Accident Analysis and Prevention 73 (2014) 252–261
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Speed limit reduction in urban areas: A before–after study using Bayesian generalized mixed linear models Shahram Heydari a, *, Luis F. Miranda-Moreno b , Liping Fu a a b
Department of Civil and Environmental Engineering, University of Waterloo, 200 University Avenue W., Waterloo, Ontario N2L 3G1, Canada Department of Civil Engineering and Applied Mechanics, McGill University, 817 Sherbrooke St. W., Montreal, Quebec H3A 2K6, Canada
A R T I C L E I N F O
A B S T R A C T
Article history: Received 26 September 2013 Received in revised form 2 September 2014 Accepted 11 September 2014 Available online xxx
In fall 2009, a new speed limit of 40 km/h was introduced on local streets in Montreal (previous speed limit: 50 km/h). This paper proposes a methodology to efﬁciently estimate the effect of such reduction on speeding behaviors. We employ a full Bayes before–after approach, which overcomes the limitations of the empirical Bayes method. The proposed methodology allows for the analysis of speed data using hourly observations. Therefore, the entire daily proﬁle of speed is considered. Furthermore, it accounts for the entire distribution of speed in contrast to the traditional approach of considering only a point estimate such as 85th percentile speed. Different reference speeds were used to examine variations in the treatment effectiveness in terms of speeding rate and frequency. In addition to comparing rates of vehicles exceeding reference speeds of 40 km/h and 50 km/h (speeding), we veriﬁed how the implemented treatment affected “excessive speeding” behaviors (exceeding 80 km/h). To model operating speeds, two Bayesian generalized mixed linear models were utilized. These models have the advantage of addressing the heterogeneity problem in observations and efﬁciently capturing potential intra-site correlations. A variety of site characteristics, temporal variables, and environmental factors were considered. The analyses indicated that variables such as lane width and night hour had an increasing effect on speeding. Conversely, roadside parking had a decreasing effect on speeding. One-way and lane width had an increasing effect on excessive speeding, whereas evening hour had a decreasing effect. This study concluded that although the treatment was effective with respect to speed references of 40 km/h and 50 km/h, its effectiveness was not signiﬁcant with respect to excessive speeding-which carries a great risk to pedestrians and cyclists in urban areas. Therefore, caution must be taken in drawing conclusions about the effectiveness of speed limit reduction. This study also points out the importance of using a comparison group to capture underlying trends caused by unknown factors. ã 2014 Elsevier Ltd. All rights reserved.
Keywords: Before–after studies Speed limit reduction Speeding Excessive speeding Bayesian generalized mixed linear models
1. Introduction In road safety studies, quantifying the effectiveness of safety treatments through observational before–after studies is a crucial task. These studies allow transportation authorities and municipalities to adopt adequate strategies in the selection and implementation of safety countermeasures. In this paper, we investigated the effect of a 10 km/h reduction (from 50 km/h to 40 km/h) in posted speed limits on speeding behaviors using a sample of local streets in the city of Montreal.
* Corresponding author. Tel.: +1 5198884567. E-mail addresses: [email protected]
(S. Heydari), [email protected]
(L.F. Miranda-Moreno), [email protected]
(F. Liping). http://dx.doi.org/10.1016/j.aap.2014.09.013 0001-4575/ ã 2014 Elsevier Ltd. All rights reserved.
Operating speeds have been widely considered as an important measure of trafﬁc efﬁciency and road safety. Several studies have conﬁrmed the importance of speed in crash frequencies and severities (Aarts and Van Schagen, 2006; De Pelsmacker and Janssens, 2007; Elvike, 2009; Speed Check Services, 2010; Cascetta and Punzo, 2011; Soole at al., 2013; Wu et al., 2013; De Pauw et al., 2013). For example, Aarts and Van Schagen (2006) reviewed a number of empirical studies that aimed at investigating the link between operating speeds and crash rates. Similarly, Elvik (2009) provides a comprehensive discussion about the relationship between speed and road safety. A number of studies have also examined the effect of speed enforcement on safety (Stephens, 2007; Cameron, 2008; Soole et al., 2013) by studying various strategies of enforcement. These studies highlighted that speed enforcement leads to a reduction in average or 85th percentile speeds and thus crash frequencies and severities decrease. A detailed review on this subject can be found in Soole et al. (2013).
