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The theoretical basis for comparing the accident record of car models
Aecid. Anal. and Prev., Vol. 28, No. 1, pp. 89-99, 1996 Else&r Science Ltd. Printed in Great Britain
Pergamon
THE THEORETICAL BASIS FOR COMPARING ACCIDENT RECORD OF CAR MODELS*
THE
JEREMYBROUGHTON Transport Research Laboratory, Crowthome, Berkshire, RG116AU, U.K. (Received
6 April 1994;accepted 18 July 1995)
Abstract-The accident records of different models of car can be compared statistically, provided that accident data which allow the make and model of accident-involved cars to be identified are collected on a national scale: this has been done in Great Britain since 1989. This paper considers the theoretical basis for comparing safety and shows that, because of the lack of detailed exposure data, the most which can currently be achieved is to measure the level of secondary safety (also known as crashworthiness). Based on mathematical considerations, it is shown that the best measure of secondary safety of a particular model is the proportion of drivers who are injured when involved in a two-car accident where one or other driver is injured. In order to minimize bias, this proportion should be adjusted statistically to allow for the influence on the accident data of factors such as type of road and age of driver.
than 1 would imply a worse record. It would then be desirable to see whether differences could be explained by factors such as mass, to judge whether the record of a particular model was better or worse than would be expected for a car of that “size”. A sophisticated statistical model is required to achieve this ideal, but information about the accident exposure of different car models is not available at the required level of detail in Great Britain. Consequently, only a more limited goal is feasible. The most that can be achieved is to measure the level of secondary safety (“crashworthiness”) provided by each model, that is, the probability of an occupant being injured once a car has become involved in an accident. Even this is not straightforward to calculate, as there are various potential sources of bias. At present, two methods in regular use take some account of the complexities involved, one from the U.K. Department of Transport (DOT) and another from the Swedish company Folksam. The two methods lead to different safety indices, but it will be shown that they are directly linked. This paper introduces some of the factors that can influence the probability of an accident-involved driver being injured; these need to be taken into account to minimize bias when evaluating any safety index. The two existing methods for estimating such safety indices are then compared, using British data from 1989-92. A sequel to this paper (Broughton 1996) presents further developments of the modeling, and more detailed results.
INTRODUCTION In several countries, accident data are collected in sufficient detail and on a sufficient scale for it to be possible to assess the relative safety of different car models. Since 1989, the British Stats197 accident reports have included the Vehicle Registration Mark of accident-involved vehicles, which enables details such as vehicle make and model to be extracted from files at the Driver and Vehicle Licensing Agency. Thus, Great Britain is now also a member of this group. This paper considers the theoretical basis for assessing the relative safety of different models, with particular reference to the data available from the Stats19 database. The ideal basis for assessing the relative safety of any model can be stated simply. It is the ratio of the number of occupant casualties that occur in accidents involving that model relative to the number that would be expected if the accidents had instead involved cars of a “standard” model representing the current average. This could be extended to a range of ratios, to include casualties to other road users. Ratios less than 1 would imply an accident record that was better than average, while ratios greater
*Crown Copyright 1994. The views expressed in this publication are not necessarily those of the Department of Transport. tThe Stats19 form is used by all police forces in Great Britain to record details of any road accident involving human injury or death. 89
J. BROUGHTON
90
FACTORS
better than model C when severity <5 but worse when severity = 6, 7 or 8. It seems improbable that a method could be devised for estimating these injury probabilities without reliable reports of accident severity. Nonetheless, the following example demonstrates the importance of allowing for differences in accident severity as far as possible. Suppose that the accident records of car models D and E are to be compared, that both models actually offer the same level of protection and that model D is involved in relatively more high-speed impacts than model E because these cars travel more on rural roads. As model D is involved in accidents of higher severity, relatively more of its occupants will be injured, so a ndive comparison of the accident records will find that D offers a lower level of protection than E. Thus, failing to take account of accident severity can lead to misleading conclusions if different car models do actually experience different ranges of accident severity. While the change in velocity during an impact can be estimated with some success by trained investigators, it is not routinely available from police accident reports. The speed limit at the accident site is available from the Stats19 reports and can be used as a proxy for accident severity.
