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Roundabout crash prediction models

June 2009

S A Turner

A P Roozenburg

A W Smith

Beca Infrastructure, PO Box 13-960, Christchurch

NZ Transport Agency research report 386

ISBN 978-0-478-35220-7 (PDF)

ISBN 978-0-478-35221-4 (paperback)

ISSN 1173-3764 (PDF)

ISSN 1173-3756 (paperback)

NZ Transport Agency

Private Bag 6995, Wellington 6141, New Zealand

Telephone 64 4 894 5400; Facsimile 64 4 894 6100

research@nzta.govt.nz

www.nzta.govt.nz

Turner, SA, AP Roozenburg and AW Smith. 2009. Roundabout crash prediction models. NZ Transport

Agency Research Report 386. 102 pp.

This publication is copyright © NZ Transport Agency 2009. Material in it may be reproduced for

personal or in-house use without formal permission or charge, provided suitable acknowledgement is

made to this publication and the NZ Transport Agency as the source. Requests and enquiries about the

reproduction of material in this publication for any other purpose should be made to the Research

Programme Manager, Programmes, Funding and Assessment, National Office, NZ Transport Agency,

Private Bag 6995, Wellington 6141.

Keywords: accident, accident prediction models, crash, cyclists, modelling, New Zealand,

pedestrians, prediction, roundabouts, sight distance, speed, visibility

An important note for the reader

The NZ Transport Agency is a Crown entity established under the Land Transport Management Act

2003. The objective of the Agency is to undertake its functions in a way that contributes to an

affordable, integrated, safe, responsive and sustainable land transport system. Each year, the NZ

Transport Agency funds innovative and relevant research that contributes to this objective.

The views expressed in research reports are the outcomes of the independent research, and should not

be regarded as being the opinion or responsibility of the NZ Transport Agency. The material contained

in the reports should not be construed in any way as policy adopted by the NZ Transport Agency or

indeed any agency of the NZ Government. The reports may, however, be used by NZ Government

agencies as a reference in the development of policy.

While research reports are believed to be correct at the time of their preparation, the NZ Transport

Agency, and agents involved in their preparation and publication, do not accept any liability for use of

the research. People using the research, whether directly or indirectly, should apply and rely on their

own skill and judgement. They should not rely on the contents of the research reports in isolation from

other sources of advice and information. If necessary, they should seek appropriate legal or other

expert advice.

Acknowledgments

Firstly, we would like to thank Professor Graham Wood of Macquarie University in Sydney for his

assistance in resolving the statistical problems associated with crash prediction modelling. Graham’s

ongoing development of the crash prediction modelling process is very much appreciated.

To Professor Bhagwant Persaud of Ryerson University in Canada, we express our gratitude for his

advice throughout this study and his in-depth knowledge of North American and international safety

analysis research.

We would also like to thank the Beca staff that worked on this project and who assisted in data

collection, surveyor coordination, data analysis and project management. We would also like to thank

the project steering group and peer reviewers who provided valuable input into the direction of this

research.

Outside of the team working on this project, we would like to thank Glenn Connelly of Palmerston

North City Council and Bruce Kelly of Christchurch City Council for providing the count data used in

this study.

Abbreviations

AADT: Annual Average Daily Traffic

BIC: Bayesian Information Criterion

CAS: Crash Analysis System

NZTA: NZ Transport Agency

S

E: Entering speed

V

10: Visibility from 10 metres back from the limit line to vehicles turning right or travelling

through the roundabout from their right.

5

Contents

1.Introduction ........................................................................................................................................................................15

1.1 Background...........................................................................................................................15

1.2Objectives ............................................................................................................................. 16

1.3Report structure .................................................................................................................... 17

2.Roundabout crash trends and previous studies ................................................................................................18

2.1General ................................................................................................................................. 18

2.2New Zealand crash data......................................................................................................... 18

2.3Influence of speed, visibility and design ................................................................................22

2.3.1Previous roundabout crash prediction model studies .................................................22

2.3.2Harper and Dunn (2005) – New Zealand .....................................................................22

2.3.3Arndt (1994, 1998) – Australia...................................................................................26

2.3.4Maycock and Hall (1984) – United Kingdom................................................................ 33

2.3.5Brude and Larsson (2000) – Sweden ...........................................................................34

2.3.6Summary of key relationships ....................................................................................35

3.Data collection...................................................................................................................................................................36

3.1Introduction .......................................................................................................................... 36

3.2Site selection.........................................................................................................................36

3.3Selection criteria....................................................................................................................36

3.4Sample size...........................................................................................................................37

3.5Motor vehicle counts .............................................................................................................38

3.6Cyclist counts........................................................................................................................ 39

3.7Pedestrian counts.................................................................................................................. 39

3.8Intersection layout................................................................................................................. 40

3.9Visibility ................................................................................................................................ 41

3.10Roundabout negotiation speed..............................................................................................42

3.11Geometric data...................................................................................................................... 43

3.12Crash data.............................................................................................................................45

4.Data analysis ......................................................................................................................................................................46

4.1Introduction .......................................................................................................................... 46

4.2Correlation among variables..................................................................................................46

Roundabout crash prediction models

6

5.Crash prediction modelling..........................................................................................................................................50

5.1Introduction...........................................................................................................................50

5.2Selecting correct functional form ...........................................................................................50

5.3Fitting crash prediction model parameters.............................................................................53

5.4Adding variables to the models .............................................................................................54

5.5Testing goodness of fit and preferred models .......................................................................56

5.6Model interpretation..............................................................................................................57

5.6.1Determining significance............................................................................................57

5.6.2Power functions..........................................................................................................57

5.6.3Exponential functions.................................................................................................59

5.6.4Covariates ..................................................................................................................60

6.Roundabout crash models............................................................................................................................................61

6.1Introduction...........................................................................................................................61

6.2Entering v circulating (motor-vehicle only).............................................................................62

6.3Rear-end (motor-vehicle only)...............................................................................................64

6.4Loss of control (motor vehicle only) .......................................................................................65

6.5Other (motor-vehicle only) ....................................................................................................67

6.6Pedestrian .............................................................................................................................68

6.7Entering v circulating cyclist ..................................................................................................70

6.8Other cyclist ..........................................................................................................................72

6.9All crashes.............................................................................................................................73

6.10High versus low speed limits .................................................................................................74

6.11Summary ...............................................................................................................................75

7.Speed models......................................................................................................................................................................77

7.1Terminology ..........................................................................................................................77

7.2Methodological considerations ..............................................................................................77

7.2.1Functional form..........................................................................................................77

7.2.2Error structure............................................................................................................77

7.2.3Data grouping............................................................................................................77

7.3Predictive models ..................................................................................................................78

7.4Analysis.................................................................................................................................80

7.5Discussion.............................................................................................................................83

8.Conclusions and recommendations..........................................................................................................................84

8.1Conclusions...........................................................................................................................84

8.2Recommendations.................................................................................................................85

7

9.References ...........................................................................................................................................................................86

Appendices ........................................................................................................................................................................................87

Roundabout crash prediction models

8

9

Executive summary

Roundabouts are a popular choice for intersection control around New Zealand, particularly for

replacing priority controlled intersections where traffic volumes are high and safety has deteriorated.

However, safety problems can occur at poorly designed roundabouts, particularly where speed is not

managed well and where cycle volumes are high.

Despite their generally good record, safety deficient roundabout designs have received considerable

attention from safety auditors over the last 10 or so years. This culminated in the publication of the

guide The ins and outs of roundabouts. This guide lists problems encountered in 50 safety audit

reports. The guide lists visibility and geometric design features, particularly inadequate deflection and

marking, as problem areas. The guide states that ‘the safe and efficient movement of traffic relies on

good unobstructed lines of sight’. The provision of good visibility at roundabouts follows the guidance

in the Austroads Guide to traffic engineering practice part 6: roundabouts. This practice, which occurs

in New Zealand and Australia, differs to practice in other parts of the world, particularly Europe, where

visibility is often restricted to reduce speeds and improve safety. This discrepancy was a major

motivator for this research project.

While roundabouts often have better safety records overall when compared with other forms of control,

they have a poor safety record with respect to cyclist crashes, particularly at large roundabouts with

multiple lanes. This higher cycle crash risk at larger and medium roundabouts is probably caused by

higher motor vehicle speeds, resulting in a larger speed differential between cyclists and motor

vehicles. In addition, the increased complexity of negotiating multi-lane and high-speed roundabouts

could be a reason why some drivers do not see cyclists.

This study, undertaken in 2006, aimed to investigate these issues by focusing on the relationship

between crashes, speed, traffic volume and sight distance for various approach and circulating

movements at roundabouts. This research extends on previous work into flow-only crash prediction

models developed in New Zealand by including key non-flow variables. Given the impact vehicle speed

is expected to have on the ‘active’ modes (walking and cycling) as opposed to the impact on motorised

modes, separate models have been developed for crashes involving these modes. Future research will

examine the impact that geometry has on various crash types.

The research team had access to existing sample sets of roundabouts that were collected in previous

studies into crash prediction models for roundabouts. The majority of the sites in this dataset were in

Christchurch and were single-lane, four-arm intersections. The researchers and steering group wanted

a more geographically diverse dataset that would produce models that could be applied nationally, so a

number of additional roundabouts were added from Auckland and Palmerston North. These

roundabouts had a more diverse range of features, including three-, four- and five-arm junctions, and

single and double circulating and approach lanes.

While a wide variety of roundabout features were included in the sample set, sites that had been

constructed within the last five years or had undergone significant modification during this period were

excluded, as their crash history over the last five years would not be representative. The broader

selection criteria were:

at least five years since installation

all approaches two-way

Roundabout crash prediction models

10

located in one of three centres (Auckland, Christchurch and Palmerston North)

urban speed limits only (70km/h or less).

Data on each of the 104 roundabouts were collected on site. This included:

manual motor vehicle, pedestrian and cyclist counts for each movement

negotiation speeds of free vehicles travelling through the roundabout as they entered and

circulated through the roundabout for each approach

the sight distance between drivers entering the intersection to vehicles approaching from their

right, measured from three locations:

– at the limit line

– 10m back from the limit line

– 40m back from the limit line

diameter

number of lanes for each approach

road markings

super-elevation direction of circulating lanes (whether inwards or outwards)

direction of the gradient of approaches

location of lighting

pedestrian and cycle facilities, where relevant

surrounding land use

features that obstruct the visibility.

Injury crash data associated with each approach for the period from 2001–2005 was obtained from the

Ministry of Transport’s Crash Analysis System database. Where roundabouts had been installed for 10

years, cyclist and pedestrian crash data were obtained for the period 1996–2001.

An additional dataset of crashes and link volumes at 17 high-speed roundabouts with speed limits of

80km/h or more was also collected to investigate the effect of higher speed limits. Given the limited

number of sites that meet these criteria, all high-speed roundabouts for which data were readily

available were included in the sample set.

We first analysed the relationships between key flow and non-flow variables, and explored the

possibility of constructing predictive speed models based on these variables. The models developed

initially did not have significantly high measures of fit, but results using statistical relationships among

variables provided good methodological bases for crash prediction modelling, and will lead to further

work on developing speed models.

11

Generalised linear models were then developed using either a negative binomial or Poisson distribution

error structure, following an analysis of the appropriate functional forms. Using the Bayesian

Information Criterion and grouped goodness of fit methodology, a preferred model for each crash type

was determined. This preferred model has a parsimonious variable set and a good fit to the data, and

comes from a large number of possible models.

Multiplicative factors were also produced for the difference in crash rate for low-speed roundabouts

(70km/h and less) and high-speed (80km/h and more) roundabouts, as shown in the following model

for the total number of crashes per roundabout approach:

HSaAAAR QA

66.04

01021.3

where:

AAAAR0 = annual number of all crashes occurring at an approach

Qa = approach flow (sum of entering and exiting motor vehicle flows)

ФMEL = factor to multiply the crash prediction by if the speed limit on the approach is greater

than 70km/h. This factor is:

HS = 1.35.

This model indicates that roundabouts with speed limits greater than 70km/h have a 35% higher crash

rate than their counterparts in the urban environment.