S. Heydari et al. / Accident Analysis and Prevention 73 (2014) 252–261
Obviously, the role of posted speed limits is important in road safety. In terms of speed management, speed limits can act as a controlling measure of vehicles' operating speeds. Research shows that a lowered speed limit in highways can reduce crash frequencies and severities (Aarts and Van Schagen, 2006). For instance, Malyshkina and Mannering (2008) found that a greater probability of injury and fatality was explained by high speed limits for some non-interstate highways in Indiana. However, the authors highlighted that a 5 mph (65 mph–70 mph) increase in speed limits on inter-state highways did not have any effect on crash-injury severities. Interestingly, Lave and Elias (1994) showed that a 10 mph (55 mph–65 mph) increase in posted speed limits improved safety by reducing the number of fatalities. Therefore, one might conclude that previous studies are not in accordance with respect to the connection between speed limit and road safety in terms of accident frequency and severity (Malyshkina and Mannering, 2008; Lahausse et al., 2010). Note that the above-mentioned studies have mainly analyzed non-urban roads. Nevertheless, the effect of speed limit reduction on serious injuries and fatalities might be more relevant in local streets (in urban and residential areas) than in arterial and rural roads because of a higher presence of pedestrians and cyclists in such environments. A study conducted in Australia (Unley Street Life Trust, 2007), for example, indicated a 46% reduction in severe injuries among pedestrians as the consequence of lowered speed limits. In this regard, Rosén et al. (2011) provide a detail discussion on the effect of car impact speed on pedestrians’ injury–fatality. In terms of before–after studies, model-based approaches are a more reliable framework; therefore, modeling speed as a function of contributing factors (site characteristics, etc.) is a main step in the assessment of speed limit changes. In general, most of the papers dealing with speed have mainly used 85th percentile or mean speed as a measure to model operating speeds or to compare speed variations in the before and after periods (Kanellaidis et al., 1990; Gattis and Watts, 1999; Fitzpatrick et al., 2003; Soole at al., 2013). Recent studies (Park et al., 2010a; Kuo and Lord, 2013; Eluru et al., 2013) have discussed various aspects of speed modeling to overcome problems (e.g., heterogeneity in data) and shortcomings associated with the classical methods. To estimate the effectiveness of safety treatments, the empirical Bayes (EB) method has been commonly used for before–after studies for many years. Note that the EB approach has been mainly used in analyzing crash data (see Elvik (2013), for a speed before– after study conducted using EB). The EB approach is able to address some of the deﬁciencies (e.g., regression to the mean problem) associated with traditional methods such as the naive before–after analysis (Hauer, 1977). However, some recent studies indicated that the full Bayes (FB) before–after estimation is a better alternative to the EB method (Lan et al., 2009; Persaud et al., 2010; Park et al., 2010b; El-Basyouny and Sayed, 2012). Three important properties of the FB paradigm are as follows: (1) while the EB provides the results in the form of point estimates, the posterior distribution of the model parameters and effectiveness index are provided by the FB analysis. As a result, the uncertainty in the estimates is fully addressed (Mitra and Washington, 2006; Park et al., 2010b; Heydari et al., 2014); (2) hierarchical and more complex statistical models can be easily employed under Bayesian statistics (El-Basyouny and Sayed, 2012); and (3) through the prior distribution, the previous knowledge related to a parameter can be used in the analysis for improved accuracy of the parameter estimates (Miranda-Moreno et al., 2013; Heydari et al., 2014). This paper, ﬁrst, employs the entire distribution of speed (instead of a single value like 85th percentile speed) by using data collected in the form of speed categories (bins). It should be noted that the use of 85th percentile speed might result in biased estimates (Tarris et al., 1996; Eluru et al., 2013). Second, analyses
are performed based on hourly observations (disaggregate data); thus, the daily variations in the speed proﬁle are accounted for. Third, the treatment effectiveness estimates are obtained by leveraging the strengths of both the FB before–after framework and generalized mixed linear models. Note that, a considerable number of studies have examined the effects of speed limit changes, to our knowledge, however the use of FB before–after analysis for speed analysis has not been attempted. Two Bayesian generalized mixed linear models (Zeger and Karim, 1992) are used for modeling purposes for which Markov chain Monte Carlo (MCMC) (Gelman et al., 2003) simulations are employed to draw the posteriors for parameters of interest. Generalized mixed linear models are capable of accounting for heterogeneity in data and correlations between observations nested in each site. The data used in this paper provides counts of vehicles for a number of speed categories (i.e., 0–20, 20–30, 30–40, etc.) before and after the implementation of the speed limit change. 2. Methodology To estimate the indices of effectiveness, our method was developed based on the examination of the variations in the speed distribution with respect to a series of predetermined reference speeds. In this study, the speed limit of 50 km/h on treated sites (in the before period) was reduced to 40 km/h, in the after period. The speed limit among comparison sites was 40 km/h. Therefore, speed limits of 40 km/h and 50 km/h were set as two reference speeds to conduct the before–after analysis. A third reference speed (i.e., 80 km/h) was also selected to examine excessive speeding. It should be noted that, in particular, offenders exceeding 80 km/h on local streets are targeted by police in Montreal, and higher ﬁnes are considered for these offenders. The use of 80 km/h allowed us to identify variables that affect excessive speeding, and consequently, to evaluate the effect of lowered speed limits on excessive speeding. Take into account that excessive speeding might be a highly inﬂuential factor in causing fatalities and serious injuries among pedestrians and cyclists in urban areas (Davis, 2001; Unley Street Life Trust, 2007; Rosén and Sander, 2011). It should be noted that a similar analysis could be conducted using reference speeds other than those mentioned above. However, a comparative analysis was beyond the scope of this paper. To analyze before–after speed data, two generalized mixed linear models were utilized under the Bayesian framework: mixedeffect binomial and Poisson models. Take into account that the adopted data was in the form of repeated measures outcomes, that is, several observations were nested in every site. It was thus necessary to account for intra-site correlations and heterogeneity in data by employing a mixed-effect approach. 2.1. Bayesian generalized linear mixed models Generalized linear models (McCullagh and Nelder, 1989) are widely used in modeling discrete and continuous outcomes. For i sites with j observations within each site, a typical random effects generalized linear model is given in Eq. (1) (Zeger and Karim, 1992) in which dij (model outcome mean) is deﬁned by a linear regression function through the link function g(.). In this paper, dij is the binomial probability or Poisson mean. gðdij Þ ¼ b0 þ bX ij þ eij þ Z ij ui
where b0 is a constant term, b is a vector of stochastic regression coefﬁcients and Xij is px1 vector of covariates. Zij is qx1 subset of Xij with random coefﬁcients. It is usually assumed that Zij is equal to one; that is, responses related to each site i have the same intercept ui + b0 (Zeger and Karim, 1992). The site speciﬁc effect ui and the error term at observational level eij are assumed to be normally
S. Heydari et al. / Accident Analysis and Prevention 73 (2014) 252–261 Table 1a Explanation of dummy variables. Variable Night hour (1 if night hour, 0 otherwise) Evening hour (1 if evening hour, 0 otherwise) Peak hour (1 if peak hour, 0 otherwise) Weekend (1 if weekend, 0 otherwise) One way (1 if one way streets, 0 otherwise) Parking (1 if parking is present, 0 otherwise) Precipitation (1 if there is precipitation, 0 otherwise) Lanes (1 if 2 lanes, 0 if 1 lane)
distributed, i.e., ui i.i.d N(0, s u2) and eij i.i.d N(0, s e2). In the Bayesian hierarchical approach, a hyper-prior distribution is also assumed for s u2 and s e2; that is, usually a non-informative gamma distribution with mean 1. For b0 and b, a non-informative Gaussian distribution with mean 0 can be assumed. Note that observations related to each site might have similar characteristics (known or not), and therefore, these might be associated. Ignoring site-speciﬁc effects assumes that all sites have exactly the same traits. It is quite possible however that this assumption does not hold (e.g., a different demography of drivers). In fact, the site-speciﬁc variance (s u2) corresponds to the covariance between observations that are related to the same site. The within-site correlation is calculated using Eq. (2) (Vittinghoff et al., 2012).
s þ s 2e 2 u
where g stands for the correlation between observations nested within the same site (intra-site correlation). Values of g close to 1 show a high correlation (g varies from 0 to 1). In other words, large between-site variability (with respect to the within-site variability) indicates a high intra-site correlation.
