category: literaturenote citekey: stipdonkexpectednumberroadtraffic2010 title: The expected number of road traffic casualties using stratified data authors: "Stipdonk, Henk; Wesemann, Paul; Ale, Ben" year: 2010 date: 2010-00-00 2010 doi: 10.1016/j.ssci.2010.04.010 publication: Safety Science url: "https://doi.org/10.1016/j.ssci.2010.04.010" zotero_key: C6GZ8NEW zotero_storage: KQ7XJRZH collections: transpordi_planeerimine folder: 001_artiklid firstAuthor: "Stipdonk, Henk"

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Safety Science

The expected number of road traffic casualties using stratified data

Henk Stipdonk a,*, Paul Wesemann ^a , Ben Ale ^b

^a Road Safety Assessment Department, SWOV, P.O. Box 1090, 2260 BB Leidschendam, The Netherlands
^bDepartment of Safety Science, Delft University of Technology, P.O. Box 5015, 2000 GA Delft, The Netherlands

article info

Article history: Received 18 August 2009 Received in revised form 6 April 2010 Accepted 11 April 2010

Keywords: Road safety Forecast Stratification Trend Model

abstract

Road safety policy plans often require robust calculation of the expected number of road casualties in a certain target year. The relevance of such estimations should be measured by their power to influence and support safety policy makers. Thus, techniques to evaluate the safety developments and the estimating methods must be sound, robust, and preferably accepted by both policy makers and the scientific community. In this paper, we concentrate on choosing an appropriate model used for the calculation, rather than on statistical techniques. We calculate a casualty rate from casualty data and mobility (distance travelled) data, which is extrapolated and subsequently multiplied by an expected future distance travelled. After correction for separately assessed effects of additional safety measures, the number of casualties is estimated. We investigate a method where this is done after both mobility data and casualty data are stratified into properly chosen subsets. Projecting these different trends generally leads to a result that differs from the projection of the aggregated data. Also, stratification enables incorporation in the estimation of explaining factors or additional measures related to a specific subset of the casualties. The principles of stratified projections are illustrated by three Dutch projections which were carried out between 2006 and 2008. Also, some preliminary results of further research on stratification are given. The results imply that the rates of change in casualty rate for different traffic modes or driver age, are not necessarily equal. We propose that these specific decreasing trends are a consequence of external influencing factors.

1. Introduction

National and international road safety policy plans often need a robust forecast of the number of casualties, fatalities or in-patients or both, in a certain target year. Towards zero, ambitious road safety targets and the safe system approach (OECD, 2007) describes the role of targets to achieve ambitious road safety goals. It illustrates how targets can be set and met by analysing data, calculating projections, estimating effects of measures and applying cost-benefit analyses. In this process, the assessment of a reliable forecast plays an important role.

As traffic crashes are a stochastical process, any forecast will always be liable to fluctuations as a consequence of chance. The best that can be done is to determine the expected number of fatalities. Strictly speaking, a forecast cannot be better than the best possible calculation of the expected number of casualties. In this paper, we refer to forecasts in this sense.

E-mail address: henk.stipdonk@SWOV.nl (H. Stipdonk).

Road safety development analyses can be used to inspire new road safety strategies (De Ceuster, 2005; Stipdonk and Wesemann, 2007), or to evaluate the consequences of a new road safety strategy or specific safety measures, either ex ante (forecasting the development) or ex post (explaining the development after it has taken place). An example of the first possibility is given by Broughton (2009), who describes a general way of making such forecasts. Another example is the assessment of the expected safety effects of road pricing in The Netherlands (Schermers and Reurings, 2009). This report illustrates how the expected reduction in mobility and, as a consequence, in the number of casualties, plays a role in the political decision making process of Dutch mobility policy. Elvik (2005) gives an example of an ex post evaluation which shows the difficulty of attributing all past developments to actual changes in safety policy.

Several difficulties are encountered in forecasting road safety casualties. It is to be decided how to choose an appropriate model which contains plausible relations between the relevant factors and the expected number of casualties to be calculated. The most important focus of this paper is to describe our current intermediate results of our research to develop a possible method to develop such a model. Also, there are important technicalities involved in

^* Corresponding author. Tel.: +31 703173370, mobile: +31646196673; fax: +31 703201261.

calculating the best forecasts. Further, it is important to determine the actual effects of specific factors that change over time and influence safety (changes in mobility, ageing population, weather effects, safety measures) on the past developments. When such relations are completely unknown, extrapolation of these trends can be meaningful if it is safe to assume that the relationships are stochastic and these relationships can be considered to be the transfer function of an otherwise black box. It remains unknown if such an assumption is solid.

In this paper, the safety assessments of either past or future road safety developments has two objectives: to reveal effects of safety measures that were taken in the past, and to substantiate additional safety measures to further improve safety in the future. To be acceptable, we believe that the technique to evaluate the development of safety and the forecasting methods needs to be sound, robust, and, preferably, accepted by policy makers and accepted by the international scientific environment.

2. Approach

2.1. The basic three step approach

The general purpose of a road safety forecast is to accurately calculate the expected number of casualties in a future year, often in the presence of some strategic road safety policy programme. Broughton (2009) distinguishes three steps to carry out such a forecast. In the first step, casualty rates (casualties per distance travelled) in the target year are estimated, based on a trend analysis of past years. In the second step, the result of the first step is multiplied by the predicted distance travelled in the target year. This results in the baseline forecast (the number of casualties in the target year i.e. the projection of the baseline trend). In the third step, the effects of new road strategy measures are estimated. In The Netherlands, SWOV has used this method in several forecasts (Wesemann, 2007, 2008; Aarts et al., 2008; Schermers and Reu rings, 2009). In this paper, we adopt this three step approach as a general principle.

To apply this method, several problems need to be solved. The first problem is how exactly to extrapolate the data from the past to forecast the expected casualty rate for the target year. Forecasting requires a mathematical expression of the casualty rate as a function of time. A priori, this mathematical expression is unknown, and the goodness of fit is not a sufficient indication of the correctness of any mathematical expression (Elvik, 2005). The second problem is a consequence of the fact that the baseline trend of the casualty rate is assumed to contain implicitly the effects of many measures that are not explicitly included in the model (Broughton, 2009). The problem is how to decide to what extent the future development of the baseline trend, which is supposed to contain a continuation of existing safety policies and measures, requires future road safety improvements which are necessary to realize this development. To put it differently: the problem is to determine the amount of effect of future safety measures that is necessary to maintain the current baseline trend. The third problem is to estimate the expected additional effect of specific measures in truly new policy.