For urban roundabouts, the most important non-flow variable was found to be vehicle speed. This is

illustrated by the model for entering versus circulating crashes that did not involve cyclists, as shown

below:

13.226.047.0

112. C

ce

-8

UMAR SQQ106A

where:

AUMAR1 = annual number of entering versus circulating crashes involving motor vehicles only

Qe = entering flow on the approach

Qc = circulating flow perpendicular to the entering flow

SC = free mean speed of circulating vehicles as they pass the approach being modelled.

This model illustrates that as the free mean speed of circulating vehicles increases, so does the

number of crashes. The relationship between increasing speeds and increasing crashes is similar for

other crash types, and is supported by international studies of roundabout safety. Another important

variable is the visibility of vehicles approaching from the right, particularly for loss of control type

crashes. Interestingly, this indicates that crashes increase with increasing visibility.

It was found that higher visibility is directly correlated to higher vehicle speeds, indicating that the

increase in crashes may be more to do with higher speeds, which are a result of greater visibility.

Crashes therefore increase because as visibility increases, so does the speed. Another important

finding is that roundabouts with multiple entry lanes have a much higher number of crashes (66%

more) than single-lane roundabouts, even when the increased volume at the former is taken into

account.

The authors recommend that:

Roundabout crash prediction models

12

further research is undertaken to determine how negotiation speed through the roundabout is

affected by roundabout geometry and visibility, and in turn how this influences safety

the models for total roundabout crashes per approach for urban and high-speed roundabouts be

included in the NZ Transport Agency’s Economic evaluation manual vol. 1, replacing the existing

product of link models.

13

Abstract

The management of speed is considered an important safety issue at

roundabouts. The approach speed and negotiating speed through roundabouts

depends on the geometric design of the roundabout and sight distance. In

New Zealand and in Australia, the design standards recommend long approach

sight distances and provision of relatively high design speeds. This is in contrast

to European roundabouts, where visibility is normally restricted and the

geometric design encourages slow approach and negotiation speeds. This work,

undertaken in 2006, extends previous research by the authors developing crash

prediction models at roundabouts to include sight distance, intersection layout

and observed speed variables.

Models have been produced for the major motor vehicles only, pedestrians

versus motor vehicles and cyclists versus motor vehicle crash types. Flow-only

models have also been produced for roundabouts on roads with high speed

limits. The models produced indicate that roundabouts with lower speeds

(observed and speed limit), fewer approach lanes and reduced visibilities have

lower crash rates.

RURAL CRASH PREDICTION MODELS

14

1 Introduction

15

1. Introduction

1.1 Background

Roundabouts are a popular choice for intersection control around New Zealand, particularly to replace

priority controlled intersections where traffic volumes are high and safety has deteriorated. However,

safety problems can occur at poorly designed roundabouts, particularly where speed is not managed

well and where cycle volumes are high.

Safety deficiencies in existing and proposed roundabouts have received considerable attention from

safety auditors over the last 10 years or more. The reoccurrence of common deficiencies in the design

of new roundabouts in New Zealand culminated in the publication of the guide The ins and outs of

roundabouts, which was published by Transfund New Zealand (2000). This guide provides a list of

problems that have been encountered in 50 safety audit reports. Visibility and geometric design

features, particularly inadequate deflection and marking, feature as problems in many of the safety

audit reports. While not specifically mentioned in this report, approach and negotiating speed have the

potential to exacerbate any geometric and other deficiencies present at a roundabout.

Roundabouts, particularly large and two-lane roundabouts, have a poor safety record with respect to

cyclists. This is illustrated in the proportion of injury crashes involving cyclists at roundabouts (25%),

compared with signalised crossroads (8%) and priority crossroads (11%). Many cycle advocates have

strong opinions on this matter and strongly oppose the use of roundabouts, particularly larger

roundabouts, on cycle routes. Two main reasons are given for this increased crash risk to cyclists:

As roundabouts become larger, with more lanes and often higher speeds, they become more

complex to negotiate by motor vehicle drivers, and motorists are less likely to see cyclists because

of the relatively small size of cyclists.

As motor vehicle speeds increase, the relative speed between cyclists and motor vehicles increases

and drivers are more likely to overtake cyclists in an unsafe manner, while cyclists are more likely

to misjudge the gap/space required for various manoeuvres.

It is expected that reduced vehicle speeds and complexity (single-lane circulating) should improve

safety for cyclists.

The research presented in this report, which was carried out in 2006, focuses on the relationship

between crashes, speed, traffic volume and sight distance for various approach and circulating

movements at roundabouts. The flow-only models developed by Turner (2000) are extended in this

study to include observed speed, sight distance and intersection layout variables in various forms.

Given the impact vehicle speed is expected to have on ‘active’ mode (walking and cycling) crashes,

compared with motor vehicle only crashes, separate models have been developed for the major crash

type for each mode.

Roundabout crash prediction models

16

1.2 Objectives

The purpose of this research is to extend the current flow-only motor vehicle crash prediction models

developed by Turner (2000) and the flow-only cyclist crash prediction models developed by Turner et

al (2006) for roundabouts to include design (eg number of through-lanes), visibility, and approach and

negotiation speed variables.

The research objectives are:

to develop crash prediction models for motor vehicle only crashes at roundabouts that

include significant flow and non-flow variables: this may include turning traffic volumes, inter-

vehicle visibility, approach and negotiating vehicle speed, and geometric variables such as

approach alignment, inscribed circle diameter, number of lanes and deflection

to develop crash prediction models for cycle versus motor vehicle crashes at roundabouts

that include significant flow and non-flow variables: this includes turning motor vehicle and

cycle volumes, how visible circulating cyclists are to approaching motorists, approach and

negotiating vehicle speed, and geometric variables such as approach alignment, number of lanes

and inscribed circle diameter

to develop crash prediction models for pedestrian versus motor vehicle crashes at

roundabouts that include significant flow and non-flow variables: this may include turning

motor vehicle flows, crossing pedestrian flows, approach and negotiating vehicle speeds,

pedestrian crossing time, and geometric variables such as pedestrian crossing facilities, number of

lanes and deflection

to provide guidance to traffic engineers (particularly safety auditors) and geometric designs

on the key design elements that influence the safety of motor vehicle occupants, cyclists and

pedestrians at roundabouts, which will enable safety auditors to prioritise design elements that

need to be fixed at existing roundabouts

to address the lack of research on the impact that visibility and negotiating speed have on

crash occurrence at roundabouts. While competent safety auditors have opinions on the influence

of such factors on the safety of road users and identify such factors in their safety audit reports,

they are unable to quantify the effect on safety of each factor.

to explore the possibility of developing predictive models for entering and circulating speeds

based on other key variables, both as an end in itself and also for integration with crash

prediction models where speed data is not available.

1 Introduction

17

1.3 Report structure

This report has been divided into five sections (chapters 2–6), excluding the introduction, conclusions

references and appendices:

Chapter 2 introduces the topic of crash types and user involvement in roundabout crashes in

New Zealand, and reviews other studies that have investigated the effect of visibility and speed on

crash occurrence at roundabouts.

Chapter 3 details the site selection criteria and the data that was collected, while chapter 4

analyses the relationships between speed, visibility and traffic volume obtained from this data.

Chapter 4 also gives an analysis of the relationships among variables that potentially contribute to

crash rates.

Chapter 5 outlines the crash prediction modelling process, the analysis of goodness of fit,

selection of the preferred models and the interpretation of the modelled relationships.

Chapter 6 presents the preferred models for each crash type and other statistically significant

relationships uncovered.

Chapter 7 develops predictive models for speed, based on key variables.

These chapters are followed by conclusions and recommendations, and four appendices:

Appendix A contains all the crash prediction models developed in the study

Appendix B presents the Crash Analysis System (CAS) codes used by the NZ Transport Agency

(NZTA).

Appendix C explains the model subscripts.

Appendix D presents the aerial photos for a selection of the roundabouts studied; these photos

were used to measure the roundabouts’ geometrics (see section 3.11).

Roundabout crash prediction models

18

2. Roundabout crash trends and previous

studies

2.1 General

The first task was to examine the crashes that occur at urban roundabouts and to investigate which

roundabout features should be included in the models as possible predictor variables.

This involved:

examining the involvement of pedestrians and cyclists in crashes at roundabouts

determining the major crash types occurring at roundabouts for the various modes

reviewing other studies on roundabouts to check that we include all important prediction variables

and to check how visibility and speed have been introduced in the crash predictions.

2.2 New Zealand crash data

The Ministry of Transport’s CAS contains details of all crashes reported by the police to the NZTA.

National crash data was extracted from CAS for all urban roundabouts and other forms of intersection

control between 2001 and 2005. Urban intersections have a speed limit of 70km/h or less on all

approaches. Most roundabouts have a 50km/h speed limit on all approaches.

Figure 2.1 shows the location of injury and non-injury combined urban intersection crashes during

2001–2005. This shows that 12% of intersection crashes occur at roundabouts.

Figure 2.1 Intersection control of urban intersection crashes (2001–2005)

The proportion of crashes at each form of intersection control is related to the number of intersections

of each type and the number of crashes occurring at each, which is a function of the form of control

and the traffic volume. The form of control also influences the severity of crashes. Figure 2.2 illustrates

the severity of crashes at each form of control and shows that roundabouts have the lowest severity of

Priority T-

j

unctions

46%

Priority

crossroads

18%

Roundabouts

12%

Signalised T-

j

unctions

7%

Signalised

crossroads

17%

2 Roundabout crash trends and previous studies

19

all intersection types examined. The intersection types with the highest severity are crossroads,

particularly priority crossroads. This is because at crossroads, crashes can occur where vehicles are

travelling perpendicular to each other at speed, resulting in an impact to the side of the vehicle, where

occupants have less protection than when hit at an acute angle or from behind.

Figure2.2 Severity of crashes by form of control (2001–2005)

The roundabout crash data was disaggregated at several levels in order to produce useful statistics for

analysis. The first step in disaggregating the crash data was to categorise reported crashes by severity.

Non-injury crashes are generally excluded from any analysis because of their generally low – and at

times highly variable – reporting rates.

The crash types within the NZTA’s crash coding system (see appendix B) were then analysed. Figure 2.3

shows that the majority of injury and fatal crash types are entering v circulating crashes, followed by

loss of control crashes.

4% 4% 2% 3% 4%

19% 23% 13% 17% 20%

77% 73% 85% 80% 76%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Priority T-

j

unctions Priority

crossroads Roundabouts Signalised T-

j

unctions Signalised

crossroads

Proportion o

f

crashes

Non-injury

Minor

Fatal and serious

Roundabout crash prediction models

20

Figure 2.3 Crash type of injury crashes at urban roundabouts (2000–2004)

Figure 2.4 shows the proportion of each crash type for non-injury crashes. This figure shows that

entering v circulating crashes are less common in non-injury crashes. The proportion of entering v

circulating crashes drops from 51% of all injury crashes to 38% of non-injury crashes, indicating that

this crash type has a higher severity than other crash types. This would be because crashes where the

side of the vehicle is struck are more severe than crashes to the rear and front of the vehicles where

the occupants have more protection.

Figure2.4 Crash type of non-injury crashes at urban roundabouts (2000–2004)

When stating the proportion of crashes involving pedestrians and cyclists at roundabouts, the

proportion of all reported crashes is often stated. This is misleading because of the generally higher

injury severity of crashes involving these modes when compared to motor vehicle only crashes,

meaning that cyclists and pedestrians are involved in a greater proportion of injury crashes. Figure 2.5

shows the relative proportions of injury crashes involving cyclists and pedestrians for different forms of

intersection control. Cyclists are involved in a much greater proportion of injury crashes at

Entering

versus

circulating

51%

Rear-end

11%

Loss o

f

control

19%

Cutting of

f

by moving

right

2%

Cutting of

f

by moving

left

4%

Pedestrian

8%

Others

5%

Entering

versus

circulating

38%

Rear-end

20%

Loss o

f

control

15%

Cutting of

f

by moving

right

9%

Cutting of

f

by moving

left

12%

Pedestrian

0% 6%

Others

2 Roundabout crash trends and previous studies

21

roundabouts compared to other intersection types. The large difference in the proportion of pedestrian

crashes between these two intersection types is likely to be because signalised intersections are

prevalent in areas of high pedestrian demand.