12 am–6 am 7 pm–12 am 6 am–10 am and 3 pm–7 pm Saturday and Sunday – – – –
2.1.1. Mixed-effect binomial logistic model The data collected in several bins were dichotomized into two bins using the predetermined reference speeds of 40 km/h and 50 km/h (e.g., two speed bins of 0–50 km/h and greater than 50 km/ h). Hence, the nature of the problem shifted into a binary response problem for which a binomial model could be used. The binomial model assumes that the outcome (i.e., driver’s choice of exceeding a reference speed) follows a binomial distribution. In other words, drivers have two choices of whether exceeding the reference speed or not. The binomial probability can be then assumed to be a function of various factors such as site characteristics (e.g., lane width), temporal attributes, etc. A mixed-effect model was used to account for unobserved and unknown factors that may vary across the sites and observations under investigation, and also to consider the potential correlation that may exist among observations nested in the same site. The binomial model for the above mentioned framework is described as follows: rij jX ij ; eij ; ui Binomialðpij ; vij Þ
logitðpij Þ ¼ b0 þ bX ij þ eij þ Z ij ui ; Eðeij Þ ¼ 0 and Eðui Þ ¼ 0
Table 1b Summary statistics of statistically signiﬁcant site attributes. Data Comparison sites (before period)
1700 observations (9 sites)
Comparison sites (after period)
1659 observations (9 sites)
Treated sites (before period)
3162 observations (19 sites)
Treated sites (after period)
3266 observation (19 sites)
Evening hour Lane width (m) Night hour One way Parking Precipitation Trafﬁc volume (vehicles/hour) Evening hour Lane width (m) Night hour One way Parking Peak hour Weekend Trafﬁc volume (vehicles/hour) Evening hour Lane width (m) Night hour Parking Precipitation Trafﬁc volume (vehicles/hour) Evening hour Lanes Night hour One way Parking Peak hour Weekend Trafﬁc volume (vehicles/hour)
0.225 4.761 0.178 0.134 0.901 0.214 25.609 0.217 5.142 0.183 0.137 0.902 0.368 0.403 23.602 0.215 6.562 0.219 0.912 0.211 32.503 0.21 0.11 0.22 0.36 0.91 0.35 0.40 31.21
0.418 1.386 0.383 0.341 0.299 0.410 62.983 0.412 2.059 0.386 0.344 0.298 0.482 0.491 51.085 0.411 1.793 0.413 0.283 0.408 39.266 0.41 0.31 0.41 0.48 0.29 0.48 0.49 41.27
0.0 3.3 0.0 0.0 0.0 0.0 1.0 0.0 3.3 0.0 0.0 0.0 0.0 0.0 1.0 0.0 4.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0
1.0 8.1 1.0 1.0 1.0 1.0 634.0 1.0 10.6 1.0 1.0 1.0 1.0 1.0 440.0 1.0 11.2 1.0 1.0 1.0 385.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 446.0
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posterior distribution, p(b|k), can be obtained. expðb0 þ bX ij þ eij þ ui Þ pij ¼ 1 þ expðb0 þ bX ij þ eij þ ui Þ
For each observation, the observed number of vehicles rij exceeding a reference speed was assumed to be binomial with trafﬁc volume vij and posterior probability (rate of exceeding a reference speed) pij. Using generalized linear mixed models with the Logit link function, pij can be obtained from Eqs. (4) and (5). The effect of the speed limit reduction with respect to 40 km/h and 50 km/h was veriﬁed using the estimates of the expected rates of exceeding these speed values. 2.1.2. Mixed-effect Poisson model The number of vehicles exceeding 80 km/h (excessive speeding given speed limits on local urban streets) was very limited compared to trafﬁc volume. Since Poisson distribution is suitable for rare count events, a Poisson model was employed to study variables that affect excessive speeding. The suitability of Poisson distribution in this case is formally tested and discussed in Section 4.3. Note that binomial distribution failed to adequately model excessive speeding since the number of vehicles with excessive speeding was extremely smaller than the total number of vehicles passing through each road segment. By the adopted approach, the expected number of vehicles (in the before and after periods) could be compared to assess the treatment effectiveness with respect to excessive speeding. A mixed-effect Poisson model can be described based on Eqs. (6) and (7). rij jX ij ; eij ; ui Poissonðmij Þ
logðmij Þ ¼ b0 þ bX ij þ eij þ Z ij ui ; Eðeij Þ ¼ 0 and Eðui Þ ¼ 0
where rij and mij are, respectively, the observed and expected number of vehicles exceeding 80 km/h. 2.2. Full Bayes (FB) inference Instead of the classical methods such as the maximum likelihood estimation (MLE), fully Bayesian approach can be utilized to draw posterior estimates of the parameters of interest. Under the Bayesian paradigm, the prior and likelihood distributions are used to obtain the posterior distribution. When the data is large (e.g., in this paper), the role of the prior density becomes less critical, and the posterior is mainly drawn from the likelihood distribution. It should be noted that Bayesian methods have several advantages. For a review of Bayesian methods see, for example, Carlin and Louis (2009) and Daziano et al. (2013). From Eq. (8), the
pðb=kÞ ¼ Z
where b = vector of parameters; k = observed data; p (k|b) = likeliR hood density; p(b) = prior density; p(k|b)p(b)d(b) = marginal likelihood. Note that because of the complexities associated with Eq. (8), Markov chain Monte Carlo (MCMC) methods are used instead of analytical methods (Carlin and louis, 2009). 2.3. FB before–after studies A before–after study on speed data can estimate the effect of an implemented safety countermeasure (e.g., reduction in speed limits) on the operating speeds or the rate of compliance with speed limits in the after treatment period. A group of comparison sites can also be used as a measure to control potential underlying trends that might vary between the before and after treatment periods (Hauer, 1977). Similar to accident data, expected rates p or expected number of vehicles m exceeding a reference speed (before and after the implementation of a treatment) can be utilized to conduct a before–after analysis. In the FB before–after framework, despite the EB method in which point estimates are estimated for parameters of interest, all parameters are treated as random variables and therefore uncertainty is fully considered. In addition, more complex and ﬂexible statistical models can be employed when using the FB approach (Persaud et al., 2010; ElBasyouny and Sayed, 2012). For detailed explanations on FB before–after studies see, e.g., Park et al. (2010)b. Under the FB context, the following steps can be applied to estimate the index of effectiveness. Here, we provide these steps based on the values of p (see Section 2.1.1); however, the same procedures are applied when m (see Section 2.1.2) is set as a measure to assess the treatment efﬁcacy. Note that the expected rates mentioned below are average values over all sites. 1. For each of the four data sets (comparison before, comparison
