Accident Prediction Modelling Down-under: A Literature Review

Transcription

1 Accident Prediction Modelling Down-under: A Literature Review Presented by: Shane Turner Beca Infrastructure Ltd, 119 Armagh Street, Christchurch, New Zealand, Phone ; Fax shane.turner@beca.com Graham Wood Department of Statistics, Macquarie University, NSW 2109, Australia, Phone: Fax: gwood@efs.mq.edu.au Abstract A large number of accident prediction models have been developed in New Zealand and Australia, particularly the former, for different road elements and for different speed limits. These models provide an insight into accident causing mechanisms, which can in turn assist engineers in diagnosing safety problems. In conjunction with other road safety research (for example, results of `before and after studies), they can also be used to predict the change in accidents that might result from an engineering improvement, whether good or bad. The modelling methods used in New Zealand are based on best practice overseas, from the UK, Canada and the USA, with some local enhancements. An overview of the statistical methods used by Wood and Turner are outlined in this paper. The research to date has produced a number of interesting and thought provoking outcomes including the `safety-in-numbers effect for cyclists and pedestrians, and that reducing visibility can lead to safety gains at roundabouts. Key words: accident prediction models, generalised linear models, speed and visibility Word count: 8255 Words + 7 Figures + 19 Tables 1. Introduction/Background/Objectives Road safety has received considerable attention in both New Zealand and Australia since the mid 1980s when both countries were performing well behind the safest countries in the developed world. In 1987 New Zealand had 795 fatalities, which was 23.8 per 100,000 population. Even in 1995 New Zealand was ranked 23rd out of 29 developed countries. In 1995 the USA was ranked 22nd and Australia had improved to 9th. The poor safety performance in New Zealand was partly attributed to significant volumes of travel on rural two-lane highways, with limited median divided highways. Many organisations and occupations are involved in the effort to reduce the number of road accidents, working across the areas of education, enforcement and engineering. The role of transport engineers and planners within this overall effort is to create as safe a road environment as possible. This is achieved by 1) applying accident remedial measures to unsafe parts of the road network, 2) mitigating the unsafe implications of non-safety related network changes, and 3) the planning and construction of safer new roads and intersections. Since the late 1980s both countries have introduced a number of processes and measures to reduce accident occurrence and trauma. This has included the widespread adoption of road safety strategies and targets (at national/federal, state, regional and local levels), accident reduction studies, road safety audits, emphasis on low cost safety improvements, and more recently, safety management systems. This has contributed to the reduction in the road toll between 1995 and 2005 from 11.2 to 8.1 fatals per 100,000 population in Australia, and from 16.0 to 9.9 fatals per 100,000 population in New Zealand (ranking has improved from 23rd to 16th), despite a significant increase in motor-vehicle travel over this period. In comparison, the road toll in the USA has only reduced from 15.9 to 14.7 fatals per 100,000 population, with the USA now ranked 28 out of 29 countries on this measure. Page 1 The information provided in this publication is for information purposes only. Neither the author nor Beca assumes any duty of care to the reader or gives any representation or warranty in relation to the contents of this publication. You should not rely on this information as professional advice or as a substitute for professional advice. Specialist professional advice should always be sought for your particular circumstances.

2 There has been considerable interest since the late 1980s in quantifying the safety effects of various engineering improvements, to enable the limited funding on road safety to achieve the greatest benefits in trauma and injury accident reduction. In the late 1980s New Zealand set up the Accident Investigation Monitoring Analysis Database, which held data on low cost safety improvement projects throughout the country. In the 1990s this database (LTSA, 1994) was used to calculate reduction rates for various improvement types using five year `before and after accident data. Following on from the work by Maycock and Hall (1984) and Hauer (1989), several New Zealand and Australian researchers also saw the potential for accident prediction models to be used to estimate the safety effects, good or bad, of engineering projects. Starting in the early 1990s a number of research projects were undertaken to produce accident prediction models for various road elements (e.g. intersection or road link), using generalised linear modelling methods. This includes the work by Jackett (1992 and 1993), Wood (1992 and 1993), Arndt (1994 and 1998), Turner (1995) and Nicholson and Turner (1996). These methods, using multiple predictors, enabled the typically non-linear relationship between a number of engineering factors and accidents to be quantified. As with most accident modelling research, care must be applied when using the research results due to correlations between variables and models only being applicable to particular ranges of variables, such as traffic volumes. This paper summarises a number of the studies undertaken down-under, along with the key findings from these studies. Much of this research has not been widely published internationally, but is available from the authors or in the proceedings of various conferences, including the IPENZ transportation group web site ( archives.htm) for papers back to 2001 and from ARRB conference proceedings (contact website is The results of the New Zealand research studies are also available in LTNZ and Transfund research reports listed at govt.nz/research/reorts/index.html (copies of most report are available from the website back to 2005). Many of the models are also available in the New Zealand economic evaluation manual (Appendix A6), which is also available on-line. Modelling Methods The accident modelling process begins by listing the critical variables influencing the accident rate, together with a clear procedure describing how they should be measured. Data is then collected for all such variables. Generalised linear models are then developed using either a negative binomial or Poisson distribution error structure. Generalised linear models were first introduced to road accident studies by Maycock and Hall (1984), and extensively developed by others (e.g., Hauer et al., 1989). These modelling techniques were further developed in the New Zealand context for motor vehicle only accidents by Turner (1995). A typical model describes the relationship between the mean number of accidents and predictors such as traffic volumes and non-flow variables. For two continuous variables (such as flows) and a single discrete variable (with two levels, say the presence or absence of a median strip) the model would take the form: A = b 0 x 1 b 1x2 b 2e ±b 3 where A is the annual mean number of accidents, x 1 and x 2 are average daily flows of vehicles, and b 1, b 2 and b 3 are model coefficients. Software has been developed in Minitab in order to fit such models (that is, to estimate the model coefficients); this can be readily done, however, in many commercial packages, such as GENSTAT, LIMDEP or SAS. In order to illustrate this first stage, consider the situation shown in Figure 1, that of a four-way intersection. Accident prediction models are presented for type HA accidents, where a vehicle travelling through the intersection on the major road (running horizontally) is hit from the right by a vehicle travelling through the intersection along the minor road. Figure 1: Explanatory Flow Variables for Number of Accidents The key variables are: A q 2 q 5 Φ STOP q 2 annual mean number of major road accidents per major road approach minor road through flow to right of major road approach major road through flow q 5 a multiplicative factor used if the approach has a stop control. Page 2

