When the variance is significantly larger than the mean, the data are said to be over-dispersed, and are successfully modeled using a negative binomial (NB) model (Washington et al., 2003). In the NB model, the difference between variance and mean is presented by V[u] = u +˛u2, where V[u] is the estimated variance of the crash frequency; u is the estimated mean crash frequency and ˛ is the NB over-dispersion parameter. The Poisson regression model is regarded as a limiting model of the NB regression model as ˛ approaches zero, which means that the selection between these two models is dependent on the value of ˛. The NB distribution has the general form of Eq. (1):

$$P(y_i) = \frac{\Gamma((1/\alpha) + y_i)}{\Gamma(1/\alpha) + y_i!} \left[ \frac{1/\alpha}{(1/\alpha) + \lambda_i} \right]^{1/\alpha} \left[ \frac{\lambda_i}{(1/\alpha)\lambda_i} \right]^{y_i} \tag{1}$$

where - (.) is a gamma function. The NB model was developed mathematically by assuming that unobserved crash heterogeneity (variation) across sites (intersections, road segments, etc.) is gamma distributed, while crashes within sites are Poisson distributed (Washington et al., 2006). Following a review of various model forms used in the literature and applying selected model forms to the data, the model form in Eq. (2) was used where y = predicted crash frequency; Xi = explanatory variables and bi = model parameters:

$$y = \exp \sum_{i=1}^{n} b_i X_i \tag{2}$$

2.4. Model calibration and validation

Model calibration, also known as model estimation, is estimating the parameters of the postulated model based on observed data. This is done by various common software packages for statistical analysis such as SAS (Hadayeghi et al., 2007), GenStat (Qadeer Memon, 2006), Stata (Noland, 2002; Wier et al., 2009), LIMDEP (Washington et al., 2006), aML (de Guevara et al., 2004), R Statistical Software (Aguero-Valverde and Jovanis, 2006), WinBUGS (Quddus, 2008), SPSS, SPlus, etc. All these softwares include modules for regression analysis and provide most of the statistical tests required for model calibration. The regression coefficients as well as the over-dispersion parameter were estimated by the maximum likelihoodmethod of the statistical software (GenStat, 2008; STATA, 2007) following a forward procedure. In this procedure, a simple model with only an intercept term is used as a starting point and the independent variables are added to the model one by one. Keeping these variables in the model was based on whether they would improve goodness-of-fit results.

Goodness-of-fit tests for model calibration and the decision on whether or not to keep a variable in the model is based on the following criteria (Washington et al., 2003; Cameron and Trivedi, 1998): (1) t-statistic significant at the 5% level; (2) ratio of deviance or Pearson chi-square over degrees of freedom in the range of 0.8–1.2; (3) Akaike's information criteria (AIC) defined as AIC = [−2 Ln(ˇ)+2b]/N, where Ln(ˇ) is the log-likelihood value of model at convergence, b is the number of variables in the model, and N is number of TAZs. The model with the minimum value of AIC is the best model.

3. Modeling results

In order to investigate how each trip purpose affects crash occurrence, different models were calibrated. These models are regarded as CGMs in correspondence with the conventional trip generation models in the four-step travel demand modeling technique. The modeling procedure complies with the methodology presented in the preceding section.

CGMs developed here are twofolds: crash production and crash attraction. The former employs trip production data and reports the relationship between number of trips produced in each TAZ and number of crashes observed in that TAZ, while the latter relies on trip attraction data for model calibration and reports the relationship between number of trips attracted to each TAZ and number of crashes observed in that TAZ. In order to capture the combined effect of total number of trips generated in each TAZ, a third model is calibrated consisting of only two variables: total trip production (all purposes) and total trip attraction (all purposes) per TAZ. For convenience, this category is designated crash generation model.

3.1. Models for total crashes

The regression results of the crash production and crash attraction models for total number of crashes per year are presented in Table 2. As shown, the relationship between crashes and selected variables is highly significant at the 5% level. Personal trips produced (pper) were highly correlated with other variables, hence omitted from the crash production models. The dispersion parameter is 0.87 for the crash production model and 0.98 for the crash attraction model, proving the significant justification of NB over the Poisson formulation. The ratios of Pearson chi-square and deviance over degrees of freedom are within the acceptable range of 0.8–1.2, another indication that the choice of the NB error structure is appropriate and the calibrated model is valid. It is also seen from Table 2 that crash production model is a better fit based on the larger loglikelihood value and lower AIC and significance (p-) values. The mean absolute deviance for this model is about 0.938.

