2104数学建模美赛A 交通流 安全性 参考资料 交通量

ROAD ACCIDENTS AND TRAFFIC FLOWS:AN ECONOMETRIC INVESTIGATION

Andrew Dickerson, John Peirson and Roger Vickerman

April 1998

Abstract

This paper develops an empirical model of the relationship between roadtraffic accidents and traffic flows. The analysis focuses on the accidentexternality which is mainly determined by the difference between themarginal and average risks. The model is estimated using a new datasetwhich combines hourly London traffic count data from automated vehiclerecorders together with police records of road accidents. The accident-flowrelationship is seen to vary considerably between different road classes andgeographical areas. More importantly, even having controlled for these andother differences, the accident externality is shown to vary significantly withtraffic flows. In particular, while the accident externality is typically close tozero for low to moderate traffic flows, it increases substantially at hightraffic flows.

JEL Classification: C14, C80, D62, R40

Keywords: Road Traffic Accidents, Traffic Flows, Accident Externalities

Acknowledgements: We would like to thank, without implication, the Department ofTransport for giving us access to the automatic count data used in this study. Particular thanksare due to Daniel Aromire for his assistance in the provision and explanation of these data.The accident data were supplied by the ESRC Data Archive at the University of Essex.Neither the original collectors of the accident data nor the Archive bear any responsibility forthe analyses or interpretations presented here. This research was funded in part by theTRENEN project of the European Commission’s Fourth Framework RTD Programme, DGVII (Project ST-96-SC-116). Finally, we are grateful to Alan Carruth for his useful commentson an earlier draft.

Correspondence address: Dr. John Peirson, Department of Economics, Keynes College,University of Kent at Canterbury, Canterbury, KENT CT2 7NP, UK. email: jdp1@ukc.ac.uk;tel: +44 (0)1227 823328; fax: +44 (0)1227 827850.

2104数学建模美赛A 交通流 安全性 参考资料 交通量

ROAD ACCIDENTS AND TRAFFIC FLOWS:AN ECONOMETRIC INVESTIGATION

1.

Introduction

The costs of road accidents are generally regarded as being considerable and they are thereforean important element in the analysis of transport projects and the formation of transport policy(Evans, 1994; Department of Transport, 1996). For the United Kingdom, the annual total costshave been estimated at between £5 billion and £26 billion (see Maddison et al. 1996; Pearce,1993; Fowkes et al., 1990; Hopkins and O’Reilly, 1993; Hansson and Marckham, 1992 andNewbery, 1988). New transport projects and polices affect the number of road accidents at themargin. Thus, it is important to investigate the marginal external accident costs of additionalroad traffic rather than total or average accident costs.

There are three principal stages in the estimation of the external costs of road accidents(Maddison et al., 1996 and Peirson et al., 1998). The first stage is to identify the functionalrelationship between accidents and vehicular flows. Vickrey (1968, 1969), Jones-Lee (1990),Newbery (1987, 1988), Vitaliano and Held (1991) and, more recently, Jansson (1994) andSchefer and Rietveld (1997) can be regarded as the significant contributors to the formalmodelling of this relationship. Second, the different elements of road accident externalitieshave to be defined. The studies cited above, together with Hansson and Marckham (1992),Jones-Lee et al. (1993) and Persson and Ödegaard (1995), have defined and discussed theseexternalities in a helpful manner. Finally, values have to be placed on these externalities. Theliterature on this subject is extensive (see, for example, Jones-Lee et al., 1985, Jones-Lee,1990 and Jones-Lee et al., 1993). This paper is primarily concerned with the first stage, that ofdetermining the nature of the accident-vehicle flow relationship.

2104数学建模美赛A 交通流 安全性 参考资料 交通量

We construct a unique and novel dataset for this study by amalgamating two disparate sourcesof data. Traffic flow data for the period mid-1993 to end-1995 are taken from the Departmentof Transport’s automated recording devices which are distributed throughout London. Thesevehicle flows are then matched with police data on all road traffic accidents in thecorresponding area during the same period.

The statistical analysis of these data reveals two important findings. Firstly, the accident-flowrelationship varies significantly between different road classifications and broad geographicalareas. This has an important implication for a number of previous studies which have failed toallow for such heterogeneity. Typically, these studies have concluded that there is a near-proportional relation between accidents and flows at all levels of traffic flow and, thus, thereis no apparent accident externality. Such a conclusion may be erroneous since it may resultfrom the aggregation of heterogeneous accident-flow relationships which do not exhibitproportionality.