3 In this example, five accident prediction models (CPMs) were fitted, of increasing complexity; these are presented in Table 1. We fit models in stepwise forward fashion. The separate flows are tried first; q 2 was found to produce the higher likelihood. Then additional variable q 5 was found to improve the likelihood more than Φ STOP. Finally both flows with the stop control variable were fitted. Table 2 shows the parameter estimates for each of the five models. A criterion was needed to decide when the addition of a new variable is worthwhile; this balances the inevitable increase in the likelihood L of the data against the addition of a new variable. We have chosen to use the popular Bayesian Information Criterion (BIC). We stop adding variables when the BIC reaches its lowest point. The BIC is given by: BIC = (-2ln(L) + pln(n))/n (where p is the number of variables included in the model and n is the total number of observations in the sample set). The model with the lowest BIC is typically the preferred model form. Addition of a new variable to a model always provides an improved fit, though this may be slight. Table 1: Accident Prediction Equations No. Description Equation (accidents per approach) 1 q 2 only A = b 0 q 2 b 1 2 q 5 only A = b 0 q 5 b 2 3 Conflicting flows only (q 2, q 5 ) A = b 0 q 2 b 1 q5 b 2 4 q 2 and control A = b 0 q 2 b 1 ΦSTOP 5 Conflicting flows and control (q 2, q 5 ) Table 2: Prediction Model Parameters A = b 0 q 2 b 1 q5 b 2 ΦSTOP No. b 0 b 1 b 2 Φ Error Structure BIC NB (k=0.7)* NB (k=0.3)* NB (k=0.9)* NB (k=0.7)* NB (k=0.9)* 1.28 *k is a parameter of the negative binomial (NB) distribution. Table 2 shows that the BIC is lowest for Model 3. Evidently use of two explanatory variables is optimal in this case. The equation for this model is displayed below. It includes vehicle flows for both conflicting flow movements. The exponents of this model indicate that the accident rate increases more quickly with increases in the minor road through traffic volume q 2 (road on which vehicles should give way) than increases in the main road through volume q 5. Goodness of Fit The BIC provides us with a model, but the model may still not fit the data well. The usual methods for testing goodness of fit of generalised linear models involve the scaled deviance G 2 (twice the logarithm of the ratio of the likelihood of the data under the larger model, to that under the smaller model). This test is not valid in our situation, because of the low mean value problem; our models are being fitted to data with very low means. This difficulty was first pointed out by Maycock and Hall (1984). In Wood (2002) a grouping method has been developed which overcomes the low mean value problem. The central idea is that sites are clustered and then aggregate data from the clusters is used to ensure that a grouped scaled deviance follows a chisquare distribution if the model fits well. Evidence of goodness of fit is provided by a p-value. For the scaled deviance a low value of p, of below 0.05, is evidence that the model does not fit well. Middling p-values indicate a satisfactory fit. Software has been written in the form of Minitab macros in order to run this procedure. This test has been carried out for our preferred model above, yielding a p value of Thus we are more secure that our model is a reasonable fit, and can proceed to the next stage, use of the model. Confidence Intervals Two key questions can now be answered: For intersections with given flows, what can be said about the average accident rate? For a new intersection with given flows, what can be said about the number of accidents that will occur? Confidence intervals for both these quantities (the first is always a lot narrower than the second, because it is capturing an average, rather than an individual value) can be produced, using formulae summarised in Wood (2005). Here we illustrate an answer to the first question. As the selected model in our example has two explanatory flow variables, the mean, and lower and upper 95% confidence limits are threedimensional surfaces. To represent these graphically, a two dimensional slice can be taken through these surfaces at median flows for the other variable. Figure 2 shows typical confidence interval slices. A = q q Page 3

4 Figure 2: The 95% confidence intervals for the mean of one flow (at median of other flow) We can now make statements of the type For intersections where the major through flow is 1648 veh/day, and the minor through flow 400 veh/day, a 95% confidence interval for the true mean number of accidents is from 0.07 to 0.12 per annum. Model Interpretation This section outlines the interpretation of model parameters and how these relate predictor variables to accidents. Caution should always be exercised when interpreting relationships as in some multiple predictor variable models two or more variables can be highly correlated. If this occurs then the exponents can be difficult to interpret. In the typical model form (as stated above) the parameter b 0 acts as a constant multiplicative value. If the number of reported injury accidents is not dependent on the values of the two-predictor variables (x 1 and x 2 ), then the model parameters b 1 and b 2 are zero. In this situation the value of b 0 is equal to the mean number of accidents. The value of the parameters b 1 and b 2 indicate the relationship that a particular predictor variable has (over its flow range) with accident occurrence. There are five types of relationship for this model form, as discussed in Table 3. Table 3: Relationship between predictor variable and accident rate Value of Exponent b i > 1 b i = 1 0 < b i < 1 b i = 0 b i < 0 Relationship with accident rate For increasing values of the variable, the number of accidents will increase, at an increasing rate For increasing values of the variable, the number of accidents will increase, at a constant (or linear) rate For increasing values of the variable, the number of accidents will increase, at a decreasing rate There will be no change in the number of accidents with increasing values of the variable For increasing values of the variable, the number of accidents will decrease Generally, accident prediction models of this form have exponents between b i = 0 and b i = 1, with most flow variables having an exponent close to 0.5, i.e. the square root of flow. In some situations, however, parameters have a value outside this range. In the case of models including a covariate (here, discrete variables with a small number of alternatives) a multiplier for different values of the variable is produced, and it is easy to interpret the relationship. This factor indicates how much higher (or lower) the number of accidents is if the feature is present. A factor of 1 indicates no effect on accident occurrence. It is important to note that these relationships apply only to models of the above form (power function models). Other model forms can be tested in the modelling process (for example polynomials, Hoerl s function and combination power and exponential functions). The interpretation of these is often more complex. The relationship between accidents and predictor variables should always be plotted at the outset. A detailed survey of the fitting of generalised linear models to accident data can be found in Wood and Turner (2008). Traffic Signals Models A number of studies have focused on accident prediction models for traffic signals. In addition to flow-only models for all vehicle types in a slow speed urban environment, models have been developed for various accident types, including those involving pedestrians and cyclists and for high speed signals. Models with non-flow variables, including visibility, signal phasing and intersection layout, including one-way streets, have also been developed. The more important studies are outlined in this section. Turner (2000) developed flow-only accident prediction models for signalised cross-roads and T-junctions in New Zealand, for all major accident types. The models were based on a sample set of Page 4

