category: literaturenote citekey: cheninvestigationmodelsrelatingroundabout2013 title: Investigation of models for relating roundabout safety to predicted speed authors: "Chen, Yongsheng; Persaud, Bhagwant; Sacchi, Emanuele; Bassani, Marco" year: 2013 date: 2013-00-00 2013 doi: 10.1016/j.aap.2012.04.011 publication: "Accident Analysis & Prevention" url: "https://doi.org/10.1016/j.aap.2012.04.011" zotero_key: AC83DEVQ zotero_storage: A5SRJK9H collections: transpordi_planeerimine folder: 001_artiklid firstAuthor: "Chen, Yongsheng"

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Accident Analysis and Prevention

j ou rnal home page: www.elsevier.com/locate/aap

Investigation of models for relating roundabout safety to predicted speed

Yongsheng Chen^a,∗, Bhagwant Persaud^a,1, Emanuele Sacchi ^b,2, Marco Bassani ^b,³

^a Department of Civil Engineering, Ryerson University, 350 Victoria Street, Toronto, Ontario M5B 2K3, Canada
^b Department of Environment, Land and Infrastructure Engineering, Politecnico di Torino, 24 Corso Duca degli Abruzzi, Torino 10129, Italy

a r t i c l e i n f o

Article history: Received 1 November 2011 Received in revised form 21 March 2012 Accepted 15 April 2012

Keywords: Roundabout Safety performance function Predicted speed Effect selection Bayesian model Zero-inflated Poisson model

a b s t r a c t

Despite widespread recognition of operating speed as a key safety-related variable for roundabouts, there is no consensus on the best models for capturing the relationship between crashes and speed, or, for that matter, on how speed can be estimated in situations where it cannot be observed (such as when a roundabout is being designed or redesigned). This paper uses US and Italian roundabout approach-level data to investigate models relating safety to various measures of predicted speed. This is an indirect approach for developing safety models for estimating the effects of design features, the premise being that these features can better predict speed, which, in turn, can be used as a predictor of crash frequency. After exploring various possibilities, the approach average speed (AAS) – defined as the average of entry, upstream circulating and exiting speeds in this study – was found to be the speed measure that best predicts safety. US data were used to develop a Bayesian Poisson-gamma safety model based on predicted AAS with random coefficients and varying dispersion parameter. This model structure was not appropriate for the Italian data used to examine whether the approach could be generalized to data for another country. For that data, a zero-inflated Poisson (ZIP) model was found to be suitable. Notwithstanding the heterogeneity of the model structure, the investigation suggests that the indirect approach for evaluating the safety of a roundaboutis a sound one in thatit can preserve model parsimony while capturing the effects of design changes that affect safety.

1. Introduction

There is now substantial evidence indicating that modern roundabouts can significantly reduce traffic crashes (Federal Highway Administration, 2008) and that the safety benefit results largely from the factthatthey are designed to controltraffic speeds. It stands to reason, therefore, that safety performance of a roundabout can be related to some measure of its operating speeds.

Recent research presented in NCHRP Report 572 (Rodegerdts et al., 2007) did attempt to establish a speed-based approach-level safety performance function (SPF) for US roundabouts, with the following structure:

$$Crashes/year = exp(intercept) \cdot AADT^b \cdot exp(cX)$$ (1)

where AADT is the average annual daily traffic, X is the independent speed-related variable, and b and c are the calibration parameters.

However, the estimated model was deemed inadequate on the basis of the weak effects of the speed variables (Rodegerdts et al., 2007), so no speed-based SPF could be recommended. By contrast, there were a number of successful non speed-based SPFs estimated for US roundabouts in that research. These models were estimated at both roundabout and approach levels. Some of the approach level models did contain geometric variables, but for the roundabout level models, the sum of entering AADTs from all approaches was typically the only variable.

Researchers have also developed SPFs for roundabouts in Great Britain, Australia, New Zealand and Sweden (Federal Highway Administration, 2000; Turner et al., 2006; Brude and Larsson, 2000; Maycock and Hall, 1984). Some ofthese efforts included, in addition to traffic exposure, variables reflecting geometric features, configuration of vehicles, and speed features (85th percentile speed, speed limits or relative speed difference). Research from New Zealand (Federal Highway Administration, 2000) also introduced a model relating speed features and factors such as diameter and visibility.