Figure2.5 Pedestrian and cyclist involvement in injury crashes for different intersection control types (2000–

2004)

The majority of pedestrian crashes at roundabouts involved pedestrians crossing perpendicular to the

vehicle direction of travel. Because of the crash coding process, it is not clear nationally whether these

occur on the exit or entering lanes. Of the cycle crashes at roundabouts, 82% of are entering v

circulating crashes, 74% of which occur when the cyclist is circulating and the motor vehicle is entering

(approximately 60% of cycle crashes).

Figure 2.6 shows the frequently listed crash causes for injury crashes at roundabouts. The high

proportion of crashes where a road user failed to give way reflects the high proportion of entering v

circulating crashes.

12% 7% 8%

14% 17%

15%

11%

25% 7%

8%

0%

5%

10%

15%

20%

25%

30%

35%

Priority T-

j

unctions Priority

Crossroads Roundabouts Signalised T-

j

unctions Signalised

crossroads

Proportion o

f

crashes

Cyclists

Pedestrians

Roundabout crash prediction models

22

Figure 2.6 Percent of crashes where a particular cause is reported

2.3 Influence of speed, visibility and design

2.3.1 Previous roundabout crash prediction model studies

A small number of studies internationally have examined the influence of roundabout design on crash

occurrence. The majority of studies on roundabout safety focus on the conversion of priority and signal

controlled intersections to roundabout control.

This section summarises the results of four studies investigating the effect of roundabout design.

These include studies undertaken in New Zealand, Australia, the United Kingdom and Sweden. The

New Zealand and Australian studies are investigated in detail, as designers in both countries generally

follow the design advice in the Austroads Guide for roundabout design (Austroads 1993). The final

section summaries the key variable relationships pertaining to the objectives of this study.

2.3.2 Harper and Dunn (2005) – New Zealand

Harper and Dunn (2005) detail research on the development of crash prediction models for

roundabouts, including geometric variables. Their 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) and this study.

Harper and Dunn (2005) developed models for individual crash types and product of link crashes using

similar crash 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

0%

5%

10%

15%

20%

25%

Failed to

give way Did not see

or look for

another

party until

too late

Alcohol or

drugs Too fast for

conditions Inattentive;

failed to

notice

Wrong

lane/turne

d

from wrong

position

Following

too closely

Percent o

f

crashes

2 Roundabout crash trends and previous studies

23

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 2.7.

Roundabout crash prediction models

24

Figure 2.7 Basic geometric measurement definition plan (from Harper and Dunn 2005)

Notes to figure 2.7:

CW = circulating width

SPLL = splitter island length

SPLW = splitter island width

ACDNA = alternative chord distance to next approach

ICR = inscribed circle radius

ICD = inscribed circle diameter

CID = central island diameter

CIR = central island radius

MCW = median circulating width

O = offset

E = entry width

V = approach half width

Harper and Dunn (2005) outlined the methodology used in developing the models. Models were

developed using generalised linear modelling techniques with Poisson and negative binomial error

structures. It was stated that model accuracy and fit were measured using the 2, R2 and

1- Pr(>|z|) statistical measures. A ‘bottom up’ process was employed to construct the models to avoid

overcomplicating the relationships and to minimise the number of explanatory variables. The model

form used for the conflicting flow models is specified in equation 2.1.

i

b

i

G

b

c

b

eeQQbA 21

0 (Equation 2.1)

where:

A = accidents (crashes) per year

Qe = entering flow on the approach

Qc = circulating flow perpendicular to the entering flow

Gi = geometric variables

bi = model parameters.

2 Roundabout crash trends and previous studies

25

It was found that the entering v circulating, rear-end and pedestrian flow-only crash prediction models

had relationships to flow that were similar to those developed in Turner (2000). It was reported that

models for loss of control and rear-end crashes 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 crash

prediction, as a number of the variables were correlated with flow.

The model for the total number of crashes included only one non-flow variable. Equation 2.2 shows

this model.

057.029.047.04

1031.5 ACWL

ceTotal eQQA (Equation 2.2)

where:

ACWL = adjacent circulating width left: The circulating width between the current approach and

the next approach in a clockwise direction (see ‘CW’ (circulating width) in figure 2.7)

Qe = entering flow on the approach

Qc = circulating flow perpendicular to the entering flow.

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 crash frequency. The parameter of this

variable indicates that as ACWL increases, so does the number of crashes.

Two geometric variables were significant in models for entering v circulating crashes. Equation 2.3

shows this model.

)52.0()057.0(73.059.05

1093.2 ELACDNA

ceEvC eQQA (Equation 2.3)

where:

AEvC entering v circulating accidents per year

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 (see figure 2.7)

EL = number of entry lanes (ie the number of entry lanes in the current approach)

Qe = entering flow on the approach

Qc = circulating flow perpendicular to the entering flow.

Harper and Dunn state that the entering v circulating model is possibly the most logical, with the

number of entry lanes and distance to the next approach having strong significance. Their model

indicates that the number of crashes of this type decrease with increasing numbers of entry lanes and

greater circulating radius.

Harper and Dunn also developed models for pedestrian crashes. Equation 2.4 shows this model. The

model includes all crossing locations, which included some geometric variables and specific ones for

crossings with kerb cut-downs only, zebra crossings and signalised crossings. The number of

approaches with each facility type is not clarified in this paper. It should be noted that the numbers of

Roundabout crash prediction models

26

pedestrians crossing each roundabout approach are not included in the model. The model indicates

that as the distance of the crossing from the intersection increases, so does the number of crashes.

This may be caused by a reduction in intervisibility between drivers exiting the roundabout and

pedestrians crossing at the crossing point, and an increase in vehicle speeds (as drivers accelerate out

of the roundabout).

)058.0(63.04

1010.4 PDG

cPed eQA (Equation 2.4)

where:

APed = pedestrian crashes per year

PDG = pedestrian crossing distance to the give way line: the distance from the give way line of

the current approach to the closest point of the pedestrian crossing

Qc = circulating flow perpendicular to the entering flow.

2.3.3 Arndt (1994, 1998) – Australia

Ardnt developed models using multiple linear regression with independent variables related to flow,

85th percentile speed, vehicle path radius and changes in 85th percentile speed (as a vehicle

progresses through the roundabout) for roundabouts in Queensland, Australia. The first study (Arndt

1994) included the first set of models, while the second (Arndt 1998) included models for additional

crash types, and was refined to include variables such as the number of approach lanes, the vehicle

path radius (the curve radius of different elements for vehicles travelling through the roundabout) and

the length of each vehicle path (distance travelled by vehicles through the roundabout).

Both rural and urban roundabouts were included in the study, with a total sample size of 100

roundabouts. Seventy-two percenthad four arms and 61% had at least one approach with multiple

entering and circulating lanes.

To determine 85th percentile speeds through a roundabout, Arndt calculated theoretical speeds 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

the largest radius. The process of calculating 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 single and multiple lanes in figure 2.8 and figure 2.9 respectively.

2 Roundabout crash trends and previous studies

27

Figure2.8 Vehicle path construction through a single-lane roundabout (Department of Main Roads 2005)

Figure2.9 Vehicle path construction through a double-lane roundabout (Department of Main Roads 2005)

Roundabout crash prediction models

28

Arndt developed linear and non-linear regression models with a Poisson error structure. Models for the

main six crash types were developed. These were:

single vehicle crash model

rear-end vehicle crash model

entering v circulating vehicle crash model

exiting v circulating vehicle crash model

sideswipe vehicle crash model

other vehicle crash model.

Two models for single vehicle crashes are presented. The models do not apply to vehicles turning left.

The models are for crashes prior to (equation 2.5) and after (equation 2.6) the give way line. Eighteen

percent of the 492 crashes in Arndt’s dataset are single vehicle crashes.

A

sp = 1.64×10-12×Q1.17×L×(S+ΔS)4.12×R-1.91 (Equation 2.5)

A

sa = 1.79×10-9×Q0.91×L×(S+ΔS)1.93×R-0.65 (Equation 2.6)

where:

Asp = number of single vehicle crashes per year per approach prior to the give way line

Asa = number of single vehicle crashes per year per approach after the give way line

Q = flow in direction considered (Qe for Asp, Qc for approach to left for Asa)

L = length of vehicle path on the horizontal geometric element (length prior to or after the

give way line)

S = 85th percentile speed on the horizontal geometric element (85th percentile speed prior to

or after the give way line)

ΔS = 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)

R = vehicle path radius on the horizontal geometric element (radius of vehicle path prior to or

after the give way line).

Equations 2.5 and 2.6 indicate that crashes increase with increased 85th percentile speeds and change

in 85th percentile speeds. Interestingly, the models predict that as radii increase, the number of

crashes decreases. This is obviously contradictory with the first finding, as speeds will be directly

correlated to radii, as would radii and segment length.

Eighteen percent of the total crashes in Arndt’s dataset are rear-end crashes that occur when vehicles

approach a roundabout. Equation 2.7 shows the model for this crash type.

2 Roundabout crash trends and previous studies

29

Ar = 1.81×10-18×Qe1.39×Qc0.65×Sa4.77×Ne2.31 (Equation 2.7)

where:

Ar = number of approaching rear-end vehicle crashes per year per approach

Qe = entering flow on the approach

Qc = circulating flow perpendicular to the entering flow

Sa = 85th percentile speed on the approach curve

Ne = number of entry lanes on the approach.

Like the models for single vehicle crashes, higher 85th percentile speeds would result in higher

numbers of crashes per year. The model also indicates that an approach with similar flows and speeds

but with a single entry lane would have 80% fewer crashes than an approach with two entry lanes.

Fifty-one percent of crashes in Arndt’s dataset are entering v circulating crashes, making it the

dominant crash type. Equation 2.8 shows the model for this crash type.

A

EvC = 7.31×10-7×Qe0.47×Nc0.9×Qc0.41×Sra1.38×tGa-0.21 (Equation 2.8)

where:

AEvC = number of entering v circulating vehicle crashes per year per approach

Qe = entering flow on the approach

Qc = circulating flow perpendicular to the entering flow

Nc = number of circulating lanes adjacent to an approach

Sra = the average relative 85th percentile speeds between vehicles on the approach curve and

circulating vehicles from each direction (km/h)

tGa = the average time taken to travel from the give way lines of the preceding approaches to

the intersection point between entering and circulating vehicles.

Equation 2.8 indicates that the number of crashes increases with increasing circulating vehicle lanes

and average relative 85th percentile speeds, and decreases with increasing average travel times

between approaches.

Arndt developed a model for exiting versus circulating crashes on multi-lane roundabouts.

Equation 2.9 outlines the model for this crash type.

Roundabout crash prediction models

30

A

EvX =1.33×10-11× Qc0.32×Qx0.68×Sra4.13 (Equation 2.9)

where:

AEvX = number of entering v exiting vehicle crashes per year per departure approach

Qx = exiting flow on the approach

Qc = circulating flow perpendicular to the departure approach being modelled

Sra = average relative 85th percentile speeds between vehicles on exiting and vehicles

circulating.

Equation 2.9 indicates that crashes will increase with greater relative exiting and circulating speeds.

A model was developed for ‘sideswipe’ crashes on roundabouts with multiple lanes. The model was

applied separately to:

road segments prior to the approach curve and on the approach curve

the circulating through-segment

the circulating right-turn segment

the departing through segment

the departing right-turn segment.

Crashes occurring on these segments contribute 4% (18 crashes) to the total number of crashes. Given

the small number of observed crashes, care should be applied when using such a model.

This model uses a product of the total flow (Q) on the particular geometric elements (see figure 2.10)

and the flow of a particular movement (Qt). The flows differ depending on which geometric element is

being considered; these are outlined in table 2.1. Equation 2.10 outlines the model for this crash type.