5 109 cross-roads (or 436 approaches) and 30 T-junctions. Figure 3 shows the conflicting movements that were used in the models for each approach. Figure 3: Conflicting and approach flow types (Cross-roads and T-junctions) The large negative parameters in the LB and JA models indicate that intersections with higher flows have fewer accidents. It is speculated that at high traffic flows the installation of right turn bays and exclusive right turn phases reduces accident occurrence. This issue was investigated for right turn against accidents in the study by Turner and Roozenburg (2004). Turner and Roozenburg (2004) considered a number of non-flow variables for right turn accidents (left turn accidents for right hand drive), including visibility, number of opposing lanes and signal phasing type. The flow-only model form was modified to allow non-flow variables to be added. The form of the extended accident prediction model is as follows: A T = b o q 2 b 1 q 7 b 2 c 1 b 3 φ where: The four major all-vehicle accident types at cross-roads and T-junctions, and the corresponding models are shown in Table 4 and 5 respectively. b o q 2 is a model constant is the daily through traffic flow opposing right turning traffic Table 4: Signalised cross-road accident prediction models Accident Crossing (No Turns) Right Turn Against NZ Accident Codes Equation (accidents per approach) HA A = q q LB A = q q Rear-end FA to FE A = Q e 1.07 Loss-of control C & D A = Q e 0.95 Others Rest of codes A = Q e 0.46 Table 5: Signalised T-junction accident prediction models Accident Right Turn Against NZ Accident Codes Equation (accidents per approach) LB A = q q Rear-end FA to FD A = Q e 1.45 Crossing (Vehicle Turning) Loss-of control Others JA A = q q C & D A = Q e 0.17 All other codes A = Q e 0.15 b 1 is a model parameter. It is applied as the power of q 2 q 7 is the daily right turning traffic flow b 2 is a model parameter. It is applied as the power of q 7 c b 3 φ is a continuous non-flow or non-conflicting flow variable such as visibility (V) or intersection depth (I), both measured in metres is a model parameter. It is applied as the power of C is a multiplicative model parameter for a discrete nonflow variable, which can take on one of two values. The φ enters in the format exp (b V), where V is the coded value (always chosen as +1 and -1 ). The discrete model parameter then has the un-logged model value of exp (b 4 code value). In addition to the three important features of visibility, number of opposing lanes and signal phasing type, other variables, such as opposing right turning flow and intersection depth (distance between opposing limit, or hold, lines) were also investigated, but were found to be not significant. Visibility and intersection depth are continuous variables and are added as c variables. Number of opposing lanes and signal phasing are discrete variables and are added as a multiplicative parameter φ. The first variable to be added to the model was visibility V. The model form is: A T = q q V The negative exponent of visibility indicates that the number of accidents would decrease as the visibility increases. Although this model does predict a change in the number of accidents with Page 5

6 varying visibility, an initial data analysis indicated that there was a stronger relationship between the number of accidents and the difference between the observed and recommended visibility (V - V R, as specified in the Austroads guides to Traffic Engineering). The resulting model was as follows: A T = q q (V V R + 100) The second non-flow variable to be included in the accident prediction model was the number of opposing through lanes (φ L ). The resulting model is as follows: Table 6: Signalised cross-road cycle accident prediction equations Accident Same Direction Right Turn Against Motor vehicle turning Equation (accidents per approach) A = Q e 0.29 C e % A = q c % Proportion of Cycle Accidents A T = q q φ L Where φ L = for a single opposing through lane and φ L = for multiple opposing through lanes. The factors for φ L indicate that where the opposing through and right turning traffic volumes are held constant, then the predicted accident rate for an intersection with two or more opposing through lanes would be nearly twice that for the same flows as a single opposing lane. Another non-flow variable that was investigated was that of the signal phasing with the model depending on whether the right turn was a filter turn or a fully or partially protected turn (green arrow) (φ P ). The resulting model is: A T = q q φ P where φ P = for a fully filtered turn and φ P = for fully or partially protected turns. The factors for φ P indicate that the predicted number of accidents is higher, but only just, for approaches with a fully filtered turn, than for approaches where there is at least a partially protected signal phase. The model indicates that a site with a fully filtered turn on average would have an accident rate 10% higher than one with a partially or fully protected turn. Turner et al. (2005) developed accident prediction models for accidents involving pedestrians and cyclists. Pedestrian and cycle accidents are fairly common at most low speed urban traffic signals. Models were developed for the two major cycle versus motor-vehicle accident types at signalised cross-roads. The first, a same direction model, predicts accidents on a single approach between cyclists either colliding with a stationary vehicle or moving motor-vehicle, travelling in the same direction. The second model is for right-turn-against accidents where a cyclist is travelling straight through the intersection and collides with a motor vehicle turning right. These two models, along with the proportion of cycle accidents by type at signalised cross-roads are shown in Table 6. Cycle movements are coded in a similar manner to motor-vehicle movements. Entering flows, for example C e, are the sum of all cycle entering flows, for example c 1 + c 2 + c 3. Figure 4 shows the movements graphically. Figure 4: Cycle model variables The small value of the exponent of cycle flows b 2 in Table 6 indicates a safety in numbers effect where the accident rate per cyclist decreases substantially as the number of cyclists increases. Models were developed for the two major pedestrian accident types at signalised cross-roads. The majority of all accidents involving pedestrians, not just those at signalised cross-roads, occur where vehicles are travelling straight along the road and the pedestrian is crossing. These accidents represent 50% of pedestrian accidents at signalised cross-roads. The second major type of pedestrian accident at signalised cross-roads is where right turning vehicles (left turn for right hand drive) collide with pedestrians crossing the side road. Table 7 present these two models. Figure 5 shows the pedestrian movements used graphically. Table 7: Signalised cross-road pedestrian accident prediction equations Accident Crossing vehicle intersecting Crossing vehicle turning right Equation (accidents per approach) A = Q 0.63 P % Proportion of Cycle Accidents A = q p % Page 6