Secondly, under the simple assumption that occupants of additional vehicles internalise theaverage risk of an accident, the excess of the marginal accident rate over the average ratedetermines the magnitude of the road accident externality. The data indicate that the ratio ofthe marginal accident risk to the average accident risk varies significantly with traffic flow,and, thus, this variation is important. In particular, we show that while the externality istypically close to zero for low to moderate traffic flows, it increases substantially at hightraffic flows. That is, the relationship between accidents and traffic flows is non-linear.

The remainder of the paper is organised as follows. Previous studies of the relation between

2104数学建模美赛A 交通流 安全性 参考资料 交通量

accidents and flows are discussed in Section 2. The data are described in Section 3 and thedegree of heterogeneity across road class and geographical area is depicted. In Section 4, weestimate the relationship between road accidents and traffic flow using an appropriateeconometric specification. The final section draws some conclusions.

2.Previous Studies of Road Accidents and Traffic Flows

Road users impose accident risks on other road users. Such accident risks may give rise toimportant externalities, but the relationship between road accidents and traffic flows is notwell understood and has been the subject of few empirical studies. Maddison et al. (1996,p.122) noted that “given the policy relevance of ... the degree of market failure in roadtransport, it is unfortunate that not more is known regarding traffic flows and accident rates”.Vitaliano and Held (1991, p.373) commented that “there is a significant gap in ourunderstanding of this important facet of highway economics”. The key economic determinantof road accident externalities is the difference between the marginal and average accidentrates. Intuitively, an extra vehicle leads to an additional risk of an accident for all vehicleswhile the extra vehicle faces only the average risk. The difference between the marginal andaverage accident rates represents the divergence between the social and private costsassociated with the extra vehicle (see also Walters, 1961).

In what follows, we construct an illustrative naïve model of road accidents and traffic flows.This model is then extended to allow for variation in road characteristics and drivingbehaviour, and is related to the extant empirical studies. Assume that all road users areidentical, the road network and its use are homogeneous and all vehicles are driven in an

2104数学建模美赛A 交通流 安全性 参考资料 交通量

identical manner whatever the level of traffic flow F1. The potential number of accidentsinvolving two vehicles2 is proportional to F(F-1)/2 ½F2 for reasonable values of F. Let theprobability of an accident be , which will be a function of road conditions, driving behaviour,road network characteristics etc. Then, the expected total number of accidents, N, is given by:

N = ½ F 2

(1)

This is a quadratic function3 and therefore the marginal and average accident rates are notequal. As noted above, the divergence between the marginal (m = F) and average (a = ½ F)number of accidents can be interpreted as the extent of the accident externality.

In general, the degree of the externality can be represented by the ratio of the marginal toaverage accident rates. For example, equation (1) yields m/a = 2, which indicates that the fullcost of additional vehicles is imposed on other road users. In comparison, a functionalspecification which generates m/a = 1 suggests that there is no externality and the effects ofadditional vehicles are effectively being completely internalised by existing road users. Thus,if the marginal and average accident rates are equal, then we can conclude that there is noaccident externality. Such a result will only emerge if the number of accidents increasesexactly proportionally with traffic flow, N F.

One effect of higher levels of traffic are to reduce average speeds and make driving behavioursafer, suggesting that is a function of F (see Bailey, 1970 and Peirson et al., 1998 for a more This model is in the spirit of Newbery (1987, 1988), Jansson (1994) and Peirson et al.(1998).2

Approximately two-thirds of all road accidents involve two vehicles (source: owncalculation from Road Traffic Accident Statistics 1992-1995).3

Shefer and Rietveld (1997) show that the number of two vehicle accidents, N, will increasequadratically with the number of vehicles, F, for a variety of different shapes of road network.

1

2104数学建模美赛A 交通流 安全性 参考资料 交通量

detailed explanation). The form of this function is important since it indicates the degree towhich the externality is internalised by road users, and therefore affects the marginal andaverage accident rates. It therefore determines the degree of the externality. Thus, for example,if 1/F, then the model in equation (1) yields m/a = 1, and hence there is no accidentexternality4.