2 Roundabout crash trends and previous studies

31

Figure 2.10 Vehicle path segments used for modelling crash rates by Arndt (Department of Main Roads

2005)

Note to figure 2.10: the abbreviations in this diagram relate to computer coding and are not relevant to this report.

Table 2.1 Parameters used for modelling sideswipe traffic flows

Movement Parameter Applicable traffic flow

Approach Q

Qt

Total approaching traffic flow

Total approaching traffic flow

Entering Q

Qt

Total approaching traffic flow

Total approaching traffic flow

Circulating through Q

Qt

Circulating through-traffic flow

Total circulating through-traffic flow

Circulating right-turn Q

Qt

Circulating right traffic flow

Total circulating through-traffic flow

Exiting through Q

Qt

Exiting through-traffic flow

Total departing traffic flow

Exiting right-turn Q

Qt

Exiting right traffic flow

Total departing traffic flow

Roundabout crash prediction models

32

Ass = 6.49 × 10-8 × (Q × Qt) 0.72 × Δf1 10.59 (Equation 2.10)

where:

ASS = number of sideswipe vehicle crashes per approach per vehicle path segment

Δf1 = difference in potential side friction (km/h2/m).

The difference in potential side friction is calculated with equation 2.11.

c

c

Sc

cc

R

S

R

SS

f

127127

2

2

1 (Equation 2.11)

where:

Sc = 85th percentile speed on the horizontal geometric element for the particular movement

for vehicles cutting lanes

ΔSc = decrease in 85th percentile speed at the start of the horizontal geometric element for the

particular movement for vehicles cutting lanes

Rc radius of vehicle path for vehicles cutting lanes

R radius of vehicle path for vehicles not cutting lanes.

For completeness, Arndt developed a model for the crashes types not included in any of the other

model categories. The model is simply the total remaining crashes divided by the total number of

vehicles entering all the roundabouts in one day. Equation 2.12 presents this model.

A

0 = 4.29 × 10-6 × ΣQe (Equation 2.12)

where:

A0 = number of ‘other’ crashes per year

ΣQe = sum of all flows entering the roundabout.

2 Roundabout crash trends and previous studies

33

2.3.4 Maycock and Hall (1984) – United Kingdom

Maycock and Hall studied 84 four-arm roundabouts in the United Kingdom using generalised linear

modelling. Maycock and Hall used traffic flow variables and geometric variables describing the

characteristics of each intersection. They also developed models for pedestrians and used pedestrian

crossing volumes in their models. They found that the traffic flow variables explained a lot more of the

variation in the crash occurrence than the geometric variables, and that, in many cases, the geometric

variables were not statistically significant and could therefore be removed from the models.

Maycock and Hall divided the crashes that occurred at the roundabouts into five crash types which

were associated with each approach of the intersection. The crash types were:

entering v circulating crashes

approaching crashes

single-vehicle crashes

other crashes (all crashes not included in other categories

pedestrian crashes (any crash involving a pedestrian).

Three different types of model were developed with varying levels of complexity. The lowest level of

complexity was the product of links model, which calculated the total number of injury crashes as a

function of vehicle and pedestrian flows. The second level models are similar to the first, but predict

crashes by crash type and use specific turning movements that are conflicting flows. The third and

highest level models are the same as the second but include non-flow variables such as geometry. It is

these third level models which are of primary interest here.

In developing the level 1 models, it was found that the numbers of crashes were higher at roundabouts

with small central islands than roundabouts with ‘normal’ central islands. In general, roundabouts with

higher speed limits on the approaches also had higher crash rates.

For entering v circulating crashes, Maycock and Hall found that crashes increased with increasing entry

width, percentage of motorcycles and increasing uphill gradient on the approach to the roundabout. It

was also found that crashes decreased with increasing angle between the approach and the approach

to the left, and increasing entry path curvature.

Maycock and Hall found that approaching crashes increased with increasing sight distances, decreasing

entry path curvature (higher radius), decreasing entry width and decreasing uphill gradient.

For single-vehicle crashes, the number of crashes increased with increasing approach width,

decreasing entry path curvature and increasing sight distances.

For ‘other’ and pedestrian crashes, no non-flow variables included in the analysis were significant.

Roundabout crash prediction models

34

2.3.5 Brude and Larsson (2000) – Sweden

Brude and Larsson (2000) surveyed 650 of Sweden’s approximately 700 roundabouts and classified

them with respect to geometric design, speed and a number of other factors. Crash data was then

collected as well as the number of vehicles, cyclists and pedestrians passing through the roundabouts

for a number of sites. Three studies were then undertaken into speed at 536 roundabouts, cyclist and

pedestrian safety at 72 roundabouts, and motor vehicle safety at 182 roundabouts.

Speed surveys were conducted by driving through each roundabout and measuring the entering,

circulating and exiting speeds through the roundabouts. A non-linear regression was then carried out

and a speed prediction model was developed. These steps revealed the following:

Speeds are higher when the general speed limit is higher than the local limit.

Speeds were higher on multi-lane roundabouts than on single-lane roundabouts.

Speed is lower if the radius of the central island is 10–20m than if it is smaller or larger.

Flaring the approach to the left reduces speeds into and through the roundabout (Sweden drives on

the right-hand side of the road).

Provision of additional trafficable area around the central island has no effect on speed.

The study investigating pedestrian and cyclist crashes included roundabouts where cyclist volumes

were assessed to be at least 100 cyclists per day. The factor that had the greatest effect on crashes

involving cyclists, apart from cyclist and motor vehicle volume, was the number of lanes. Brude and

Larsson also found that fewer cyclist crashes occurred if the radius of the central island was greater

than 10m. They found that it was safer for cyclists to travel on cycle bypasses than on the roadway.

They found that single-lane roundabouts were much safer for pedestrians than multi-lane

roundabouts.

Brude and Larsson also studied motor vehicle crashes at 182 roundabouts from 1994–1997. Crash

prediction models were developed and made several interesting findings:

The number of crashes is directly proportional to speed.

The number of injuries has approximately a quadratic relationship with speed.

The lower the speed limit, the lower the crash risk and the lower the number of injuries per crash.

Crash and injury rates are higher if the radius is large (>25m) or small (<10m). Brude and Larsson

suggest that roundabouts with large radii result in higher speeds. Where radii are small, vehicles

can travel straight through the roundabout, resulting in higher speeds and more crashes.

2 Roundabout crash trends and previous studies

35

2.3.6 Summary of key relationships

This section summarises the key relationships that pertain to this study.

Number of entry and circulating lanes

Arndt (1998) found that multiple entry lanes increased the number of rear-end crashes, and multiple

circulating lanes increased the number of entering v circulating crashes. This is consistent with Brude and

Larsson (2000), who found that multi-lane roundabouts had higher crash rates for motorists, cyclists and

pedestrians. The only study where the opposite relationship was observed was that of Harper (2005), who

found that approaches with multiple entry lanes had lower entering v circulating crash rates. This seems

contradictory with Harper’s model for total crashes, which indicated that crashes increased with increasing

circulating width for vehicles that travel straight through the intersection.

Vehicle speed

Arndt (1998) used a theoretical relationship with radii of path of travel to determine 85th percentile

speeds. He found that increasing 85th percentile speeds resulted in more crashes for nearly all crash types

and also that a change in 85th percentile speed between geometric elements resulted in more single-

vehicle crashes. Brude and Larsson (2000) found that speed was directly proportional to crashes and that

speeds were higher when the general speed limit was higher than the local limit, where the roundabout

had multiple lanes and where the radius of the central island was 10–20m. They also found that where the

radius of the central island was smaller than 10m, speeds were higher. This is consistent with Maycock and

Hall’s observation (1984) that more crashes occurred at roundabouts with higher speed limits.

Sight distance

The only study to include sight distance as an explanatory variable in the analysis was that of Maycock and

Hall (1984). The variable was found to be significant in the approaching and single-vehicle crash models.

Both of these models indicated that crashes increased with increasing sight distance. This may be because

large sight distances are correlated to higher speeds.

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36

3. Data collection

3.1 Introduction

This section discusses the site selection process; the location and types of roundabouts included in the

sample set; and the collection of motor vehicle, cyclist and pedestrian counts, speed and visibility

measurements, and crash data.

3.2 Site selection

The research team had access to an existing sample set of roundabouts that was collected in two

previous studies by Turner (2000) and Turner et al (2006). The majority of the sites in the latter study

were in Christchurch, and were single-lane four-arm intersections. A number of additional sites were

added from Auckland and New Plymouth to increase the sample size and to include other roundabout

types.

3.3 Selection criteria

A roundabout is made up of a series of ‘give way’ controlled T-junctions, where the through (or

circulating) route is one-way. Roundabouts can be large or small, and can have one or more circulating

and entry lanes. New Zealand has a significantly diverse variety of roundabouts because of changes in

design practices over many years.

The most common roundabout type in New Zealand has four arms and one circulating lane. Previous

studies on roundabouts by Turner (2000) and Turner et al (2006) concentrated on this common

roundabout type. Even this common roundabout type has a lot of variety in terms of central island

diameter, approach design and overall roundabout shape.

The research steering group and research team decided that a broader sample of roundabouts should

be included in this study, so that the effects of speed, visibility and layout on crash occurrence could

be examined. The sample set for this study includes three-, four- and five-arm roundabouts with both

single and dual entering and circulating lanes. As in the population of roundabouts, the sample set has

considerably more roundabouts with single entering and circulating lanes.

While a wide variety of roundabout features were included in the sample set, sites that had been

constructed within the last five years or had undergone significant modification during this period were

excluded, as their crash history over the last five years would not be representative. The broader

selection criteria were:

at least five years since installation

all approaches two-way

located in one of three centres (Auckland, Christchurch and Palmerston North)

urban speed limits only (70km/h or less).

3 Data collection

37

3.4 Sample size

Experience in other studies of this type indicates that a sample set of at least 100 sites is generally

necessary to develop crash prediction models for the major crash types. A large sample size is

particularly important in this study, as it considers a variety of intersection types and uses a lot of non-

flow variables as predictor variables, and because the study develops models for less common modes,

such as cyclists and pedestrians.

In total, a sample set of 104 roundabouts were selected in Auckland, Christchurch and Palmerston

North. Table 3.1 shows a breakdown of the sites by location and roundabout type.

Table 3.1 Roundabout locations and types

Location Type

Christchurch Auckland Palmerston

North Total

Single-lane circulating

three-arm – 2 2 4

four-arm 35 22 8 65

Two-lane circulating

three-arm – 4 – 4

four-arm 4 21 3 28

five-arm – 3 – 3

TOTAL 39 52 13 104

No database lists all the roundabouts in New Zealand, so it was not possible to use a formal sampling

procedure to select a sample of sites that meet the criteria. Instead, the sites were selected so that a

variety of different layouts and sizes were included in the sample from around the country.

A smaller sample set of 17 high-speed roundabouts was also selected from around the country. This

included sites in Christchurch, Auckland, Hamilton and Tauranga. A high-speed roundabout must have

one through-road that has a speed limit of 80km/h or more. Given the limited number of sites that

meet these criteria, all high-speed roundabouts for which data was readily available were included in

the sample set.

Roundabout crash prediction models

38

3.5 Motor vehicle counts

The flow variables used in the urban roundabout intersection models were first defined for four-arm

intersection in Turner (1995). Each vehicle movement is numbered in a clockwise direction starting at

the northernmost approach. Approaches are also numbered using the same technique and are

numbered in a clockwise direction (see figure 3.1).

Figure 3.1 Numbering convention for movements and approaches

Individual movements are denoted as a lower case character for the user type (eg qi). Totals of various

movements are denoted with an upper case character (eg Qi). Models are developed for each approach

and are defined using the totals of various movements. These are:

Qe entering volume for each approach

Qc circulating flow perpendicular to the entering flow

Qa approach flow (the sum of the entering and exiting flows for each approach).