7 Figure 5: Pedestrian model variables Unlike the cycle models, the safety in numbers effect is not as pronounced, with exponents of flow being similar to those observed for motor vehicle flows. Accident prediction models for high speed signalised cross-roads and T-junctions were developed by Turner (2006) from high speed limit sites in New Zealand and from the State of Victoria, in Australia (i.e. traffic signals in Melbourne). The variable Φ VIC, a multiplication factor, was included in the models to show the differences between New Zealand and Victoria, Australia. The models for high speed cross-roads and T-junctions are shown in Tables 8 and 9. Q major and Q minor are the major (highest volume) and minor roads that intersect respectively. Table 8: High-speed signalised crossroad accident prediction models Accident Equation (accidents per approach) Φ VIC Error Structure Crossing A = q q11 Φ VIC 3.95 Poisson Right-turn-against A = q 2 Φ VIC 1.43 NB (K=0.9)* Rear-end A = Q e Φ VIC 9.91 NB (K=1.7)* Loss of Control A = Q e Φ VIC 1.15 NB (K=5.7)* Others A = Q e Φ VIC 4.44 NB (K=1.7)* *K is the Gamma shape parameter for the negative binomial (NB) distribution. Table 9: High-speed signalised T-junction accident prediction models Accident Equation (accidents per approach) Φ VIC Error Structure Right-turn-against A = q q5 Φ VIC 2.85 NB (K=2.3)* Rear-end (Major Road) A = Q Major Φ VIC 0.89 Poisson Crossing (Vehicle turning) A = q Φ VIC 1.67 Poisson Loss-of-control (Major Road) A = Q Major Φ VIC 1.06 Poisson Other (Major Road) A = Q Major Φ VIC 2.81 Poisson Other (Minor Road) A = Q Major Φ VIC 5.04 NB (K=8.0)* The Victorian factor/variable indicates that for most accident types the number of accidents at similar high-speed traffic signals in Victoria is higher than in New Zealand. It is unclear why the accident risk in Victoria is higher than New Zealand. It could be due to differences in reporting rates (the models provided are for reported injury accidents) or due to different road layout standards, although there is not a lot of difference in roading standards between the countries, or due to different driver behaviour. Page 7

8 Roundabout Models Flow-only accident prediction models for 4-arm roundabouts were first developed in New Zealand by Turner (2000). The models are presented in Table 10. The circulating flow (Q c ) is the traffic to which the entering flow (Q e ) at each roundabout approach must give-way. Table 10: Roundabout Accident Prediction Equations Accident Entering vs Circulating NZ Accident Codes HA, LB, JA, MB, KA &KB Equation (accidents per approach) A = 4.46E-04*Q e 0.42 *Q c 0.45 Rear-end FA to FD A = 2.88E-06*Q e 1.19 Loss-of control C & D A = 1.51E-03*Q e 0.55 Arndt developed linear and non-linear regression models with a Poisson error structure. Models for main six accident types were developed. These were: Single Vehicle Accident Model Rear-End Vehicle Accident Model Entering versus Circulating Vehicle Accident Model Exiting versus Circulating Vehicle Accident Model Sideswipe Vehicle Accident Model Other Vehicle Accident Model. The models developed by Arndt (1994) are shown in Table 11. Figure 6: Vehicle Path Construction through A Double Lane Roundabout Others A = 1.14E-02*Q e 0.26 Arndt (1994) developed accident models in Queensland, Australia, using multiple linear regression methods. Independent variables related to flow, 85th percentile speed, vehicle path radius and changes in 85th percentile speed (as a vehicle progresses through the roundabout). Arndt published two documents, the first published in 1994 (Arndt, 1994) included the first set of models. The second was published in 1998 (Arndt, 1998) and included models for additional accident types and refined to include variables such as the number of approach lanes, the vehicle path radius (curve radius of different elements for vehicles travelling through the roundabout), and the length of each vehicle path (distance travelled by vehicles travelling through the roundabout). Both rural and urban roundabouts were included in the study, with a total sample size of 100 roundabouts; 72% had four arms and 61% had at least one approach with multiple entering and circulating lanes. The 85th percentile speeds through a roundabout were calculated from the theoretical speeds, which were based on curve radii using a modified version of a method to calculate speeds for various curve radii on rural roads. To do this, curve radii through the roundabout from each approach had to be measured. Curve radii were measured assuming a vehicle path that would allow the highest possible speed and therefore largest radii. The process to calculate the approach, circulating and departure curve radii is described in the Road Planning and Design Manual (Department of Main Roads, 2005) and is summarised for roundabouts with multiple lanes in Figure 6. Page 8