NCHRP Report 572 (Rodegerdts et al., 2007) also quoted and tested the following speed prediction models documented in the Federal Highway Administration Roundabout Guide (Federal Highway Administration, 2000):

$$V = 8.7602R^{0.3861}$$ , for $e = +0.02$

$$V = 8.6164R^{0.3673}$$ , for $e = -0.02$ (2)

^∗ Corresponding author. Tel.: +1 647 281 4937.

E-mail addresses: yongsheng.chen@ryerson.ca (Y. Chen), bpersaud@ryerson.ca (B. Persaud), emanuele.sacchi@polito.it (E. Sacchi), marco.bassani@polito.it (M. Bassani).

¹ Tel.: +1 416 979 5000x6464.

² Tel.: +39 011 564 5625.

³ Tel.: +39 011 564 5636.

where V is the predicted speed for left-turn circulating,through circulating, exit or entry movements (km/h), R is the radius of vehicle path (m), and e is the super-elevation (m/m) (inner edge of curve is lower than the outer when e is positive).

Recent research by Bassani and Sacchi (2011) developed a multiple linear regression model for Italian roundabouts shown in Eq. (3).

$$V{85} = 0.4433 \cdot D{\text{INT}} + 0.8367 \cdot W{\text{CR}} + 3.2272 \cdot W{\text{ENL}}$$ (3)

where V⁸⁵ is the 85-percentile operating speed at circulating roadway (km/h), DINT is the diameter of the central island (m), WCR is the width of the circulatory roadway (m), and WENL is the width of the entry lane (m).

The two speed prediction models above are different in key aspects. Eq. (2) was fitted based on fundamental functions of vehicle dynamics, while Eq. (3) is an empirically derived function. In this last case, according to the authors, it was developed without a constant term to logically force an estimate of zero speed when there is a value of zero for all covariates. Moreover, Eq. (3) pertains to 85-percentile circulating speed while Eq. (2) is presumed to pertain to predicted circulating design speed.

In summary, international research does suggest that speed can be related to safety performance of roundabouts. However, there is a wide spectrum of definitions for the speed variables, especially in the European literature, with no clear indication ofthe best variable specification.

This paper aims to address this issue by investigating and comparing possible choices of speed variables, and making a recommendation for the optimal one, with design features as inputs.

The paper further investigates the development of a roundabout SPF with predicted speed as the key input.

2. Sample data

2.1. Summary statistics of raw data

The study used approach-level data for 139 roundabout approaches from eight States in the US, and 34 roundabout approaches from three cities in Italy. US roundabouts are in a mixture of urban, suburban and rural environments, while all Italian roundabouts are in urban and suburban areas. Four observed speeds – approach, entry, upstream (left side of approach) circulating and upstream exiting speeds – and three types of speed differential between each pair of adjacent speeds were available for 34 of the US approaches and 6 of the Italian ones. These are all median speeds. The US sample database is the same one used in the earlier NCHRP study (Rodegerdts et al., 2007). Table 1 shows the summary statistics of the data. These indicate that the US and Italian data are generally comparable.

Fig. 1 depicts the geometric characteristics and the approach locations where speeds were obtained. Location A is for approach speed (measured at least 200 ft upstream of the yield line) (Rodegerdts et al., 2007), B for entry speed, C for upstream circulating speed and D for upstream exiting speed. US speed data were collectedby radar guns (Rodegerdts et al., 2007). For the Italiandata, the collection of speed and positional data of the vehicles in traffic were acquired with a speed gun and a video camera. The acquisition system was located in positions not visible to drivers and only isolated vehicles were considered, thereby excluding information that could be affected by factors such as those linked to the dynamics of traffic flow. In the case of video measurements, a high-speed digital video camera was employed. From the subsequent analysis of captured frames, and knowing the distance between selected sections, the average speeds of isolated vehicles were calculated.

Fig. 1. Geometric characteristics ofthe approach-level area and locations for speeds.

2.2. Derived data

The average value of measured entry, upstream circulating and upstream exiting speeds was considered as a new speed variable (referred as "approach average speed", AAS). Logically, AAS represents the average operating speed inside or at the periphery of one approach. The sum of absolute values of the three differentials in Table 1 (referred as SDSum) could be used as another measure, representing the overall level of speed gaps. The third derived data item is the speed differential of approach vs. AAS (referred as SDApproachAAS). Table 2 illustrates the summary statistics of the derived speed measures.