The first two problems are, in our view, connected. If we ever succeed in correctly expressing the past and expected developments of the casualty rate mathematically, the precise effects of past safety measures must be known as a function of the relevant factors. In that hypothetical and ideal case, time would no longer be a necessary parameter in the model: the casualty rate follows from the factors and the estimated parameters. However, as long as we have not reached this ideal situation, the development of the number of fatalities will contain an unexplained component which represents the unexplained development in time. The exact functional form of this component can only be guessed, and the safety measures that are responsible for the development of this component are, by definition, unknown.

This paper will discuss a possible approach to estimate this unexplained component, such that the baseline forecast depends on unexplained components as little as possible. Our general approach is based on stratification (i.e. disaggregation) of crash data by relevant factors such as traffic mode, age or road type.

Conceptually, the third problem is the most simple one, although when put into practice it is laborious and difficult to manage. The effect of safety measures, either in the past or in the future, can be determined by thoroughly analysing the nature of the corresponding measures, and their quantitative relation to the expected number of casualties. In principle, any measure affects a subset of the total number of casualties only. For this subset, which we will call the domain of the measure, an analysis should reveal the expected effect of the measure on the number of casualties in that domain. Thus, every measure requires a separate analysis that results in a specific partial model: a quantitative relation between some parameter that quantifies the measure and the number of casualties saved. In traffic safety literature, many safety measures and their effects have been described. For example: Evans (1986) calculates the risk reduction as a result of seat belt use. The outcome can be used to estimate the effect of enhanced seat belt use in relation with a gradual increase in seat belt use due to seat belt reminders (Lie et al., 2008). In this paper, we will not discuss modelling the effect of specific safety measures. The issue is well described by Sigrist (this issue). Instead, we will attempt to deal with the first two problems.

2.2. Possible improvements of the method

Apart from suitable stratifications, the forecasts can be further improved if it is known how to add the effects of explanatory factors (such as safety measures, demographic changes, weather influence) in the past. In fact, this means replacing part of the obscure and unexplained trend by an external factor. This way the baseline becomes more accurate. A possible approach could be to formulate and test hypotheses, corresponding to specific stratifications, that show a decreasing trend. Another possibility is to use existing and well established relations between a specific Safety Performance Indicator (SPI) and a specific road safety domain (e.g. the proportion of car occupants wearing seat belts vs. the number of casualties among car occupants in crashes where a seatbelt can be effective). This also requires sufficiently accurate data on the SPI.

Stipdonk and Berends (2008) have shown that between 1950 and 2005 the development of the number of any subset of casualties in The Netherlands generally depended strongly on the conflict type, characterised by the traffic mode of the casualty and (if any) the traffic mode of the other vehicle. They showed that both the development of the number of casualties and the casualty rate in single vehicle car crashes differed considerably from the development of that of, for example, car–car crashes, or pedestrian-motorcycle crashes. Thus, any expression that describes such development should preferably reflect these differences, and contain the parameters that explain these differences. Thus, stratification of crash data by traffic mode, or rather, by casualty type (defined by the combination of traffic modes involved), makes it possible to establish completely different trends. These results indicate that the total number of casualties is a sum of many components, each following substantially different trends. Maier and Ahrens (2009) have followed these lines to estimate the expected number of casualties in Germany for 2015 and 2020.

Even when using simple negative exponential trends to model the different components, there are two arguments in favor of projecting stratifications as compared to just fitting a simple negative exponential trend to the sum directly. Firstly, stratification enables using data on distance travelled, stratified by traffic mode. Secondly, if the resulting casualty rates show very different trends (e.g. some are constant whereas others are decreasing), these differences can be taken into account when calculating any expected number of casualties. A third advantage could occur when different trends coincide with differences in the way these trends are understood (e.g. when motorcycle casualty rates decrease less than car casualty rates as a consequence of an increased seat belt use affecting car occupants only). This way, stratification allows us to add explanatory factors that are specific for a specific stratification. Our suggestion to solve problems one and two, therefore, is to sufficiently stratify crash data, such that different risk developments are mathematically described accordingly, by different trends.

2.3. An indication of the improved correctness of the model

By applying properly chosen stratifications, we hope to find better forecasts, in the sense that the calculated uncertainty of these forecasts is as little as possible. As an illustration of the effect of which method is chosen for the projection, on the outcome and uncertainty of the projection, we calculated the expected number of fatalities in 2020 in three ways. The results are shown in Fig. 1. The mobility data used are motor vehicle data extracted from Statistics Netherlands (2000), which are based on a combination of surveys, traffic counts and other sources. The results are obtained with state space modelling as described by Commandeur and Koopman (2007), applying the above approach, and without any stratification.

The three different ways to calculate the expected number of fatalities in 2020 are:

1. We assume that the expected mobility is a stochastic quantity, its uncertainty being defined by the fluctuations in the mobility data. The model further contains a mathematical intervention, to take care of a sudden decrease in the number of fatalities in The Netherlands in 2004 of more than 20%. The purpose of this intervention is to demonstrate its effect on the calculated forecast.
2. As 1, but here we assume that the projected mobility for 2020 is exact.
3. As 2, but the mathematical intervention is removed.

The results are shown in Table 1.

The uncertainty in the expected number of fatalities decreases by more than a factor of 3 when assuming that the expected mobility is an external and exact factor. The results further show how the presence of an intervention in the model influences both the forecast and its uncertainty.