9 Table 11: Arndt Roundabout Accident Prediction Equations Accident % accidents Equation (accidents per approach) Single Vehicle prior to give-way line 18% A sp = Q 1.17 L (S + ΔS) 4.12 R Single Vehicle after give-way line A sa = Q 0.91 L (S + ΔS) 1.93 R Rear-end 18% A r = Q e 1.39 Q c 0.65 S a 4.77 N e 2.31 Entering versus circulating 51% A EC = Q e N c Q c S ra t Ga Side-swipe models# 4% A ss = (Q Q t ) 0.72 Δf Others 9% A 0 = ΣQ e # see Table 12 and Figure 5 for the definition of Q and Qt for each road element A list of the variable definitions follows: Q e Q c Q ΣQ e L S ΔS S a N e N c S ra t Ga Δf 1 R Entering flow on the approach Circulating flow perpendicular to the entering flow flow in direction considered (Qe for Asp, Qc for approach to left for Asa) Sum of all flows entering the roundabout length of vehicle path on the horizontal geometric element (length prior to or after the give-way line) 85th percentile speed on the horizontal geometric element (85th percentile speed prior to or after the giveway line) decrease in 85th percentile speed at the start of the horizontal geometric element (decrease in 85th percentile speed prior to or after the give-way line) 85th percentile speed on the approach curve Number of entry lanes on the approach Number of circulating lanes adjacent to an approach The average relative 85th percentile speeds between vehicles on the approach curve and circulating vehicles from each direction (km/h) The average travel time taken from the giveway lines of the preceding approaches to the intersection point between entering and circulating vehicles Difference in potential side friction (km/h2/m), which is defined in Arndt (1994) vehicle path radius on the horizontal geometric element (radius of vehicle path prior to or after the give-way line). The single vehicle accident models show that accidents increase with increased 85th percentile speeds and change in 85th percentile speeds. Interestingly, the models also predict that as radii increase the number of accidents decrease. This clearly contradicts the first finding, as speeds will be directly correlated to radii as would be radii and segment length. The rear-end model shows that an approach with similar flows and speeds but with a single entry lane would have 80% fewer accidents than an approach with two entry lanes. The number of entering versus circulating accidents increases with increasing circulating vehicle lanes, average relative 85th percentile speeds and decreases with increasing average travel times between approaches. The sideswipe model uses a product of the total flow (Q) on the particular geometric element and the flow of a particular movement (Q t ). The flows differ depending on what geometric element approaching and through the roundabout is being considered. Harper and Dunn (2005) details research on the development of accident prediction models for roundabouts, include geometric variables. The models were developed using a dataset of 95 urban roundabouts throughout New Zealand. A number of the roundabouts used in this study were common to the study undertaken by Turner (2000). Harper and Dunn (2005) developed models for individual accident types and product of link accidents using similar accident types to those used by Turner (2000). They found that in most cases the inclusion of geometric variables improved the predictive accuracy of the models. Scaled aerials were used to measure a number of geometric variables. The measurements were taken in respect to each approach. Harper and Dunn (2005) noted that sight distance could not be accurately calculated from aerial photos and therefore excluded this from their analysis. Also, deflection was excluded from the analysis as no apparent standard had been established for defining the deflection path. The geometric characteristics used in the study are illustrated in Figure 7. Page 9

10 Figure 7: Basic Geometric Measurement Definition Plan (from Harper and Dunn, 2005) Harper and Dunn stated that the significance of the ACWL variable seemed to be a strange result and argue that the circulating width at this point constricts all vehicles entering and circulating the roundabout, and therefore has a significant influence on the accident frequency. The parameter of this variable indicates that as ACWL increase so does the number of accidents. Two geometric variables were significant in models for entering versus circulating accidents. Equation 2.3 shows this model. Equation 2.3 A EvsC = (ACDNA )+(EL -0.52) Q e Q c e where, ACDNA Alternative Chord Distance to Next Approach: The distance between the tip of the splitter island of the current approach and that of the next approach in a clockwise direction, based on the average inscribed circle radius of both approaches The general model form used for the (conflicting) flow models is specified as follows: EL Number of Entry Lanes: The number of entry lanes in the current approach. where the variables are: Q e Q c G i b i A = b 0 Q e b 1 Q c b 1 e Gibi Entering flow on the approach Circulating flow perpendicular to the entering flow Geometric variables Model parameters. It was found that the entering versus circulating, rear end, and pedestrian flow-only accident prediction models had similar relationships to flow as those developed in Turner (2000). It was reported that models for loss of control and rear end accidents could not be enhanced by the addition of any of the 28 geometric variables tested. Harper and Dunn (2005) stated that this is not surprising, as the traffic volume variables make many of the geometric variables redundant for the purposes of accident prediction, as a number of the variables were correlated with flow. The model for the total number of accidents included only one non-flow variable. where, ACWL A Total = ACWL Q e Q c e Adjacent Circulating Width Left: The circulating width between the current approach and the next approach in a clockwise direction. Harper and Dunn state that the entering versus circulating model is possibly the most logical, with the number of entry lanes and distance to next approach having strong significance. Their model indicates that the number of accidents of this type decrease with increasing numbers of entry lanes and greater circulating radius. Harper and Dunn also developed models for pedestrian accidents. The model includes all crossing locations, which included some geometric variables and specific ones for crossings with kerb cutdowns only, zebra crossings and signalised crossings. It is unclear from the paper the number of approaches with each facility type. The model indicates that as the distance of the crossing from the intersection increases so does the number of accidents. This may be due to a reduction in inter-visibility between drivers exiting the roundabout and pedestrians crossing at the crossing point. where, PDG A Ped = (PDG 0.058) Q c e Pedestrian Crossing Distance to Give Way Line: The distance from the give way line of the current approach to the closest point of the pedestrian crossing. Table 13 shows the cycle versus motor-vehicle models developed for roundabouts by Turner et al. (2006). The models are based solely on the volume of motor vehicles and cycles using each facility. Again the safety-in-numbers effect is evident by the lower parameter for the cycle volume. Page 10