In Table 1, speed differential is the arithmetic difference of two adjacent speed measures (e.g., speed differential of approach vs. entry = approach speed − entry speed). The speed differential data were obtained only when both of speed observations were available. As result, the frequency of speed differentials was sometimes smaller than the minimum frequency of two relevant speed measures.

In Table 2, SDSum is the addition of the three speed differentials from Table 1. The sum was obtained only when all three speed differentials were available. Therefore, the frequency of SDSum is less than those of individual speed differentials.

3. Selection and estimation of speed prediction model

For a speed measure to be representative of design features in an approach-based crash prediction model, it must be reflective of speeds in the vicinity of the approach. In earlier research (Chen, 2010; Chen et al., 2011), the authors did try to model individual speeds and individual speed differentials, but that proved fruitless.

The final determination of the most appropriate measure was achieved by running the "effect (variable) selection" procedure within the framework of generalized linear models (GLMSELECT Procedure) in the SAS software (SAS Institute Inc., 2011). Based on a pre-set collection of variables, the procedure of "effect selection" iterates the entry or removal of effects until selection stops at a minimum value of the model optimization criterion (the Schwarz Bayesian information criterion, SBC).

Table 1 Summary raw data statistics (Note: These are in US customary units reflecting the origin of the US data).

Country	Variable	Unit	Min.	Max.	Mean	SDa	Freq.b
USA	Total collisions		0	29	4.4	6.0	139
	Collisions with severe injury		0	7	0.5	1.1	139
	Number of years of collision data		0.33	8	3.8	2.3	139
	Entering AADT	veh./day	220	19,593	4637	3706	139
	Inscribed circle diameter, DINS	feet	85.3	300	144.1	49.1	139
	Central island diameter, DINT	feet	19.7	214	77.7	44.8	134
	Entry width, WENL	feet	12	49	22.2	8.3	138
	Circulating width, WCR	feet	11.5	45	26.1	8.3	138
	Exit width, WEXL	feet	12	51	23.0	8.5	128
	Approach speed	mph	18.2	52.2	34.5	8.8	36
	Speed differential of approach vs. entry	mph	5.8	34.5	17.8	7.4	36
	Entry speed	mph	11.7	26.2	17.0	3.3	49
	Speed differential of entry vs. circulating	mph	−1.5	6.6	2.2	2.2	39
	Upstream circulating speed	mph	10.1	23.7	15.1	3.6	40
	Speed differential of circulating vs. exiting	mph	−9.8	0.3	−4.8	2.1	34
	Upstream exiting speed	mph	14.5	30.3	19.3	3.7	40
Italy	Total collisionsa		0	252	24.7	49.3	34
	Collisions with severe injuryc		0	65	6.8	12.9	34
	Number of years of collision data		0.8	8.5	6.2	2.1	34
	Entering AADTc	veh./day	5000	39,000	20,506	9455	34
	Inscribed circle diameter, DINS	feet	54.1	529.9	151.7	92.1	34
	Central island diameter, DINT	feet	13.1	492.0	98.1	93.0	34
	Entry width, WENL	feet	6.3	45.7	19.0	7.5	34
	Circulating width, WCR	feet	17.2	36.7	26.0	5.4	34
	Exit width, WEXL	feet	6.5	38.3	18.8	7.5	34
	Approach speed	mph	20.9	30.8	24.7	3.9	6
	Speed differential of approach vs. entry	mph	0.4	10.0	4.6	4.0	6
	Entry speed	mph	14.3	23.8	20.0	3.2	6
	Speed differential of entry vs. circulating	mph	1.5	5.5	2.9	2.0	6
	Upstream circulating speed	mph	12.8	18.7	17.2	2.3	6
	Speed differential of circulating vs. exiting	mph	−10.5	0.4	−5.3	4.5	6
	Upstream exiting speed	mph	12.4	27.3	22.5	5.9	6

^a Standard deviation.

Table 2 Summary statistics of derived speed measurements.