after, treatment before, and treatment after), obtain the expected rate or number of vehicles exceeding a given reference speed; 2. Compute the adjustment factor, AF, as the ratio between the posterior expected rate of exceeding for comparison sites in the after period and the posterior expected rate of exceeding for comparison sites in the before period, i.e., AF = pca/pcb. 3. Multiplying the adjustment factor by the posterior expected rate of vehicles exceeding the reference speed in the before period
Table 1c Summary statistics of response variables. Data
Vehicles Vehicles Vehicles Vehicles Vehicles Vehicles Vehicles Vehicles Vehicles Vehicles Vehicles Vehicles
exceeding exceeding exceeding exceeding exceeding exceeding exceeding exceeding exceeding exceeding exceeding exceeding
40 km/h 50 km/h 80 km/h 40 km/h 50 km/h 80 km/h 40 km/h 50 km/h 80 km/h 40 km/h 50 km/h 80 km/h
ENTITY NOT DEFINED !!!Unit of response variables is vehicles per hour.
5.744 1.375 0.147 20.265 11.034 0.233 16.269 5.685 0.281 25.99 15.70 0.48
9.458 2.052 0.459 48.244 28.389 0.561 24.766 10.572 0.658 35.65 23.10 1.03
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
93.0 21.0 4.0 431.0 249.0 5.0 326.0 167.0 6.0 413.0 343.0 14.0
S. Heydari et al. / Accident Analysis and Prevention 73 (2014) 252–261 Table 1d Descriptive statistics of statistically signiﬁcant dummy variables. Data
Comparison sites (before period) 1700
Evening hour Night hour One way Parking Evening hour Night hour One way Parking Peak hour Weekend Evening hour Night hour Parking Evening hour Lanes Night hour One way Parking Peak hour Weekend
Comparison sites (after period) 1659 observations
Treated sites (before period) 3162 observations
Treated sites (after period) 3266 observations
The uncertainty around u* – which is given directly when using the MCMC algorithm – is estimated through Bayesian credible intervals at different levels of signiﬁcance. Take into account that the Bayesian intervals, instead of the conﬁdence intervals estimated for the EB method, provide the probability with which a posterior mean occurs in a given interval. A safety treatment is considered effective (it has a positive effect on safety or has a reducing effect on operating speeds in this study) if the upper limit of the credible interval is smaller than 1, which also means that the posterior mean of u * is smaller than 1. Note that if this interval includes the value 1, a deﬁnitive inference about the effectiveness cannot be drawn. It is important to mention that (as discussed in Section 1) we are aware of the fact that a reduction in operating speeds does not necessarily imply a reduction in accident frequencies or severities. In this paper, an implemented reduction in speed limits is considered effective when speeding behaviors are decreased. Note that the primary goal of lowering speed limits is decreasing operating speeds and speeding behaviors. Obviously, a more robust conclusion on safety can only be made once accident data by severity are also used in a before–after speed analysis. In case accident data are not available, estimating the variation in the speed proﬁles can still be useful, particularly, when considering excessive speeding in urban areas. 3. Data A data collection campaign was designed and conducted in years 2009 and 2011 following the Montreal transportation plan in 2008. This plan aimed particularly at improving the safety of pedestrians – as the most vulnerable road users – in urban areas. A sample of speed data from a set of local streets in Montreal with observations collected in the before and after treatment periods was used to illustrate the proposed methodology.
77.5 82.2 86.6 9.9 78.3 81.7 86.3 9.8 63.2 59.7 78.5 78.1 8.8 78.7 89.3 78.4 64.3 9.3 65.0 60.3
22.5 17.8 13.4 90.1 21.7 18.3 13.7 90.2 36.8 40.3 21.5 21.9 91.2 21.3 10.7 21.6 35.7 90.7 35.0 39.7
The adopted case study, which is only a part of the entire study area in Montreal, consisted of 19 treated sites and 9 comparison sites. The data were provided by the Montreal Department of Transportation. Note that each site included several observations. Treated and comparison sites were selected based on a series of criteria such as (1) the street length should be at least 200 meters so that drivers were able to reach a relatively high speed, (2) the street should not be in school or park areas where the speed limit of 30 km/h applies, and (3) the street should have a speed limit of 40 km/h or 50 km/h. The selected sites were then visited by experts to validate the collected data. Comparison sites were selected based on their similarity (e.g., in geometry, trafﬁc ﬂow, etc.) to treated sites. This similarity is a crucial task in before–after studies, and contributes directly to the quality of such studies. Thus, the comparison sites were selected with caution to have similar characteristics to treated sites. One should take into account that comparison sites act as a controlling measure to account for underlying trends in the before and after periods. Generally, a larger control group (with respect to the treated sites) improves the reliability of the estimated index of effectiveness in before–after studies. In this paper, the data relating to 9 comparison sites (against 19 treated sites) were only available. Note also that some sites have two directions (e.g., north and
for treated sites, compute p that is “what would have been the rate of vehicles exceeding a given reference speed in the after period if speed limit reduction had not been applied”, i.e., p = AF x ptb. 4. Obtain the index of effectiveness, u*, as the ratio between the posterior expected rate of vehicles exceeding the reference speed in the after period (for treated sites) and p, i.e., u* = pta/p.