Bruce Kelly of the Christchurch City Council and Glenn Connelly of Palmerston North City Council

provided manual turning movement data for these two cities. In Auckland, turning movement counts

North

6 5

4 3

12

Approach

1

Approach

2

Approach

3

North

6

5

4

3

12

Approach

1

Approach

2

Approach

3

Approach

4

8

9

12

11

10

7

13

24

5

678

20

19

18

17

10 9

111

2

15

14

13

16

North

Approach

1

Approach

2

Approach

3

Approach

4

Approach

5

3 Data collection

39

had to be collected. Three one-hour manual turning volume counts were either provided or collected at

each site, in the morning, evening and at mid-day.

All volume counts were factored up to the annual average daily traffic using the weekly, daily and

hourly correction factors given in the Guide to estimation and monitoring of traffic counting and traffic

growth (Traffic Design Group 2001). The hourly factors were calculated from flow profiles for the

different road types.

For the analysis of high-speed roundabouts, approach volumes (Qa) have been used. The volumes for

the high-speed intersections have been estimated from the link volumes collected through tube

counting programmes.

3.6 Cyclist counts

Manual cyclist movement counts were collected at each site for the morning and evening peaks, and at

mid-day. Like motor vehicle counts, daily and hourly correction factors were used to estimate annual

averaged daily volumes. Seasonal factors were also applied. These took into account the secondary

school terms and holidays. Three separate profiles were used. These were applied based on the

location of the roundabout and the vicinity of schools. The three profiles were ‘commuter’, ‘school/off-

road’ and a combination of both. The commuter profile was always used for dual-lane roundabouts, as

it was not expected that many school cyclists would travel through these. These factors are updated

versions of those found in the Cycle network and route planning guide (Land Transport New Zealand

2004).

The cyclist flow variables are defined by movement in the same way that motor vehicle movements are

defined: they are numbered in a clockwise direction at intersections, starting at the northernmost

approach. Individual cyclist movements are denoted as a lower case character for the user type (eg ci).

Totals of various movements are denoted with an upper case character (eg Ci).

3.7 Pedestrian counts

Manual pedestrian counts were collected at each site in conjunction with cyclist counts. Pedestrians

were counted as they crossed each arm of the intersection. These counts were also factored to average

annual daily flows. Three profiles were used: ‘suburban’, ‘CBD’ and ‘combined’. In most cases, the

‘suburban’ profile was used, with the exception being roundabouts in a commercial area. These factors

were developed using data collected in a previous study (Turner et al 2006). The total crossing volume

for each approach is denoted as an upper case P. The approaches are numbered from the

northernmost approach for consistency with cyclist and motor vehicle movements.

Roundabout crash prediction models

40

3.8 Intersection layout

Data on the layout of each roundabout was collected on site. This included such items as:

road markings

diameter

superelevation direction of circulating lanes (whether inward or outward)

direction of the gradient of approaches

location of lighting

pedestrian and cycle facilities provided

surrounding land use

features that obstruct visibility.

An example of the data collection form is shown in figure 3.2.

Figure 3.2 Example of intersection layout information collected on site (in this case, the Riccarton

Road/Deans Ave roundabout in Christchurch)

From the information collected, variables were developed to represent different roundabout features

where a large number of roundabouts had the feature present. These variables were discrete, unlike

vehicle flows, which are continuous, and were incorporated into the crash prediction models as

covariates. The covariates are represented by multiplicative factors that are used to adjust the

prediction if the feature is present. The covariates used in the modelling process and their definitions

are listed in table 3.2.

3 Data collection

41

Table3.2 Intersection layout covariates incorporated into crash prediction models

Variable Description

ФMEL Multiple entering lanes

ФMCL Multiple circulating lanes

ФTJUN Intersections with three arms

ФGRADD Downhill gradient on approach to intersection

3.9 Visibility

The visibility between drivers entering the intersection to vehicles approaching from their right was

collected on-site for all approaches. The visibility was measured from three locations:

at the limit line

10m back from the limit line

40m back from the limit line.

Table 3.3 contains definitions of these three visibilities and figure 3.3 shows an example for the

measurement of the visibility from 10m back from the limit line.

Table 3.3 Visibility variables used for crash prediction models

Variable Description

VLL Visibility from the limit line to vehicles turning right or travelling through the roundabout

from their right

V10 Visibility from 10m back from the limit line to vehicles turning right or travelling through

the roundabout from their right

V40 Visibility from 40m back from the limit line to vehicles turning right or travelling through

the roundabout from their right

Figure3.3 Measurement of

V

10

(visibility for drivers 10m from limit line to vehicles on their right)

Roundabout crash prediction models

42

3.10 Roundabout negotiation speed

The average free speed of vehicles entering and circulating on all approaches was calculated using

observed data. The entry speeds were the speeds measured as vehicles crossed the limit line and the

circulating speeds were taken from circulating vehicles adjacent to the approach’s splitter island. Only

the free speeds of vehicles travelling straight through (not turning) were collected, as these vehicles are

involved in the major crash type (entering v circulating). Collecting speeds at the two locations (entry

and circulating) provided speed data for each conflicting vehicle stream in the entering v circulating

crash type.

A target of 30 speed observations was collected at each location on each approach using a radar gun.

Only the free speeds of vehicles (where vehicles did not have to give way) were recorded, so that

speeds could be related to the design of the roundabout and not to the traffic conditions at the time of

the survey.

Table 3.4 contains definitions of the speed variables used in the modelling exercise. Figure 3.4

illustrates the location (entry and circulating) where speeds were collected.

Table3.4 Speed variables used for modelling

Variable Description

SE Average free mean speed of entering vehicles travelling through the roundabout at the

limit line

SC Average free mean speed of circulating vehicles travelling through the roundabout as

they pass each approach (adjacent to splitter island)

SSDE Standard deviation of free speeds of entering vehicles at the limit line

SSDC Standard deviation of free speeds of circulating vehicles as they pass the approach being

modelled

Figure3.4 Entering and circulating vehicle speeds

Entering

vehicle speed

(SE)

Circulating

vehicle

speed (SC)

3 Data collection

43

3.11 Geometric data

A Computer Aided Design program was used to measure geometrics for each roundabout from aerial

photographs. Figure 3.5 shows one example of these photographs; a sample of other photographs

used for this study is shown in appendix D. The aerial photographs were obtained from either local

councils or Google Earth.

Figure 3.5 Aerial photo of the Buchanans Road/Carmen Road roundabout in Christchurch with overlaid

measurement lines

Roundabout crash prediction models

44

The aspects of each roundabout that were measured included:

average diameter of central island

difference between the maximum and minimum diameter

entry path radius

exist path radius

circulating path radius

total width of approach traffic lanes

distance to the upstream approach.

The surveyed roundabouts were found to have circular and oval central islands. An average of and the

difference between the maximum and minimum diameters was recorded for oval central islands.

The entry path radius is the radius of an arc that is:

tangent to a line 1.5m offset from and parallel to the approach centreline,

tangent to an arc 1.5m offset from and concentric to the kerb line to the left of an approach

tangent to a circle passing halfway between the central island and splitter islands and concentric to

the central island.

The exit path radius is measured similar to the entry path radius, but for the roundabout exit directly

across from the corresponding approach.

The circulating path radius is the radius of an arc that is tangent to the entry path radius, the exit path

radius, and a circle 1.5m offset from and concentric to the central island.

The total width of the approach traffic lanes was measured and divided by the number of traffic lanes

to find the approach lane width.

The path travelled through roundabout (following the entry, circulating and exit path radii) between the

limit line to the right of the approach and the approach splitter island was measured to find the

distance to the upstream approach.

Figure 3.6 below illustrates the roundabout geometric measurements.

3 Data collection

45

Figure 3.6 Entry, circulating, and exit path radii

3.12 Crash data

Crash data for each roundabout was extracted from the Ministry of Transport’s CAS for 1 January 2001

to 31 December 2005. The sample set crash data was compared with national crash data to assess

whether similar crash trends were evident. During this period, 1202 injury crashes were reported at

urban roundabouts, including 7 fatal and 154 serious crashes (13% of injury crashes). This compares to

365 reported injury crashes, including 2 fatal and 44 serious crashes (13% of injury crashes) at the 104

urban roundabouts included in the sample set.

Models were developed from the major crash types, with the remaining crashes being grouped as

‘other’. The crash types used in the modelling exercise are as follows:

entering v circulating (motor vehicle only)

rear-end (motor vehicle only)

loss of control (motor vehicle only)

other (motor vehicle only)

pedestrian

entering v circulating cyclist

other motor vehicle v cyclist.

Roundabout crash prediction models

46

4. Data analysis

4.1 Introduction

To understand the relationships between crashes and explanatory variables observed in the crash

prediction models, it is necessary to know how these variables are related to each other. This section

analyses the relationships between the key non-flow variable, speed, and the other explanatory

variables in the dataset.

The relationships examined in this section include:

traffic volume and speed

visibility and speed

diameter and speed.

It is important to note that the speed is free speed and not that of all entering vehicles, which would

depend on the traffic volumes at the time of the speed survey, where speeds would be lower in periods

of high traffic flows.

4.2 Correlation among variables

Correlation coefficients between two variables measure the linear dependence between them. Zero

indicates independence; -1 and 1 indicate complete negative and positive dependence respectively.

Coefficients for relevant variables are listed in table 4.1.

Table 4.1 Coefficients for the variables used in the modelling

Variable 1 Variable 2 Correlation

coefficient

Entering volume Entering speed (SE) 0.30

Circulating volume Circulating speed (SC) 0.23

Sight distance – VLL Entering speed (SE) 0.33

Sight distance – V10 Entering speed (SE) 0.40

Sight distance – V40 Entering speed (SE) 0.37

Diameter Entering speed (SE) 0.49

These results show that speed is positively correlated with flow volume, though not strongly. This is

probably a result of roundabouts with high traffic volumes being designed to have higher speeds to

improve capacity. Of the sight distance variables, speed is most strongly correlated with the sight

distance from 10m behind the limit lines. Speed is even more strongly correlated with diameter. Plots

of these relationships are shown in figures 4.1 to 4.6.

4 Data analysis

47

Figure 4.1 Relationship between entering volume and entering speed (SE)

0

10

20

30

40

50

0 5,000 10,000 15,000 20,000 25,000 30,000 35,000

Entering volume (AADT)*

* AADT: Annual Average Daily Traffic

Figure 4.2 Relationship between circulating volume and circulating speed (

S

C)

0

10

20

30

40

50

0 5,000 10,000 15,000 20,000 25,000 30,000 35,0

0

Entering speed (SE) (km/h)

Circulating speed (SC)

(km/h)

Roundabout crash prediction models

48

Figure 4.3 Relationship between sight distance (

V

LL) and entering speed (

S

E)

0

10

20

30

40

50

0 100 200 300

Figure 4.4 Relationship between sight distance (

V

10) and entering speed (

S

E)

0

10

20

30

40

50

0 50 100 150 200 250 30

0

Entering speed (SC) (km/h)

Entering peed (SE) (km/h)

4 Data analysis

49

Figure 4.5 Relationship between sight distance (

V

40) and entering speed (

S

E)

0

10

20

30

40

50

0 50 100 150 200 250 300 3

Figure 4.6 Relationship between diameter and entering speed (

S

E)

0

10

20

30

40

50

01020304050

Entering peed (SE)

(km/h)

Entering speed (SE)

(km/h)

Roundabout crash prediction models

50

5. Crash prediction modelling

5.1 Introduction

The aim of crash prediction modelling in this case is to develop relationships between the flow

variables (mean number of crashes, and traffic, cycle and pedestrian flows) and the non-flow predictor

variables such as visibility and speed. The models are called generalised linear models and typically

have a negative binomial or Poisson error structure. Generalised linear models were first introduced to

modern road crash studies by Maycock and Hall (1984) and extensively developed in Hauer et al

(1989). These models were further developed and fitted using crash data and traffic counts in the

New Zealand context for motor vehicle only crashes by Turner (1995).

Over recent years, the process has been refined to allow for incorporating non-flow variables, which

allow different functional forms, improved goodness of fit statistics and the selection of ‘preferred’

models. This chapter outlines the current modelling process used, which is:

1. selecting the correct functional form for model parameters

2. fitting crash prediction models

3. selecting models for goodness of fit testing

4. testing goodness of fit and selecting preferred models

5. interpreting crash relationships and significance.