It is important to note that simple exponential accident functions of the form

N = F

(2)

imply a fixed ratio between the marginal and average accident rates of for all levels of F.The estimation of accident-flow relationships using equation (2) is therefore very restrictive.Moreover, such relationships are forced to pass through the origin, whereas the (expected)number of accidents may be zero at some positive flow.

As noted in the introduction, there has only been a limited number of studies that havemodelled road accidents and the consequent externalities. Vickrey (1968) inferred fromevidence on Californian freeway driving in the early 1960s that the marginal accident rate was1.5 times the average rate. However, Vickrey’s work has been criticised for his use of simplearithmetical calculations rather than any formal estimation of an econometric model(Vitaliano and Held, 1991).

Vitaliano and Held (1991) is one of the few studies that econometrically estimates therelationship between road accidents and traffic flows. They investigated annual road accidentand traffic flows on urban and rural roads in New York State in 1985. Control variables4

The marginal and average accident rates are both equal to .

2104数学建模美赛A 交通流 安全性 参考资料 交通量

included the number of lanes, annual snowfall, speed limit, length of road segment studied,etc, although the slope of the accident-flow relationship was assumed to be constant across alltypes of roads. They conclude that the relationship between accidents and flows is nearlyproportional and thus the external accident effect is close to zero, although when the 19highest flow observations with more than 50,000 vehicles per day were separately analysed,there is some evidence of a small negative external effect5. However, sample separation byvehicular flow rather than road characteristics would appear to be inappropriate, and it is notpossible from their results to determine whether the difference is significantly different fromthe estimated external effect for the remaining 380 observations.

The Department of Transport (1996) COBA manual assumes that, on average, the number ofaccidents on road links between junctions is proportional to traffic flow. This requires thatdrivers adjust their driving behaviour to achieve a fixed risk of an accident per vehicle-kilometre travelled at differing traffic flows. For traffic flows at junctions, the accident-flowrelationship differs according to the type of junction. Exponential functions of the form ofequation (2) were estimated for different types of junction. The estimated ’s range between0.44 and 1.77, but, as noted above, the functional form of equation (2) is very restrictive.

Rather than attempting to estimate the accident-flow relationship, the other notable studieshave typically made assumptions about the accident-flow relationship or have drawn onprevious research. Citing the above evidence and the US Federal Highway Cost AllocationStudy (US Federal Highway Administration, 1982), Newbery (1987) argued that the numberof road accidents is proportional to flow. Thus, the average accident rate per vehicle-kilometre5

Note that this subsample represents only 5% of their 399 sites.

2104数学建模美赛A 交通流 安全性 参考资料 交通量

travelled is constant and is equal to the marginal accident rate. By contrast, in a later paper,Newbery (1988) took the ratio of the marginal to average accident rate to be 1.25 - an averageof the Department of Transport’s (1981) and Vickrey’s (1968, 1969) views about the relationbetween road accidents and traffic flow. Jones-Lee (1990) considered the value of theexternality from road accidents where the accident rate per vehicle-kilometre is fixed, but thepedestrian accident rate is proportional to the distance travelled by vehicles. Pearce (1993)argued that a minimum estimate of external costs should include the cost of pedestrian andcyclist fatalities and injuries. Though Pearce discussed other possible external accident costs,no formal model of road accident externalities was developed. Maddison et al. (1996) used asimilar approach.

There are two important criticisms of the empirical studies discussed above. First, differentroad types are not distinguished in the analysis. It seems inappropriate to assume that drivers’behaviour, and thus accident rates, do not differ between, say, motorways and small sidestreets. While some of the differences are undoubtedly random (see Fridstrøm et al., 1995),some are likely to be systematic. In addition, roads have particular design characteristicswhich determine their ability to carry different levels of traffic flows safely. Thus, it isdifficult to believe that the relationship between accidents and flows is the same for all roadtypes. One solution is to specify dummy variables to capture any fixed effects deriving fromdifferences in roads and their impact on behaviour. However, such specifications cannotcapture any differences in the slopes of the relationship between accidents and flows, and it isthese which are of greatest importance given that transport policy is only ever likely to affecttraffic flows and accident rates at the margin.