45.0 40.0 35.0 30.0 25.0 20.0 15.0 10.0 5.0 0.0
Before Treatment After Treatment
Speed Categories Fig. 1. Speed histogram for a generic site.
S. Heydari et al. / Accident Analysis and Prevention 73 (2014) 252–261
85th percentile Speed (kmph)
65.0 Before Treatment 60.0
45.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Hour Fig. 2. Hourly speed variation for a generic site.
south). This can be captured by variable one way. Regardless of the number of comparison sites, the potential underlying trend between the before and after periods could be captured since comparison sites included a large sample with more than 1600 observations for each of the before and after periods. Obviously, the methodology discussed in this paper is not affected by the size of
the comparison group. Recall that the focus of this paper was on providing a methodological framework to conduct before–after speed studies, and discussing the data collection issues (including the suitability of comparison sites) was out of the scope of this study. The data were collected during a ﬁve-day period (Thursday, Friday, Saturday, Sunday, and Monday) in 2009 (before period) and 2011 (after period). The implementation of the project started in fall 2009 and ended in January 2011. Drivers were informed about the upcoming change in speed limits via public awareness campaigns (videos, signs, etc.). For instance, instructive signs were posted in different locations of neighborhoods. Moreover, a speed limit map was published on the Web in the PDF format and new speed limits were displayed by signs. To collect operating speeds, trafﬁc sensors were installed on the pavement of roads (one sensor on each lane). This data collection process allowed recording the speed of vehicles passing through installation points. The hourly counts of vehicles were provided directly by trafﬁc sensors for various speed categories, i.e., 0–20, 20–30, 30–40, etc. In this paper, the analyses were conducted based on the number of vehicles for every speed category despite using aggregate speed values such as 85th percentile speed. It should be noted that speed is a continuous random variable.
Table 2 Estimation results (Binomial model) – models for posterior rates of speeding (vehicles exceeding reference speeds of 40 km/h and 50 km/h). Variables
Comparison sites (before period)
Constant Lane width Night hour Parking Precipitation
s 2e s 2u p
g Comparison sites (after period)
Constant Night hour One way Parking Peak hour Weekend
s 2e s 2u p
g Treated sites (before period)
Constant Lane width Night hour Parking Precipitation
s 2e s 2u p
g Treated sites (after period)
Constant Lanes Night hour Parking Peak hour Weekend
s 2e s 2u p
40 km/h Posterior mean
Bayesian intervals 2.50%
50 km/h Posterior mean
Bayesian intervals 2.50%
1.583 0.177 0.316 0.021 0.05 0.194 0.678
0.401 0.08 0.078 0.019 0.036 0.023 0.462
2.327 0.047 0.16 0.07 0.134 0.152 0.225
0.886 0.359 0.468 0.001 0.002 0.241 1.853
3.437 0.209 0.346 0.104 0.04 0.153 1.117
0.73 0.136 0.118 0.077 0.034 0.017 0.834
4.963 0.011 0.112 0.287 0.126 0.122 0.363
2.253 0.501 0.576 0.004 0.001 0.188 3.202
0.345 0.778 2.221 0.08 0.285 1.182 0.067 0.032 0.194 0.94 0.767 0.829 0.623 0.057 0.141 0.134 0.121 0.169 0.751 0.438 0.816 1.399 0.978 0.084 0.154 0.066 0.009 0.261 0.475
0.004 – 0.31 0.058 0.265 0.083 0.038 0.025 0.023 0.657 0.004 – 0.238 0.037 0.04 0.072 0.029 0.01 0.284 0.002 – 0.173 0.466 0.045 0.081 0.029 0.008 0.015 0.181
0.337 – 1.502 0.004 0.008 1.346 0.005 0.001 0.152 0.301 0.759 – 1.156 0.003 0.063 0.286 0.177 0.15 0.383 0.434 – 1.07 0.103 0.008 0.332 0.012 0 0.233 0.237
0.354 – 2.725 0.215 0.938 1.019 0.149 0.092 0.241 2.618 0.775 – 0.186 0.137 0.22 0.013 0.064 0.189 1.467 0.443 – 1.742 1.866 0.181 0.016 0.124 0.031 0.291 0.93
0.11 0.88 0.229 0.347 0.38 0.878 0.079 0.049 0.153 0.986 0.409 0.866 2.089 0.044 0.318 0.203 0.116 0.164 0.703 0.155 0.811 0.086 0.966 0.34 0.183 0.07 0.01 0.166 0.635
0.003 – 0.149 0.071 0.43 0.063 0.037 0.031 0.017 0.892 0.004 – 0.311 0.038 0.046 0.089 0.034 0.013 0.264 0.002 – 0.077 0.48 0.04 0.066 0.023 0.009 0.009 0.259
0.105 – 0.013 0.208 0.008 0.995 0.011 0.003 0.122 0.282 0.401 – 2.934 0.001 0.229 0.387 0.183 0.139 0.353 0.152 – 0.002 0.128 0.262 0.317 0.025 0 0.148 0.309
0.117 – 0.604 0.487 1.61 0.751 0.152 0.115 0.188 3.162 0.418 – 1.581 0.143 0.409 0.036 0.049 0.189 1.363 0.159 – 0.278 2.002 0.42 0.053 0.115 0.034 0.184 1.289
p = expected probability/rate (averaged over observations) of exceeding the reference speeds and g = intra-site correlation.