Country	Variable	Unit	Min.	Max.	Mean	Standard deviation	Frequency
USA	AAS	mph	12.4	25.7	17.0	3.1	33
	SDSum	mph	12.7	45.3	25.5	7.7	32
	SDApproachAAS	mph	5.8	35.4	18.1	7.8	32
Italy	AAS	mph	13.2	23.2	19.9	3.6	6
	SDSum	mph	5.9	25.0	12.9	7.5	6
	SDApproachAAS	mph	−1.2	11.2	4.8	4.6	6

With AAS, SDSum and SDApproachAAS as response variables, and six features as impact factors, including country code (CCODE, defined as categorical variable for the US or Italy data), inscribed circle diameter (DINS), central island diameter (DINT), entry width (WENL), circulating roadway width (WCR), and exit width (WEXL), the effect selection procedures were established. This process first excluded "sum of absolute speed differentials" and "speed differential of approach vs. AAS" since even the optimized final models with these measures had very low coefficients of determination (R2), as shown in Table 3. The results were

more favorable with AAS as the response variable, as shown in Table 4.

As seen in Table 4 the results initially indicated that only central island diameter and exit width should be included (as indicated by the results for "Effect Selection Run 1"). However, a better model was obtained when the procedure was repeated with the average of the inscribed circle and central island diameter (DAV), and the average of entry, circulating and exit width as derived variables (WAV). Those parameters of the final model are indicated by the column labeled "Effect

Table 3 Speed predictive model with SDSum and SDApproachAAS as responses.

Items	SDSum model	SDApproachAAS model
Full collection of covariates	CCODE, DINS, DINT, WENL, WCR, and WEXL	CCODE, DINS, DINT, WENL, WCR, and WEXL
Covariates for selected model	CCODE	CCODE
Adjusted R2	0.2559	0.2972

^b Frequency, number of sites with data.

^c For Italian data, collisions and entering AADT are for the whole roundabout.

Table 4 Basic parameters of speed predictive models considered with AAS as response.

Items	Effect Selection Run 1	Effect Selection Run 2
Full collection of covariates	CCODE, DINS, DINT, WENL, WCR, and WEXL	CCODE, DAV, and WAV
Covariates for selected model	CCODE, DINT, and WEXL	CCODE, DAV, and WAV
Adjusted R2	0.8432	0.8297
Representativeness	Narrow	Wide
Final recommendation		√

Table 5 Results for recommended speed predictive model.

Parameter	Estimate	Standard error	t value	Pr > t
	Model form: AAS = a + b0 + b1 × DAV + b2 ×WAV
a	13.015958	0.775832	16.78	<0.0001
b0: USA	−3.088964	0.620977	−4.97	<0.0001
: Italy	0
b1	0.034074	0.009867	3.45	0.0015
b2	0.142936	0.053786	2.66	0.0118

Selection Run 2" in Table 4, while the model results are shown in Table 5.

While both runs in Table 4 have satisfactory statistical performance in terms of adjusted R2, the covariates of "Effect Selection Run 2" provide greater representativeness. Consequently, the model of "Effect Selection Run 2" was deemed preferable.

Summary statistics of the predicted speeds with this model are shown in Table 6 for the larger dataset of 138 US and 34 Italian roundabout approaches. The distribution of these predicted speeds is shown in Figs. 2 and 3.

Fig. 2 demonstrates how the model for predicting AAS can expand sample data size by allowing the inclusion of sites without observed speeds. Fig. 3 pertains only to the 39 sites with both speeds, and is intended to visually illustrate the accuracy of the speed prediction model in Table 5.

Table 6 Summary statistics for predicted speeds.

Country	Variable	Unit	Min.	Max.	Mean	Standard deviation	Frequency
USA	Predicted AAS	mph	14.2	24.5	17.2	2.5	138
Italy	Predicted AAS	mph	15.5	35.6	20.3	3.7	34

Fig. 2. Predicted vs. observed AAS for all sites.

4. Selection and estimation of safety models based on predicted speed

4.1. Model selection for SPF based on predicted speed

4.1.1. Model identification

Collision counts, which are non-negative integers with overdispersion, are well described by a mixed Poisson distribution family (Chou and Steenhard, 2009). By generalizing some forms prevailing in the road safety domain (Chou and Steenhard, 2009; Lord et al., 2010; Persaud et al., 2010), the common form can be derived as follows.

The number of collisions Yit for a particular ith site and time period t is Poisson distributed about its mean it:

$$Y{it}|\mu{it}\sim Po(\mu_{it})$$ (4)

where i = 1, 2, . . ., I, and t = 1, 2, . . ., T.

The mean of the Poisson distribution is structured as:

$$\mu_{it} = \exp(x_i \beta + e) = f(X, \beta) \cdot \varepsilon_i$$ (5)

where ^f(·) is the function of the covariates (^X or xi), ^ˇ is the vector of unknown coefficients, and e, εⁱ are the random error terms.