INFLUENCING A MODEL’S ACCIDENT RECORD
The “Stats19 database” contains statistical details of all road accidents reported to the police in Great Britain in which one or more people are injured. A report is published annually to summarize the latest data (e.g. Department of Transport 1994). Injuries are classified as fatal, serious or slight, and all police forces supply data using the uniform Stats19 report form. The accident data analysed in this paper come from the Stats19 files for 1989-1992, so all empirical results relate to injury accidents. The accident record of a specific car model is likely to be affected by various factors whose presence is recorded in the Stats19 reports. These are examined below, and ways of using this information to achieve the index of safety with the lowest practicable level of bias will be discussed later. Accident severity
A sound conceptual basis from which to approach the definition of an index of safety is set out in various Folksam reports (e.g. HBgg et al. 1992). This uses the concept of “accident severity”, which is not exactly specified but seems to be best represented by the change in velocity during an impact. For any particular make of car, the probability of an occupant being injured increases with accident severity, but the rate of increase will differ between models. Figure 1 presents three hypothetical examples, using 10 levels of accident severity. No driver of any of the three models is injured at level 1, at level 10 all drivers of each of the three models are certain to be injured, but there are differences between the models at intermediate levels. Model B protects occupants better than models A and C; model A protects occupants
Diferences in accident distributions
Table 1 examines the extent to which models’ accident-involvement distributions varied in 1991, using a list of 74 models derived from a recent report (Department of Transport 1993a) which included all models with more than 20,000 vehicles registered. Older and newer cars (those first registered before and after 1 January 1989) are treated separately, and
1.0
1
2
3
4
5
6
Accident
severity
Fig. 1.
I
8
9
10
The theoretical
basis for comparing
Table 1. Distribution
the accident record of car models
of injury accident involvements,
91
1991
Size of car
Cars registered pre-1989 Number % with: male drivers drivers aged 17-24 drivers aged 25-34 drivers aged 35-54 drivers aged 55 and older % on nbu roads Cars registered 1989-91 Number % with: male drivers drivers aged 17-24 drivers aged 25-34 drivers aged 35-54 drivers aged 55 and older % on nbu roads
Small
Small/ medium
Medium
Large
All
44284
59615
42787
10189
156875
49 42 24 21 12 28
69 35 27 25 12 28
79 25 30 34 12 27
81 11 25 48 15 30
67 33 27 28 12 28
13938
19513
15177
3415
52043
53 26 29 29 15 30
71 20 32 34 14 34
81 12 31 46 12 40
86 6 21 60 13 44
70 18 30 38 14 35
the table includes the percentage of cars that were involved in accidents on non built-up (nbu) roads (‘non built-up’ is the term applied in Great Britain to any road with a speed limit exceeding 40 mph). Results are presented according to the size of car as defined in the DOT report; as an approximate guide, ‘small’ cars are 140-152 inches long, ‘small/medium’ cars are 153-167 inches long, ‘medium’ cars are 168-180 inches and ‘large’ cars are over 180 inches long. The table represents 67.8% of all cars involved in accidents in 1991, the others being excluded because the model could not be identified or was not on the list. The table shows clear variations between model groups in their accident circumstances. Among newer cars, larger cars are involved in relatively more accidents on nbu (high speed) roads, but the range of variation is much less among older cars: this presumably reflects the travel patterns of company-owned cars (which in Great Britain are rarely more than 3 years old and tend to be larger than privately-owned cars). Variations are even wider among individual models-one half of the newer Jaguars involved in accidents were travelling on nbu roads, compared with only one sixth of older Hyundai Stellars (the Stellar is a medium-sized model). These variations presumably indicate differences in exposure since, for example, it seems unlikely that Jaguars are involved in an especially high number of accidents per mile travelled on nbu roads, rather that a high proportion of their mileage is on these roads. A driver’s age and sex affect the likelihood of becoming involved in an accident, but they also affect the likelihood of being seriously injured and hence
the relationship between accident severity and injury probability. An elderly driver is more vulnerable in an accident than a younger driver, so models with a high proportion of elderly drivers will tend to have relatively many seriously injured drivers. One third of drivers of newer Toyota Carinas (a medium-sized model) are at least 55 years old, compared with one twelfth of drivers of older Fiat Pandas (a small model). Similarly, the injury distributions for men and women differ, and there are clear differences in the driver sex distributions: only one ninth of newer Ford Granadas (a large model) were driven by women, compared with three fifths of older Nissan Micras (a small model). Thus, age and sex of driver also need to be taken into account when comparing secondary safety. All cars involved in accidents (with the trivial exception of parked cars) have exactly one driver, so a consistent basis for comparing models is achieved by studying only driver casualties. The inclusion of passenger casualties appears attractive because it increases numbers and so raises the likelihood of identifying apparently significant differences. The comparison will, however, be biased against those models which carry relatively many passengers unless there is reliable evidence for the number of passengers carried per model. This does not exist in Great Britain and, moreover, the Stats19 data show wide variations in the ratio of passenger to driver casualties which cannot be explained simply by car size. Thus, for this country at least, it would not be safe to include passenger casualties when calculating a safety index.
92
J.BROUG~N
EXISTING
SAFETY
INDICES
Three series of publications have compared car safety on the basis of accident data: (a)The US Highway Loss Data Institute (HLDI) routinely publishes analyses of insurance claims data, principally from injury claims paid to the occupants of insured cars and collision claims paid to the owners of insured cars. The data are supplied by a group of 11 or 12 major motor insurers, with exposure defined as the number of “insured vehicle years” covered by policies with these companies (see, for example, Highway Loss Data Institute, 1993). Various types of claim are analysed, for example of injury claims, theft and collisions claims, also repair costs. Only basic allowances are made to compensate for the relatively high number of claims for cars with “youthful operators” and for policies with higher “deductible amounts”, however, so this method will not be considered further. (b)The Swedish insurance company Folksam has published comparisons for a decade, using a relatively sophisticated method to analyse its claims data. Its method will be considered in detail. (c) The U.K. Department of Transport has published three annual reports of analyses of the Stats19 data enhanced with vehicle details from the DVLA files. The analytical method has developed in the successive reports, and the method from the latest report (Department of Transport 1993a) will be considered. In addition, a recent German report (Engels and Lupsen 1993) has presented a safety ranking which is similar to that used by the DOT. It is described below. The Folksam method This discussion of the Folksam method is based upon a recent report (HBgg et al. 1992), although previous reports have contained the identical description. The method is applied to statistical data compiled from accident reports made by motorists insured with the company. Experience in Great Britain suggests that these may be completed rather less consistently than police accident reports, but it is the actual method rather than the reliability of the Swedish results that will be considered here. The method analysts only two-car accidents. It is claimed that this avoids bias due to variations in accident severity, on the grounds that in such accidents both cars experience the same potential risks and so the results are independent of any differences between accident severity distributions. The Folksam report deals with integrals over the range of accident
severities, but then returns to sums over discrete severity intervals. It seems simpler to adopt a discrete approach throughout. Let a, the range of accident severities, be divided into n intervals a,.~, of increasing severity, and let tti be the probability of the driver of a car of model c being injured in an accident of severity cli. The set of injury probabilities (t,-i) is central to studying the relative safety of model c, and Fig. 1 has illustrated the sets for three hypothetical cars. In addition, let fci be the proportion of the involvements of a car of model c in accidents of severity Ui$ so that f is the severity distribution for model c. If model c cars are involved in N accidents, the expected number of injured drivers of these cars is simply Z,=N* C t,i*fi I, is approximately equal to the actual number of injured drivers, but it is not clear how to identify (t,J To avoid this difficulty, consider only accidents of severity Cli where cars of model 1 and model 2 collide, then Drivers of in these N Drivers of in these
tli*fli
x
model 1 cars injured accidents model 2 cars injured accidents
The report claims that this ratio has the same value for all severity ranges and hence, summing over all ranges, that Z&z
+ = 2
Driver casualties in model 1 cars Driver casualties in model 2 cars
is independent of the severity distributions. Further, since for most model-pair., there will be rather few collisions on which to base the calculation, it is assumed that the distribution of “other” cars is the same for each model and the index actually calculated for model 1 is RI=
Driver casualties in model 1 cars Driver casualties in other cars
This is the “Folksam index”, which is also claimed to be unaffected by differences between severity distributions. These claims will now be reviewed. To see that R,, can in fact vary with the severity range, consider the following simple example. There are only two levels of accident severity, low and high; drivers of model 2 cars are injured in all accidents while model 1 provides better protection and its drivers are only injured in high severity accidents.