The crash generation model consisting of total trips produced and attracted to each TAZ was developed too, which was a good fit based on the log-likelihood and deviance/df ratio results. This model is represented in Eq. (3):

$$y = \exp(3.38057) + \exp(.000104 \times \text{ptotal}) + \exp(.000044 \times \text{atotal})$$

Table 2 Results of total crash generation models.

| | Crash production model | | | | Crash attraction model | | | | | Crash generation model | | | | | | |-------------------------------------------------------------|-----------------------------------------------------------------------------------|-----------------------------------------------------------------------------|-----------------------------------------------------------------|--------------------------------------------------------------|--------------------------------------------------------|-------------------------------------------------------------------------|--------------------------------------------------------------------|--------------------------------------------------------|------------------------------------------------------|---------------------------|------------------------------|----------------------------|-----------------------|----------------------|--| | Var. | Coeff. | Std err | t-stat | Sig. | Var. | Coeff. | Std err | t-stat | Sig. | Var. | Coeff. | Std err | t-stat | Sig | | | cons.
pw
pedu
peduu
psh
pre
ppilg
pnhb | 3.54888
.00010
−.00145
.00367
.00609
−.00222
−.00479
.000276 | .12031
.00004
.00053
.00133
.00233
.00103
.00159
.0001 | 29.5
2.46
−2.72
2.76
2.60
−2.15
−3.00
2.76 | .000
.014
.007
.006
.009
.031
.003
.006 | cons.
awa
aedua
ash
area
anhba
apera | 4.04834
−.77e−4
.81e−4
.64e−4
.000103
−.00011
.000339 | .07451
.17e−4
.35e−4
.34e−4
.53e−4
.48e−4
.15e−3 | 54.3
−4.43
2.30
1.87
1.93
−2.30
2.29 | .000
.000
.022
.062
.053
.021
.022 | cons.
ptotal
atotal | 3.38057
.000104
.44e−4 | .11785
.21e−4
.12e−4 | 28.69
4.93
3.46 | .000
.000
.001 | | | Goodness of fit | | | | | | | | | | | | | | | | | | No. of observations | | 380 | | No. of observations | | | 380 | | No. of observations | | | 380 | | | | | Degree of freedom (df) | | 372 | | Degree of freedom (df) | | | 373 | | Degree of freedom (df) | | | 377 | | | | Deviance/df | | | 1.2102 | | Deviance/df | | | 1.3569 | | Deviance/df | | | 1.1954 | | | | | Pearson/df
0.8857 | | Pearson/df | | | 1.0608 | | Pearson/df | | | 0.8928 | | | | | | ˛ | 1.02 | | ˛ | | | 1.14 | | ˛ | | | 1.12 | | | | | | Log-likelihood | | | −1915.3 | | Log-likelihood | | | −1937.1 | | Log-likelihood | | | −1934.2 | | | | AIC | | | 10.12 | | AIC | | | 10.24 | | AIC | | | | 10.19 | |

Table 3 Results of PDO crash generation models.

Crash production model						Crash attraction model	Crash generation model
Var.	Coeff.	Std err	t-stat	Sig.	Var.	Coeff.	Std err	t-stat	Sig.	Var.	Coeff.	Std err	t-stat	Sig
cons. pw pedu peduu psh pre ppilg	3.38287 .000105 −.00167 .004085 .007012 −.00250 −.00622	.12482 .43e−4 .00056 .00138 .00245 .00108 .00166	27.1 2.43 −2.98 2.96 2.86 −2.31 −3.74	.000 .015 .003 .003 .004 .021 .000	cons. awa aedua ash area anhba apera	3.88208 −.87e−4 .99e−4 .66e−4 .00011 −.00012 .000377	.07776 .18e−4 .37e−4 .36e−4 .56e−4 .51e−4 .16e−3	49.9 −4.72 2.66 1.86 1.97 −2.37 2.37	.000 .000 .008 .065 .049 .018 .018	cons. ptotal atotal	3.1828 .00011 .48e−4	.1245 .22e−4 .13e−4	25.56 5.01 3.55	.000 .000 .000
pnhb	.000293	.00010	2.83	.005
Goodness of fit
	No. of observations		380		No. of observations			380		No. of observations			380
	Degree of freedom (df)		372		Degree of freedom (df)			373		Degree of freedom (df)			377
Deviance/df			1.2078		Deviance/df			1.3515		Deviance/df			1.2009
	Pearson/df 0.9102		Pearson/df			1.0877		Pearson/df			0.9388
˛	1.10		˛			1.24		˛			1.23
	Log-likelihood −1851.9		Log-likelihood			−1877.7		Log-likelihood			−1875.8
AIC			9.79		AIC			9.92		AIC				9.89

