Time series analysis in road safety
research using state space methods
Time series analysis in r
oad safety r
esear
ch using state space methods
Frits Bijleveld
Time series analysis in road safety
research using state space methods
SWOV–Dissertatiereeks, Leidschendam, Nederland.
In deze reeks is eerder verschenen:
Jolieke Mesken (2006). Determinants and consequences of drivers emotions.
Ragnhild Davidse (2007). Assisting the older driver: Intersection design and in car devices to improve the safety of the older driver.
Maura Houtenbos (2008). Expecting the unexpected. A study of interactive driving behaviour at intersections.
Dit proefschrift is mede tot stand gekomen met steun van de Stichting Wetenschappelijk Onderzoek Verkeersveiligheid SWOV.
Uitgever:
Stichting Wetenschappelijk Onderzoek Verkeersveiligheid SWOV Postbus 1090 2262 AR Leidschendam E: info@swov.nl I: www.swov.nl ISBN: 978-90-73946-04-0 c ° 2008 Frits Bijleveld
Alle rechten zijn voorbehouden. Niets uit deze uitgave mag worden verveel-voudigd, opgeslagen of openbaar gemaakt op welke wijze dan ook zonder voorafgaande schriftelijke toestemming van de auteur.
VRIJE UNIVERSITEIT
Time series analysis in road safety
research using state space methods
ACADEMISCH PROEFSCHRIFT
ter verkrijging van de graad Doctor aan de Vrije Universiteit Amsterdam, op gezag van de rector magnificus
prof.dr. L.M. Bouter, in het openbaar te verdedigen ten overstaan van de promotiecommissie
van de faculteit der Economische Wetenschappen en Bedrijfskunde op dinsdag 4 november 2008 om 15.45 uur
in de aula van de universiteit, De Boelelaan 1105.
door
Frederik Deodaat Bijleveld
promotor: prof.dr. S.J. Koopman
Contents
1. Introduction 9
1.1. The main ideas of this research 9
1.2. Important issues in time series analysis of road safety data 12
1.2.1. Time dependence 12
1.2.2. Multiple road safety outcomes 16
1.2.3. Exposure data 17
1.2.4. Explanatory variables 19
1.2.5. Conclusions 23
1.3. Structure of this thesis 24
2. Safety, exposure and risk 28
2.1. Introduction 28
2.2. Risk exposure in road safety analysis 30
2.2.1. Statistical distributions 31
2.2.2. The distribution of accident counts 33
2.2.3. Over-dispersion 33
2.2.4. Gaussian approximations 34
2.2.5. The distribution of victim counts 34 2.2.6. The relation between trials and exposure 35 2.3. Traffic volume and accident occurrence 36
2.3.1. The relation between ‘traffic volume’ and the number of
accidents 36
2.3.2. A remark on traffic volume and multiparty accident
oc-currence 38
2.4. Summary and discussion 38
3. Multivariate structural time series models 41
3.1. Introduction 41
3.2. The concept of state and its observation 42 3.3. The latent risk time series model 45 3.3.1. A basic latent risk observation model 45 3.3.2. The role of the dynamic relation among states 47 3.3.3. Specification by means of linear structural models 54 3.3.4. Linear measurement equations 60 3.3.5. General state space model specification 62 3.3.6. Estimation of parameters and latent factors, missing data 62 3.3.7. Kalman smoother, auxiliary residuals 64
3.4. Applications 65 3.4.1. State space DRAG-similar models 65 3.4.2. Estimating the registration level of accidents involving
hospitalised victims 71
3.5. Non linear extensions 77
3.5.1. Introduction 77
3.5.2. Mixing additive and multiplicative models 78
3.5.3. Further generalisations 79
4. The covariance between the number of accidents and victims 80
4.1. Introduction 80
4.1.1. The need for multivariate modelling of influences on road
safety 80
4.1.2. The issue of dependence among outcomes 81
4.1.3. An approximating solution 82
4.1.4. Overview of the paper 84
4.2. The covariance structure of road safety related
outcomes 85
4.2.1. Introduction 85
4.2.2. Results 85
4.3. Simulation studies 86
4.4. Examples 89
4.4.1. The mortality ratio 89
4.4.2. Multivariate state space modelling and the Kalman filter 91 4.4.3. The relative error of the variance estimate of the
loga-rithm of a Poisson distributed random variable 91
4.5. Conclusions 92
5. Model-based measurement of latent risk in time series 94
5.1. Introduction 94
5.2. The statistical framework 96
5.3. Case I: a two-dimensional insurance LRT model 100 5.4. Case II: a three-dimensional credit card LRT model 103 5.5. Case III: a multiple exposure LRT model 106
5.6. Conclusions 108
6. Multivariate nonlinear time series modelling of exposure and risk in
road safety research 109
6.1. Introduction 109
6.2. Data description 112
6.3. The multivariate nonlinear time series model 113 6.3.1. Specification of model and assumptions 113
6.3.2. Unobserved stochastic local linear trend factors 115
6.3.3. Observation equation 115
6.3.4. Nonlinear state space model formulation 116 6.4. Estimation of parameters and latent factors 118 6.5. Empirical results: estimation and model selection 121 6.5.1. Parameter estimation results 122 6.5.2. Signal extraction: trends for exposure and risk 123
6.5.3. Model fit 124
6.5.4. External validation 126
6.6. Implications for road safety research 127
6.7. Conclusions 128
7. The likelihood filter: estimation and testing 130
7.1. Introduction 130
7.2. Maximum likelihood approach to filtering 132 7.2.1. Gaussian maximum likelihood approach to filtering 132 7.2.2. General maximum likelihood approach to filtering 132 7.3. Laplace approximation of the likelihood 133
7.4. Simulation studies 134
7.5. Applications 141
7.5.1. Volatility: pound/dollar daily exchange rates 141 7.5.2. The effects of precipitation on road safety 142
7.5.3. Conclusions 155
7.6. Discussion and conclusions 155
8. Conclusions 157 References 163 Author index 173 Appendix A. 177 Appendix B. 187 Appendix C. 194 Samenvatting 199 Dankwoord 207
1.
Introduction
1.1.
A short description of the main ideas of this research
In this thesis we present a comprehensive study into novel time series models for aggregated road safety data. The models are mainly intended for analysis of indicators relevant to road safety, with a particular focus on how to measure these factors. Such developments may need to be related to or explained by external influences. It is also possible to make forecasts using the models. Rel-evant indicators include the number of persons killed per month or year. These statistics are closely watched by government agencies and the public, and their relevance to society is not disputed. A large body of research is devoted to the improvement of road safety. To that end, changes in the number of accidents or victims are often attempted to be explained by (changes in) factors such as exposure, policy, driving under the influence of alcohol, speeding by drivers. Some factors such as policy changes can be directly observed (although com-pliance with policy and law may not). Other factors can be observed in theory but in practice their measurement is either difficult or very expensive. Exam-ples of such factors are exposure, which is measured using surveys and vehicle counting systems, and percentage of drivers exceeding the legal blood alcohol concentration limit, which is measured using road side surveys. Finally, some factors are even harder to observe such as driver skill or experience.