The theoretical basis for comparing the accident record of car models
For the two accident severities, RI2 (low) = O/number of low severity accidents = 0
R,,(high)=
number of high severity accidents =l number of high severity accidents
and overall R,,=proportion of accidents of high severity. This example shows that the ratio can vary with injury distribution t, and the previous example involving models D and E can be developed to show that it can also vary with the severity distribution f and differences between the distributions of “other” cars. Although these examples are clearly artificial, they prove that the ratio R is not independent of the severity distributions. Perhaps in the real world the bias will not be as marked as in these examples, but it will undoubtedly exist to some degree. The restriction of the analysis to two-car accidents is probably effective in reducing bias, but it is still desirable to take account of accident severity whenever a suitable proxy exists. The DoTmethod
The latest DOT report dealing with car accident rates (Department of Transport 1993a) has two sets of tables, one relating to primary safety and the other to secondary safety. Comparisons by model are only made for secondary safety; the method used will be referred to as the DOT method and is now described. The DOT method is conceptually simpler than Folksam method. Again, only two-car accidents with at least one injured driver are included, and the index (to be denoted by D) which corresponds to the Folksam R is the proportion of drivers of cars of a particular model who are injured when involved in one of these accidents, i.e. numbers of drivers who are injured in these accidents D= number of drivers who are involved in these accidents A second index is the proportion of these accident-involved drivers who are killed or seriously injured (ksi). The DOT report makes no attempt to justify its approach theoretically, in contrast to the Folksam report. This and other differences in presentation tend to create the impression that the two indices differ markedly. It will be shown, however, that the two are directly related, and that analyses of a single data set by the two methods yield very similar rankings. The restricted set of accidents used to evaluate the index is intended to standardize the risks faced by the drivers, but the DOT method goes further by
93
introducing factors which are likely to influence the likelihood of a driver being injured. As described in detail in the DOT report, a logistic regression model is used to represent the independent influence of: speed limit of road sex of driver
first point of impact age of driver
Clearly, speed limit is only a proxy for the variable of interest, accident severity. In order to compare D and R in the following sections, only the basic definitions will be used and D will not be adjusted to allow for these influences. The accompanying paper (Broughton 1996) presents a statistical model which provides a theoretical justification for the DOT index. This approach also allows the index to be evaluated rather more rapidly than is possible with the logistic regression method currently used. The treatment of serious accidents
The Stats19 slight injury category covers a wide range of relatively minor types of injury, such as sprains, bruises and slight shocks. This might suggest that attention should be focused on serious and fatal injuries, which have greater personal and economic consequences than slight injuries. The Folksam and DOT indices can both be applied to this restricted range, but unfortunately the number of driver casualties is thereby reduced by approximately six-sevenths, making it less likely that statistically significant differences between models will be identified. The DOT report exemplifies the problem: with the index based on all driver casualties, 29 out of 90 models differ significantly from the appropriate size group means but only 2 differ significantly with the index based on drivers ksi. If it can be shown that the two DOT indices are sufficiently closely linked then the ‘all casualty’ index will indicate reliably the protection provided in fatal and serious accidents. The ksi index quoted in the DOT report for model m cars is: DA= Proportion
of drivers of model m cars who are ksi when involved in two-car accidents where at least one driver is injured.