11 Table 13: Accident prediction models for Cyclists Accidents (Flow-only) Accident Equation (accidents per approach) Error Structure Significant Model Roundabouts Entering versus circulating (HA, LB, KB & KA) A = Q e 0.79 C c 0.32 NB (K=0.8)* Yes *K is the Gamma shape parameter for the negative binomial (NB) distribution. where, A Q e C c Annual number of accidents for an approach or mid-block section Motor vehicle flow entering the intersection for an approach Circulating cycle flow at an approach. Turner et al. (2006) developed roundabout models including a number of non-flow variables. The key variables that were found to be significant included: V 10 S c S LL φ MEL Visibility 10 meters back from the limit/hold line Circulating speed Speed at limit/hold lines Multiple entry lanes. The accident prediction models developed are shown in Table 14. Table 14: Urban roundabout accident prediction models Accident Equation (accidents per approach) Error Structure GOF** Entering-vs-Circulating (Motor-vehicle only) A = Q e Q c S c NB (k=1.3)* 0.26 Rear-end (Motor-vehicle only) A = Q e e 2.42Qe NB (k=0.7)* 0.25 Loss-of-control (Motor-vehicle only) A = Q a V 10 NB (k=3.9)* 0.25 Other (Motor-vehicle only) A = Q a φ Poisson 0.17 MEL φmel = 2.66 Pedestrian A = P 0.60 e 0.67Qa NB (k=1.0)* 0.17 Entering-vs-Circulating (Cyclist circulating) A = Q e C c S LL NB (k=1.2)* 0.61 Other (Cyclist) A = Q a C a Poisson 0.50 All Accidents A = Q a φ NB (k=2.2)* 0.28 MEL φmel = 1.66 *k is the gamma distribution shape parameter for the negative binomial (NB) distribution. **GOF (Goodness Of Fit statistic) indicates the fit of the model to the data. A value of less than 0.05 indicates a poor fit whereas a high value (above 0.5) indicates a very good fit. Page 11

12 Priority Control Intersection Models Models for low speed (urban) priority intersections were also developed by Turner (2000). These accident prediction models are presented in Table 15. At priority cross-roads the straight through flows are differentiated into those with priority (q p ) and those which have to give-way (q g/w ), or stop. For the crossing (no turns) accidents, both the q 2 and q 11 flows are used as predictors, but their order in the equation depends on their priority. Table 15 Priority Cross-road Accident Prediction Equations Accident Crossing (No Turns) Right Turn Against Crossing (Vehicle Turning) Loss-of control NZ Accident Equation (accidents per approach) HA A = 1.95E -3 *q g/w 0.38 *q p 0.37 LB A = 3.75E -3 *q *q JA A = 5.40E -7 *q *q C & D A = 5.22E -3 *Q e 0.30 Others A = 1.87E -3 *Q e 0.57 Table 17 Uncontrolled T-junction Accident Prediction Equations Accident Right Turn Against NZ Accident Equation (accidents per approach) LB A = 1.49E -3 *q *q Rear-end FA to FD A = 8.69E -8 *Q e 1.5 Crossing (Vehicles Turning) Loss-of control JA A = 3.62E -4 *q *q C & D A = 2.51E -3 *Q e 0.31 Others A = 6.27E -3 b 0 *Q e 0.41 The parameters for each of these junction types for the major accident types tend to be close to the square root, except rearend accidents where the exponent is greater than one, indicating that the risk per vehicle increases as traffic volumes increase. The work on low speed priority intersection was extended by Turner (2006) to high speed (typically rural) intersections. The typical mean-annual numbers of reported injury accidents for rural T-junctions can be calculated using turning movement counts and the accident prediction models in Table 18. The accident rates at priority T-junctions are predicted by accident type and approach using the equations in Table 16. Table 16 Priority T-junction Accident Prediction Equations Accident Right Turn Against NZ Accident Codes Equation (accidents per approach) LB A = 3.33E -6 *q *q Rear-end FA to FD A = 1.45E -6 *Q e 1.18 Crossing (Vehicles Turning) Loss-of control JA A = 3.60E -5 *q *q C & D A = 8.22E -3 *Q e 0.30 Others A = 2.49E -3 *Q e 0.51 The accident rates at uncontrolled T-junctions (that is T-junctions that have no give-way, stop or signal controls) are predicted by accident type and approach using the equations in Table 17. Page 12