S. Heydari et al. / Accident Analysis and Prevention 73 (2014) 252–261
However, by discretizing the speed distribution into various speed categories, the nature of the problem changed to discrete data. For the scope of this paper, three scenarios of speed references (40 km/h, 50 km/h, and 80 km/h) were set and the data were classiﬁed accordingly. For each scenario of speed reference, four data sets were analyzed: (1) treated sites in the before period (TB); (2) treated sites in the after period (TA); (3) control sites in the before period (CB); and (4) control sites in the after period (CA). A variety of variables were also taken into consideration for modeling purposes as follows. 1. Site characteristics including number of lanes, lane width, one
way street, vertical and horizontal alignments (low, medium, and high), sight distance (smaller or greater than 150 m), presence of parking and bicycle route, pavement condition (poor, average, good), etc. 2. Weather conditions including precipitation and wind speed. 3. Temporal variables including night hour, evening hour, peak hour, weekend, etc. Table 1a provides an explanation of dummy variables. Tables 1b and 1c provide a summary statistics of the data. In addition, Table 1d reports a descriptive statistics for dummy variables using percentage information to better represent this type of variables. Fig. 1 shows a speed histogram for a generic site. The hourly variation of speed is shown in Fig. 2. It is important to mention that there has been a general decrease in drivers’ compliance with posted speed limits in the after period despite the implemented speed limit reduction. This decreasing trend that was caused by
unknown reasons was observed among both treated and comparison sites. The importance of using comparison sites to account for such underlying trends is thus highlighted in this paper. In this regard, a discussion is provided in Section 4.2. 4. Results and discussions The computation process consisted of two steps: (1) estimation of model parameters and the expected rate or number of vehicles exceeding the reference speeds (i.e, mij and pij) and (2) calculation of the indices of effectiveness to evaluate the implemented safety countermeasure with respect to each reference speed. Statistical software OpenBUGS (Lunn et al., 2009) was used to run MCMC simulations to obtain posteriors of the model parameters. A total of 40,000 iterations were performed using two chains with different initial values. The ﬁrst 10,000 iterations were discarded as burn-in; thus, 60,000 samples were used to obtain the posterior distributions of the stochastic parameters. The presence of two chains, suitably, allows the veriﬁcation of the convergence; for instance, via graphical options available in OpenBUGS such as the Gelman-Rubin test (Brooks and Gelman, 1998). In addition, the convergence can be veriﬁed by monitoring trace and history plots in OpenBUGS. When samples are drawn from more than one chain simultaneously, the above-mentioned plots are shown in a different color for each chain. Therefore, the user is able to observe how chains overlap. Convergence is satisfactory when chains mix or overlap well. After the convergence, as a rule of thumb, the samples should be drawn until Markov chain errors (these are provided by OpenBUGS for all
Table 3 Estimation results (Poisson model) – models for posterior frequencies of excessive speeding (speed > 80 km/h). Variables
Comparison sites (before period)
Constant Evening hour Night hour One way
s 2e s 2u m g Comparison sites (after period)
Constant Evening hour Lane width Night hour Parking
s 2e s 2u m g Treated sites (before period)
Constant Evening hour Night hour Parking
s 2e s 2u m g Treated sites (after period)
Constant Evening hour Night hour One way Parking
s 2e s 2u m g
2.639 0.585 2.370 0.916 0.778 2.009 0.147 0.721 1.386 0.392 0.126 1.302 1.052 0.383 0.635 0.234 0.624 0.098 0.420 1.528 1.185 0.359 1.377 0.285 0.793 0.111 0.425 1.223 0.197 0.690 0.373 2.062 0.482 0.847
0.540 0.187 0.413 0.784 0.225 1.794 0.009 – 0.532 0.145 0.080 0.217 0.219 0.128 0.492 0.013 – 0.099 0.095 0.139 0.218 0.078 0.665 0.009 – 0.108 0.078 0.099 0.185 0.198 0.051 0.945 0.013 –
m = expected hourly exceeding frequency (averaged over observations) with respect to 80 km/h, and g = intra-site correlation.
Bayesian intervals 2.50%
3.923 0.961 3.258 0.035 0.330 0.480 0.130 – 2.498 0.682 0.007 1.744 1.486 0.164 0.173 0.210 – 0.003 0.607 1.805 1.599 0.194 0.557 0.267 – 0.003 –0.580 –1.419 0.005 –1.085 0.284 0.879 0.459 –
1.700 0.224 1.630 2.846 1.238 6.490 0.166 – 0.412 0.113 0.310 0.892 0.630 0.646 1.886 0.260 – 0.349 0.236 1.262 0.742 0.510 3.076 0.304 – 0.390 –0.272 –1.031 0.689 –0.281 0.481 4.444 0.506 –
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model parameters) are smaller than 5% of the standard deviations for each parameter (the readers are referred to the software's user manual). Note also that the statistical signiﬁcance of variables can be veriﬁed at different levels (e.g., 5%) monitoring Bayesian intervals, which are provided for every parameter of interest. Those estimates that are away from zero are statistically signiﬁcant. The variables that do not satisfy the signiﬁcance criterion should be dropped from the model, and simulations should be run again. The above-mentioned procedures were thoroughly considered in this paper to assure the dependability of the estimates. Note that a sensitivity analysis was also conducted to verify the effect of prior choice on the posterior estimates. Since each data set included hundreds of observations (see Table 1b), non-informative priors have performed adequately. Research studies have shown that the effect of the prior distribution on the posterior density is negligible when a large sample of data is used (Daziano et al., 2013; Heydari et al., 2013). 