There is an alternative definition, however: D,=
Proportion of drivers of model m cars who are ksi when involved in two-car accidents where at least one driver is ksi.
D, is the result of raising the “severity threshold” for D by excluding drivers with only slight injuries, so it is more closely related than DA to D. It has another simple advantage over DA-the values of D
J. BROUGHTON
94
and DB lie in the same range, so they can be compared directly and differences between models emerge more clearly. The relationship with the “all casualties” index is more important, however, and this can be checked statistically with accident data for 1989-92. Linear models have been fitted to see whether D, or D, is more closely related to D. The fitted line for D, is D,= l.l9D-0.14
@X2=0.87, N=91, t=21.6)
rather than a causal relation. There are devices which can protect in serious accidents but not in slight, in particular those which are triggered at a certain speed, and models equipped with these may not conform to the general relationship. This analysis has shown the current relationship to be reliable, but it will need to be rechecked in future. For the present, rankings based on data from all car accidents should also be valid for the more severe accidents. If, however, DB indicates a statistically significant difference between one model and other comparable models (only likely among the most popular models), this should be relied on rather than the comparison based on D.
(1)
while R2 =0.63 when the model is refitted for D,: Fig. 2 compares D, and D, and includes the fitted line. Hence, DB is also more closely related empirically than D, to D, and there is a simple explanation for this. Some models will be exposed to greater risk of fatal and serious accidents than the average because of the types of condition under which they are normally driven, so their drivers will be at above-average risk of being killed or seriously injured. The calculation of D, automatically takes this into account, as it is based solely on data from fatal and serious accidents; D,, however, includes all accidents and so is affected by atypical exposure to risk. The residuals from (1) are generally greatest for those car models which are involved in the fewest accidents, so deviations are probably statistical in origin rather than the result of systematic differences with accident severity in the protection provided by these models. Thus D, (the proportion of drivers involved in fatal or serious accidents who are killed or seriously injured) correlates well with D (the proportion of drivers involved in injury accidents who are injured). It is important to realize that this is a statistical
An alternative
treatment
If the relation between D, and D had not proved to be so strong, a method involving casualty costs could have been used to emphasize data from fatal and serious accidents. This is considered now because it has recently been applied by researchers at the University of Cologne (Engels and Ltipsen 1993) to compare secondary safety in cars on German roads. The Department of Transport estimates the cost of road accident casualties in order, for example, to assess proposed highway investments; an article (Department of Transport 1993b) has explained the recent revision of the methodology. At 1991 prices, the costs quoted are E683,2OO/fatal casualty, ~71,OOO/seriouscasualty and f5,8OO/slight casualty. A new safety index can be calculated using these figures: M = mean cost of driver casualties per model m car involved in an accident
0.8 -
G x E z z 'p %
O.l-
z ._ r z e a
0.5 -
0.6 -
0.4 -
0.3 -
0.2 ' 0.3
0.4
0.1
0.6
0.5 Proportion
of drivers Fig. 2.
injured
0.8
0.9
The theoretical basis for comparing the accident record of car models
This is, in effect, a version of the DOT index D that is heavily weighted towards fatal casualtieseach fatality is the equivalent of 118 slight casualties, each serious casualty is the equivalent of 12.3 slight casualties. Its chief advantage is to emphasize the more severe casualties, but the higher correlation found above means that rankings by the two indices will be very similar. This new index has some potential drawbacks:
(4 relatively few drivers are killed, but the ranking is heavily influenced by the distribution of these fatalities by model; hence, the ranking is influenced to some extent by chance; U-9if, as happened recently in Great Britain, casualty costs were to be revised, a ranking based on M would also have to be revised (although the high correlation between D and DB suggests that in practice the ranking would be little changed); (4 other countries have different approaches to casualty costing, so that rankings could not be compared internationally.