The precise form of the distribution of the safety model depends ultimately on the specific choice of the probability distribution of the error term εi.

Among the mixed Poisson distribution family members, the Negative Binomial (NB) distribution is the default choice for crash prediction models. This is actually a special case of the Poissongamma distribution when ^εⁱ ^∼ Gamma (ϕ, (1/ϕ)) (Lan et al., 2009), or in other words, when the distribution's mean is it and variance is (it ⁺ ˛it) (Lord et al., 2010). Its probability density function (PDF) is given (Lord et al., 2010) as:

$$f(y{it}, \alpha, \mu{it}) = \frac{\Gamma(y{it} + a^{-1})}{\Gamma(a^{-1})y{it}!} \left(\frac{\alpha^{-1}}{\mu{it} + a^{-1}}\right) \left(\frac{\mu{it}}{\mu{it} + \alpha^{-1}}\right)^{y{it}}$$ (6)

Fig. 3. Predicted vs. observed AAS for 39 sites with both speeds.

Fig. 4. Empirical integral function analysis for AAS.

where yit is the response variable for observation i and time period t, it is the mean response for observation i and time period t, and ˛ is the dispersion parameter of Poisson-gamma distribution.

4.1.2. Determination of functional form

The form of the roundabout speed-based SPF can be specified generally as follows:

$$Y{it}|\mu{it}\sim Po(\mu_{it})$$

$$\mu{it} = f(AADT, AAS{predicted}; \beta_0, \beta_1, \beta_2)$$

Dispersion parameter = $$\alpha = \frac{k}{e^{(\gamma0 \cdot \text{ASS}{\text{predicted}})}}$$ (7)

where f is the functionalform ofthe mean of collisions, with choices shown in Table 7, and k is the theoretical maximum threshold of the dispersion parameter.

While a power function is universally used for AADT, the power, gamma or exponential functions could be considered for predicted AAS (Persaud et al., 2002; NCHRP Report 572, 2008; Chou and Steenhard, 2009), as depicted in Table 7. The dispersion parameter is specified as varying rather than constant to improve the fit, in accord with current thinking (Lord et al., 2010).

The empirical integral function (EIF) method (Hauer and Bamfo, 1997; Persaud et al., 2002) was applied to preliminarily identify the functional form of AAS. This approach plots a cumulative probability graph against one specific covariate, and then compares this empirical cumulative curve with standard curves of typical function forms in order to indicate the most appropriate relationship between the dependent variable and the candidate covariate.

Fig. 4 shows the results ofthe EIF analyses for US and Italian data. Here the ordinate is the logarithm of the relevant original ordinate of the EIF curve, such that, if the plotted line is linear, the power function is suggested. For the US data, a reasonably linear trend can be seen from ln(AAS) = 2.7 (i.e., for AAS approximately 24 km/h (14.9 mph) and above), which is consistent with range of US predicted AAS shown in Table 6, suggesting that a power function for AAS may be appropriate. For the Italian data, the trend is globally linear, indicating that a power function for AAS is notinappropriate for Italy.

Notwithstanding this preliminary indication, all of the forms in Table 7 were actually attempted, with the final recommendation based on a comparison of model performance.

4.1.3. Modeling approach and platform

A Bayesian modeling approach, which combines prior knowledge from a reference population and information from sitespecific observations, was the preferred one, consistent with the state of research. This is because model parameters with this approach are treated as unknown random variables with inferences based on the posterior distributions of the parameters. This gives more flexibility than fixed parameter estimations (Persaud et al., 2010), a property that will increase the odds of achieving a successful model.

The general purpose Markov chain Monte Carlo (MCMC) simulation procedure in SAS (SAS Institute Inc., 2011) was applied for Bayesian model estimations.

"Quasi-Newton" was chosen as the optimization method of the MCMC procedure and the simulation results were assessed using the following two criteria:

model optimization, as a whole, is indicated by convergence (output of the iterative optimization history is ultimately displayed as "Convergence criterion satisfied", that is, the relative function convergence criterion, FCONV< = 1E−6);
all parameter estimates are significant as judged by the Heidelberger-Welch Diagnostics of Markov chain convergence. This requires that both the "Stationarity Test" (verifying that the iteration process is finally stationary) and the "Half-width Test" (verifying that sample size is adequate) be passed for all parameter estimates.