The methodology used by the novel approach introduced in this thesis is de-signed to address potential inaccuracies of data, both in dependent variables and in explanatory variables. The methodology also addresses the potential multivariate nature of road safety analysis problems due to multiple depen-dent road safety outcomes like the number of accidepen-dents and victims. The first aspect results in non-homogeneous observation error variances and the needs for a multivariate approach to modelling. The second aspect introduces struc-tural but time varying covariance among (multivariate) observation errors.
Both issues are accounted for by readily available statistical techniques derived from the Kalman filter (Kalman, 1960). In this thesis a special form of Kalman (1960)’s model which is referred to as a structural time series model is further developed. Structural time series models originate from Muth (1960), and were made popular by Harvey (1983), and applied in multivariate form by Harvey and Koopman (1997). A special form of the latter model designed for road safety risk analysis is developed in this thesis and was published as Bijleveld, Commandeur, Gould, and Koopman (2008). This model is combined with an
approach to estimating the structural covariance among accident related data in Chapter 4, which was published in Bijleveld (2005).
Structural time series models were first applied in road safety analysis by Har-vey and Durbin (1986). In HarHar-vey and Durbin (1986) the consequences of the introduction of the seat belt law in the United Kingdom in 1983 is evaluated. The same methodology was later applied to a seat belt use change in West-Germany by Ernst and Br ¨uning (1990) and to a re-analysis of the introduction of the seat belt law in the Netherlands by Bos and Bijleveld (1991). Other ap-plications in road safety analysis based on this method are by Lassarre (2001), Scuffham and Langley (2002), and COST329 (2004), and in recent PhD theses the method is applied by Scuffham (1998), Christens (2003), Gould (2005) and Van den Bossche (2006).
Given the fact that time series are analysed, the choice for structural time series models was mainly made because the time series can then be decomposed into interpretable components. This allows for the interpretation of risk and other developments while such developments are not directly observed.
In addition, estimating interpretable components also allows for limited vali-dation of their development, as the interpretable parts should at least have a reasonably plausible developments. In case additional information is available pertaining to the development of interpretable components, such information can be included in the model. Adding such additional information allows the researcher to use as much available information as possible. The possibility of a limited form of validation of the results is a substantial advantage of the structural time series approach over more black-box like analysis alternatives. One of such alternatives are ARIMA models as applied in Box and Tiao (1975), see also Box and Jenkins (1976) and many textbooks.
The structural approach presented in this thesis allows the researcher to distin-guish factors that affect road safety from factors that affect the way road safety is observed. A change in a travel survey is not likely to change travel patterns, it is more likely to change travel data. Furthermore, it is also possible to specify on which component (or components) a particular factor should have an effect according to theory or hypothesis, which can then be further verified.
As a side effect, the multivariate approach introduced in this thesis in which traditional dependent variables as well as variables traditionally treated as ex-planatory variables are simultaneously treated as dependent variables has an additional benefit. A regression coefficient associated with the relation
be-tween an explanatory variable and a dependent variable can be absorbed in the model.
The special case where exposure is the explanatory variable is given promi-nent attention in this thesis. In the log linear context, as used in Chapter 3 and Chapter 5, a regression coefficient as described in the handbook by Elvik and Vaa (2004, p. 49) and many other studies, is absorbed in the model. Elvik and Vaa (2004)’s approach has the advantage of (approximately) accounting for a non linear relation between traffic volume and the number of accidents, as suggested by for instance Hauer (1995). However, Elvik and Vaa (2004)’s approach has the disadvantage of limiting the comparability of its results be-tween models that have different coefficients. The model developed in Chap-ter 3 and ChapChap-ter 5 estimates development for risk as the ratio of the number of accidents per vehicle kilometre, which should be comparable between mod-els. Other models, described in Chapter 3 are inspired by and share properties of the DRAG (demande routi`ere, accidents et leur gravit´e) framework by Gaudry (1984) and Gaudry and Lassarre (2000).
The first study within the context of this thesis was Bijleveld (1999). The objec-tive of Bijleveld (1999) was to improve the reliability of short-term prognosis of general road safety outcomes, to be used as part of an annual review of the development of road safety in the Netherlands. Specifically, such prognoses were intended to be used to determine whether or not road safety outcomes in the reviewed year were in line with what could be expected from road safety developments just before that year. A comprehensive analysis of changes in the development of road safety related indicators could help the road safety researcher detect recent general changes in road safety conditions, if any. After Bijleveld (1999), the objective was extended to the analysis of the development of aspects of road safety in general, resulting in this thesis.
A primitive form of the model was developed during work on the COST329 (2004) report in the second half the 1990’s. The simplicity of the implemen-tation of the EM algorithm (Expecimplemen-tation Maximisation, e.g. Dempster, Liard, and Rubin, 1977; McLachlan and Krishnan, 1997) for state space estimation found in Fahrmeir and Tutz (1994) and others, which easily allowed for a gen-eral multivariate implementation of the approach taken by Harvey and Durbin (1986), was also of importance. The final publication of COST329 (2004) was delayed, and as a result the approach was first published in Bijleveld (1999).
The results presented in this thesis are aimed at providing better and statis-tically more reliable options for time series analysis of road safety data. The
analyses performed in this thesis are not intended to answer specific road safety questions, but are intended to illustrate the application of the methods introduced in this thesis.
1.2.
Important issues in time series analysis of road safety
data
In this section four central issues involved in time series analysis of road safety data are presented: time dependence, multiple road safety outcomes, exposure data, and other explanatory variables.
1.2.1. Time dependence
When a specific condition in road traffic suddenly changes at a certain time point, it is often to be determined whether (or not) a relevant road safety indi-cator changed at about the same time point. The opposite also occurs: when a specific road safety indicator changed at a certain time point, it is often to be determined whether (or not) a relevant road traffic condition changed at about the same time point. A classical approach to statistical analysis in this situation would be to select a type of accident that should be affected by the change (which is called the experimental group), and a type of accident that should not be affected by the change (which is called the control group). Then both accident counts for a period before and after the change are compared in a 2×2 table:
Count before after experimental group eb ea
control group cb ca
In a typical before/after study, the rate before eb/cb and after ea/ca are
com-pared. It has to be assumed that the rates would remain constant if the condi-tion in road traffic had not changed. If the rate e/c was constantly decreasing, eb/cb would be larger than ea/ca, if only for that reason. This drop could be
falsely attributed to the sudden change in road traffic conditions. There are numerous reasons why the rate could change with time. For instance, when the experimental group is moped victims, and the control group is bicycle vic-tims, the rate will change when bicycles are getting preferred over mopeds for travel. Therefore it is wise to determine the rate e/c for a number of periods in the before and after period. Then verify that the rate e/c is reasonably constant
in the before and after period, before a change in this rate can be attributed to the change in road traffic conditions. If this analysis is performed, and a se-ries of rates e/c is available for a period of time, it is also wise to determine whether the drop in the rate occurred about the time of the change in road traffic conditions or not. If this is not the case, some other influence may have caused the drop (this possibility can never be excluded). Furthermore it can be determined whether or not the change in the rate is exceptional. If similar drops in the rate occur regularly and cannot be explained, there is no reason to assume that this particular drop is not coincidental but caused by the change in road traffic conditions, while others are considered coincidental.