Nonetheless, the German researchers have ranked the secondary safety of 125 car models based on police accident reports from several German Ltinder for 1991 and (in part) 1992. These reports include the manufacturer and model of car, also an assessment of which driver was responsible for the accident. The analysis is restricted to the cars in twocar accidents whose drivers were judged to be not responsible for the accident. For each model, the mean casualty cost is calculated (including injured passengers), an adjustment is made for the mean number of passengers per model (the standard German accident report form includes the number of vehicle occupants, including those not injured) and the models are ranked accordingly. No allowance is made for factors such as age of driver or type of road, as with the Folksam reports, and this may introduce bias. The introduction of the question of which driver was responsible is a curious feature of the German approach. Even if responsibility can be attributed reliably (and such factors are excluded from the Stats19 report form in part because of doubts about the reliability of police reporting of such subjective data), there is little reason to expect that the “responsible” and “not responsible” drivers face different risks in the typical accident. The exclusion of the “responsible” drivers simply halves the number of cars included in the analysis and reduces the precision of the results.
COMPARISON An aggressivity
95
OF FOLKSAM INDICES
AND DOT
index
The question of whether certain models are relatively “aggressive” in accidents, in the sense that they are associated with unexpectedly many casualties in the “other” car, will be examined as a prelude to comparing the DOT and Folksam indices in more detail. Ideally, an index of secondary safety should be independent of aggressivity and measure purely the occupant protection provided by the model being evaluated, but it will be shown that neither D nor R achieves this ideal. Suppose that models 1 and 2 are identical except that the latter possesses a feature which leads to greater damage to the ‘other’ car in two-car accidents, and that there are two sets of two-car accidents which are identical except that the first involves model 1 cars and the second involves model 2 cars. The following table shows the hypothetical numbers in the two sets of accidents: a and b represent those accidents in which drivers of the other cars are injured by the particular feature of model 2 (non-injury accidents are included): Driver in model
Driver in other car
Injured Not injured
1
Injured
Not injured
n2
n3
n1
n4 Driver in model 2
Driver in other car
By definition,
Injured Not injured
Injured
Not injured
n,+a n,-a
n,+b n,-b
the DOT indices for the two models are D, = (nl+ nz)/(n, +
n2 +
~1
D2 = (nl + n2)/(n1 + n2 + n3 + b)
so D,/D2 = 1 + b/(nt + n2 + n3)
The Folksam indices for the two models are R, = (~1 +nz)hz
+
~1
R, = (nl + n2)/(n2 + a + n3 4 b)
so RI/R2 = 1 +(a + b)/(n, + n3)
Thus, with either the DOT or Folksam index, if both models are involved in identical sets of accidents, the index will be less for model 2 than for model 1. Although, by assumption, both models protect their
J BROUGHTON
96
which is also shown in the figure. The fact that there are no clear outliers suggests that no models are markedly more or less aggressive than other models which provide a similar level of occupant protection. When the calculation is repeated for drivers ksi, a similarly close relation is found between A, and 0;
drivers to the same extent, the model which is more aggressive (i.e. it “inflicts more injuries”) has the lower index. The DOT index comes closer than the Folksam index to the ideal of being independent of the casualties in the “other” car. An aggressivity Index will now be defined to complement the DOT and Folksam “Protection” indices. The natural definition is, by analogy with the DOT index, A=
AB = 0.92 -0.040,
- 0.860;
(R2=0.91, N=91,
Proportion of drivers of cars in collision with model m cars who are injured.
t=0.2, 4.5)
Thus, the relationship found for drivers ksi differs slightly from (2), having rather more curvature and, predictably, the data are slightly more scattered. There are no clear outliers.
This proportion is calculated over the same category of two-car accidents that was used before, so it can be written
The Folksam and DoTindices compared
Using the notation introduced in the previous section, the Folksam index can be written
As with the DOT index, A is influenced by differences between accident distributions; these could be allowed for using a statistical model but for the present the index will be calculated directly from the accident data. The index can also be calculated for more severe accidents by restricting n,, nz and n3 to drivers who were killed or seriously injured: this index will be written A, as its definition parallels that of D,. Figure 3 compares A and D. The data lie in a narrow pencil, with the aggressivity index A falling as the ‘protection index’ D rises. The relationship is not quite linear, however, and is well represented by the quadratic A=0.95-0.180-0.66D2
R=(nl+n2Mn2+n3)
while the DOT index is D = (nr + Mnl+
n2 + n3)
so D/R=(n2+n3)/(n,+n2+n3)=A.