13 Table 18: Rural priority T-junction accident prediction models Accident Equation (accidents per approach) Error Structure Crossing Vehicle Turning (Major Road approach to left of Minor Road) A = q q (V RD + V LD ) 0.33 NB (K=8.3)* Right Turning and Following Vehicle (Major Road) Other (Major Road approach to right of Minor Road) Other (Major Road approach to left of Minor Road) A = q q S L 11.0 A = (q 5 + q 6 ) 0.91 A = (q 3 + q 4 ) 0.51 NB (K=1.4)* NB (K=1.0)* NB (K=3.0)* Table 18 shows that when the sum of the visibility deficiency to the left and right of the minor road (V RD + V LD ) increases, the number of crossing-vehicle turning accidents also increases. The model for accidents involving vehicles turning right from the major road and vehicles travelling in the same direction is strongly influenced by the approach speed S L. The exponent for this variable is positive, indicating that accidents increase with increased speed. The typical mean-annual numbers of reported injury accidents for rural cross-road intersections can be calculated using the accident prediction models in Table 19. The model for right-turning and following vehicles includes a variable for the presence of a right-turn bay. If a bay is present the prediction is multiplied by the value of this variable (0.22). Table 19: Rural priority crossroad accident prediction models Accident Equation (accidents per approach) Error Structure Crossing (Minor Road vehicle hit from left) A RMXP1 = q q NB (K=0.9)* Crossing (Minor Road vehicle hit from right) Right Turning and Following Vehicle (Major Road) A = q q A = q q Φ RTB NB (K=2.0)* ΦRTB = 0.22 NB (K=2.6)* Other (Major Road) A = (q 4 + q 5 + q 6 ) 0.76 NB (K=1.1)* Other (Minor Road) A = (q 1 + q 2 + q 3 ) 0.27 NB (K=0.2)* Table 19 indicates that in high speed environments the right-turn-bay will reduce the number of right-turning and following vehicle accidents by 78%. Arndt (2004) and Arndt and Troutbeck (2005) also developed accident prediction models for Queensland, Australia, for urban and rural priority cross-road and T-junctions intersections, taking into account a large number of non-flow variables. This study includes a sample set of 63 cross-roads and 143 T-junctions, with main road operating speeds varied from 40kph to 110kph. A large number of non-flow variables were considered. Urban and Rural Mid-block Models A number of models have been developed for urban and rural mid-block section in New Zealand. Space does not permit all these models to be presented, so instead a brief summary of the work undertaken down-under, with suitable reference for those interested in more detail, is provided. Jackett (1992 & 1993) was an early developer of accident prediction models using generalised linear modelling methods in New Zealand. This work included models for urban mid-block sections, based on a sample set of 523 urban routes in Auckland, Wellington and Christchurch. For each route data was collected on accidents, traffic volume and several geometric and environmental factors. Page 13

14 For each route he developed models for mid-block, intersection and total accidents. Many of the mid-block sections passed through intersections. The intersection accidents were removed for the mid-block models, but a factor was still included in the model for number of intersection in each section. Models for urban arterial, collector and local mid-block accidents were also developed by Turner (2000). Models for average injury accident rates were produced based on speed limit, roadside development, and for arterials, the presence of a solid median. Accident models have also been developed for accidents involving cyclists through commercial (shopping) areas by Turner et al. (2006). The models are based solely on the volume of motor vehicles and cycles using each facility and are only for cycle accidents that involved a motor vehicle. Basic flow-only accident prediction models for rural mid-block highway sections were developed by Turner (2000) using generalised linear modelling methods. Because the number of rural highway mid-block accidents tends to be correlated with the terrain type (flat, rolling and mountainous) a terrain factor was included as a covariate in the accident prediction models, with sites being classified as flat or rolling. Insufficient data was available in the sample set for highways in mountainous terrain. Cenek et al. (2004) developed accident prediction models for rural highways using RAMM data available from Transit NZ for the State Highway network. Variables that Cenek investigated include: Traffic volume Road geometry (horizontal curvature, gradient and cross-fall) The preferred model produced for all reported injury and fatal accidents follows: where, A Q T L A = Q T e L Annual number of reported injury and fatal accidents Two-way traffic volume (AADT) Weighted sum of the values of various road characteristics (as defined in the paper). Turner (2004) investigated the impact of roadside hazards on the occurrence and severity of rural single vehicle accidents. A detailed roadside hazard inventory was undertaken of 850 x 1km sections of rural roads. Models were developed for three subsets of accidents: total single-vehicle hazard accidents, fatal and serious hazard accidents, and accidents with a specific hazard involved. As proposed by Turner et al. (2006) a new model is currently being developed in New Zealand for rural roads that will consider all the key variables, including: traffic volume, access density, horizontal geometry, horizontal geometry consistency, seal width, shoulder environment, roadside hazards, region and SCRIM coefficient. Turner (2000) also developed some basic models for motorways (freeways), for the more frequent accident types, and total accidents, based on traffic flow only. The annual reported injury accident rate (per direction) is given by the following equation: Road surface condition (roughness, rut depth, texture depth and skid resistance) Carriageway characteristics (Transit region, urban/rural environment). Cenek et. al. investigated four subsets of road accidents: All reported injury and fatal accidents Selected injury and fatal accidents, covering loss of control events Reported injury and fatal accidents occurring in wet conditions Selected injury and fatal accidents occurring in wet conditions. where, Q O L A T = Q L is the daily single direction traffic volume (AADT) on the link, and is the length of the motorway link. Page 14