Failure to fully control for the effects of differences in road types can also seriously bias

2104数学建模美赛A 交通流 安全性 参考资料 交通量

estimates of the underlying accident-flow relationship. Intuitively, observations from low-flowroads will typically be associated with a small number of accidents and thus will be close tothe origin, whilst observations from high-flow roads with a greater number of accidents willbe some distance away from the origin. Simple aggregate regression techniques which do notdistinguish these two road types are therefore likely to produce near-proportional relationshipsbetween accidents and traffic flows. This will give near-equality between the marginal andaverage accident risk rates and will lead to the spurious conclusion that there is no significantaccident externality. To avoid this potential bias, empirical models should carefullydistinguish the accident-flow relationships for different types of road.

The second point to be noted concerning the previous empirical studies is that the accident-flow relationship should be flexibly specified. In particular, the functional form selectedshould allow for the possibility that the marginal and average accident rates may vary withtraffic flow, rather than imposing the constraint that the marginal to average accident rate ratiois the same at all levels of traffic flow.

3.3.1

Data Description and Preliminary AnalysisData Description

The data used in this study derive from two sources. First, data on hourly vehicular flows havebeen provided by the Department of Transport from their automatic count data sites in Londonfor the period July 1993 to December 1995 inclusive. Data are available from a total of 54sites which are geographically spread fairly evenly throughout the London boroughs. At eachsite, an automatic data recorder counts each vehicle as it passes and the hourly totals are thentransmitted to a central recording device. The data are fairly complete in that there are very

2104数学建模美赛A 交通流 安全性 参考资料 交通量

few breakdowns in the recording apparatus, but where these do occur, the Department ofTransport imputes values from previous flows at the site. Given that we have hourly countsfor 30 months, measured in both directions, for over 50 sites, our data on flows comprise inexcess of two million hourly observations.

Data on accidents are taken from the Road Accident Data deposited at the ESRC DataArchive by the Department of Transport Road Accident Statistics Branch. These data recordevery accident and its circumstances occurring on the public highway where at least one roadvehicle and one casualty are involved. Details include the date and time (to the nearest hour),the local authority/police force area, road class (although not exact location), the number andtype(s) of vehicle(s) involved and some information on the number and severity of casualties6.From these data, we select only those accidents that occurred within the London area andduring the period covered by the automatic count data. This yields a total of 96,440 accidents.

In order to investigate the relationship between accidents and flows, we need to amalgamatethese two sources of data. While there are a number of ways in which this can beaccomplished, given the discussion above and the results of previous studies, we choose touse year-month-hour as our basic time dimension on which to merge the two datasets. That is,each observation in the combined dataset comprises the total number of accidents, Ahmy, thatoccurred during a particular hour of the day, h, in a given month, m, and year, y, together with

6

Certain information, such as the exact location of the accident, is missing, so as to maintainthe confidentiality of the victims.

2104数学建模美赛A 交通流 安全性 参考资料 交通量

the average vehicular flow, Fhmy, in that year-month-hour from the automatic count data7. Weinvestigate the sensitivity of our results to this particular choice of unit of observation.

3.2Preliminary Analysis

The review of the literature on the relationship between accidents and traffic flows inSection 2 suggests that econometric specifications should be capable of allowing fordifferences between road types and for variations in the ratio of the marginal to average riskrates. Some preliminary evidence on the necessity of the former can be gleaned from anexamination of Figure 1. This plots the total number of accidents against the average vehicularflow, where both total accidents, Ahmy, and average flows, Fhmy, are calculated separately forfour different road classifications (A-roads, B-roads, C-roads and Unclassified roads) and arealso dichotomised by location into Inner London and Outer London areas8. We choose todisaggregate on these two dimensions of the data since accident rates and traffic flows exhibitconsiderable heterogeneity across both road classification and area as can be seen in Table 1.

<< Table 1 about here >><< Figure 1 about here >>

Figure 1 reveals that there are distinct “clusters” of observations. These can clearly be seen tocorrespond to the eight different road-area classifications as shown in Figure 2, which also7

Note that we cannot match the two data sources by location since accidents are only locatedon a road class (A-roads, B-roads, C-roads or Unclassified roads) within a borough rather thanat an exact location. Moreover, even if the exact geographical location of each accident wasknown, they could only ever be imperfectly matched by location (presumably using theirgeographical proximity to one of the limited number of automatic count sites).8

Thus there are 2.5 (years) x 12 (months) x 24 (hours) x 4 (roads) x 2 (areas) = 5760 accident-flow observations in Figure 1. The definitions used for Inner London and Outer London are inthe Appendix.