4.1. Model parameter estimates A wide range of variables were used to build models to determine factors that cause excess in operating speeds with respect to the reference speeds of 40 km/h, 50 km/h, and 80 km/h. The analyses outcomes indicated that contributing factors and their associate posterior estimates vary across the scenarios of reference speeds and data sets (TB, TA, CB, and CA). The posterior mean and credible (Bayesian) intervals related to the parameter estimates are shown in Tables 2 and 3. Table 2 implies that variables one way, lane width, number of lanes, weekend, peak hour, and night hour have an increasing effect on the rate of vehicles exceeding 40 km/h and 50 km/hr. On the other hand, parking and precipitation have a decreasing effect on the choice of drivers in exceeding these reference speeds. Interestingly, the inﬂuence of night hours on the rate of vehicles exceeding 40 km/h is reduced in the after treatment period for both comparison and treated sites. But it remains almost the same for the rate of exceeding 50 km/h. Table 2 also reports the posterior expected rates of exceeding the reference speeds of 40 km/h and 50 km/h. These rates are higher for both treated and comparison sites in the after period for the aforementioned speeds. From Table 3, it can be inferred that variables lane width and one way affect excessive speeding positively while parking, night hour, and evening hour have a decreasing effect on the expected number of vehicles exceeding 80 km/h. Note that these expected numbers are higher in the after period for both comparison and treated sites (Table 3). Tables 2 and 3 also show the estimates of variances for between- and within-site random effects. Using Eq. (2), the intrasite correlation (g ) for each scenario of data and reference speed
Table 4 Speed limit reduction effectiveness estimates (u*) – proposed method. Bayesian intervals Reference speed
2.222 0.974 0.805 3.71 0.58 0.82 1.588 0.452 1.075
0.031 0.015 0.012 0.11 0.02 0.03 0.13 0.04 0.1
2.163 0.946 0.782 3.5 0.54 0.77 1.348 0.378 0.895
2.282 1.003 0.829 3.93 0.61 0.87 1.857 0.536 1.282
p u* 50 km/h
p u* p u*
excessive speeding: u* is not statistically signiﬁcant.
can be estimated. In general, the values of s u2 are higher than the values of s e2 resulting in high g values (see Tables 2 and 3). Recall that values of g close to 1 indicate large intra-site correlations, which in turn conﬁrm the need to accommodate site-speciﬁc effects. Take into account that the estimated values of s 2u are extremely higher compared to s 2e values for treated sites when modeling excessive speeding. This interesting ﬁnding indicates how underlying attributes of sites may vary with respect to various reference speeds. In such cases, a detailed investigation would be required to detect factors causing large between-site variability. By employing the classical methods (e.g., 85th percentile speeds), such ﬁndings might not be possible as the entire distribution of speed is represented by a single point estimate. 4.2. Treatment effectiveness evaluation Table 4 reports estimates related to the assessment of the reduction in speed limits for different scenarios of reference speed. While the index of effectiveness is signiﬁcant with respect to 40 km/h and 50 km/h, the implemented treatment does not reduce excessive speeding cases. Note that excessive speeding is a major risk of severe injuries and fatalities especially in local urban streets because of a higher concentration of vulnerable road users: pedestrians and cyclists. Moreover, the perception of safety among residents might be highly correlated with excessive speeding in residential areas. Therefore, caution is required in making conclusive inferences – based on the realized reduction in speed limits. One should take into consideration that the rate of vehicles exceeding 40 km/h and 50 km/h has surprisingly increased in the after period in all sites including treated sites in which speed limits have been reduced by 10 km/h. Nevertheless, this rate would have been much higher without the application of the speed limit reduction (compare p values in Table 4 with p related to treated sites in the after period in Table 2). This ﬁnding highlights the vital role of comparison sites in conducting before–after studies. By using comparison sites, in fact, the existing underlying trend derived from unknown factors can be revealed. This trend caused an absolute increase in the rate or number of vehicles exceeding reference speeds in the after period. The estimated adjustment factors, AFs, being greater than 1 (Table 4) indicate that further investigation is necessary to ﬁnd the reason for the observed decreasing trend in drivers' compliance with posted speed limits in the after period. In addition to estimating the treatment effectiveness employing the proposed method, the treatment effect was estimated based on a simple before–after study with a comparison group considering the variation in mean and 85th percentile speeds in the before and after periods. The estimated indices of effectiveness obtained from the latter approach were 0.954 CI[0.95, 0.96] and 0.99 CI[0.98,0.995] for mean and 85th percentile speed, respectively (Table 5). The results estimated from a simple before–after study show that there has only been a marginal positive effect on reducing operating speeds. In contrast, the analysis outcomes calculated on the basis of the proposed methodology imply a signiﬁcant improvement (positive effect) in reducing operating speeds with respect to 40 km/h and 50 km/h (Table 4).