The analysis steps described above, are regularly performed in time series analysis. In the first step, a trend is determined, in the second and third step a so-called structural break is identified (both its location (where and when it occurred) and whether it is significant).
For this reason alone it can be suggested to perform a more elaborate time series analysis than a before/after study, which itself is a rudimentary analysis of time ordered data, with just two time points. More reasons can be suggested to make this choice.
When a specific condition in road traffic does not change suddenly but changes gradually it is not trivial to use a before/after study. In such situations (time series) regression analysis is currently most often applied.
There are other ways in which time dependence may affect the analysis of road safety data. For example, time dependence implies some structure among ob-servations. There is sufficient reason to at least consider time dependence in road safety analysis. If data collected over a longer period of time are con-sidered, the general road safety situation is likely to have changed as, among other conditions, road and vehicle design may have improved. If this is the case, observations close in time will resemble each other more than observa-tions further apart in time. This phenomenon is reflected in the development of many road safety related features like the number of fatally injured victims in road accidents in Figure 1.1. The road safety situation in 1970 will say lit-tle about the road safety situation in 2000, while the road safety situation in 2007 may give a rather accurate idea of what the road safety situation in 2008 probably will be.
1950 1960 1970 1980 1990 2000 1000 1500 2000 2500 3000
Figure 1.1. The development of the number of police recorded fatally injured victims in road accidents (1950–2000) in the Netherlands. Source: CBS (2000).
Most statistical models require that the difference between the model and data is purely coincidental and no two differences are related1. Technically this means that the so-called disturbances (the difference between the observed and the prediction by the true model, which is not observed) are required to be independent of each other. Failure to satisfy this requirement may lead to over or under estimation of model uncertainty, which again may lead to statistical tests being too conservative or worse, not being conservative enough. This in turn may lead to falsely positive identification of relationships or interven-tions in road safety analysis. See, for instance Scheff´e (1967, Chapter 10) for a discussion on violations of assumptions on the disturbances in a linear model, which also includes uniformity of the variance of the disturbances. This poten-tial problem cannot be ignored, and accounting for it is the second way time dependence affects the analysis of road safety data.
Model residuals are differences between observed values and the predicted values from the estimated model, as depicted at time point “4” on the left hand side of Figure 1.2. Model residuals are observed in contrast to the distur-bances. The residuals in this figure are positive for the first two time points, the next residual is approximately zero, then three residuals are negative, the next three residuals are positive, the following three residuals are again negative, etc. Most models require that disturbances are independent of one another, which roughly speaking means that knowing one residual (which estimates a disturbance) should not help in predicting the next. In the example shown in Figure 1.2 the requirement of independence of the disturbances is most likely violated.
1Models exist which require an independent source of error, not necessarily describing the
0 5 10 15 20 16 17 18 19 20
Figure 1.2.Theoretical development of the number of accidents (hy-pothetical development is 20−t/5+sin(t)for t = 1, . . . , 20) and a linear regression over the first 16 observations (to the left of the vertical reference line) plus a forecast (to the right of the vertical reference line). The differences between the dots and the (straight) line (to the left of the vertical reference line) are called the model residuals. The differences to the right of the vertical reference line are technically not model residuals, as they were not included in the regression. It can be seen that consecutive residuals tend to share the same sign.
An example of the first way in which time dependence may affect the analy-sis of road safety data is correcting for time dependencies in model residuals for (short-term) prognosis. This can be understood from the example devel-opment presented in Figure 1.2. It is not uncommon to have a develdevel-opment of the number of accidents similar to Figure 1.2, where there is a linear trend (in this case fixed at 20−t/5) and some fluctuation around it, (for example sin(t)), yielding the function 20−t/5+sin(t) for t = 1, . . . , 20. From Fig-ure 1.2 it is clear that the forecast for t = 17, . . . , 20 obtained by extending the linear regression line (as depicted by the straight line in Figure 1.2) can be substantially improved by using the knowledge that the observations follow a pattern of being positioned over and under the regression line. Roughly, this is what considering ‘time dependence’ of model residuals amounts to: account-ing for an empirically revealed structure in residuals. In general, the dynamic structure is unknown, and much like in this example it is attempted to build a description of the dynamic structure. First a linear trend (or another structure suggested by theory) is fitted. Then the residuals are studied. If those residu-als do not reveal a structure, the model may be adequate. If not, the dynamic structure is adapted. There are a number of approaches to adapt the dynamic structure, one of them is chosen later in this thesis.
1.2.2. Multiple road safety outcomes
One important aspect of road safety (time series) analysis is that road safety cannot be measured unambiguously. There is no unique measure of road safety. Usually, road safety is measured in terms of the amount of ‘lack of road safety’, for instance the number of accidents occurring per time unit. Even if the number of accidents is selected as the measure of road safety, it could still be all accidents, injury accidents, serious accidents or fatal accidents, or other types of accidents. But even then the number of victims per accident may be of interest, as well as the number of fatalities per accident.
It should further be considered that influences on road safety may primarily affect certain parts of the road safety process. For instance, it is sometimes claimed (and disputed) that the use of seat belts primarily has an effect on ac-cident consequences, not on acac-cident occurrence. If it is true that the use of seat belts primarily has an effect on accident consequences, it would be sufficient to study the number of victims. Risk adaptation theories (such as for exam-ple Wilde (1994) and Summala and N¨a¨at¨a¨anen (1988)) state that developments that could be expected from theory may be counteracted due to behavioural adaptation, in this case possibly increased speeding by drivers. If this is true, not only the accident consequences in terms of the number of injuries need to be considered, but also the number of accidents. Even if the original theory is assumed to be true, it is sensible to study both the development of the number of accidents and the number of victims.
Assume a study into the effect of the introduction of a seat belt law on road safety is to be conducted. It is possible that in the period in which the seat belt law was introduced, other influences had an effect on road safety. Such influences may have had an effect on the indicators that are considered to be relevant to the safety effect of seat belts. If the effect of the seat belt law is to be determined, one may need to correct for other influences. Therefore the mod-elling approach should be able to disentangle multiple effects. These effects may have had an impact on the number of accidents or victims of a certain type, or both, which is best done by modelling them simultaneously. How-ever, the number of accidents and the number of victims resulting from these accidents are correlated, and this correlation should be accounted for in the analysis. In summary, the modelling approach should be capable of simulta-neously treating at least two dependent variables (in case of the example above these would be the number of accidents and victims), and their covariance.
Another reason to consider multiple road safety outcomes is that although road safety interventions may be introduced to reduce certain accident out-comes, they may also – hopefully to a lesser extent – increase certain other accident outcomes. In general, the accumulated effect of road safety interven-tions is considered most important as it indicates the net effect to society. In specific applications the differentiated effect of road safety interventions needs to be studied, for instance to test hypotheses on theories.