In the ideal case of a model that explains, and correctly predicts the development of the number of fatalities in terms of all relevant external factors, the variance r² of the prediction should be equal to the prediction. Our aim is to find a model that brings down this variance by applying stratifications and subsequently searching for relevant external factors. This does not guarantee that we will eventually find the perfect model and thus solve the first two problems mentioned above. However, the variance of the prediction is an independent indication of model correctness that may help to find improvements of the model. This can be done by adding stratifications and by introducing the influence of relevant external factors.

stochastic projection of distance travelled M,

deterministic distance travelled M mathematical intervention on risk in 2004

deterministic distance travelled M no mathematical intervention on risk in 2004

Fig. 1. State space projection for the crash rate and the mobility in 2020, based on crash data between 1950 and 2007, and mobility data from traffic counts, between 1950 and 2000, with its 68% uncertainty interval. Top and middle panel: in 2004 a mathematical intervention was added to the model to allow for the sudden decrease in 2004. Lower panel: no intervention in 2004. In the top panel, the future mobility data were projected as well, in the middle and lower panel the future mobility data were assumed to be exact.

Table 1

Results of three state space projections for the crash rate and the mobility in 2020, based on crash data between 1950 and 2007, and mobility data from traffic counts, between 1950 and 2000, with its 68% uncertainty interval. In the first result a mathematical intervention was added to the model to allow for the sudden decrease in 2004. The expected mobility was considered as a stochast. The second result was similar to the first, but the expected mobility was considered exact. The third result was similar to the second, but the mathematical intervention in 2004 was removed.

	1st result	2nd result	3rd result
Expected number of fatalities	480	480	440
Expected number +r	930	600	560
r Expected number	250	390	340
68% confidence interval	680	210	220

2.4. How much stratification is necessary and how much is sufficient?

It is difficult to say a priori which stratifications are necessary.

causes, as it is likely that different causes indicate that they are sensitive to different measures (Hale, 2001) and therefore show different trends. Unfortunately we know little of the actual causes of road accidents. Thus, other criteria for choosing the proper stratifications are required. There are several ways to achieve this. One consists of looking for stratifications with different trends, where we compare the parameters associated with these trends, and decide if their difference is statistically significant. It must be noted that in this case an arbitrary choice must be made concerning the preferred significance level. Another way is to look for systematic misspecifications in the trend for aggregated data: if a supposedly negative exponential trend is actually a sum of two different trends (e.g. a constant + a negative exponential), the aggregate trend analysis will in principle show a misspecification, which indicates that some (unknown) stratification may be necessary.

At present no theoretical framework is available that helps to choose the factors for which a stratification improves the model. To decide on stratification for different crash groups, we consider two indications, and two conditions:

1. Crash groups with different developments of the crash rates over time.
2. Crash groups with different crash rates, combined with different development of the mobility, e.g. in the case of a shift in mobility between different crash groups over time.
3. Availability of stratified mobility data, as a condition for stratification.
4. A sufficient number of crash data, as a condition for stratification.

In The Netherlands, stratified mobility data can be obtained from Statistics Netherlands (NTS, 1985–2008). These data are based on an annual household survey, currently with a response of approximately 30,000 households, corresponding to ca. 70% of the current size of the survey. The response decreased from 60,000 in 1999 to 30,000 in 2005, due to a decrease in the size of the survey. The results of the survey are used by Statistics Netherlands to calculate the mobility for the entire Dutch population. Traffic of non-inhabitant drivers is not included, nor is freight

Fig. 2. Annual per capita mobility by age, for male and female bicyclists, mopedists and passenger car drivers. The left panel shows the data for 1995 and 2008, the right panel shows averaged 3 year data (1995–1997 and 2006–2008).

traffic. Under the assumption that the fraction of mobility by noninhabitants is constant, indeed the accuracy of the data enables stratification by traffic mode, age and gender, at least to some extent. To illustrate this, Fig. 2 shows the estimated per capita mobility by age and gender of drivers of three traffic modes (passenger car, bicycle and moped), and for three consecutive years. The figure indicates that the estimated data indeed fluctuate, partly due to chance. These fluctuations can be considered as an upper limit of the inaccuracy of these estimates, as these variations may also stem from fluctuations in the actual mobility. The figure also shows a consistent age-dependent pattern, and differences between male and female mobility by age and gender, that are significantly more pronounced than the size of the fluctuations. Thus, these age- and gender-dependent patterns are informative about actual differences in male and female age-dependent mobility. The majority of these data are available through the SWOV-website (NTS, 1985–2008).

3. Development of the method, applied to Dutch data

Between 2006 and 2009, Wesemann et al. (this issue) carried out three road safety forecasts for The Netherlands. In 2006, Wesemann (2007, 2008) forecasted the expected number of road fatalities and hospitalized injuries for the year 2020, based on road safety and mobility data up to 2005. Knowing that stratification was important, he experimented with three different stratifications: by traffic mode, by age and by road type. Each method resulted in a different expected number of casualties in 2020. This result is a consequence of the fact that the different subsets of casualty types show different developments in time, which indicates that the development is indeed not uniform. This result can be seen as an indication of the necessity of stratification.

At that time, a recent drop in the number of fatalities in 2004 (-20% as compared to 2003) complicated the forecasts. Thus, Wesemann had to decide how to incorporate this drop in his analysis. Three different scenarios were analysed. In scenario 1 the drop was considered to be an incidental, temporal drop, to be undone during the next year or years. In scenario 2 the drop was considered to be a unique and incidental, but permanent downward step in an otherwise steadily decreasing trend. In scenario 3 the drop was attributed to chance: it was treated as the outcome of the combination of known and unknown processes, to be considered as a regular part of the traffic process. These different scenario's are illustrated in Fig. 3, which shows simple negative exponential trends of the annual number of fatalities (hence not corrected for mobility) between 1987 and 2005. For reasons of completeness, data of 2006–2008 are added to the figure. The figure shows that the projection for scenario 2 is the most optimistic for 2020, although scenario 3 can be expected to be the most optimistic in the long run.

In 2008, two more forecasts were carried out, both using stratification of casualty rate by traffic mode. One forecast was aimed at estimating a new ambitious and realistic target for the number of fatalities in 2010 (Aarts et al., 2008), and the other was an analysis of the road pricing effects (Schermers and Reurings, 2009), aimed at calculating the expected number of fatalities in 2020, with and without road pricing to be introduced in 2012. The ministry considers many different pricing schemes, such as pricing specific routes, or pricing roads to and from the main cities, either during peak hours only or around the clock, and, for all variants, whether they should only be applied to the main infrastructure or to rural roads also. In all variants, road pricing is expected to decrease the mobility during peak hours. Assuming an invariant casualty rate, the safety increases on the relevant roads.