The Bayesian Poisson-gamma approach was not successful for the Italian data so alternative approaches were considered for that data.

4.2. Model estimation for SPF based on predicted speed

4.2.1. Bayesian Poisson-gamma model for US data

Prior distributions for allparameters (ˇ0,ˇ1,ˇ2, 0) are assumed as non-informative N(0, 103), to reflect the lack of precise knowledge of the value of the coefficients (Persaud et al., 2010).

Table 7 Possible function forms of it.

AADT	AAS
	Power function	Gamma function	Exponential function
Power function	ˇ0 ·AADTˇ1 ·(AASpredicted)ˇ2	ˇ0 ·AADTˇ1 ·(AASpredicted)ˇ2 · exp(ˇ3AASpredicted)	ˇ0 ·AADTˇ1 · exp(ˇ2AASpredicted)

Table 8 Estimation of recommended Bayesian Poisson-gamma SPF for US data.

Parameter	Mean	Standard deviation	Monte Carlo standard errors	Heidelberger-Welch diagnostics
Posterior statistics,	it =ˇ0 ·AADTˇ1 ·(AASpredicted)ˇ2
lnˇ0	−16.3755	2.1801	0.2991	PASS
ˇ1	0.5094	0.1245	0.0122	PASS
ˇ2	4.3314	0.7765	0.0899	PASS
Dispersion parameter, ˛ =	k (0·AASpredicted) = e	3 exp(0.0618×AASpredicted)
DIC (smaller is better) = 652.641
Mean Absolute Deviance, aMAD = 1	n n i=1	yˆi − yi = 1.22

^a n is the sample size and yi and yˆⁱ are the observed and estimated mean values at site i respectively (Lord et al., 2010).

In the case of multiple significant outputs, the preferable model is recommended by the Bayesian model selection indicator, i.e., the Deviance Information Criterion (DIC) (Claeskens and Hjort, 2009; SAS Institute Inc., 2011), defined as:

$$DIC = \overline{D(\theta)} = pD = D(\overline{\theta}) + 2pD \tag{8}$$

where ¯ is the posterior mean of parameter , D() is the posterior mean of the deviance, D() is the deviance evaluated at ¯, and pD is the effective number of parameters.

In SAS the smaller DIC indicates the better fit to the data set (SAS Institute Inc., 2011).

For the US data, models with power and exponential forms for AAS passed the two above-mentioned criteria, while the model with gamma form failed. On the basis of the DIC (652.641 for the power function, 652.923 for the exponential function), the model with the power form is the best, albeit by a small margin. As a result, the model with the power form for AAS is recommended. Details are listed in Table 8. These indicate that the Mean Absolute Deviation (MAD) is of the same order as the average observed collisions per year (=1.54), another indication that the model is adequate.

4.2.2. Alternative modeling for the Italian data

As noted above, for the Italian data, the Bayesian Poissongamma modeling approach was unsuccessful based on all assessment criteria, so an alternative model had to be attempted. After unsuccessful attempts for other Bayesian mixed Poisson family members, zero-inflated Poisson (ZIP) and zero-inflated negative-binomial (ZINB) models were considered. The ZIP model was successful. For this model:

$$P(y_{it} = 0|x_i) = F_i + (1 - Fi) \exp(-\mu{it})$$

$$P(y_{it}|x_i) = (1 - Fi) \frac{\exp(-\mu{it}) \cdot \mui^{y{it}}}{y{it}!}, \quad y{it} > 0$$ (9)

where yit, it, xi have the same meaning as those in Eqs. (5) and (6), it ⁼ˇ⁰ ·AADTˇ¹ ·(AASpredicted)^ˇ² or it ⁼ˇ⁰ ·AADTˇ¹ · exp(ˇ² ·AASpredicted), and Fi is the probability for yit = 0.

ZIP modeling was accomplished with the "COUNTREG procedure" in SAS (SAS Institute Inc., 2011). The rule for deeming a model as significant was stipulated as requiring all parameter estimates to be significant as judged by their p-values, at the 10% level. Multiple significant models were ranked by Akaike's information criterion (AIC) (Claeskens and Hjort, 2009), defined as:

$$AIC(M) = 2 \log -likelihood_{max}(M) - 2 dim(M)$$ (10)

where dim is the number of estimated parameters for model M.