1.2.3. Exposure data
In France more accidents occur in road traffic than in the Netherlands, but does that necessarily mean that road traffic is safer in the Netherlands than in France? Is it not the case that France is a much larger country than the Neth-erlands, and thus has more potential to have accidents in road traffic than the Netherlands? One would expect an imaginary country twice the Netherlands in every respect and otherwise completely equal to have twice as many acci-dents as the Netherlands. This reasoning is often used to justify using accident rates in terms of the number of accidents per unit of scale when comparing dif-ferent entities such as road sections or countries. In this example, the number of accidents for the imaginary country would be divided by two as the coun-try potentially has twice as many accidents. The potential to have accidents (or victims) is generally referred to as exposure in road safety analysis.
Accounting for differences in exposure is not always straightforward. For in-stance, when comparing the number of fatal accidents in France to the Neth-erlands (which is about 6.5 to 1), the difference in country size (about 552.000 km2for France and about 42.000 km2for the Netherlands, including water sur-face) could be used to account for differences between France and the Nether-lands. This would make the Netherlands in this respect less safe than France. Such a figure would ignore differences in land use (notably population den-sity), which could be considered a disadvantage. Alternatively, population size could be used, which was about 61 million for France and about 16 mil-lion for the Netherlands in 2007. Using population figures as a measure of exposure would present France as less safe than the Netherlands. A drawback of using population figures may be that such figures may not sufficiently ac-count for differences in road use: in a large ac-country like France, the population may have to travel longer distances. In order to improve on such figures, traf-fic volume (the number of kilometres or miles driven on the road by vehicles) or travel volume (the number kilometres or miles travelled on the road by per-sons) are often used when available. Such figures may better represent the exposure of a country than its size or number of inhabitants.
It should be noted that, although traffic volume is mostly preferred as a mea-sure of expomea-sure, it is the research question that determines the optimal expo-sure meaexpo-sure. In practice the researcher not just selects one available expoexpo-sure measure, rather, exposure measures are selected for a specific purpose. The number of fatalities per unit of population per year is sometimes used specifi-cally to be compared with other mortality rates. For similar reasons, the number of victims per unit of population per year can be used to compare with other incidence rates. When road accidents are compared with work accidents, then the time spent in travel is probably the preferred choice.
It should further be noted that, no matter how accurate traffic or travel volume appears to be measured, such measurements cannot be considered an exact es-timate of exposure. As some information on traffic or travel volume is obtained through travel surveys, these data are by nature subject to random error. An-other reason is that it is not only the amount of travel that is important to road safety, but also the conditions under which the travel took place.
As an example of the uncertainties concerning traffic volume data, consider the Dutch travel data using mopeds presented in Figure 1.3. In the left hand panel of this figure the number of person kilometres2is presented for mopeds, together with the number of police registered accidents with killed or hospi-talised victims between mopeds and cars. The grey area depicts the point wise 95% percent confidence intervals for the person kilometres. These intervals are based on an estimate of the error due to sampling only – an estimate of the error due to respondents providing erroneous data is not available – therefore the actual error is likely to be larger. In the right hand panel of Figure 1.3 the rel-ative error based on Slootbeek (1993) and CBS (2003) (right hand panel, solid line, left hand axis) is presented together with a plot of 1/pnumber of trips (dashed line). This plot reveals that the relative sampling error for the to-tal moped travel in 2005 is about 16% (left hand scale). The relative error of moped travel for separate age groups will be substantially larger. In 1994 and 1995 the survey was substantially extended. In 1999/2000, the survey struc-ture has changed. Over the last few years the survey size has been reduced while the use of mopeds has also decreased. This resulted in the relative accu-racy of moped data being at about the same level as it was near the end of the 1980’s.
2Driver kilometre data are not available, but the development of driver kilometres should
be similar to the development of passenger kilometres. Moped occupancy appears to be rela-tively constant based on a moped helmet survey (Ermens and van Vliet, 2006) for 2002–2005, where it was found that on about 11% of the mopeds evaluated a passenger was present.
0.6 0.8 1.0 1.2 1.4 1.6 1985 1990 1995 2000 2005 800 1000 1200 1400 1600 0.015 0.020 0.025 0.030 0.035 1985 1990 1995 2000 2005 0.08 0.10 0.12 0.14 0.16 0.18
Figure 1.3.Traffic volume and accident data for mopeds in the Netherlands 1985–2006. Left hand panel, left hand axis, dots: the number of police registered accidents with killed or hospitalised victims between mopeds and cars. Left hand panel, right hand axis, solid line: the number of person kilometres (billion) using mopeds in the Netherlands. The grey area depicts the point wise 95% percent sampling confidence intervals for the person kilometres based on Slootbeek (1993). Right hand panel, left hand axis, solid line: relative sampling error in person kilometres based on Slootbeek (1993). Right axis, dashed line: 1/pnumber of trips.
In the left hand panel of Figure 1.3, the traffic volume appears to go up and down by a substantial amount near the end of the 1980’s, while the accident counts seem relatively stable. Ignoring the fact that the traffic volume data in this case are not accurate, one may conclude that both the traffic volume and the risk (being the ratio of the number of accidents to the traffic volume) fluctuated substantially in this period, which was probably not the case.
The topic of exposure is further discussed in Chapter 2, which also discusses whether exposure affects road safety linearly or non linearly, as for instance argued by Hauer (1995).
1.2.4. Explanatory variables
Besides exposure, the development of road safety can be influenced by devel-opments in many areas such as road design, vehicle technology, education, de-mography, weather, economy, etc. Quantitative information on such develop-ments is regularly obtained from separate research results. The studies which provide such results can be regularly and consistently conducted surveys, as is the case with the travel survey in the Netherlands, or population figures ob-tained from censuses or registers. However, studies may come from different disciplines, may have different viewpoints, and are often limited by design to some subsection of the complete road safety field. As road safety time series analysis typically considers a longer period of time, it is likely that study de-sign and purpose have changed over time, although such studies generally
still measure the same phenomenon. It is possible that such changes could influence analysis results, the impact of which should be minimised.
Example: drink driving data
One example of a case where data collection may potentially affect analysis results is data on the percentage of drivers exceeding the legal blood alcohol concentration limit (drink driving). It is commonly assumed that drink driv-ing is a risk increasdriv-ing factor. When the consequences of drink drivdriv-ing for road safety are to be determined, it is important to know how many drivers are ac-tually exceeding the legal blood alcohol concentration limit. In Figure 1.4 the percentage of car drivers tested to have a Blood Alcohol Concentration (BAC) larger than 0.5 g/l (0.5 grammes per litre) in the Netherlands is given. The re-sults are obtained from a number of surveys intermittently conducted during autumn weekend nights, starting in 1970. This example demonstrates another case of an important explanatory variable that in general should measure the same phenomenon (the percentage of drivers exceeding the legal blood alcohol concentration limit). Due to changes in measurement and scale of the survey, the series of data is not fully consistent and not systematic in its accuracy. Fur-thermore, the measurement for one year is distorted, possibly as a result of the fact that the focus of the survey that year was directed at the introduction of a new drinking driving law. Finally, the studies are justifiably focused on assessing the worst extent of the problem by measuring drink driving in a pe-riod, weekend nights, where the percentage of drivers under the influence of alcohol is expected to be largest. The measure is therefore unlikely to represent drink driving in general road traffic.