5.2 Selecting correct functional form

When crash prediction models were developed for conflicting flow-only variables, only one model was

generally developed for each crash type. The form of the functional form of the crash model was

assumed to be a power function as shown in equation 5.1.

2

2

1

1

0bb xxbA (Equation 5.1)

However, with the inclusion of non-flow variables and the realisation that a power function may not

always be appropriate, a tool was needed to determine potential functional forms for all predictor

variables being included in the model. Also, if the functional form does not match the relationship

between the predictor variable and crashes then the fit of the model is likely to be poor and the model

may be misleading, particularly over certain ranges of the variable. Hauer and Bamfo’s (1997)

integrate-differentiate method is such a tool that assists in identifying possible functional forms.

The integrate-differentiate method has been used in this study with three different functional forms;

these were: power functions (equation 5.2), exponential functions (equation 5.3) and Hoerl’s functions

(equation 5.4).

1

1

0b

xbA (Equation 5.2)

11

0bx

ebA (Equation 5.3)

5 Crash prediction modelling

51

2

11

1

0bxb exbA (Equation 5.4)

where:

A = annual mean number of crashes

x1 = continuous flow or non-flow variable

b0, b1 and b2 = model parameters.

The first step in the integrate-differentiate method consists of determining the empirical integral

function. The method for determining the empirical integral function is calculated as follows (Hauer

and Bamfo 1997):

1. Sort the crash and predictor variable data by the predictor variable of interest, eg the data could be

sorted by flow (Q).

2. Determine the ‘bin width’ of each data point. If this were flow, then it would be the difference in

flow between the next higher and next lower flow divided by two.

3. Calculate the ‘bin area’ by multiplying the bin width by the crash count for each data point.

Continuing the example, the bin area for each approach would be the number of crashes at the

approach multiplied by the bin width.

4. Calculate the sum of all bin areas from the lowest value of the predictor variable up and plot this

against the predictor variable as shown in figure 5.1.

Figure5.1 Example showing the estimate of the empirical integral function

Assuming that a function f(Q) exists for the relationship between the predictor variable (Q) and crashes

(A), the definite integral of f(Q) from Q=0 to Q=x (ie the area under the curve f(Q)) will be the integral

function, F(Q). The summing of the bin areas to determine the empirical integral function is therefore

an estimate of the integral function.

0

200

400

600

800

1000

1200

1400

1600

1800

0 2000 4000 6000 8000 10000 12,000 14,000

AADT (

Q

)

Cumulative 'bin area' (

F(Q

)

)

Roundabout crash prediction models

52

By inspecting the empirical integral function (as shown in figure 5.1), the relationship can be inferred

by comparing it with the graphs in figure 5.2 for (from Hauer and Bamfo 1997). In the case of

figure 5.1, the relationship is unclear.

Figure 2.2 Corresponding functional form (

f(x)

) and integral function (

F(X)

) (from Hauer and Bamfo 1997)

To determine which functional form may be suitable, the empirical integral function can be

transformed. In the case of the power function, this can be done by plotting the natural log of flow

against the natural log of the integral function. Figure 5.3shows this transformation has a linear trend.

This indicates that the power function is the appropriate functional form. If a linear trend is not

observed then the functional form is inappropriate.

5 Crash prediction modelling

53

Figure 5.3 Transformed F(Q) indicating that a power function is the appropriate relationship

5.3 Fitting crash prediction model parameters

Once the functional form for each variable has been determined, generalised linear models can then be

developed using either a negative binomial or Poisson distribution error structure. Generalised linear

models were first introduced to road crash studies by Maycock and Hall (1984) and extensively

developed by others (eg Hauer et al 1989). These modelling techniques were further developed in the

New Zealand context for motor vehicle only crashes by Turner (1995).

Software has been developed in Minitab in order to fit such models (ie to estimate the model

coefficients); this can be readily done, however, in many commercial packages, eg GENSTAT, LIMDEP or

SAS.

9.5 10

ln(

Q

)

ln(

F(Q

)

)

y = 1.7282x - 8.4462

R2= 0.773

0

1

2

3

4

5

6

7

8

9

6 6.5 7 7.5 8 8.5 9

Roundabout crash prediction models

54

5.4 Adding variables to the models

Given the large number of possible variables for inclusion in the models for a particular crash type, a

criterion is needed to decide when the addition of a new variable is worthwhile; this balances the

inevitable increase in the maximum likelihood (ML) of the data against the addition of a new variable

(where p is the number of variables included in the model and n is the total number of observations in

the sample set). We chose to use the popular Bayesian Information Criterion (BIC). We stop adding

variables when the BIC reaches its lowest point. The BIC is given by equation 5.5.

BIC = (-2ln(ML) + pln(n))/n (Equation 5.5)

The model with the lowest BIC is typically the preferred model. Addition of a new variable to a model

generally provides an improved fit, though this may be slight and may therefore not reduce the BIC. In

figure 5.4, the BIC values indicate that the parsimonious number of parameters is two. However, if the

analyst considers that model with three parameters includes an important variable that the model with

two parameters does not then he/she could justifiably select the model with three parameters,

depending on the outcome of goodness of fit testing (see section 5.5).

Figure5.4 Graph used to determine the number of parameters yielding the optimal BIC

Modelling every possible combination of variables to determine which has the lowest BIC would be

time-consuming and inefficient. The process used in this study is to introduce each non-flow variable

to a model with the main flow variables. Many studies have shown that flow variables are generally

more important predictor variables than non-flow variables. The variables that maximise the log-

likelihood (and therefore minimise the BIC) are then added to the flow-only model in a forward

substitution process and the BIC is calculated. This process is repeated for a number of variable

combinations (but not all combinations), taking into account that some variables may be correlated, as

this is fairly common, particularly for layout/design variables.

BIC

1 665

1.670

1.675

1.680

1.685

1.690

1.695

5 Crash prediction modelling

55

Where variables are correlated, the ‘best’ two variables may not result in a better model. The

correlation between different variables can be determined by examining the correlation matrix. The

correlation matrix is a matrix of correlation coefficients between the variables used for modelling.

Correlation coefficients indicate the strength and direction of a linear relationship between two random

variables, where a value of one indicates a perfect positive correlation between two variables and a

value of zero indicates statistical independence. Figure 5.5 illustrates an example of different values of

linear correlation

Figure 5.5 Examples of linear correlation

Roundabout crash prediction models

56

5.5 Testing goodness of fit and preferred models

After the model with the lowest BIC has been obtained, the models are ranked in order of lowest (best)

to highest (worst) BIC. A number of models are then selected for goodness of fit testing, because

although the BIC provides us with models based on a parsimonious variable set and maximum

likelihood, the models may still not fit the data well. Additionally, likelihood and goodness of fit are not

directly related, meaning that the model with the best likelihood or BIC may not be the model with the

best goodness of fit.

The models that are selected for goodness of fit testing are those that have a low BIC and have the

variables that professional knowledge deems necessary. These ‘necessary’ variables are usually limited

to the conflicting flow variables, such as entering and circulating flows in predicting entering v

circulating crashes.

The usual methods for testing goodness of fit for generalised linear models involve using the test

statistics: scaled deviance G2 (twice the logarithm of the ratio of the likelihood of the data under the

larger model to that of the data under the smaller model) or Pearson’s

2 (the sum of squares of the

standardised observations). These statistical tests are not accurate for testing goodness of fit for crash

prediction models, except at an aggregate level (total crashes) at higher flow intersections where crash

rates are relatively light. In most cases, the models are fitted to data with very low crash means, and

this results in the ‘low mean value’ problems. This problem 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

2 distribution if the model fits well. Evidence of

goodness of fit is provided by a p-value. If this value is less than 0.05, say, this is evidence at the 5%

level that the model does not fit well. Software has been written in the form of Minitab macros in order

to run this procedure.

Once the goodness of fit has been calculated for the models selected for testing, the ‘preferred’ model

is identified. This is the model that maximises the goodness of fit.

If the model fits poorly over a certain range of predictor variables (for example high or low volumes),

this can be identified using the grouping technique by plotting predicted crashes against reported

crashes. A poor fit is illustrated by a group that has a different predicted and reported number of

crashes (where the plotted point is furthest from the 45 degree line). The site features of approaches in

any outlier groups can then be examined to determine where the model relationship may not apply.

5 Crash prediction modelling

57

5.6 Model interpretation

5.6.1 Determining significance

Once models have been developed, the relationship between crashes and predictor variables can be

interpreted from the parameter values in most cases. However, caution should always be exercised

when interpreting such relationships when multiple predictor variables are used because two or more

variables can be correlated (see section 5.4). Where variables are correlated or where a variable appears

twice in the model (Hoerl’s function), it is advisable to plot the model to understand the relationship

between the predictor variables and crashes.

When examining the relationships with non-flow variables, it is important to determine whether they

are significant. The significance of the model parameters is determined by examining the 95%

confidence interval for the model parameter to identify if the relationship changes in trend over the

range of the confidence interval. For example, a relationship may be significant if the both the upper

and lower limits of the confidence interval indicate crashes increase with increases in the value of the

predictor variable.

In the following sections, guidance is given on interpreting crash relationships for:

power functions

exponential functions

covariates.

5.6.2 Power functions

Equation 5.6 presents a model with a single variable (such as a flow or speed) with a power function

form. This section examines interpretation of the relationship between crashes and a predictor variable

in a model of this type. The method can also be used to examine a single variable with a power

function form in a multiple variable model.

1

1

0b

xbA (Equation 5.6)

where:

A = annual mean number of crashes

x1 = continuous flow or non-flow variable

b0 and b1 = model parameters.

In this model form, the parameter b0 acts as a constant multiplicative value. If the number of reported

injury crashes is not dependent on the value of predictor variable (x1), then the model parameter b1

would be zero. In this situation, the value of b0 is equal to the mean number of crashes. The value of

the parameter b1 indicates the relationship that the predictor variable has (over its range) with crash

occurrence. Five types of relationship exist for this model form, as presented in figure 5.6 and

discussed in table 5.1.

Roundabout crash prediction models

58

Figure 5.6 Relationship between crashes (

A

) and predictor variable

x

for different values of model exponents

(

b

1)

Table 5.1 Relationship between predictor variable and crash rate

Value of exponent Relationship with crash rate

bi > 1 For increasing values of the variable, the number of crashes will increase at

an increasing rate

bi = 1 For increasing values of the variable, the number of crashes will increase at

a constant (or linear) rate

0 < bi < 1 For increasing values of the variable, the number of crashes will increase at

a decreasing rate

bi = 0 The number of crashes will not change with changes in the predictor

variable

bi < 0 For increasing values of the variable, the number of crashes will decrease

Generally, models of this form have exponents between bi = 0 and bi = 1, with most flow variables

having an exponent close to 0.5, ie the square root of flow. In some situations, however, parameters

have a value outside this range.

0

2

4

6

8

10

12

14

12345

Predictor variable (

x

)

Crashes

(

A

)

b = 1.5

b = 1.0

b = 0.5

b = 0

b = -0.5

5 Crash prediction modelling

59

5.6.3 Exponential functions

Equation 5.7 presents a model with a single variable (such as a flow or speed) with an exponential

function form. As with power functions, the interpretation can also be used to examine a single

variable in a multiple variable model.

11

0bx

ebA (Equation 5.7)

where:

A = annual mean number of crashes;

x1 = continuous flow or non-flow variable; and

b0 and b1 = model parameters.

The value of the parameter b1 indicates the relationship that the predictor variable has (over its range)

with crash occurrence. Three types of relationship can be seen for this model form, as presented in

figure 5.7 and discussed in table 5.2.

Figure 5.7 Relationship between crashes (

A

) and a predictor variable

x

for different values of model

parameter (

b

1)

Table5.2 Relationship between predictor variable and crash rate

Value of parameter Relationship with crash rate

bi >0 For increasing values of the variable, the number of crashes will increase at

a increasing rate

bi = 0 The number of crashes will not change with changes in the predictor

variable

bi < 0 For increasing values of the variable, the number of crashes will decrease at

a decreasing rate

0

2

4

6

8

10

12

14

16

1 2 3 4 5

Predictor variable (

x

)

Crashes

(

A

)

b = 0.5

b = 0

b = -0.5

Roundabout crash prediction models

60

5.6.4 Covariates

In the modeling exercise, covariates are different b0 parameters for various features which, in this

study, are discrete variables with a small number of alternatives such as the number of entry lanes. As

all crash prediction models include multiplicative b0 parameters regardless of the functional form of the

predictor variables (section 5.2), covariates can be applied to all models.