Fig. 3. Negative exponential trends of the yearly number of fatalities (hence not corrected for mobility) between 1987 and 2005, for three different scenarios to incorporate the sudden decrease in 2004. For scenario 1 the trend was calculated without the use of the 2004 and 2005 data. For scenario 2 the trend of scenario 1 was multiplied by an appropriate factor less than 1, to adjust for the sudden decrease in the number of fatalities in 2004. For scenario 3, no special treatment was given for the data of 2004 and 2005. For reasons of completeness, data of 2006– 2008 are added to the figure.

In these forecasts, it was chosen to stratify by casualty type: for single vehicle crashes the casualty type is defined by the traffic mode of the casualty; for crashes between more than one vehicle, the casualty type is defined by the traffic mode of the casualty and the most important other vehicle. Thus, the casualty type is determined only by the vehicles involved. Although other stratifications (e.g. by road type or by age of driver) are still considered important, they have as yet not been applied. This is because we have experienced that all relevant stratifications should actually be carried out simultaneously (Wesemann, 2007, 2008), and not separately, as was done earlier by Wesemann. Combined stratifications are indeed technically feasible, as was shown by Stipdonk (2008) and Van Norden (2009).

In Aarts et al. (2008), only fatalities were analysed, stratified into several casualty types or groups of casualty types. Ten casualty types, each contributing significantly to the total number of fatalities, were analysed separately, following the approach described above. The remaining casualty types were grouped with the same dominant traffic mode for which there was forecasted mobility data available (passenger car, van, motorcycle, truck, moped). For fatalities of casualty types that could not be related to any dominant traffic mode, such as bicycle–bus or pedestrian-bicycle, the casualty rate could not be analysed. Instead, the total number of fatalities was projected for this group. As the number of fatalities, 117 annually on average, has approximately been constant for more than 30 years, this number was kept constant in the forecast. This was the only group for which the number of casualties was analysed instead of casualty rates. Hence, the number of fatalities was considered to be independent of mobility changes. For all other casualty types or groups, a negative exponential trend was estimated. A similar approach was followed by Maier and Ahrens (2009).

Schermers and Reurings (2009) altered the applied method. This was necessary because the Dutch Ministry of Transport requested a distinction be made between different road types. In general, this would require using mobility data by road type, and such data is not available. For this specific analysis, the authors estimated the mobility development, stratified by road type and traffic mode. We will not give a detailed description of the methods used here.

The several approaches mentioned above thus show different results for different partially stratified analysis. Partial stratification by age leads to a different expected number of casualties than partial stratification by transport mode. This indicates that some or all of the partial stratifications are in itself suboptimal. We propose to solve this problem by carrying out all relevant stratifications simultaneously. Thus, each separate subset is extrapolated separately. Therefore, there is no longer any discrepancy between different stratification schemes.

4. Further stratifications to be applied in the near future

4.1. Some preliminary results

Van Norden (2009) analysed several casualty types, for example bicycle–car crashes, which she further stratified by age of both drivers. Due to the unavailability of mobility data stratified by individual age years at that time, their methods were not yet suitable to be applied to all relevant casualty types taken together (which is necessary when forecasting the total number of fatalities or in-patients). However, mobility data for individual driver age have become available recently, enabling further refinement and application of this method. The method implies that single vehicle crashes are stratified by driver age, and two-vehicle crashes are stratified by both the driver age of the traffic mode of the casualty, and the driver age of the opponent; crashes with a casualty in each vehicle are counted twice. In the assessment, the mobility, stratified by driver age, is in fact a mobility per capita, thus enabling the usage of past and future development of demographics in the calculations. In case of two-vehicle crashes, this method takes the mobility of both vehicles involved into account. The method involves smoothing techniques, assuming gradual changes in per capita mobility, and in casualty rates over time and for driver age. For two-vehicle crashes, the smoothing is applied to the number of crashes divided by the relevant distance travelled by both traffic modes, i.e. crash rate per km (i.e. crashes per km per km). Thus, the number of casualties is implicitly assumed to be locally proportional to distance travelled by age of driver of both modes.

By analysing the per capita mobility, we correct for changes in the mobility over time that are caused by demographic changes. In Stipdonk (2008) it was shown that for many driver ages and traffic modes the per capita mobility is not very time dependent. Thus, for forecasts of the mobility of pedestrians or mopeds reasonable forecasts could possibly be achieved by just applying the demographic development to a constant or slowly changing age-dependent per capita mobility. Furthermore, it is assumed that in general mobility is not very different for similar ages (e.g. 42 year old drivers are supposed to drive approximately equal distances as 43 year old drivers). Thus by smoothing the per capita mobility data, we are able to estimate the actual mobility stratified by age group and calender year.

Using stratified mobility data makes it possible to take different mobility and risk behaviour into account. By doing so, we make use of available information from the mobility database and the accident database (i.e. information on the age of drivers) that would be otherwise undisclosed and unused. Even if stratification results in small numbers with high variance in the database, this should generally enable an improvement of the model.

In addition, Van Norden (2009) showed that the development in time of the crash rate associated with crashes of drivers of a specific age, depends on driver age. To calculate this crash rate, she smoothed the risk with respect to age, and allowed for a (negative) exponential change in risk over time. In general, for young drivers, the resulting crash rate decreases less than for experienced drivers. This suggests that safety measures are not effective for all drivers. This may be related to the nature of the safety measures involved.

In this analysis it was not possible to give confidence intervals. For an accurate calculation of confidence intervals we would have to analyse data by adding several calender years and age groups. It would then be possible to present results of risk changes by driver age over time by calculating this risk for periods of 5 years, and for different age groups. This way, it would become possible to show the same results, for discrete age groups and time intervals of several years, together with their variances. In principle, the results would be the same. We have chosen to stress the principles of disaggregation instead of calculating accurate results and confidence intervals.