On the first of November 1974 a new law introducing the 0.5 g/l BAC legal limit became effective in the Netherlands. At the same time, chemical test tubes for road side testing were introduced. This time point is marked by the first vertical reference line in Figure 1.4. SWOV (1978) reports that the measurement for that year (1.5 %) was based on the average of observations specifically taken one weekend immediately before the introduction of the law (the weekend of 25–27 October 1974, 12% violations) and two larger surveys in weekends immediately after the introduction of the law (the weekends of 8–10 November and 22–24 November 1974, 1% violating the law). Given the ob-servation in 1975 and the fact that 12% violations were recorded the weekend before the introduction of the law (and 15% in 1973), it may not be realistic to consider the observation of about 1.5% for 1974 as being representative for the percentage of drivers exceeding the 0.5 g/l BAC limit in the whole of 1974.
1970 1975 1980 1985 1990 1995 2000 2005 0 2.5 5 7.5 10 12.5 15
Figure 1.4.Percentages of car drivers having a Blood Alcohol Con-centration (BAC) exceeding 0.5 g/l in the Netherlands based on sur-veys taken in the autumn during weekend nights (see, Mathijssen, 2004). The survey was not conducted every year. Dots mark avail-able data points.
In 1984 (marked by the second vertical reference line in Figure 1.4) electronic alcohol breath test devices for selection purposes were introduced (blood tests were still needed for legal confirmation). Starting in 1985 a gradual change from selective to random police alcohol controls took place, which changed the population sampled. As of the first of January 1987 (marked by the third ver-tical reference line), results of alcohol breath tests could be used for evidential purposes (in addition to blood sample tests). As of the first of November 1992, heavier fines for drink-driving were introduced. The survey initially consisted of about 3,000 observations, by the early 1990s this number increased to about 15,000, and at the end of the series there are about 30,000 observations. More-over, the survey has not been conducted each year. Missing data are interpo-lated in Figure 1.4. However, the percentages not necessarily dropped linearly starting in 1984, the first of three years in which no surveys were conducted (as is noted by Mathijssen (2004)). If accident occurrence is indeed related to alcohol use by drivers, a drop in alcohol use by drivers could be reflected by a drop in accident occurrence. A drop in accident occurrence at a later year may indicate that alcohol use could have dropped later, but may not be conclusive. An estimate of the missing values based on the accident development is likely more reliable than the linear interpolation.
The example concerning drink-driving data as well as the discussion on expo-sure data suggest that explanatory variables should not be considered at face value. Each explanatory variable should be carefully considered and weighed.
In both examples the survey size varies over time, effectively meaning that the accuracy of the data is not the same for all time points. As a result, het-eroscedasticity among observation errors should be considered.
Further issues
Apart from the reliability of an explanatory variable, another important issue to consider is its validity, that is, whether or not it actually represents what it is supposed to represent. For instance, in the drink driving example, the data actually refer to autumn weekend nights, not full days. This means that the scope of the data should be considered. In road safety research one quite often is forced either not to use an explanatory variable or to assume that the ‘true’ explanatory variable (in this case drink driving on average days) has a ‘similar’ development to the one actually available, or to try and find confirmation of this assumption from other studies. Exposure data are subject to similar prob-lems. The exposure data are obtained from household surveys CBS (2003) and AVV (2005). As the sampling unit is households3, the persons in the survey are almost exclusively residents of the Netherlands (but not necessarily Dutch nationals). This implies that travel data for non-residents of the Netherlands is not included in the survey, thus the survey does not represent all travel in the Netherlands.
The scale of studies providing explanatory variables may vary between the mi-croscopic level – at the level of individual accidents – and the (supra) national macroscopic level of aggregated data. Generalisations of many such ‘pieces’ of information may be necessary to complete the ‘puzzle’ of road safety. A microscopic level study may reveal the effect of seat belts on victims, while macroscopic level studies may establish the effect a law on seat belt use has on society.
As the type of analysis targeted in this research tends towards macroscopic (aggregated) level analysis rather than microscopic level analysis, consequen-ces of using results from lesser aggregated studies should be considered. For instance, while a microscopic level study into the influence of weather on road safety may reveal that the average temperature explains some variation in acci-dent counts, the average temperature over a year may not. As a second exam-ple, Eisenberg (2004, p. 637) finds that “in a typical state-month pair in the US from 1975 to 2000, increased precipitation is associated with reduced fatal road traffic crashes. More precisely, an additional 10 cm of rain in a state-month is associated with a 3.7% decrease in the fatal crash rate”. Later he states: “First, when the regression analysis is conducted with the state-day, rather than the
state-month, as the unit of observation, the association between precipitation and fatal crashes is estimated to be positive and significant, as in the literature.” (Eisenberg, 2004, p. 637). Eisenberg (2004) continues to explain the importance of lagged precipitation data in his (daily) model, effectively introducing a time series model. This shows that different aggregation levels may yield opposite results.
1.2.5. Conclusions
In this chapter it is demonstrated that travel volume data and data on the per-centage of drivers exceeding the legal blood alcohol concentration limit (both derived from surveys) have to be considered as observed under error. How-ever, it is not just travel or alcohol surveys that are observed under error. Sim-ilar arguments would hold for data derived from surveys like crash helmet use on mopeds (Ermens and van Vliet, 2006), and many others. If a variable is measured under error this means that instead of the true value, by coinci-dence a different value is used, which can be considered random fluctuation from the true value. In case of traffic volume data, the true value would be the number of kilometres driven, while the value actually used would be the number of kilometres driven based on the randomly selected respondents of a survey, instead of the entire population. In general, the fluctuations are on average (expected to be) nil. However, its variance, which is a measure of the statistical accuracy of the data is larger than nil.
The issue of the potential random fluctuations in exposure and other explana-tory data is mostly ignored in road safety analysis, probably as often no infor-mation with respect to the statistical accuracy of the data is available. In many cases, however, the consequences of ignoring statistical inaccuracy of expo-sure or explanatory data may be negligible compared to other inaccuracies. Neglecting the statistical accuracy of the data is not always warranted. For instance disaggregate traffic volume data (traffic volume data for subgroups) may be subject to substantially larger sampling errors than aggregate data (as described in the example on moped travel), up to more than 100% sampling error. Furthermore, there is no reason not to account for the inaccuracy of the data when it is possible to do so.
Therefore it is important to consider the possibility of random fluctuations in the explanatory data as well as random fluctuations in the accident data. Considering possible random error in explanatory variables as well as in de-pendent variables implies an ‘errors-in-variables’ approach (see, Seber and Wild, 1988, Chapter 10). This approach essentially treats explanatory varia-bles (which are assumed to have error) as dependent variavaria-bles alongside the
original dependent variables. As a result, models are multivariate in the sense of multiple dependent variables. Besides the ‘errors-in-variables’ argument, there are further reasons to consider road safety analysis problems multivar-iate. It is argued that road safety cannot be measured unambiguously as no unique measure of road safety is available. Depending on the research ques-tion road safety can be measured in terms of the number of accidents or vic-tims, and combinations of these.
In this thesis road safety is therefore considered inherently a multivariate prob-lem, that should preferably be analysed accordingly. Furthermore, the conse-quences of time dependence should be considered, not only in view of reli-ability of statistical tests, but also in view of making forecasts of future road safety indicators. It will be demonstrated in Chapter 3 that considering time dependence allows for an intuitive treatment of missing data as well.