3.2. Models for PDO crashes

Property Damage Only (PDO) crashes include non-injury crashes that involve monetary loss and are reported by the police. The regression results of the crash production and crash attractionmodels for PDO crashes per year are presented in Table 3. As can be seen both by log-likelihood and AIC, the results of the PDO crash models indicate goodness-of-fit improvements over the total crash models. It is interesting to see the same set of variables as those of the total CGMs became significant here. Coefficient estimates are also close and sign of variables are the same in both models. This is due to the large influence of number of PDO crashes on total number of crashes.

3.3. Models for severe crashes

Severe crashes include fatal and injury crashes but exclude PDO crashes. The regression results of the crash production and crash attraction models for severe crashes per year are presented in Table 4. As can be seen, the results of the severe CGMs indicate major goodness-of-fit improvements over the total crash models and PDO crash models. This is partly because data on severe crashes are more accurately collected and maintained by the police. On the other hand, number of variables proved significant to be kept in the model has decreased. Still, the model includes major trip purposes (i.e. work, shopping, educational and non-home-based trips) that are mostly influenced by travel demand-management scenarios. The dispersion parameter, ˛, is also half that of the total and PDO CGMs, which also indicates less dispersed and more accurate data.

3.4. Models for injury crashes

The regression results of the crash production and crash attraction models for injury crashes per year are presented in Table 5. As can be seen, the results of the injury crash models indicate goodness-of-fit improvements over the previous models. The injury models show mostly the same characteristics as that of the severe crash models. As would have been expected, work, educational, shopping and non-home-based trip purposes were the significant variables in thesemodels. This indicates themost important trip purposes that may be affected by future policies and travel demand-management plans. Coefficient estimates here are almost the same as the severe CGMs, indicating the significant influence of number of injury crashes on number of severe crashes. In fact, the small number of fatality crashes seems to have little effect on parameter estimations for severe and injury crash models.

Table 4 Results of severe crash generation models.

Crash production model						Crash attraction model			Crash generation model
Var.	Coeff.	Std err	t-stat	Sig.	Var.	Coeff.	Std Err	t-stat	Sig.	Var.	Coeff.	Std Err	t-stat	Sig
cons.	1.63069	.09769	16.7	.000	cons.	1.9756	.07698	25.7	.000	cons.	1.686	.09588	17.6	.000
pw	.68e−4	.32e−4	2.18	.029	aw	.000137	.41e−4	3.35	.001	ptotal	.607e−4	.18e−4	3.44	.001
pedu	−.00022	.00012	−1.88	.060	aedua	−.46e−4	.21e−4	−2.16	.031	atotal	.208e−4	.98e−5	2.12	.034
psh	.000889	.00314	2.83	.005	ash	.42e−4	.25e−4	1.67	.095
pnhb	.000209	.85e−4	2.47	.014	anhba	−.56e−4	.26e−4	−2.16	.031
Goodness of fit
	No. of observations		380		No. of observations			380		No. of observations			380
	Degree of freedom (df)		375		Degree of freedom (df)			375		Degree of freedom (df)			377
Deviance/df			1.1917		Deviance/df			1.3493		Deviance/df			1.1740
	Pearson/df 1.0065			Pearson/df			1.0592	Pearson/df			1.0009
˛	0.71		˛			0.73		˛			0.72
	Log-likelihood −1182.3			Log-likelihood			−1186.3		Log-likelihood			−1184.5
AIC 6.25		AIC			6.28		AIC			6.25

Table 5 Results of injury crash generation models.