The results of Van Norden (2009) can be illustrated by the following examples:

1. The risk for young car drivers of being involved in a single vehicle car crash is high and it hardly decreases, while there still is some decrease of the risk for elderly drivers. The same holds for car–car crashes. The risk of being involved in a car–car crash is higher for young drivers, and does not decrease much, whereas for elderly drivers, the risk of being involved is decreasing. This effect may be due to the fact that measures such as enhanced enforcement mainly have an effect on experienced drivers. Young drivers first need to be exposed to some of this enforcement, while elderly drivers, based on their experience, simply respond to a general announcement of enhanced enforcement. In addition, the average experience of elderly drivers is likely to have increased over the last decades, whereas young drivers always have had hardly any experience at all. This is illustrated in Fig. 4, which shows the estimated casualty rate for car drivers (KSI/km) in single vehicle car crashes, by age of driver, between 1987 and 2007. The figure illustrates that during these 20 years the estimated casualty rate has been decreasing less for young drivers (a risk reduction of less then 20% for driver ages below 25) as compared to more experienced drivers (a reduction of almost 50% for driver ages between 40 and 70). The data are based on Van Norden (2009).
2. For cyclists, the risk to be involved in a bicycle–car crash was seen to have decreased most for children (6–12 years), and much less for other cyclist ages. In 1987, the children's risk was much higher than that for other ages, and between 1987 and 2007 it decreased by more than 70% for cyclists younger than the age of 10. For 30-year-old cyclists, this reduction was 13%. Thus, if this decrease was a consequence of safety measures, they were not uniformly effective for all cyclist ages. To achieve further improvement of cyclist safety in bicycle–car crashes, it is likely that new measures that also affect 30-year-old cyclists are to be thought of. Today, the risks for 10-year-old cyclists is about 1.3 times that for 30-year-old

Fig. 4. Estimated casualty rate for car drivers (KSI/km) in single vehicle car crashes, by age of driver, for 1987 (solid line) and 2007 (dashed line).

cyclists. In 1987 this difference amounted to a factor of 4.3 This may be due to the safety around schools which has been improving during the last 15 years. This is illustrated in Fig. 5, which shows the estimated casualty rate for cyclists (KSI/km) in bicycle–car crashes, by age of cyclist, between 1987 and 2007. The data are based on Van Norden (2009).

Furthermore, the involvement in serious crashes by age is influenced by different mechanisms, such as vulnerability and experience. This can be illustrated by comparing the risk to be involved in a car–car crash as an in-patient (Fig. 6, solid lines) with the risk to be involved in a car–car crash as a driver of the opponent car (Fig. 6, dashed lines), for crashes with at least one injured driver. It shows the involvement of elderly drivers in car–car crashes as the injured driver strongly increases with age, whereas this is much less the case for their involvement in car–car crashes as the opponent driver. For younger drivers this difference is much less pronounced. The distinction between these two ways a driver can be involved in a car–car crash, supports our approach which intends to apply this distinction in the future models. Mark that for younger drivers, the risk of being involved in a car–car crash as an injured driver is clearly lower than being involved in a single vehicle car crash. For elderly drivers the car–car risk is higher.

4.2. Our stratification scheme for the near future

It is our intention to improve our forecasts where possible. One approach could be to stratify the data by groups for which better –

Fig. 5. Estimated casualty rate for bicycle drivers (KSI/km) in bicycle–car crashes, by age of cyclist, for 1987 (solid line) and 2007 (dashed line).

Fig. 6. Estimated casualty rate for car drivers in a car–car crash, either when they are involved as injured driver (solid lines) or as the driver of the opponent car (dashed lines). Curves are given for 1987 and 2007.

or more plausible – forecasts can be made. A balance has to be found between the advantages of further stratifications and data reliability and model complexity.

There is no theoretical objection against forecasts using casualty rates and mobility data stratified for many variables. However, in practice the level of stratification will, in practice, be determined by the available mobility data, and the available crash data. In The Netherlands, the resolution of the available mobility depends on the traffic mode. For several traffic modes including cycling and car driving, National Traffic Survey (NTS) data are available. For those traffic modes, some further stratification by driver age and gender is also possible. This stratification is also desirable because of the different casualty rates. For other traffic modes, no mobility data are available. Further stratification, such as by road type is more difficult to carry out due to lack of mobility data by road type.

We want to apply stratifications that meet the indications given in Section 2.4, i.e. either significantly different crash rate development or a combination of significantly different crash rates and significantly different mobility development over time. Also, the two conditions, i.e. availability of mobility and crash data, have to be met. If these conditions allow some further stratification, we intent to research if the indications show that stratification is indeed called for. This gives rise to the following scheme, which is derived from the availability of the mobility and crash data. The first part of the scheme goes into the available mobility data, and the second part describes how to cope with small numbers of crashes or small mobility survey samples.

1. The available mobility data enable the level of stratification. This availability divides the traffic modes into three groups, described below. A group 1a for which NTS-mobility data is available, a group 1b for which no NTS-mobility data is available, but useful substitutes can be used instead, and a group 1c for which no mobility data are available. For each group, the mobility data is incorporated as follows: For casualties in single vehicle crashes with traffic modes from group 1a or 1b, the mobility data are used to calculate the casualty rate which is to be forecasted. For casualties in two-vehicle crashes in which one of the traffic modes involved are from groups 1a or 1b, the mobility data of that mode is incorporated in the analysis, where the number of casualties is divided by the mobility of that mode to arrive at a casualty rate which is to be forecasted. For casualties in two-vehicle crashes in which both of the traffic modes involved are from groups 1a or 1b, the mobility data of both modes are incorporated in the analysis, where the number of casualties is divided by the mobility of that modes to arrive at a casualty rate per km, which is to be forecasted.
- 1a. Traffic modes, for which NTS-data are available. These are pedestrian, bicycle, moped (including slow moped), motorcycles and passenger cars. For those traffic modes, stratification by age and gender is also possible, although it depends on the traffic mode if the available data allow stratification by individual age year or if perhaps age groups are to be made. The majority of fatal and serious crashes in The Netherlands involve at least on of these modes, e.g. today in 70% of all fatal crashes a passenger car is involved (Stipdonk and Berends, 2008).
- 1b. Traffic modes for which no mobility data are available in the NTS household survey, but for which an approximation of distance travelled can be used (road traffic volume statistics, fleet size data, sales data). These are small vans or trucks. For those traffic modes, one of these approximations will be used, assuming that a suitable approximation is proportional to the actual mobility. As a consequence, the resulting casualty rate forecasts concerning these modes will be

vulnerable to unknown changes in the relation between the used approximation and mobility itself. For example, if fleet size is used as an approximation of mobility, a change in the mean annual distance travelled per vehicle will show as a change in the casualty rate.