In this report, instead of having multiple b0 values, a b0 value is presented for the most common case

(eg single entry lanes) and a multiplier for other situations (eg multiple entry lanes). This multiplier

factor indicates how much higher (or lower) the number of crashes is for sites with a particular value of

the covariate. For example, for a specific crash type, a covariate analysis may indicate that irrespective

of traffic volume and other key predictor variables, roundabouts with multiple entry lanes have 66%

more crashes than roundabouts with single entry lanes.

6 Roundabout crash models

61

6. Roundabout crash models

6.1 Introduction

The following sections present the crash prediction models developed for the following major crash

types at urban roundabouts:

entering v circulating (motor vehicle only)

rear-end (motor vehicle only)

loss of control (motor vehicle only)

other (motor vehicle only)

pedestrian

entering v circulating cyclist

other cyclist.

A model for all crashes is also presented in section 6.9. This ‘all crash’ model type has been developed

so that it is possible to predict the total number of crashes at a roundabout where only link volumes

are available. We strongly recommend that analysts collect turning volume count data for at least

motor vehicles at roundabouts and that they use the models by crash type, as this will give more

accurate predictions.

Roundabout crash prediction models

62

6.2 Entering v circulating (motor vehicle only)

Models were developed for entering versus circulating crashes involving all motor vehicle classes but

excluding crashes with cyclists. The NZTA crash types included in this dataset are crash codes H, J, K

and L1.

The models were developed in accordance with the process outlined in chapter 5. In this analysis, 22

models were developed for this crash type before setting in a preferred model. Appendix A outlines the

full set of predictor variables and model parameters that were calculated for each of the 22 models.

Equation 6.1 presents the preferred model form, which includes the entering and circulating volumes

and the mean speed of the circulating traffic.

13.226.047.0

112. C

ce

-8

UMAR SQQ106A (Equation 6.1)

where:

AUMAR1 = annual number of entering versus circulating crashes involving motor vehicles only

(subscript denotes model type – see appendix C)

Qe = entering flow on the approach

Qc = circulating flow perpendicular to the entering flow

SC = free mean speed of circulating vehicles as they pass the approach being modelled.

Equation 6.1 implies that the European approach to the design of roundabouts, where circulating

speeds are reduced, has safety benefits. For example, the model suggests that if a mean circulating

speed of 26km/h was reduced by 20% then the resulting reduction in crashes of this type would be

38%. Examination of the correlation matrix indicates that the speed of circulating vehicles is correlated

to the flow of circulating vehicles. This may be a result of roundabouts at higher volumes being

designed for higher speeds.

Equation 6.1 has a p-value of 0.26, indicating a model with good fit (values below 0.05 indicate a poor

model). The goodness of fit can be illustrated by comparing the predicted mean number of crashes and

the reported number of crashes for ‘grouped’ (approaches) data (as outlined in Wood 2002). Figure 6.1

presents this comparison between ‘grouped’ reported and predicted crashes for the preferred model. A

poor fit is illustrated by a group that has a different predicted and reported number of crashes (where

the plotted point is furthest from the 45 degree line). If we find no evidence of poor fit, this gives us

valid grounds for increased confidence in the model. Figure 6.1 indicates a generally good fit for most

approach groups.

1 A copy of the NZTA’s crash type coding matrix is included in appendix B.

6 Roundabout crash models

63

Figure 6.1 Relationship between predicted and reported crashes for the

A

UMAR1

model

A number of other models were developed in the modelling process. Apart from circulating vehicle

speed, the following crash relationships are significant:

presence of multiple entering lanes

entering vehicle speed (SE)

variation in entering vehicle speed

presence of multiple circulating lanes.

The models showed that the number of crashes increased with increasing circulating and entering

vehicle speeds, provision of multiple entering and circulating lanes, and greater variation in vehicle

speeds and increasing visibilities (see appendix A). These results are consistent with those of Arndt

(1998), and Brude and Larsson (2000).

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20

Predicted grouped mean (crashes/five years)

Reported grouped mean (crashes/five years)

Roundabout crash prediction models

64

6.3 Rear-end (motor vehicle only)

Models were developed for rear-end crashes involving motor vehicles only. The NZTA crash types that

are included in this dataset are crash codes FA–FO, GA and GD (see appendix B).

The models were again developed in accordance with the process outlined in chapter 5. Fifteen models

were developed in total. Appendix A outlines the predictor variables and the parameters for each of the

models developed. Equation 6.2 presents the preferred model.

e

Q

e

-2

UMAR eQA 00024.0

38.0

21063.9 (Equation 6.2)

where:

AUMAR2 = annual number of rear-end entering crashes involving motor vehicles only

Qe = entering flow on the approach.

Non-flow variables were included in a number of the crash prediction models developed. However,

these did not feature in the preferred model. Equation 6.2 is also different from the typical power

function crash prediction models developed in previous research studies: it has a Hoerl’s function as its

functional form. Given the functional form, this model should only be applied over the flow ranges for

which data was available. At low and high volumes, the model forms will produce unreliable and

deceptive crash predictions.

Figure 6.2 Relationship between predicted and reported crashes for the

A

UMAR2

model

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20

Predicted grouped mean (crashes/five years)

Reported grouped mean (crashes/five years)

6 Roundabout crash models

65

The model has a p-value of 0.25, indicating a good fit. Figure 6.2 presents the comparison between

the predicted and reported number of crashes for the preferred model. Figure 6.2 indicates a generally

good fit. However, the model appears to underestimate crashes at sites with higher traffic volumes.

Although no non-flow variables were present in the preferred model, a number of models were

developed to include non-flow variables with relationships that are significant. These are:

variation in entering vehicle speed

entering vehicle speed (SE)

presence of multiple entry lanes

visibility measured from 10m back from the limit line (V10).

The models showed that crashes increased with increasing entering volumes, increasing speeds and

variation in speeds, presence of multiple entering lanes and visibilities (see appendix A). The only

reviewed study to investigate visibility (Maycock and Hall 1984) found this to be an important variable

in predicting crashes of this type (approaching) and produced a model that predicts more crashes with

greater visibility.

6.4 Loss of control (motor vehicle only)

Models were developed for loss of control crashes involving motor vehicles entering and exiting the

roundabout. The NZTA crash types that are included in this dataset are crash codes CA–CO, DA–DO, AD

and AF (see appendix B).

Twelve models were developed in total. Appendix A outlines the predictor variables and the parameters

of each of the models developed. Equation 6.3 presents the preferred model, which includes the

approach flow and visibility.

68.0

10

59.0

31036.6 VQA a

-6

UMAR (Equation 6.3)

where:

AUMAR3 = annual number of rear-end entering crashes involving motor vehicles only

Qa = approach flow (sum of entering and exiting flows

V10 = visibility 10m back from the limit line to vehicles turning right or travelling through the

roundabout from the approach to the right.

The model indicates that as traffic volume or visibility increases, the number of loss of control crashes

also increases. The model has a p-value of 0.25, indicating a good fit.

Figure 6.2 presents the comparison between the predicted and reported number of crashes for the

preferred model. Figure 6.3 indicates a generally good fit.

Roundabout crash prediction models

66

Figure 6.3 Relationship between predicted and reported crashes for the

A

UMAR3

model

A number of other models were developed in the modelling process. Apart from visibility (V10), other

significant non-flow relationships are:

visibility measured from the limit line

visibility measures from 40m back from the limit line (V40)

entering vehicle speed.

Like rear-end crashes, where visibility has a significant relationship with crash rates, the models

indicate that as visibility increases, so does the number of crashes. For this same crash type, Maycock

and Hall (1984) found visibility to be an important predictor variable and observed a similar

relationship. The models also show that the number of crashes increased with increasing entering

vehicle speeds.

While the models show that reducing visibility on the roundabout approach (V10) seems to reduce crash

rates, design standards (and drivers) will have a minimum acceptable visibility. This is an area requiring

further research.

0.00

0.20

0.40

0.60

0.80

0.00 0.20 0.40 0.60 0.80

Predicted grouped mean (crashes/five years)

Reported grouped mean (crashes/five years)

6 Roundabout crash models

67

6.5 Other (motor vehicle only)

Eleven models were developed for ‘other’ motor vehicle only crashes at roundabouts. The crash types

include all those not covered by the three previous models that do not involve pedestrians or cyclists.

Appendix A outlines the predictor variables and the parameters of all the models developed for this

crash type. Equation 6.4 presents the preferred model that includes the entering flow and number of

entry lanes.

MELa

-5

UMAR QA

71.0

41034.1 (Equation 6.4)

where:

AUMAR4 annual number of ‘other’ crashes involving motor vehicles only

Qa approach flow (sum of entering and exiting flows)

ФMEL factor to multiply the crash prediction by if multiple entry lanes are present. This factor is

ФMEL.

The model indicates that as traffic volumes increase, the number of crashes also increases. It also

indicates that intersection approaches with multiple entering lanes have an ‘other’ crash rate 2.66

times higher than those with single entry lanes. The model has a p-value of 0.17, indicating a good fit.

Figure 6.4 presents the comparison between the predicted and reported number of crashes for the

preferred model. This indicates a generally good fit.

Figure 6.4 Relationship between predicted and reported crashes for the

A

UMAR4

model

0.00

0.20

0.40

0.60

0.00 0.20 0.40 0.60

Predicted grouped mean (crashes/five years)

Reported grouped mean (crashes/five years)

Roundabout crash prediction models

68

Apart from presence of multiple entering lanes, this crash type had no other significant relationships

between non-flow predictor variables and crashes.

6.6 Pedestrian

Models were developed for all crashes involving pedestrians and motor vehicles. The NZTA crash types

that are included in this dataset are crash codes N and P (see appendix B).

The models were developed in accordance with the process outlined in chapter 5. Sixteen models were

developed in total for this crash type. Appendix A outlines the predictor variables and the parameters

of all the models developed. Equation 6.5 presents the preferred model form, which includes the

approach volume and the number of pedestrians crossing each approach.

a

Q

-4

UPAR ePA 000067.0

60.0

11045.3 (Equation 6.5)

where:

AUPAR1 = annual number of pedestrian crashes;

Qa = approach flow (sum of entering and exiting flows)

P = pedestrians crossing the approach in either direction.

Non-flow variables were included in the crash prediction model development but did not feature in the

preferred model. Equation 6.5 differs from the typical crash prediction models in that it includes both

exponential and power functions (a Hoerl’s function?). Given the functional form, this model should

only be applied over the flow ranges for which data was available. At low and high volumes, the model

form will produce unrealistic and deceptive crash predictions.

The model has a p-value of 0.17, indicating a good fit. Figure 6.5 presents the comparison between

the predicted and reported number of crashes for the preferred model. This indicates a generally good

fit. However, it appears to underestimate crashes at sites with a combination of high pedestrian and

high traffic volumes.

6 Roundabout crash models

69

Figure 6.5 Relationship between predicted and reported crashes for the AUPAR1 model

Although no non-flow variables were present in the preferred model, two variables have significant

relationships. These are:

presence of multiple entry lanes

variation in entering vehicle speed.

The models show that the number of crashes increases with increasing vehicle and pedestrian flows,

the presence of multiple entry lanes and greater variation in entry speeds (see appendix A).

0.00

0.50

1.00

1.50

2.00

2.50

0.00 0.50 1.00 1.50 2.00 2.50

Predicted grouped mean (crashes/five years)

Reported grouped mean (crashes/five years)

Roundabout crash prediction models

70

6.7 Entering v circulating cyclist

Models were developed for entering v circulating crashes involving motor vehicles (entering) and

cyclists (circulating). A much smaller percentage of crashes involve cyclists entering and motorists

circulating. Therefore these crashes are included in the ‘other cyclists’ crash type. The NZTA crash

types that are included in this dataset are crash codes H, J, K and L (see appendix B).