In SAS a smaller AIC indicates a better model (SAS Institute Inc., 2011). Since the ZIP model with a power function for AAS had a smaller AIC (=782.5) than the model with the exponential function for AAS (=786.0), it is recommended as the best for the Italian data, and is shown in Table 9. As seen, the Mean Absolute Deviation (MAD) is of the same order as the average observed collisions per year (=3.90), another indication that the model is adequate.

5. Discussion of model results

5.1. Assessment of practical significance

The practical significance of the predicted speed-based SPF was further assessed by comparing the implied collision modification factors (CMFs) for changes in average speed with those recently derived and presented in NCHRP Report 617 (Harkey et al., 2008). The CMFs in that report were related to initial mean travel speed and change in mean travel speed. The speed related CMFs implied from the models in this paper are estimated from:

$$CMF{speed} = \left(\frac{AAS{changed}}{AAS_{initial}}\right)^{\beta_2}$$ (11)

where the calibrated value of ˇ² is given in Table 8 or Table 9.

Table 9 Estimation of alternative ZIP SPF for Italian data.

Parameter	Estimate	Standard error	t value	Pr > t
Parameter estimates,	it =ˇ0 ·AADTˇ1 ·(AASpredicted)ˇ2
lnˇ0	−29.5239	1.2224	−24.15	<0.0001
ˇ1	2.8623	0.1302	21.98	<0.0001
ˇ2	0.6339	0.1628	3.89	<0.0001
AIC(smaller is better) = 782.5364
Mean Absolute Deviance, MAD = 2.67

Fig. 5. Comparisons of implied CMFs with NCHRP Report 617 CMFs.

Fig. 6. Curves of predicted collisions vs. AAS (at the level of mean AADT).

Fig. 5 plots the CMFs implied by models in Table 8 or Table 9, along with the NCHRP Report 617 CMFs. The comparison indicates that the CMFs implied by the US SPF for total crashes are reasonably consistent with those in the NCHRP report, even though the latter pertains to fatal plus injury crashes. On the other hand, CMFs implied by the Italian SPF are not as consistent, which is understandable, considering that the Italian driving environment is different from that in the US.

5.2. Model comparisons

Notwithstanding differences in random structure, the model functional forms of US and Italy are consistent. They each have power functions for both AADT and AAS. This will give practitioners a consistent structure for roundabout safety models. The difference between US and Italian models lies on coefficients. Compared with Italian SPF, the US SPF has smaller coefficient for AADT but a larger coefficientfor AAS. This can be interpreted as indicating that US SPF shows more sensitivity for a change of AAS than the Italian model. This difference in sensitivity is also suggested by Fig. 6 which plots curves of predicted collisions vs. AAS at the level of mean AADT for both countries.

6. Conclusion

This paperhas contributed to developing roundabout SPFs based on predicted speed. Supported by the development of a speed predictive model, a Bayesian Poisson-gamma SPF was successfully developed for US, while for the Italian data an alternative ZIP model provided a better fit. Notwithstanding this fundamental structural difference, the developed US and Italian SPFs both suggest that the predicted speed approach seems to be promising for indirectly estimating roundabout safety performance from a model that first predicts speeds. This approach is conceptually attractive in that predicted speed can be used rather than observed speed, thereby expanding the sample size and producing more robust models.

The research has successfully investigated a "two-stage" tool for roundabout safety analysis. The first stage is to predict AAS from geometric features and the second is to use predicted AAS in an SPF that can be applied to roundabouts that are still in the planning stages. Such an approach can be superior to the SPFs directly containing all design variables. This is not only because the latter is difficult to estimate, given that roundabouts tend to have few crashes and design features that have little variation, but also because the "two-stage" approach preserves model parsimony while ultimately capturing the effects of design changes that affect safety.

The results suggest that roundabouts with larger average diameter and average width would tend to have higher AAS. The higher AAS, both by US and Italian SPFs,the higher is the expected collision frequency, with the US SPF showing more sensitivity to AAS than that for Italy.

Considering the differences between the Italian and US driving environment (roadway systems, weather, terrain, collision patterns, and driver factors such as training and behavior), the differences in random structure and sensitivity for the US and Italian SPFs are understandable. That the approach investigated in this paper works both countries and that both countries have a similar SPF functional form is indeed encouraging and suggests that the approach can be considered in the large number of countries that have roundabouts.

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