1c. Crashes between traffic modes for which no mobility data or approximation is available. These are crashes such as between buses and miscellaneous vehicles. For those traffic modes the number of casualties is modelled without any mobility. As a consequence any change in mobility of these modes that consequently affects the number of casualties involving that mode, will remain unnoticed. The calculated casualty rates and numbers of casualties cannot be corrected for those changes.
2. For some small groups, it is necessary to add groups of stratified data together, either by aggregating age years, traffic modes, gender, or even calendar years. This is the case for small population groups, with either few crashes, or where the sample size of the mobility data becomes to small and the in-sample variance cannot be trusted to give a reasonable indication of the population variance. This will depend on the sample error variance of the available mobility survey data.
- 2a. For age groups that are few in population (e.g. for very old persons) and thus have very little distance travelled and are involved in a low number of crashes, stratification by age is not useful, in which case these age groups have to be aggregated.
- 2b. For many two-vehicle casualty types, the annual number of casualties in The Netherlands does not allow stratification by age and gender for individual age years for each separate casualty type. In those cases, it is usually possible to analyse the driver age dependence for several casualty types simultaneously. An example are casualty types with a passenger car as opponent. In The Netherlands, the age dependence of those casualty types is independent of the traffic mode of the casualty. This is illustrated in Fig. 7, where the number of seriously injured in a crash with a passenger car as the opponent, is given for the driver age of the opponent. The age dependence seems to a large extent of the same form, except for a different constant between different traffic modes. We have as yet not decided which casualty types will be analysed simultaneously this way.
- 2c. Not all casualty types will be extrapolated separately. Casualties in crashes where all traffic modes involved are a member of group 1c will be analysed as one group. Crashes

Fig. 7. Number of fatalities and serious injuries in The Netherlands, between 1976 and 2005, in a crash with a passenger car as the opponent traffic mode, by age of the driver of the opponent passenger car. Data are shown for nine different casualty transport modes, on a logarithmic scale.

between both traffic modes from group 1c and one other traffic mode, will be analysed as one group. Some more casualty types might be analysed together. This still has to be decided.

This scheme describes some stratifications that, to our view, are necessary: we know that traffic mode, by age and gender, shows many different mobility developments, and we also know the associated casualty rates may differ strongly. However, other stratifications may be necessary too. Stratification by road type may be a good candidate, if mobility develops differently for different road types.

5. Discussion

5.1. Decreasing trend

When forecasting a decreasing number of casualties by estimating trend lines for the number of casualties or for the casualty rate, the use of the negative exponential trend line has some important consequences. One is that the trend always tends to zero casualties in the (distant) future, although it will never reach zero. To this end, the negative exponential trend line has an important advantage as compared to a linear trend line which indeed will cross the zero casualty rate and become negative at some future date; this is unrealistic, both for extrapolations of casualties or casualty rates. However, a negative exponential trend as such may also be unrealistic. Many examples (OECD, 2007) illustrate that part of the casualties cannot be associated with a decreasing trend; this becomes apparent after stratification. This implies that the common choice of a negative exponential trend line may be considered fundamentally incorrect, as a trend of the form:

$$Y(t) = A + Be^{-C \cdot t}$$

would be more appropriate. Here, Y(t) denotes the casualty rate (or, if mobility is constant, the number of casualties) in year t. The constant A denotes some unchanging component in Y, for instance because part of the unsafe conditions seemed immune to safety measures in the past. Stratification of data by whichever relevant dimension (traffic mode, age, road type) helps to identify the candidates for contribution to this constant. However, subsets of casualties, obtained by stratification that still show a clearly decreasing trend, may nevertheless be obscuring constant components that will only be revealed after further stratification. An example: in Aarts et al. (2008) the number of fatalities was shown to contain a constant of approximately 117 fatalities, which had been independent of time for a period of more than 30 years. This constant became apparent after stratification by traffic mode.

5.2. Explaining the overall trend

In principle, stratification results in different time series of the number of casualties by conflict type, road type, age of driver, etc. These stratified results enable a more straightforward way to assess the effect of entirely different safety measures and other relevant influencing factors, if these measures and factors apply to specific subsets. Different trends in crash rate may be the consequence of different explanatory factors (measure or development). Thus, the overall decreasing trend in casualties or casualty rates may be the result of a decreasing trend in a limited set of subsets of crashes for which explanatory factors were relevant. For all practical purposes, one may at first restrict oneself to subsets which show the most dominant decreasing trends.

5.3. Unexplained components

Ideally, the development of the casualty rate could be entirely understood from the introduction of safety measures or the development of influencing factors. Decreasing crash rates should as much as possible be identified with explicit external factors and safety measures. In practice, this goal is not achievable yet, either because the relation between the measures or factors and the number of casualties is not sufficiently well known yet, or because the development of the factor itself is not sufficiently accurately known. Thus, if there is a decreasing trend, it is uncertain if this trend will continue without additional effort. The continuation of the known measures and factors will not necessarily ensure a continuation of the decreasing baseline trend. Therefore it is important to make an explicit inventory of the measures that are supposed to have been responsible for the development of the trends in the past. Thus, at a later stage, it will be possible to correct the assessment for the future; at the same time it prevents that measures are inadvertently double-counted.

Following the same idealistic principles, sudden major changes in the number of casualties demand an explanation in terms of a change in distance travelled or casualty rate as well. In that respect, the 2004 drop in fatalities in The Netherlands should eventually be connected to external factors. At this moment, this drop cannot be explained satisfactorily. Research on Dutch crash data has shown that a considerable part of the drop can be associated with young car drivers. The drop is only manifest in accidents in which at least one car driver of at most 35 years of age is involved.