15 Discussion/Conclusions/Findings This paper outlines a number of research projects that have focused on developing accident prediction models for various road elements in New Zealand and Australia. Research has been undertaken on most of the key road elements, including intersections and mid-block locations. Some of the key findings from research down-under include. 1. The accident models for pedestrian and cycle accidents indicates that there is a safety-in- numbers effect, so that the accident rate per cyclist and pedestrian reduces when volumes of these modes increase. 2. Loss-of-control accidents at roundabouts can be reduced by reducing the visibility on each approach of the roundabout. 3. Accident rates at rural priority junctions can be improved by reducing approach speeds on the main road and improving visibility. 4. That high speed traffic signals in Melbourne, Australia are generally less safe than similar intersections in New Zealand, despite the state of Victoria having a better safety record than New Zealand overall. 5. The individual accident risk for drivers typically reduces as traffic volumes increase. This is not the case for rear-end accidents with the risk per driver is increasing. Acknowledgement We would like to acknowledge our employers, Beca Infrastructure Ltd and Macquarie University and their support for this area of research over many years. We would also like to acknowledge the New Zealand Transport Agency (formerly Land Transport NZ) for funding many of the research projects outlined in this paper. Shane would like to acknowledge Associate Professor Alan Nicholson who was his PhD supervisor and was instrumental in getting him involved in this area of research. He would also like to acknowledge Aaron Roozenberg who produced a number of the models. Shane and Graham would also like to acknowledge Ezra Hauer and his ground-breaking work in the development of the analysis methods which are the basis for our modelling methods. References Arndt, O.K (1994). Relationship between roundabout geometry and accident rates, Masters of Engineering thesis, School of Civil Engineering, Faculty of Built Environment and Engineering, Queensland University of Technology, Queensland, Australia. Arndt, O.K (1998). Relationship between roundabout geometry and accident rates, Department of Main Roads, Report Number ETD02, Brisbane, Australia. Arndt, O. K. (2004), Relationship Between Unsignalised Intersection Geometry and Accident Rates, Doctor of Philosophy Thesis, Queensland University of Technology and Queensland Department of Main Roads, Brisbane, Australia. Arndt, O.K and Troutbeck, R (2005) Relationship between unsignalised intersection geometry and accident rates, 3rd International Symposium on Highway Geometric Design. Cenek, P.D, Davis, R.B & Henderson, R.J. (2004). The effect of skid resistance and texture on accident risk, Towards Sustainable Transport Conference, Wellington, NZ Department of Main Roads, (2005), Road Planning and Design Manual, Queensland, Australia. Hall, R.D. (1986) Accidents at 4-arm single carriageway urban traffic signals, Transport and Road Research Laboratory Contractor Report CR65, UK. Harper, N.J & Dunn, R.B (2005), Accident Prediction Models at Roundabouts, Proceedings of the 75th ITE Meeting, Melbourne, Australia. Hauer, E., Ng, J.C.N., Lovell, J. (1989). Estimation of safety at signalised intersections. Transportation Research Record 1185: pp Jackett, M (1992) Accident Rates for Urban Routes, IPENZ Conference Proceedings, pp Jackett, M (1993) Accident Rates on Urban Routes 1992 Update, IPENZ Transactions Vol 20 (1) pp Land Transport NZ (now NZTA) (2005) Economic Evaluation Manual: Volume 1 Appendix A6 (Accident Analysis Procedures), NZ Transport Agency, Wellington, NZ. Land Transport Safety Authority (now NZTA) (1994) Accident Investigation Monitoring Analysis, New Zealand Transport Agency guideline, Wellington, NZ. Maycock, G. & Hall, R.D (1984) Accidents at four-arm roundabouts Transport and Road Research Laboratory Report, LR1120, UK. Nicholson, A.J & Turner, S.A (1996) Estimating Accidents in a Road Network, Proceedings ARRB conference, Part 5, pp , Christchurch, NZ. Page 15

16 Persuad, B. & Lyon, C, (2006) Empirical Bayes Before-and-After Safety Studies: Lessons Learned from Two Decades of Experience TRB Annual Meeting, Washington, USA. Turner, S.A (1995) Estimating Accidents in a Road Network PhD Thesis, Department of Civil Engineering, University of Canterbury, NZ. Turner, S.A. & Nicholson, A.J. (1998). Intersection Accident Estimation: The Role of Intersection Location and Non-Collision Flows, Accident Analysis & Prevention, Vol 30, No 4, pp , UK. Wood, G. R. (2002). Generalised Linear Accident Models and Goodness-of-fit Testing. Accident Analysis and Prevention No. 34 pp Wood G.R. (2005) Confidence and prediction intervals for generalised linear accident models, Accident Analysis and Prevention, No. 37, pp Wood G.R and Turner S.A. (2008) Towards a `start-to-finish approach to the fitting of traffic accident models, Transportation Accident Analysis and Prevention, Chapter 11, Ed. Anton De Smet, Nova Publishers, Turner, S.A (2000), Accident Prediction Models, Transfund NZ Research Report No 192, Transfund NZ, Wellington, NZ. Turner, S.A & Roozenburg, A.P (2004), Accident Prediction Models at Signalised Intersections: Right-Turn-Against Accidents, Influence of Non-Flow Variables, Unpublished Road Safety Trust Research Report, Wellington, NZ. Turner, S. A. (2004). Assessing the Accident Risk Implications of Roadside Hazards. Unpublished Land Transport Safety Authority Report, Wellington, New Zealand. Turner, S. A, Durdin P. & Roozenburg A., Prediction Models for Cycle Accidents NZ Cycle Conference, Lower Hutt, New Zealand (2005). Turner, S.A, Wood, G.R & Roozenburg, A (2006), Accident Prediction Models for Roundabouts, 22nd ARRB Conference, Canberra, Australia. Turner, S.A, Wood, G.R and Roozenburg, A.P (2006), Accident Prediction Models for High Speed Intersections, 22nd ARRB Conference, Canberra, Australia. Turner S. and Roozenburg, A.P (2006). Roundabout Accident Prediction Models. Land Transport NZ Research Report (Draft), Standards NZ. Turner, S. A., Roozenburg, A. P., Francis, T. (2006). Predicting accident rates for cyclists and pedestrians, Land Transport New Zealand Research Report 289, NZ. Turner, S. A., Roozenburg, A. P., Tate, F. & Wood, G. R (2006) Rural Accident Prediction Model A New Generation Stage 1 (Scoping) report Unpublished LTNZ Research report, Wellington, NZ. Turner, S.A, Turner, B & Wood, G.R (2008), Accident Prediction Models for Traffic Signals, 23nd ARRB Conference, Adelaide, Australia. Wood G.R. (1992) Development of models relating frequency of collisions to traffic flows, Stage I, Transit New Zealand Research Report, 50pp. Wood G.R. (1993) Development of models relating frequency of collisions to traffic flows, Stage II, Transit New Zealand Research Report, 50pp. Page 16