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Roundabout Crash Prediction Models

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Abstract and Figures

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, pedestrian 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 visibility have lower crash rates.
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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.
Roundabout crash prediction models
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.