A sufficiently flexible general framework to statistical time series analysis is already available, based on (derivations of) the Kalman filter (Kalman, 1960). This framework also handles non-homogeneous observation error variances in a straightforward manner. In this thesis a special form of (Kalman, 1960)’s model called a structural time series model is further developed in a multivar-iate dimension, specifically designed for road safety risk analysis.
Given the fact that time series are analysed, the choice for structural time series models was mainly made because the time series can then be decomposed into interpretable parts. This allows for the interpretation of risk developments – see Chapter 2 for further details, while risk itself is not actually observed. This applicability becomes even more important when as in Section 3.4 the risk relates to multiple dependent road safety outcomes. The resultant model is a multivariate unobserved components model, which is a special case of Harvey and Koopman (1997).
1.3.
Structure of this thesis
This introductory chapter provides the background of the research presented in this thesis, including how it originated and the main issues that require close attention when analysing developments in road safety: time dependence, the multivariate nature of road safety, and the problems associated with exposure data and other explanatory variables.
In Chapter 2, background definitions and statistical properties known in road safety research are provided for the three central concepts in the analysis of
road safety: safety, exposure and risk. In practical terms, Chapter 2 is about how road safety is observed at each time point.
Chapter 3 first introduces the novel multivariate structural time series frame-work. By using this framework developments in accident and victims counts, exposure and other explanatory variables can be analysed simultaneously, thus considering the multivariate nature of road safety. Their developments are modelled using structural components for exposure, risk and other factors. This approach not only allows to consider time dependencies, but also allows the researcher to interpret the development of these structural components. The latter can lead to new insights, for instance by assessing the significance of changes in risk. It can also be used for validation purposes, which may be important in limited data situations. By using the combined framework of ad-vanced state space and Kalman filter techniques, traffic volume data and other data can be treated stochastically, thus taking care of measurement errors in explanatory data and allowing to consider the covariance between accident related outcomes. Chapter 3 starts with the concept of ‘state’ in Section 3.2. The state is an unobserved vector containing parameters of the important parts (aspects) of road safety. For instance the state can be assumed to contain the parameters that define traffic volume and risk, as well as other parts consid-ered important to the particular road safety analysis. The modelling frame-work can then be used to estimate these parameters and thereby quantify these important aspects. In Section 3.3.1 the basic form of the measurement of the state of the linear models in this thesis is explained, which is used as a starting-point for the time development of the models. Thereafter the approach of how the dynamics are treated in this thesis is outlined, which coincides with the structural time series approach. In Section 3.3 the main linear multivariate structural time series model framework developed in this thesis is described. The framework allows the risk to be treated as a latent variable, and the asso-ciated model is therefore called the latent risk time series model. In Section 3.4 two applications are discussed, which extend the models discussed in Chap-ter 5 by integrating results from ChapChap-ter 4 and by including alChap-ternative source victim data. In the first example an extended LRT model is used to compare the development of two accident severity indices, the number of killed or hos-pitalised victims per serious accident and the number of fatalities per victim for rear-end accidents to the same indices for all accident types. These two appear to have different developments. In particular it is noted that the num-ber of killed or hospitalised victims per serious accident is not constant over time. This result is used in the next example, where the registration level of ac-cidents involving hospitalised victims is used as a common factor to estimate the number of accidents corrected for incomplete registration. In this example,
two sources of accident victim data are used: police records, which include de-tailed accident information, and hospital records, which have dede-tailed infor-mation on road individuals admitted to hospital, but do not include detailed accident information. Both sources are used to estimate the ‘true’ number of hospitalised victims. Under the hypothesis that the police either register all hospitalised victims or none, the ‘true’ number of accidents with hospitalised victims can be estimated using the LRT model by assuming all accidents with hospitalised victims and police recorded accidents with hospitalised victims share the same latent factor describing the number of hospitalised victims per accident. The advantage of the LRT approach over averaging is its acknowl-edgement that registration rates and the number of hospitalised victims per accident change with time. These figures are also estimates and are thus not accurately measured. This approach should yield more reliable results than calculations based on averages.
In Chapter 4, a variance-covariance structure for accident related outcomes is established, thus allowing for a proper treatment of their inter-dependencies in a multivariate time series analysis. The approach describes a straightforward way to estimating the covariance matrix of the number of accidents, victims and killed, and possibly other accident outcomes. These results are important when more than one of such variables are used in the model, see also Bijleveld (2005).
In Chapter 5, a comprehensive and technically detailed overview is presented of the main linear multivariate structural time series model framework devel-oped in this thesis. Estimation details are given, and example applications are given based on Australian and Dutch data. The examples demonstrate that the applicability of the model is not limited to road safety time series analysis. This chapter was published as Bijleveld et al. (2008).
Chapter 6 presents a nonlinear extension of the multivariate structural time se-ries model framework, based on Gaussian error distributions. The estimation procedure applies the extended Kalman filter instead of the classical Kalman filter used in the linear models discussed in Chapter 3 and Chapter 5. The model is applied to the analysis of the development of road safety disaggre-gated into inside and outside urban areas. This example is typical for disag-gregated data where not all relevant data is available in disagdisag-gregated form. In this case disaggregated traffic volume is not available for all observations. However, the total traffic volume, traffic volume for inside urban areas plus traffic volume for outside urban areas is available for all observations. Struc-tural components are estimated for risk inside and outside urban areas, which
are compared, and exposure for risk inside and outside urban areas. As one example of how the structural nature of the framework can be used to validate a model, the last of these structural components is further compared to an es-timate of traffic volume outside urban areas based on road length and traffic intensity measurements. The result of this comparison appears to support the validity of the model.
Chapter 7 discusses a further generalisation of Chapter 6 which allows for the specification of non-Gaussian error distributions. The estimation procedure in this Chapter can be regarded as a generalisation of the iterated extended Kalman filter using Laplace approximations. Apart from an example appli-cation on well known data, a simulation study is reported in Chapter 7. The approach is applied to road safety in an example. In this application, precipita-tion duraprecipita-tion is used to estimate the relative contribuprecipita-tion to risk of fatal single car accidents due to precipitation. The example model is based on two daily accident counts (with and without precipitation according to the police) traffic volume data derived from the travel survey (thus small samples, which should be accounted for) and individual precipitation duration data of 10 weather stations distributed over the Netherlands, acknowledging the consistency of weather patterns.
2.
Safety, exposure and risk: definitions and
some statistical properties
2.1.
Introduction
A philosophical discussion covering the topic of “unsafety” or the lack of safety is beyond the scope of this thesis. This thesis is focused on practical time series modelling aspects of aggregate road safety data. It is assumed that the results of “unsafety” are accident consequences such as accident or victim counts, or combinations of both. The precise type of accident to be considered is deter-mined by the research question of a study. Other accident consequences such as monetary consequences of road accidents may also be considered.
A primary assumption in road safety analysis is that accident related road safety outcomes are non-predictable, non-deliberate consequences of entities (vehicles, persons) taking part in traffic. The precise definition of what a road accident (sometimes called a crash) is, for example, has no relevance for the research presented in this thesis. In short, this thesis is concerned with the analysis of collected outcomes of non-predictable, non-deliberate accident-like events in road traffic.