| Crash production model | | | | | | Crash attraction model | | | | | Crash generation model | | | | | | |------------------------|------------------------|---------|---------|------|---------------------|------------------------|---------|--------|------------|---------------------|------------------------|---------|--------|---------|--|--| | Var. | Coeff. | Std err | t-stat | Sig. | Var. | Coeff. | Std err | t-stat | Sig. | Var. | Coeff. | Std err | t-stat | Sig | | | | cons. | 1.61185 | .09781 | 16.5 | .000 | cons. | 1.9576 | .07719 | 25.4 | .000 | cons. | 1.6676 | .09588 | 17.4 | .000 | | | | pw | .68e−4 | .31e−4 | 2.17 | .030 | aw | .000139 | .41e−4 | 3.38 | .001 | ptotal | .61e−4 | .18e−4 | 3.45 | .001 | | | | pedu | −.00022 | .00012 | −1.90 | .057 | aedua | −.46e−4 | .21e−4 | −2.14 | .032 | atotal | .21e−4 | .98e−5 | 2.13 | .033 | | | | psh | .000899 | .00314 | 2.86 | .004 | ash | .42e−4 | .25e−4 | 1.66 | .097 | | | | | | | | | pnhb | .000212 | .85e−4 | 2.50 | .012 | anhba | −.57e−4 | .26e−4 | −2.15 | .031 | | | | | | | | | Goodness of fit | | | | | | | | | | | | | | | | | | | No. of observations | | 380 | | No. of observations | | | 380 | | No. of observations | | | 380 | | | | | | Degree of freedom (df) | | 375 | | | Degree of freedom (df) | | | 375 | | Degree of freedom (df) | | | 377 | | | | Deviance/df | | | 1.1904 | | | Deviance/df | | | 1.3487 | | Deviance/df | | | 1.1735 | | | | | Pearson/df
1.0003 | | | | Pearson/df | | | 1.0554 | Pearson/df | | | 0.9957 | | | | | | ˛ | 0.71 | | ˛ | | | | 0.73 | | ˛ | | 0.72 | | | | | | | Log-likelihood | | | −1176.7 | | | Log-likelihood | | | −1180.9 | | Log-likelihood | | | −1179.0 | | | | AIC | 6.22 | | AIC | | | | 6.26 | | AIC | | | 6.22 | | | | |

3.5. Models for fatal crashes

Efforts to develop crash production/attraction/generation models for fatal crashes failed. This is in part because only 42 out of 380 TAZs had non-zero fatal crash observations. Inability to calibrate fatal crash prediction models may not be a big issue, because the climate of crashes is described well enough by crash prediction models developed for the severe/injury crashes.

4. Discussion

As was mentioned before, previous researches suggested very similar coefficients for the spatial and non-spatial models (Quddus, 2008). So the choice of NB formulation to develop CGM was based on the stated interest in developing a model for practical application that can readily be applied and interpreted. CGMs developed in previous sections provide distinct characteristics that would be discussed here:

(1) Trip purposes have directly entered the crash production models, while they have been converted to trips per square kilometer of TAZ in crash attraction model. This was because trip attraction models were calibrated based on areas of different attractions (e.g. shopping area) and thus became correlated in the process of estimation of crash attraction models. To compensate for this, trip attractions by purpose were normalized as trips per unit area. Levine et al. (1995) also suggested the same approach to account for the differences between areas of census blocks.

(2) Compared to total/PDO crash models, injury/severe crash models were less dispersed based on lower ˛ values and better fit based on smaller AIC and larger log-likelihood values. This is in part because data on injury/severe crashes are more accurate. Not all PDO crashes are captured/reported by the police, which may result biased data for some TAZs and consequently increased over-dispersion in total/PDO crash models.
(3) In order to cover for the over-dispersion, more independent variables became significant in the total/PDO crash models. As was mentioned, multi-colinearity can be tolerated in prediction models, but use of the models for explanation purposes must be with cautious.
(4) Trip productions and trip attractions by the purposes of work, shopping, educational (university) and none home-based trips show a positive effect on number of crashes. This was expected because: (a) they are mostly made by private vehicles, (b) in the rush hours, and (c) on a routine basis, and thus bear a more exposure to risk.
(5) Trip productions and attractions by the purposes of education and recreation show a negative effect, because: (a) recreational trips are made with less hurriedness and in off-peak hours and (b) because of proximity of schools to residential areas, educational (school) trips are mostly made on foot/public transit and so they are less involved in crashes.
(6) Although in the crash attraction models, trip attractions per unit of area by the purposes of work, shopping and none homebased trips show a negative sign, itmust be noted that the sign is in fact induced by the denominator (i.e. TAZ area). The negative effect of TAZ area was checked by calibrating a separate model

including only TAZ area and the intercept and is according to previous researches (Washington et al., 2006;Wier et al., 2009). The model is estimated as ^y = [exp(4.702 ⁻ .2302)/area m2]. As can be seen, the sign of inverse TAZ area is negative in this model. This also explains the positive sign of trip attractions per unit of area by the purposes of education, recreation and personal. The possible meaning for the negative effect of area on crashes is that increase in area normally results decreasing density in population and other trip producing/attracting variables.