Thus, R and D are directly linked by the aggressivity index, and if the DOT and aggressivity indices are known for a particular model, the Folksam index can be calculated directly (the same applies with the ksi indices provided DB is used). Figure 4 illustrates the close relationship between the two indices, and
(2)
(R2=0.96, N=91, t= 1.4, 5.1)
0.8
0.2
0.3
0.4
0.5
0.6 DOT index
Fig. 3.
0.7
0.8
The theoretical
basis for comparing
the accident
0.5
record
of car models
0.7
0.6
97
0.8
0.9
DOT index Fig. 4.
includes the fitted line implied by (2)
Driver in model I
R=D/[0.95-O.lSD-0.66D2]
The figure relating the Folksam ksi index and D, is similar, but the fitted line is more curved and the maximum value of R is 5. As Fig. 4 suggests, the safety ranking based on R is very similar to the ranking based on D. The rankings differ for a few models, all of which have relatively high or low aggressivity indices. It has been seen that D is less sensitive than R to differences in aggressivity, so the ranking provided by the DOT index should give the more reliable guide to the relative protection provided by the various models.
The e#ect of safety improvements The effect on the indices D and R of improving
the safety of a particular model of car is now investigated with the new notation. First, suppose that model I is an improved version of model 1, in the sense that the probability of drivers of model I cars being injured in a collision is 0 times the probability for drivers of model 1 cars; 0 is less than 1 but in practice only a little less, unless there is a major breakthrough in secondary safety. If all other models are unchanged, and model I cars are involved in identical collisions to those in the previous example, the expected distribution of accidents involving model I cars will be:
Driver in other models
Injured Not injured
Injured
Not injured
en, On,
(i-e)n,+n, (i-e)n,+n4
Applying the definitions of D and R with these data gives: D(1) = B(n, + n2)/(Onl + n2 + n3) =8(n,+n2)/(n,+n2+n3-(l-8)n,)z0D(l)
since (1 - 0) is small. R(1) = 0(n, + n2)/(n2 + n3) = BR( 1) (Collisions between two model I cars can be ignored as they will be relatively infrequent, given the low proportions of the car fleet accounted for by any single model.) Thus, D(1) is almost exactly equal to O.D( 1) and R(1) is exactly equal to O.R( 1). This is an important result: when the safety of one model is improved, all other models being unchanged, D reflects the improvement almost exactly and R reflects the improvement exactly. This has assumed that only model I is improved, but consider now the situation where all models are improved to the same degree. In collisions involving model I cars, the drivers of the other cars will also have a reduced risk of injury, so the expected distribution of accidents will now be:
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98
Driver in model I
Driver in other models
Injured Not injured
Injured
Not injured
@n, O(l -@z~+&,
8(1-B)n,+Bn3 (1 -8)Sz2+ (l-WQ+~,)+~,
Hence, in this new scenario
R(I)=R(l) Thus, if in future the secondary safety of each model is uniformly better than its current analogue, as represented by 0, the indices will be effectively unchanged. This apparent paradox is easily explained. A number of collisions that currently involve driver casualties will in future involve no casualties and the numerators and denominators will decline to leave D virtually unchanged and R exactly unchanged. Thus, the indices measure the relative secondary safety of current car models, but cannot evaluate the general progress in improving secondary safety. They compare models with the average for a particular year but other measures are needed to determine whether this average changes over the years. The ideal measure would incorporate information about damage-only accidents to see whether the proportion of accidents involving driver casualties has varied, but such information is not collected systematically at present in Great Britain and the position is unlikely to change. CONCLUSIONS This paper has considered the ways in which national accident data can be analysed to compare the safety of different car models. It has been written in the context of the data available in Great Britain, but the conclusions should be generally applicable. In addition to vehicle design, various factors influence the number of accidents and casualties involving a particular model, the principal one being exposure to accident risk. As there is virtually no information about the relative levels of exposure of different models, only simple comparisons of accident involvement rates are possible. The main index of model safety which does not require exposure data, and so can be calculated with the data available, is the proportion of car occupants involved in accidents who are injured. This index of secondary or passive safety (also known as “crashworthiness”) has been the main subject of this paper. The number of accidents and casualties is also influenced by accident severity. There is no consensus