Inspection of basic road safety data for the Netherlands (see Figure 2.1) re-veals that the number of police recorded fatal accidents increased from 969 in the year 1950 to a maximum of 2984 fatal accidents (which resulted in 3264 fatalities) in the year 1972. It then started to decrease to 1006 fatal accidents in the year 2000. As the number of fatal accidents in the year 1950 is approx-imately equal to the number of fatal accidents in the year 2000, the question
1950 1960 1970 1980 1990 2000 1000 1500 2000 2500 3000
Figure 2.1. The development of the number of police recorded fatal road accidents (1950– 2000) in the Netherlands. Source: CBS (2000).
1950 1960 1970 1980 1990 2000 10 11 12 13 14 15 16 1950 1960 1970 1980 1990 2000 0 20 40 60 80 100 120
Figure 2.2. Left hand panel: the number of inhabitants in the Netherlands (by 1 January, in millions) for 1950–2000. Right hand panel: the number of motor vehicle kilometres (in billions) in the Netherlands for 1950–2000. Source: CBS (2007) and CBS (2003).
1950 1960 1970 1980 1990 2000 75 100 125 150 175 200 225 250 1950 1960 1970 1980 1990 2000 0 25 50 75 100 125 150
Figure 2.3. Left panel: the number of police registered road accident fatalities per million inhabitants (as of 1 January) for 1950–2000. Right hand panel: the number of police registered fatal road accidents per motor vehicle kilometre (in billions) for 1950–2000. Source: DVS (2003) and CBS (2007).
arises whether all efforts to improve road safety in the period 1950–2000 only resulted in reducing safety to the level of 1950. The answer to this question depends on how one assesses the scale of the road safety problem.
In Figure 2.2 the development of the number of inhabitants and the develop-ment of (motorised) traffic volume is given for the same period of Figure 2.1. It is shown in Figure 2.2 that the population in the Netherlands increased by about 60% in that period. Traffic volume, on the other hand, was 20 times larger in 2000 than it was in 1950 (this refers to motorised traffic only, but non-motorised traffic volume, which consists of pedestrian, bicycle and (light-) moped travel is minor compared to motorised traffic volume in this demon-stration).
From the perspective of increased population and traffic volume, it is inter-esting to consider the relative ‘unsafety’ in terms of the rate of the number of fatalities per inhabitant (a public health perspective) and fatal accidents per
motor-vehicle kilometre (a traffic performance perspective). These develop-ments are displayed in Figure 2.3. The huge increase in motorised traffic vol-ume resulted in a (continuing) decrease in the number of fatal road accidents per motor vehicle kilometre, similar to Appel (1982). Even by looking at the number of inhabitants, the number of fatalities per inhabitant is lower (at about 67%) in 2000 than it was in 1950. Given the fact that road traffic substantially increased over that period, this may be considered as a remarkable result.
Which kind of exposure can best be used however is not clear from these fig-ures. The following quotes by Hauer (1995): “Thus the question is not ‘what is exposure?’, but ‘What is the accident rate good for when VMT, ADT and the like serve as exposure?’ ”4 and by (Hakkert and Braimaister, 2002, p. 7):“It will be shown that there is no general definition of exposure and of risk and that these terms should be defined within the context of the issue studied.” seem to position this issue in road safety analysis. When the probability of a person dying in a road accident is compared with the probability of a person dying of cancer, then the number of inhabitants is an appropriate measure of exposure. When road accidents are compared with work accidents, then the time (hours) spent in travel is probably the preferred choice, while comparisons between different transport modes (e.g., car, train, aeroplane) often involve the use of kilometres travelled.
In aggregate models, road safety is often studied in terms of failures per unit performance. Because of the numerous possibilities for a sensible choice of the combination of the road safety indicator and the exposure measure (Yannis et al., 2005) this thesis is not focused on one particular type of combination. As stated in Yannis et al. (2005), traffic volume is usually the preferred measure for exposure, and the examples in this thesis are therefore mainly oriented at the use of vehicle kilometres as scale factor for the road safety problem.
2.2.
Risk exposure in road safety analysis
5As the basic distributional properties of road accident statistics play a central role in road safety analysis this section first discusses this topic. A textbook level derivation of the statistical distribution of accidents is described, which is further used as a starting point for a discussion of the nature of exposure.
4VMT is vehicle-miles travelled, ADT is average daily traffic
5This section is adapted from section 2.1 of the SafetyNet WP2 state-of-the-art report
2.2.1. Statistical distributions
This section is devoted to a discussion of the statistical distribution of aggre-gated accident counts, with some reference to the distribution of victim counts. Accident distributions refer to the distribution of the number of accidents and not to the spatial distribution of the accidents over an area or temporal distri-bution over time.
An introduction to a discussion of the basic concepts of road accident statistics is the work by the French mathematician Poisson (see, Feller, 1968, page 153). Poisson investigated the properties of Bernouilli trials. A Bernouilli trial is an experiment that has two possible outcomes: success or failure. This type of experiment seems to be a useful building block for modelling road safety. For instance, the crossing of a road by a pedestrian can be conceived of as an exper-iment with a (fortunately) minimal probability of a ‘success’ (i.e., an accident occurring). A similar argument could be used for a vehicle passing through a road section, a vehicle driving past a road side obstacle, or two vehicles en-countering each other on the road. Many other examples could be considered. The concept of a trial in this chapter is different from the concept of a conflict in Hauer (1982), which is at a much later – almost final – stage of the development of an accident.
The original work of Poisson assumed the probability of success to be the same at each trial. Poisson could then prove that the distribution of the sum of all successes would tend to a Poisson distribution. The restriction Poisson used that the probability of success has to be the same value, say p, at each trial has since been relaxed (see Feller, 1968, page 282). Let N denote the number of trials, it is not necessary that all probabilities of success pi are equal to each other for i = 1, . . . , N. Rather the sum of all N probabilities should tend to a finite λ (which serves as the expected number of accidents), and its maximum (e.g. Feller, 1968, page 282) or sum of squares (e.g. Shorack, 2000, page 367) should tend to nil:
lim N→∞ N
∑
i=1 pi =λ lim N→∞1max≤i≤Npi=0 Nlim→∞ N∑
i=1 p2i =0, (2.1) where N is the number of trials, and piis the probability of an accident in triali.
For the practice of road safety analysis this result has the following conse-quence: if the number of accidents can be regarded as the sum of the outcomes of many independent conceptual events, each having a small probability piof
turning into an accident, then the distribution of the sum of those events that turned into accidents – thus the number of accidents – tends to the Poisson distribution with parameter equal to the sum of the probabilities of events re-sulting in an accident. Therefore the expected number of accidents is equal to the sum of all probabilities, which is λ in the limiting case.
It should be noted that:
1. This result applies to the distribution of the number of accidents, not to the distribution of the number of victims (unless there happens to be at most one victim per accident) or of other outcomes of accidents.