Table 5 Speed limit reduction effectiveness estimates (u*) – simple before–after study Reference speed Average 85th
Conﬁdence intervals 2.5%
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4.3. Assessing model ﬁt – posterior predictive distribution We assessed the adequacy of Poisson assumption for modeling excessive speeding incidents using posterior predictive distributions (Gelman et al., 1996). This assessment consists in computing Bayesian p-values on the basis of some discrepancy measures (e.g., chi-squares, deviance, etc.) calculated for both observed and predicted data sets. Bayesian p-values of around 0.5 indicate the suitability of the adopted model for a given data. In this paper, Bayesian p-values based on chi-squares X2 (Eq. (9)) were obtained for all four data sets, and are 0.42, 0.47, 0.52, and 0.47 for CB, CA, TB, and TA, respectively. Therefore, we have not found any evidence indicating lack of ﬁt of the model. The x2 discrepancy between the ^ is given in observations y = (y1,..., yn) and their expectations y Eq. (9). ^Þ ¼ X 2 ðy; y
n X ðyi y^i Þ Varðy^i Þ i¼1
5. Conclusions This paper proposes a methodology to estimate the effects of lowered speed limits on vehicle operating speeds and speeding. The capability to address uncertainty and accommodating more complex and ﬂexible statistical models are some of the advantages of the FB framework, which is used in this paper. The applicability of the employed methodology is illustrated using a sample of local streets from the city of Montreal with observations collected before and after the implementation of a 10 km/h reduction (from 50 km/h to 40 km/h) in speed limits. Through the data collection process using trafﬁc data collection sensors, the number of vehicles was counted for various speed bins (e.g., 30–40, 40–50, etc.). Hence, observations were in the form of discrete data. It should be noted that the use of speed bins allows employing the entire distribution of speed. In contrast, classical methods usually represent the speed distribution at an aggregate level with a point estimate (e.g., 85th percentile speed) that may cause bias in the analysis outcomes. In particulare, this paper investigates the variation in the expected rate or number of vehicles exceeding the predetermined reference speeds of 40 km/h, 50 km/h, and 80 km/h. This way, it was possible to estimate the effectiveness of the implemented reduction in speed limits. For the case of 40 km/h, for instance, the speed bins (e.g., 0–20, 20–30, 30–40, etc.) were dichotomized into two bins: (a) 0–40 and (b) greater than 40. By setting the abovementioned reference speeds, the nature of the data (discretized into multiple bins) shifted into a binomial problem. Two Bayesian generalized mixed linear models – with binomial and Poisson distributions as the random component – were adopted for modeling purposes. The heterogeneity in speed data and intra-site correlations were addressed properly via these ﬂexible Bayesian models. By the aim of the aforementioned models, the effects of various site characteristics, temporal attributes, and environmental exposure on the expected rate or number of vehicles exceeding three different reference speeds were examined. The mixed-effect binomial model was used to estimate the posterior rates of vehicles exceeding 40 km/h and 50 km/h (speeding behaviors). The mixed-effect Poisson model was utilized to estimate the expected number of vehicles exceeding 80 km/h (excessive speeding). For the sample used in this study, the analyses results highlighted that one way streets, lane width, and the number of lanes affect the expected rates or number of vehicles exceeding the reference speeds positively, indicating an increasing effect. On the other hand, the results showed that precipitation and presence of
parking have a decreasing effect. Regarding the rates of exceeding 40 km/h and 50 km/h, factors such as night hour, weekend, and peak hour had an increasing inﬂuence. While, night hour and evening hour were found to have a decreasing effect on the expected number of vehicles exceeding 80 km/h. The estimated variances related to random effects at observational and site levels indicated large intra-site correlations in all cases. Hence, the use of mixed-effect statistical models, in this paper, is justiﬁed. In regard to the treatment efﬁcacy, we found that the reduction in speed limits was effective in reducing the expected probability of exceeding 40 km/h and 50 km/h. However, this countermeasure was not found to be effective with respect to excessive speeding. This ﬁnding suggests that caution must be taken in drawing conclusions regarding the effectiveness of speed limits reduction. This paper thus highlights interesting policy implications and the need to ﬁnd efﬁcient solutions to target excessive speeding behaviors. Obviously, by adopting a classical approach, it was not possible to verify how lowered speed limits affect excessive speeding. It should be mentioned that the analyzed sample in this study does not include all sites from the entire area (in Montreal) subject to the reduction of speed limits. As a result, the estimated indices of effectiveness may not be representative of the entire study area. Recall that the key scope of this research paper was to show the appropriateness of the proposed methodology. We would also like to underline the need to test alternative model frameworks such as time-series analysis, which will be investigated in future research. Acknowledgements The authors would like to thank the city of Montreal (Department of Transportation) for providing the data. The views expressed in this paper are exclusively ours. We would also like to acknowledge four anonymous reviewers for their insightful comments. References Aarts, L., Van Schagen, I., 2006. Driving speed and the risk of road crashes: a review. Accident Anal. Prevent. 38, 215–224. Brooks, S.P., Gelman, A., 1998. Alternative methods for monitoring convergence of iterative simulations. J. Comput. Graph. Stat. 7, 434–455. Cameron, M.H., 2008. Development of Strategies for Best Practice in Speed Enforcement in Western Australia, Supplementary Report. Monash University Accident Research Centre, Melbourne. Carlin, B.P., Louis, T.A., 2009. Bayesian methods for data analysis. Chapman & Hall/ CRC. Taylor & Francis Group, Boca Raton, Florida. Cascetta, E., Punzo, V., 2011. Impact on vehicle speeds and pollutant emissions of an automated section speed enforcement system on the Naples urban motorway. Presented at the TRB 2011 Annual Meeting . Daziano, R.A., Miranda-Moreno, L.F., Heydari, S., 2013. Computational Bayesian statistics in transportation modeling: from road safety analysis to discrete choice. Transport Rev. 33 (5), 570–592. Davis, G.A., 2001. Relating severity of pedestrian injury to impact speed in vehicle pedestrian crashes. Transport. Res. Rec. 108–113. De Pauw, E., Daniels, S., Thierie, M., Brijs, T., 2013. Safety effects of reducing the speed limit from 90 km/h to 70 km/h. Accident Anal. Prev. 54. De Pelsmacker, P., Janssens, W., 2007. The effect of norms, attitudes and habits on speeding behavior: Scale development and model building and estimation. Accident Anal. Prev. 39, 6–15. El-Basyouny, K., Sayed, T., 2012. Measuring safety treatment effects using full Bayes non-linear safety performance intervention functions. Accident Anal. Prev. 45, 152–163. Eluru, N., Chakour, V., Chamberlain, M., Miranda-Moreno, L.F., 2013. A panel mixed ordered Probit fractional split model: Modeling vehicle speed on urban roads in Montreal. Accident Anal. Prev. 59, 125–134. Elvik, R., 2009. The Power Model of the relationship between speed and road safety. Update and new analyses. Norwegian Centre for Transport Research, Oslo: Institute of Transport Economics. Elvik, R., 2013. A before–after study of the effects on safety of environmental speed limits in the city of Oslo, Norway. Safety Sci. 55, 10–16. Fitzpatrick, K., Carlson, P., Brewer, M., Wooldridge, M., Miaou, S., 2003. Design Speed, Operating Speed, and Posted Speed Practices. NCHRP Report 504. Transportation Research Board, Washington DC, USA.
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