As an example, we will look at the average number of fatalities in single vehicle car accidents for a specific driver age group, and divide this number by the population of this age group. We call the resulting quotient the number of single vehicle car fatalities per driver, or fpdsvc. When we compare the fpdsvc for drivers of up to 35 years of age, and drivers older than 35, the results show a striking difference. For the younger drivers, the fpdsvc decreased from 4.7 10-⁶ to 3.1 10-. The largest drop shows between 2003 and 2004. For the older drivers, the fpdsvc is about 0.9 10-⁶ and it is steadily decreasing by 3.7% yearly. There is not a hint of a specific drop in 2004 for this age group. This is illustrated in Fig. 8, top panel. This difference in fpdsvc may be related to a change in mobility per car driver, or mpdc. The development of mpdc for the same two age groups shows a steady average increase of about 75 km yearly, whereas the mpdc for the younger group indeed has decreased considerably in the last 5 years. However, this is a gradual decrease, not a strong decrease in a short period. In 2004, hardly any decrease in mpdc is seen at all in 2004. So the question is if the mobility data nevertheless can explain the fatality data?

The answer to this question is not known, but it might be found in the effect of driver experience. In Stipdonk and Berends (2008) it was suggested that a change in the number of new car drivers could have a large effect on the safety, but a much smaller effect on the mobility. To verify this hypothesis, we need data on the experience of car drivers, involved in a crash. At this moment, no such data is available. The explanation of the sudden decrease in the number of fatalities in 2004 in The Netherlands can therefore not be answered yet. Still, we think that this example shows that a stratification by conflict type and age (group) may reveal interesting targets for possible new safety policies.

In the past, the lack of quantitative explanations for the overall decreasing trend has given rise to the suggestion that there is an autonomous process that improves the safety level (Oppe, 1991). This theoretical concept refers to a process of continuous 'learning' by road users, car manufacturers, road authorities, etc. In our view it is uncertain if such a process, for which there may well have

Fig. 8. Comparison of the number of fatalities per driver in a single vehicle car crash (fpdsvc, top panel), and the mobility per driver (mpdc, bottom panel), for two age groups: below and above 35 years of age. The top panel shows the average fpdsvc for the younger group in the two most recent periods of 5 years, and also the average annual relative decrease of the fpdsvc for the elderly group. The bottom panel shows the average annual linear increase in the mpdc for the elderly group. The points in this figure are connected with thin solid lines that serve as a guide to the eye only.

been evidence in the past, is indeed autonomous, i.e. that it can indeed be held responsible for a permanent and lasting annual decrease. Firstly, many subsets of the number of casualties (traffic modes, crash types, driver age groups) show little or no decreasing trend. Secondly, other factors, such as demographic changes, do provide explanations for some subsets with a decreasing trend. Nevertheless, specific forms of´learning processes' could explain decreasing trends in some other subsets. For example, look at elderly car drivers who nowadays have far more driving experience than the same age group 20 years ago, whereas 18 year old drivers have no driving experience at all, which is no different from the situation 20 years ago. However, this learning process may come to an end when all car drivers have learned to drive at the same early age of 18. Thus, application of a theoretical concept of general learning, as an explanation of the decreasing trend in the casualty rate, bears the risk that policy makers believe that little needs to be done, whereas in reality, the decreasing trend will eventually come to a stop. simultaneous stratifications is still feasible. We suggest that this fatalities per driver [10-6]

5.4. Limits to stratification

A perpetual drive to stratify data further and further is, from a conceptual perspective, preferable. As an illustration, consider stratification of crashes on junctions into crashes on either roundabouts or intersections. Such a stratification could reveal that A. the normalized casualty rates (i.e. the casualty rate per junction) of roundabouts and intersections are both constant, and that B. A decrease in the casualty rate on junctions can be explained by an increase in the number of roundabouts.

However, given a finite number of casualties in a period of N years, there seems to be a limit to the level at which modelling problem is less serious than it seems, according to the following reasoning. When we stratify crash data by traffic mode, age, gender, road type, etc., we no longer have a single time series. Instead, our aggregated single time series is replaced by multiple time series, each available for a period of N years, and each of which can be modelled with a specific small set of parameters for that time series. Thus, a stratified model not only explains the aggregated number of casualties, but also the separate stratified subsets. Also, by stratifying, extra information from the data is used. This justifies the use of additional parameters.

This reasoning also holds for data on distance travelled. Stratification by traffic mode and age group allows for the use of more parameters, which enables estimating different time series for different traffic modes and age groups. However, subsets in the stratification with a high crash rate, such as moped or motorcycle crashes, or young car drivers, do present a difficulty. For these traffic modes with a relatively small total distance travelled, the sample size of the traffic survey is small in relation to the associated number of crashes, because of the high crash rate. This means that the calculated high crash rates have a relatively low accuracy. This, however, is no reason for not stratifying distance travelled by traffic mode. By not stratifying the data it is a priori not possible to take into account changes in distance travelled by these groups.

Thus, by stratification into subsets, the relative uncertainty of the crash data is higher than the relative uncertainty for the aggregate. For poisson distributed crash data the standard deviation of the data is as a first approximation proportional to the square root of the observed values. On the other hand, the amount of information (the number of bits in each value) is proportional to the sum of the logarithms of the stratified numbers. This follows from the length of the binary representation of any number (Shannon, 1948). Thus, if some value v is stratified into v¹ and v2, then var(v1) + var(v2) = var(v), but log(v1) + log(v2) > log(v¹ + v2). The same reasoning holds for mobility data, although the accuracy of these data is a much more complex function of the observations that underly the estimated mobility. Still, as stratification introduces extra information, this enables development of a better model (given that the extra information can be interpreted to a sufficient extent).

However, care should be taken that the model remains feasible, and does not grow into an unnecessary complex of formula's. Additional factors and parameters should improve the understanding of the safety development, or should else be avoided. In general, only well established relations between factors and the number of casualties should be taken into account.

5.5. Past vs. future

Models that help explain the past traffic safety development, and models that enable estimating the future number of casualties, have a lot in common. It is clear that forecasting models and explanatory models are used differently.

An explanatory model aims to describe the fluctuations in the number of casualties in the past by finding relations between these numbers and external factors (such as mobility, SPIs and other safety measures). These relations, when parameterized, are then optimized to find the maximum likelihood of the known data, given the model. Thus, for an explanatory model, in principle all data is available, and the better these data, the more accurate the model will be.