2. The role of independence is important in this result. It should be quite reasonable to assume that the outcomes of the different events are inde-pendent, otherwise the result may not hold.6
3. When accident registration problems are to be considered, the concept of ‘a small probability of resulting in an accident’ can be replaced by ‘a small probability of resulting in an accident and being registered’. The reg-istration should not be selective.
4. A different but no less important accident registration issue is that usu-ally only accidents exceeding a certain level of severity are considered. In that case ‘a small probability of resulting in an accident’ can be replaced by ‘a small probability of resulting in an accident with a certain severity and being registered’. Even if these probabilities are different for each trial, the distribution of the resulting number of accidents still tends to the Poisson distribution.
5. An alternative approach to deriving the Poisson distribution for counts, based on counting processes (in real-time), requires that the (real-time) registration system cannot be saturated by the accident process. Although this is mostly relevant to Geiger-M ¨uller counter like systems, its potential effects should not be ignored in road safety analysis. For instance, police districts may allocate limited resources to less severe accidents, and may simply stop registering them once a certain threshold is exceeded, thus truncating distributions.
6Outcomes resulting from the same event, such as the number of persons killed, seriously
injured, lightly injured, and unharmed in one accident, are likely to be dependent (see Chap-ter 4 in this thesis, or see, Bijleveld, 2005). Furthermore, it should be noted that it is the events that should be independent, not the probabilities, which may depend on N. Accidents that are cause by other accidents are in most cases considered part of the initial accident.
2.2.2. The distribution of accident counts
The statistical properties of accident counts mentioned in the previous section only apply for large numbers of trials. For road safety analysis this means that the distribution of accident counts will become indistinguishable from a Pois-son distribution only in the limiting case. Thus, in practice accident counts will never be precisely Poisson distributed. The limit character of the properties of accident counts is due to the large number of trials on which it is based. If a count is based on many, many trials, it is likely that its distribution is indistin-guishable from a Poisson distribution. For instance annual, national counts of a general type of accidents will practically be Poisson distributed. However, a problem arises when the actual number of trials is not so large. This is the case when a rare accident type is studied for example, or road sections with small traffic volumes. For more discussion in the situation in which the number of trials is not very large, see in particular Lord, Washington, and Ivan (2005).
2.2.3. Over-dispersion
As mentioned in Hauer (2001) over-dispersion is commonly encountered in road safety analysis: “After the unknown model parameters are estimated, one usually finds that the accident counts are ‘overdispersed’. That is, that the differences between the accident counts and model predictions, are larger than what would be consistent with the assumption that accident counts are Poisson distributed” (Hauer, 2001, p. 799). This phenomenon also occurs in settings where one would consider the distribution to be practically identical to the Poisson distribution. The problem is with the replications used in the generic model as described by Hauer (2001). Even if the accident distribution would be indistinguishable from the Poisson distribution, replications would never be under identical conditions. In other words: replications will be drawn from a different Poisson distribution each time and the replications will there-fore vary more than would be expected when the replications are sampled from the same (Poisson) distribution. A more extensive discussion from the viewpoint of different probabilities can be found in Lord et al. (2005). See e.g. Hauer (2001) and the references therein for more on how overdispersion can be estimated. The methods applied in this thesis never assume the prediction to be fixed, rather the methods assume the predictions to be subject to error. This situation is comparable to assuming that “replications would never be under identical conditions.” as remarked just above. In a general context, this approach is called a mixture approach to generalised count models, of which the negative binomial model (a Poisson-Gamma mixture) is one example. In all cases in this thesis it appears that no overdispersion parameter in addition to the mixture needs to be estimated. The approach where the amount of
dis-persion in addition to the prediction error is estimated is taken in this thesis. More general forms and other distributions can be considered in Chapter 7.
2.2.4. Gaussian approximations
The distribution of the number of accidents is often approximated by the Gaus-sian distribution. This approximation is also used in the models presented in this thesis, except for those in Chapter 7. The common procedure is to assume (first approximation) a Poisson distribution with parameter λ, and then to ap-proximate (second approximation) the Poisson distribution with a Gaussian distribution with mean parameter and variance parameter equal to λ. In mod-elling situations, the expected value λ is often estimated by the model predic-tion of the observed count. When no statistical model is available, the expected value λ is usually estimated by the observed count. Sometimes an amount of ‘overdispersion’ is added to the variance parameter, that is a constant value is added to λ.
It should be noted that the approximation of the Poisson distribution by a Gaussian distribution deteriorates when the accident counts are getting smaller. There is no general rule as to what value the counts should exceed in order for the approximation to be sufficiently reliable since that depends on the applica-tion and the required accuracy. It should also be noted that for many types of statistical models count data versions are available. Therefore in many cases a Gaussian approximation is no longer needed.
2.2.5. The distribution of victim counts
Given that an accident occurs, determining the distribution of the number of victims resulting from that accident is difficult. Obviously the distribution is dependent on the number of persons involved in that accident7. When done at all, approximations can be made based on compound distributions. It can however be assumed that the victim counts are overdispersed, more so than accident counts. The amount of overdispersion depends on the variation of the number of victims per accident (see Chapter 4 in this thesis, or Bijleveld, 2005). This means that victim counts from accidents that rarely involve more than one victim, will be less ‘extra’ overdispersed than victim counts from accidents that (more) often involve more than one victim, as compared to the overdispersion of the number of accidents.
7Which is unfortunately not known in the Netherlands, since unharmed participants in an
Generally, the distribution of victim counts has no influence on the distribu-tion of accident counts. In practice however often accidents exceeding a cer-tain severity level are registered or used in an analysis. If the distribution of victim counts changes in a way that the probability of exceeding the severity level decreases, the expected number of accidents will decrease, and thus the accident count distribution will change.
2.2.6. The relation between trials and exposure
As discussed above, the number of trials N plays a dominant role in the ex-pected number of accidents. Assuming the pivalues to be sufficiently regular,
the expected number of accidents is proportional to the number of trials since
λN = ∑iN=1pi. The number of trials is therefore probably closest to the true
exposure we can get. Unfortunately, the value of N is generally unknown.
Since N and the pi are unknown all need be estimated. Given the fact that es-timation of each individual pi is impractical, we assume a homogeneous dis-tribution of the pi. In addition, the data are used in aggregate models, which means that aggregate counts of accidents are available as well as aggregate es-timates of exposure. This means that given and estimate of N, only the average of R (the pi) can be determined.
No general guidelines are available on how to estimate either N or R. As N is obviously somehow dependent on the scale of road traffic, and the number of accidents is dependent on both N and R, the approach taken in this thesis is to estimate both N and R by means of two (approximate, effectively stochastic) equations:
(
Scale of road traffic ≈N
Number of accidents ≈N×R. (2.2)
See Chapter 3 for further details on how N and R are estimated in this the-sis, an approach which allows for nonlinear relations. The nonlinear nature of the relations is suggested by the discussion in the next section. Note that (2.2) implies that any alternative estimate of N proportional to N cannot be distin-guished from N.
The research question determines for which kind of accident the ‘Number of accidents’ needs to be analysed. The research question also determines, given available data, the optimal choice of what quantity can best be used to measure the ‘Scale of road traffic’ (see also Hauer (1995) and Hakkert and Braimaister (2002)). All methodology presented in this thesis is independent of choices for