(7) As a usage example, if work trips produced in a TAZ are forecasted to increase by 1000 in the future, given all other trip purposes are constant, number of total crashes in that TAZ may be calculated using crash production models. This results approximately 10% increase in total crashes of that TAZ (exp[.0000997 × 1000] = 1.10) and 6% increase in severe crashes (exp[.0000614 × 1000] = 1.06). Knowledge of this growth in crashes in essential in order to develop long-term safety plans and establish counter measures of a specific type.

5. Conclusions and recommendations

The primary purpose of this paper was to investigate how travel demand in urban areas contributes to crash frequencies at TAZlevel. To this end, a series of macro-level ACPMs designated CGMs, were developed hiring number of trips by purpose generated per TAZ as the primary variables affecting crash occurrence. Thesemodels provide the basic tool for evaluation of TDM scenarios in urban transportation planning. They would predict number of crashes based on number of trips produced or attracted to a TAZ. A certain TDM scenario may be developed to reduce trip productions of a specific purpose and the related number of crashes could be predicted. These models are also appropriate for forecasting future number of crashes because no statistical model could capture changes in future trends better than the number of trips itself. Though proven a good measure of exposure (Hadayeghi et al., 2007; Quddus, 2008), VKT and traffic flow are both second-hand results of trip generation models, so it seems more appropriate to use the first-hand results, namely trip productions and attractions by purpose.

Other note-worthy issues include:

(1) The NB formulation for CGMs provides good consistency with travel demand models, which are normally calibrated by the least square method in the linear form.
(2) Aggregate models deal with macroscopic data and their results may not to be considered to evaluate a single corridor/project. On the other hand, disaggregate models are at microscopic level and are developed for the purpose of corridor/project evaluation. In other words, aggregate (macro) models are for transportation (safety) planning and disaggregate (micro) models are for traffic (safety) engineering.
(3) CGMs are suited for predicting/forecasting future trends in crashes of specific types. This helps targeted focus of stakeholders towards the problematic areas of transportation safety.
(4) CGMs may be used cautiously on identifying TAZs that suffer more crashes of a specific type in the future, and developing countermeasures based on approved methods.
(5) CGMs may not be used for developing safety countermeasures based on the coefficients of model variables (Washington et al., 2006). This needs careful considerations and further investigations utilizing micro-level or disaggregated crash prediction models.
(6) Although the underlying process of CGMs is somewhat the same as trip generation models, their ability to forecast into
the future yet needs to be verified. Hadayeghi et al. (2006) put an step in that direction.
(7) Definition of other steps in four-step safety planning needs further investigation. In essence, the role of modal share should be investigated. This would help evaluation of TDM scenarios involving introduction/increase in specific modes of travel.
(8) In order to cover for the over-dispersion, redundant and/or multi-collinear variables may become significant in the crash production/attraction models. If the models are to be used for explanatory purposes, a multi-colinearity analysis must be performed for models at TAZ-level to account for this.
(9) Spatial correlation and heterogeneity among adjacent TAZs should be tested to justify the use of standard NB models and/or determine the spatial effects on model calibration results. Again, that is if the models are to be considered for explanatory purposes.

Acknowledgments

This paper is part of an ongoing research on Ph.D. dissertation at Faculty of Civil Engineering, Iran University of Science and Technology, Tehran, Iran. We would like to thank Prof. Hamid Behbahani, Dr. Mahmud Ameri and Dr. M. Ahmadinejad, Faculty of Civil Engineering, Iran University of Science and Technology and Dr. Amir-Abbas Rasafi, Imam Khomeini International University, Faculty of Engineering, as members of the dissertation steering committee for their helpful comments. We also acknowledge invaluable comments from Dr. Kamal Shoar, Tarhe Haftom consulting engineers. Part of data was provided by Mashhad Transportation and Traffic Organization. The writers wish to extend their thanks and appreciation to the two reviewers for their constructive comments to enhance this paper.

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