2104数学建模美赛A 交通流 安全性 参考资料 交通量

reveals significant differences in the marginal accident-flow relationship according todifferent road types and area9. Estimating an aggregate accident-flow relationships for the datain Figure 1 can obviously yield misleading conclusions given the heterogeneity evident inFigure 2. Any aggregate relationship looks to be near-proportional, whereas the disaggregatedrelationships would appear to be: (i) different from one another; (ii) non-proportional; and (iii)(perhaps) non-linear. To illustrate the problem, the fitted values from both linear and cubicpolynomial regressions are super-imposed on Figure 110. Both yield functional relationshipswhich indicate that the average and marginal accident rates are very similar and fairly constantover the range of data. Of course, this exactly mirrors the results of previous studies. Table 2reports the results of polynomial regressions of this kind. Evaluated at the mean accident andflow rates, the ratio of marginal to average accident rates are 1.042, 0.834 and 0.861 for thelinear, quadratic and cubic regressions respectively, from which one might easily concludethat the accident externality is negligible (or even slightly positive)11. These results thereforeencompass previous studies which have used aggregated data.

<< Figure 2 about here >><< Table 2 about here >>

It is useful to demonstrate formally that the accident-flow relationship differs significantlybetween the various road and area classifications depicted in Figure 2. This can be9

Compare, for example, Inner London A-roads (top left) and Outer London Unclassifiedroads (bottom right).10

The econometric methodology should really take account of the fact that the number ofaccidents is bounded below by zero (effectively we are estimating the realisation of theaccident propensity and should therefore use a Tobit model). However, despite the high levelof disaggregation used here, only 7% our observations have zero accidents and hence werestrict ourselves to classical least squares estimation.11

Note that only the cubic polynomial passes the RESET misspecification test.

2104数学建模美赛A 交通流 安全性 参考资料 交通量

accomplished in a number of ways. First, note that simple polynomial regressions as inTable 2 with intercept dummy variables capturing the road-area combination are alwaysrejected against a more general specification in which the slope as well as the intercept of theaccident-flow relationship is allowed to differ between the different roads and areas12. Table 3reports the results of these regressions for the quadratic polynomial. The dummy variablespecification in column 1 in which we allow only the intercept to differ between the 4 2 = 8road-area combinations fails the RESET test, while the fully interactive specification (whichis equivalent to estimating the quadratic relationship separately for all 8 road-areacombinations) satisfies this criterion. Moreover, the interactive effects are individually andjointly significant; a test for the equality of the linear F terms across all 8 road-areacombinations yields a F-statistic of F(7,5736) = 19.98 [p=0.00], while the equality of the F2terms is also comprehensively rejected (F(7,5736) = 32.15 [p=0.00]). Additionally, theseinteractive terms also differ from each other - that is, not only are the accident-flowrelationships jointly different from that for A-roads in Inner London (the base category inTable 3), but pairwise tests reveal that B-roads in Inner London differ from B-roads in OuterLondon etc, and that Inner London roads are collectively different from Outer London roads.These differences are presumably because of the rather differ composition and densities oftraffic flows and the design and structure of the different types of roads in Inner and OuterLondon. These results thus demonstrate that the accident-flow relationship differssubstantively and statistically significantly between the 8 different road-area classifications.

<< Table 3 about here >>

12

Separately distinguishing roads and areas is not adequate (statistically) since the data revealthat it is the combination of these two characteristics which is important.

2104数学建模美赛A 交通流 安全性 参考资料 交通量

4.

Specification of the Accident-Flow Relationship

We could simply use the results in the second column of Table 3 to calculate the marginal andaverage accident rates at, for example, the average traffic flow in each road-area combination.However, such a specification has two undesirable properties. First, it is well known thatpolynomial specifications are very sensitive to observations at the ends of the data range (see,for example, Royston and Altman, 1994a). Even fractional polynomial13 estimation is notimmune to such end-effect and/or outliers (Royston and Altman, 1994b). Secondly, thespecification of simple polynomials does not allow the relationship between marginal andaverage accident rates to vary in an unrestricted manner with traffic flow as discussed inSection 2 above. In particular, polynomials force the marginal accident rate to be eitherincreasing or decreasing between turning points. For example, they do not allow a constantmarginal rate over some range of traffic flow.