The models were developed in accordance with the process outlined in chapter 5. Twenty-two models

were developed in total. Appendix A outlines the predictor variables and the parameters of the models

developed. Equation 6.6 presents the preferred model form, which includes entering motor vehicle

volumes, circulating cyclist volumes and the mean speed of the entering motor vehicles.

49.038.043.0

188.3 E

ce

-5

UCAR SCQ10A (Equation 6.6)

where:

AUCAR1 = annual number of entering v circulating cyclist crashes

Qe = entering flow on the approach

Cc = circulating cyclist flow perpendicular to the entering motor vehicle flow

SE = free mean speed of vehicles as they enter the roundabout.

Equation 6.6 has a p-value of 0.61, indicating a good fit for the model. Figure 6.6 presents the

comparison between reported and predicted crashes of the preferred model. Figure 6.6 indicates a

generally good fit, except for an outlier with a reported grouped mean of 2.0 and a predicted grouped

mean of 0.73. This outlier comprises of a group of three approaches with high entering motor vehicle

volumes and high cyclist circulating volumes.

6 Roundabout crash models

71

Figure6.6 Relationship between predicted and reported crashes for the

A

UCAR1

model

Apart from entering vehicle speed, other significant relationships between non-flow variables and

crashes are:

presence of a downhill gradient on the approach to the roundabout

circulating vehicle speed.

The models showed that the number of crashes increases with increasing circulating and entering

vehicle speeds, and with the presence of a downhill gradient (see appendix A).

0.00

0.40

0.80

1.20

1.60

2.00

0.00 0.40 0.80 1.20 1.60 2.00

Predicted grouped mean (crashes/five years)

Reported grouped mean (crashes/five years)

Roundabout crash prediction models

72

6.8 Other cyclist

Twelve models were developed for ‘other’ crashes involving cyclists entering and exiting the

roundabout. The crash types that are included in the dataset are those involving both cyclists and

motor vehicle but exclude crashes where the cyclist is circulating and the motor vehicle is entering, as

this is covered by a separate model.

Appendix A outlines the predictor variables and the parameters of all the models. Equation 6.7

presents the preferred model, which includes both the motor vehicle and cyclist approach flows.

23.004.1

207.2 aa

-7

UCAR CQ10A (Equation 6.7)

where:

AUCAR1 = annual number of ‘other’ crashes involving cyclists

Qa = approach flow (sum of entering and exiting motor vehicle flows)

Ca = cyclist approach flow (sum of entering and exiting cyclist flows).

The model indicates that as traffic volumes or cyclist volumes increase, the number of crashes also

increases. The number of crashes is influenced more by an increase in the motor vehicle volume than

an increase in the cyclist volume. Increasing the cyclist volume has a ‘safety in numbers’ effect, where

the per-cyclist crash risk drops. More evidence of this effect can be found in Turner et al (2006).

The preferred model has a p-value of 0.50, indicating a good fit. Figure 6.7 presents the comparison

between the predicted and reported number of crashes for the preferred model.

Figure 6.7 Relationship between predicted and reported crashes for the

A

UCAR2 model

0.00

0.20

0.40

0.60

0.00 0.20 0.40 0.60

Predicted grouped mean (crashes/five years)

Reported grouped mean (crashes/five years)

6 Roundabout crash models

73

No significant relationships were noted between non-flow variables and crashes for this crash type.

However, the relationships observed are similar to the ‘other’ motor vehicle only models while being, at

the same time, different from all other crash types for motor vehicles only, pedestrians and cyclists.

For example, visibility variables in both ‘other’ crash types indicate that as visibility increases, the

number of crashes decreases, while the opposite is true for other crash types. Also, only these ‘other’

crash types have a Poisson error structure, while all other crash types have a negative binomial error

structure, indicating either over-dispersion or else more variability in the data.

6.9 All crashes

Typical crash prediction models for all crashes are normally ‘product of link models’. These models use

two-way link volumes collected by tube counts on the ‘major’ and ‘minor’ roads. Models have been

developed for roundabouts in the past using these ‘major’ and ‘minor’ flows. However, unlike most

traffic signals and priority intersections, ‘major’ and ‘minor’ roads are not easy to define and often, the

main movement may be between two adjacent arms of the intersection. For this reason, models have

been developed on an approach basis, using approach volumes (attainable from link counts), with the

total number of crashes found by adding the crashes occurring on each intersection approach.

The models were developed in accordance with the process outlined in chapter 5. Eleven models were

developed in total. Appendix A outlines the predictor variables and the parameters of the models

developed. Equation 6.8 presents the preferred model form, which includes the approach volume and

the presence of multiple entry lanes.

MELa

-4

UAAR QA

58.0

01011.6 (Equation 6.8)

where:

AUAAR1 = annual number of all crashes occurring at an approach

Qa = approach flow (sum of entering and exiting motor vehicle flows)

ФMEL = factor to multiply the crash prediction by if multiple entry lanes present are. This factor

is:

MEL = 1.66.

This model indicates that approaches with multiple entering lanes will have 66% more crashes than

approaches with single entering lanes. No matter which crash type was being modelled, every time this

variable was included, the covariate was always greater than 1.0. This strong result indicates the

reduced safety of multi-lane roundabouts compared to single-lane roundabouts, irrespective of the

traffic volumes.

Equation 6.8 has a p-value of 0.28, indicating a good fitting model. This fit can be illustrated by

comparing the predicted mean number of crashes and the reported number of crashes, as shown in

figure 6.8. Figure 6.8 differs from previous graphs of this type because the higher number of crashes

results in smaller group sizes and a larger number of groups, using the Wood (2002) method. The

median group size is three and 40% of the groups include only two approaches.

Roundabout crash prediction models

74

Figure 5.1 Relationship between predicted and reported crashes for the

A

UAAR0

model

Other variables were included in the modelling process. Apart from the number of entry lanes, the only

other significant relationship was entering vehicle speed (SE), with the model indicating that as entry

vehicle speed increases, so does the crash rate (see appendix A).

6.10 High versus low speed limits

Using the smaller sample set of 17 high-speed roundabouts (with speed limits on at least two

approaches being greater than 70km/h), the influence of high speed limits was investigated. As this

data consisted only of the approach volume and number of crashes, no non-flow variables could be

examined for this dataset other than the speed limit.

Using the link flow data, a covariate analysis of the effect of higher speed limits on crashes was carried

out. Equation 6.9 presents the model that contains approach flows and approaches with a speed limit

above 70km/h.

HSaAAAR QA

66.04

01021.3 (Equation 6.9)

where:

AAAAR1 = annual number of all crashes occurring at an approach

Qa = approach flow (sum of entering and exiting motor vehicle flows); and

ФHS = factor to multiply the crash prediction by if a speed limit on the approach is greater than

70km/h. This factor is

HS = 1.35.

0

1

2

3

4

5

0 1 2 3 4 5

Predicted grouped mean (crashes/five years)

Reported grouped mean (crashes/five years)

6 Roundabout crash models

75

The model has a good fit, with a p-value of 0.16. The covariate for the higher speed sites indicates that

at speed limits of 80km/h or greater, 35% more injury crashes are reported than at a roundabout with

an urban speed limit, for a given traffic volume.

6.11 Summary

This section summarises the models for each crash type. The typical mean annual numbers of reported

injury crashes at an urban roundabout can be calculated using turning movement counts; data for

various non-flow variables such as visibility, speed and geometry; and the crash prediction models in

table 6.1. The total number of crashes can be predicted by summing the individual predictions for each

crash group which are calculated for each approach. Where turning movement counts and/or non-flow

variable data are unavailable, the total number of crashes can be estimated using the model outlined in

section 6.9. However, we strongly recommend the use of the crash models by type, particularly where

volumes of cyclists and pedestrians are likely to be high.

Table 6.1 Urban roundabout crash prediction models

Crash type Equation (crashes per approach) Error

structure GoFa

Entering v

circulating (motor

vehicle only)

13.226.047.08

11012.6 C

ceUCAR SQQA

NBb

(k=1.3)c 0.26

Rear-end (motor

vehicle only) e

Q

e

-2

UMAR eQA 00024.0

38.0

21063.9 NB

(k=0.7)* 0.25

Loss of control

(motor vehicle

only)

68.0

10

59.0

31036.6 VQA a

-6

UMAR NB

(k=3.9)* 0.25

Other (motor

vehicle only) MELa

-5

UMAR QA

71.0

41034.1

66.2

MEL

Poisson 0.17

Pedestrian a

Q

-4

UPAR ePA 000067.0

60.0

11045.3 NB

(k=1.0)* 0.17

Entering v

circulating cyclist 038.043.05

11088.3 eceUCAR SCQA

NB

(k=1.2)* 0.61

Other (cyclist) 23.004.1

207.2 aa

-7

UCAR CQ10A Poisson 0.50

Notes to table 6.1

a 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 indicates a good fit.

b NB = negative binomial

c k is the gamma distribution shape parameter for the negative binomial distribution.

The models in table 6.1 can be compared with those developed in previous studies to determine

whether crash rates per vehicle have changed or whether the importance of particular variables has

changed for the entering v circulating crash models developed in Turner (2000), Turner et al (2006)

and this study. The ‘flow-only’ models developed for this study are shown in table 6.2 along with the

model for circulating cyclist crashes from Turner et al (2006) and the model for crashes involving all

wheeled road users (cyclists and motor vehicles) in Turner (2000).

Roundabout crash prediction models

76

Table 6.2 Entering-versus-circulating crash prediction models

Flow only models Study Equation (crashes per approach)

Motor vehicle only crashes This study 37.048.0

149.2 ce

-5

UMAR QQ10A

Motor vehicle and cyclist

Crashes Turner 2000 41.042.0

114.1 ce

-4

UWXR QQ10A

Circulating cyclist crashes This study 38.046.0

151.1 ce

-4

UCAR CQ10A

Circulating cyclist crashes Turner et al

2006 32.079.0

140.2 ce

-5

UCXR CQ10A

A comparison between the preferred models (motor vehicle only) in table 6.1 and the flow-only models

in table 6.2 illustrates the effect of the correlation between circulating flow and mean circulating

speed. The lower exponent for the circulating flow (Qc) in table 6.1 (enters v circulating), when

compared with the Qc in the first model in table 6.2, shows the correlation between circulating flow and

circulating speed.

Table 6.2 shows that the relationships between the flow variables and motor vehicle crashes appears in

this current study and the Turner (2000) study. The higher b coefficient for the earlier study

(1.14 x 10-4) compared with this study (2.49 x 10-5) is likely to be the result of a downward trend in

crashes in New Zealand over recent years, and the inclusion of cyclist crashes in the Turner (2000)

study. It is interesting that the models for cyclist crashes have similar exponents on the circulating flow

variable to the models for motor vehicle only crashes. This indicates that similar relationships between

flows and crashes may exist for both road user groups.

7 Speed models

77

7. Speed models

7.1 Terminology

Chapter 4 showed that speed is most strongly correlated with sight distance from 10m behind limit

lines (V10; for the remainder of this chapter, this variable will be denoted as visibility) and diameter. We

will explore speed models where the independent variables consist of these two quantities.

7.2 Methodological considerations

7.2.1 Functional form

In ascertaining the most appropriate functional form for diameter (taking the average of the speed over

the site) and visibility, a power curve produced the best relationship by the methodology outlined

earlier in this report. Therefore we present results considering power relationships.

7.2.2 Error structure

It is not clear which error structure should be assumed in the development of a speed model. The

frequency distribution of speed has a skewness of 0.33 and a kurtosis of 3.16, which are not outside

expected ranges for skewness and kurtosis of Normal datasets of this size (n = 309). Therefore, in the

absence of any indication to the contrary, we have assumed a Normal error structure for speed.

7.2.3 Data grouping

Data exists for all approaches to surveyed roundabouts. Diameter, however, is a property of the

roundabout site, not of the approach; it is therefore necessary to consider a second dataset: the

original set grouped by site.