For a forecasting model, just as for an explaining model, we need a, preferably accurate, mathematical description of the relation between the number of casualties, and the factors this depends on. In principle, the explanatory model should provide us with all we need to forecast road safety. However, there are some drawbacks. For example, to build a perfect explanatory model for forecasts, if this can be done at all, we need to know the values of all relevant factors in the model for the target year. If the explanatory model for instance uses motorcycle mobility, safety belt use and the number of roundabouts as explanatory factors, we need to know or estimate these factors in the target year explicitly.

Also, we need to know what additional measures will be taken, and what their effects will be. Notwithstanding these drawbacks, we know of no arguments for the idea that in its mathematical description, an explanatory model is different from a forecasting model, and the best possible explanatory model should be a good start for a forecasting model. The more these explanatory models contain constant parameters and time dependent measured quantities such as mobility, and SPIs, the more their time dependence will be determined by the time dependent observations of external factors. At this moment, the models still hardly contain constant parameters and measured quantities, but are dominated by time dependent trends. As these trends may come to a halt, such explanatory models should be handled with care when used to estimate a future number of casualties.

5.6. Additional road safety measures

In this paper, we did not go into an analysis of methods to assess the effect of additional measures, because such assessments are, to a large extent, dependent of the measure. There is a large number of measures for which such assessments are available (Elvik and Vaa, 2004). As the domain of each measure usually is only a (small) subset of the total number of casualties, stratification helps identify that subset. Hence, it may be that a general problem of assessing the effect of safety measures, namely the assessment of a combination of different safety measures, can be partly overcome by stratification. Elvik (2009) showed that estimates of the simultaneous effect of measures on the total number of casualties, tends to be less than would be expected according to the separate effects. Once logic attaches the effect of each separate safety measure to the correct domain of the crashes that are affected by the measure, it may turn out that the expected effect of each separate measure becomes smaller, and that a more accurate estimate of a set of simultaneous measures becomes possible.

5.7. Availability of mobility data

Countries that do not have stratified mobility data (by traffic mode, by driver age) at their disposal, cannot easily stratify their analysis. In that case, it may be an idea to hypothesize a constant (or slowly increasing) per capita distance travelled (by traffic mode), and to try to check if this is a reasonable assumption, for instance by analysing driving licence data, fleet ownership data, public transport use data, traffic offence data or any other dataset that may serve as an approximation of per capita mobility. Once it has been established that a constant (or slowly changing) per capita distance travelled is a reasonable assumption, it is possible to analyse stratified casualty trends.

5.8. The dependence between mobility, crash rate and the severity of the crash

Analysing road safety data implies careful treatment of crash data, crash severity data and mobility data, as was extensively described by Bijleveld (2008). We give two examples of the subtleties involved.

In our approach we prefer to analyse mobility and crash rate simultaneously, rather than independently. This is because, in fact, mobility and crash rate are not independent factors. This is illustrated by the fact that in two-traffic mode accidents, the crash rate of one traffic mode depends on the mobility of the other mode.

In The Netherlands, underreporting of crashes is known to be a serious problem. Even for fatal crashes only about 92% is reported by the police, and of the seriously injured (MAIS2+) in-patients only about 30% are reported. The underreporting of accidents without injuries is assumed to be even more serious. As a consequence, there is a correlation between the registered number of crashes and their severity. This implies that crash severity and crash occurrence are not independent (at least, as far as registered crashes are concerned), and that it is therefore not possible to correctly consider the number of casualties as the outcome of separate processes for the number of crashes and the probability of a serious injury. This is because crashes involving vehicles carrying more than one passenger are more likely to be registered and more likely to have more casualties. To overcome this problem in our approach as described in Section 5, we analyse casualties among drivers only and do not consider casualties among passengers yet.

6. Conclusions

The three step approach described by Broughton (2009), is a suitable method to carry out road safety forecasts. In principle, by introduction of the intermediate quantity casualty rate, the number of casualties per distance travelled, this approach enables a correct analysis of the effects of mobility and other factors.

As casualty rates differ widely between traffic modes, it is necessary to stratify both casualty data and mobility data by traffic mode. Essentially, the same holds for any other factor for which the casualty rate differs significantly, such as driver age or road type.

As the decreasing trends of casualty rates may also differ widely, trend extrapolations demand stratification as well; even if risks do not differ strongly, but (decreasing) trends do, it is necessary to stratify, to prevent the forecast from being incorrect.

A further reason for stratification of mobility and casualty data is that influencing factors such as SPIs may be effective in a small subset of the total number of casualties only. We have called this subset the domain of the influencing factor. Therefore, stratified models enable incorporation of explanatory factors by offering explicit domains of the explanatory factors (assuming that the stratifications are appropriate). The same holds for implementation of additional safety measures; their future effects, if limited to a specific subset, can conveniently be calculated if this subset is specified by appropriate stratifications, such that the projection of this subset is separately carried out in the model.

Often, it appears that negative casualty trends, when stratified, decompose into trends that decrease quickly and trends that decrease more slowly or not at all. This suggests that the trend for the aggregated set is not, as is often assumed, negative exponential, but is in fact the sum of several negative exponential trends and possibly a constant. The assumption that trend lines generally contain a constant allows model parameter optimizations to generate solutions that do not tend to decrease to zero, in cases where this is in fact incorrect.

The fact that, in practice, so many subsets of casualty rates show very differently decreasing trends, indicates that the sometimes suggested general social developments that cause a general decreasing casualty rate, may not exist. Assuming such developments to be the implicit cause of a decreasing trend, may lead to the incorrect conclusion that less needs to be done to ensure a further improvement of traffic safety. If it were true that the learning society can be seen as a cause for a continually improving safety system, then one would expect this to be effective for different traffic modes, driver ages, etc. Without actual proof of such an effect, the assumption that there might be some autonomous decreasing casualty rate could be misleading.

Acknowledgement

The authors wish to thank Frits Bijleveld and Jacques Commandeur, for their contribution to the analyses described here, and for sharing their valuable thoughts and knowledge on this subject.

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