Therefore, in order to facilitate the identification of an appropriate functional specification, wefirst employ a non-parametric (kernel) estimator of the relationship between accidents andtraffic flows (Härdle, 1990). Effectively, this simply fits a “smooth” curve through theobservations, where the degree of smoothness depends on the particular kernel methodchosen. While the choice of smoothing algorithm is essentially arbitrary, we choose a methodthat is robust to outliers but has “locality” - that is, it tends to follow the data closely. Such amethod is provided by Cleveland (1979)14. The kernel estimates of the relationship between1314

That is, non-integer exponents in the polynomial function.

This particular smoother is based on the fitted values from a locally weighted regression. Acentred subset of the data is used in a (polynomial) regression, with each observation in thissubset weighted according to the inverse of its distance from the central point. The estimatedregression is then used to predict the smoothed value at the central point only and theprocedure is repeated for each observation in the dataset (see Härdle, 1990, for further detailsof the algorithm).

2104数学建模美赛A 交通流 安全性 参考资料 交通量

the number of accidents and traffic flows for each of the road-area classifications are shown inFigure 3. The bold line is the kernel estimate while the light lines are 95% confidence boundson this curve. As can be seen, in almost all cases, the relationship looks near-proportional forlow to moderate traffic flows (although of course the marginal rates differ substantially giventhe different scales on the axes). However, at high traffic flows, there is evidence to suggestthat the accident-flow relationship changes15. Any parametric specification therefore needs tobe able to accommodate this observation. In particular, smooth (i.e. continuouslydifferentiable) polynomial functions may not be sufficiently flexible to capture thischaracteristic of the data.

<< Figure 3 about here >>

We therefore need to select a flexible specification in which the relation between the marginaland average accident rates is allowed to differ across traffic flows. A simple, semi-parametric,specification is to use piecewise linear splines. These maintain the continuity of the accident-flow functional relationship and also provide the necessary flexibility. Moreover, they permitthe average and marginal rates to be directly calculated. The visual inspection of the dataprovided by the kernel estimates, together with some pre-testing, indicates that a maximum ofthree spline segments (i.e. two nodes) is sufficient for all road-area combinations.

The results from estimating spline functions are illustrated in Figure 4. All 8 estimatedregressions pass the RESET test for misspecification, and thus can be regarded aseconometrically satisfactory. We scale the graphs for ease of reference, but it should be noted15

The use of a narrower bandwidth increases the slope of the functions at high flows. Thus,the computed accident-flow relationship deviates even further from proportionality.

2104数学建模美赛A 交通流 安全性 参考资料 交通量

that the slopes of the spline functions are generally significantly different between the road-area classes. As can be seen, the third segment in each road-area classification typically has aslope greater than that of the first two segments. Thus at high vehicle flows, the marginalaccident rate is greater than the average accident rate. The exception is C-roads in InnerLondon where the third segment is downward sloping. This is purely a site-specific problemin that we only have one representative site for this particular road-area combination. Hencethe results for this road-area classification should be treated with some caution16.

<< Figure 4 about here >>

In order to better evaluate the differences between the marginal and average accident rates atdifferent flows, Table 4 tabulates the marginal accident rate, m, (the slope of each splinesegment), the average accident rate, a, (evaluated at the mid-point of each spline segment),and the ratio of the marginal to average accident rate, m/a, for each road-area combination.That is, we present estimates of m/a at the first, third and fifth sextiles of the data.

<< Table 4 about here >>

16

The distribution of the 54 sites between road types and area is as follows:

Hence, the results for A-roads and Unclassified-roads are based on traffic flows averaged

across rather more sites than those for B-roads and C-roads and thus can perhaps be regardedwith greater confidence, although the Department of Transport chooses all the sites for itsautomatic recorders to be fairly representative of traffic flows throughout London. Anadditional problem is that there is some evidence that there are inconsistencies in therecording of accidents on C-roads and U-roads (particularly in urban areas as in this study),perhaps because the exact classification of any road is largely unknown to the police whenthey complete the Road Accident Report Form (STATS 19) (for example, road classificationsare not recorded on OS maps; see Lupton et al., 1997).