as to how this should be measured, although change in velocity during an impact is a reasonable proxy. Different car models are likely to have different distributions of accident severity, but ideally a safety index should be free of bias arising from differences between the distributions. As no measure of accident severity is routinely available in accident reports, this ideal is unlikely to be achieved, but bias can be reduced by using speed limit as a proxy and by restricting the analysis to two-car accidents. Three indices of model safety are in regular use, in Sweden, Great Britain and U.S.A., and in addition an analysis has recently appeared in Germany using an index whose basic formulation is essentially similar to that used in Great Britain. The American. index makes no serious attempt to represent accident severity, so it has not been studied in detail. This paper has concentrated on the approaches of the Swedish Folksam insurance company and the British DOT, both of which measure the protection provided to a car driver in an accident. It has been seen that their basic formulations are directly linked by an aggressivity index which measures the infhience of each model on the risk of injury to the drivers of the cars with which it collides. As a result of this direct link, the two indices give very similar rankings, with differences occurring only where models are more or less aggressive than usual. The DOT ranking is preferable. The Folksam index is claimed to be unaffected by differences between models in their accident severity distributions, but this has been seen to be incorrect. No such claim is made for the DOT index, and instead a’statistical procedure is used to minimise bias by taking account explicitly of speed limit and driver characteristics. In this respect as well, the DOT index is preferable. Both indices are relative rather than absolute measures. They will compare the safety of different models at one time but cannot assess the changes in safety over a number of years, as new models with improved safety features enter the car fleet. Two DOT indices are published annually, one based on all driver casualties in two-car accidents and the other on drivers killed or seriously injured (ksi). The latter may seem the more pertinent because it deals with more severe accidents, but the former is more discriminating because of the much larger number of accidents used in its calculation. The two indices have been shown to be highly correlated, once the ksi index has been redefined slightly, so they give model rankings that are virtually identical. Thus, with most models it is sensible to concentrate on the more discriminating “all casualties” index in the knowledge that it also represents the ranking of secondary safety in the more severe accidents. For the most popular
The theoretical basis for comparing the accident record of car models
models, however, the ksi index will be sufficiently precise for its ranking to be preferred.
An
accompanying
paper
(Broughton
1996)
develops certain aspects of the analysis, examines the stability of the DOT index and demonstrates a method for adjusting it to take account of the greater occupant protection provided by heavier cars. work described in this paper was carried out in the Safety and Environment Resource Centre of TRL.
Acknowledgement-The
REFERENCES Broughton, J. The British index for comparing the accident record of car models. Accid. Anal. and Prev. 28: 101-109; 1996.
99
Department of Transport. Cars: make and model: injury accident and casualty rates: Great Britain: 1991. London: HMSO; 1993a. Department of Transport. Road Accidents Great Britain 1992. London: HMSO; 1993b. Department of Transport. Road Accidents Great Britain 1993. London: HMSO; 1994. Engels, K.; Ltipsen, H. Rangreihenfolge von Pkw-Typen hinsichtlich ihrer passiven bzw. inneren Sicherheit (Ranking of passenger car models with respect to their passive and interior safety). University of Cologne; 1993. Hlgg, A.; KamrCn, B.; v Koch, M; Kullgren, A, Lie, A;
Malmstedt, B; Nygren, A; Tingvall, C. Folksam car model safetv rating 1991-92. Stockholm: Folksam Research; 1992. Highway Loss Data Institute. Insurance collision report R93-2; 1993.
Pergamon
THE THEORETICAL BASIS FOR COMPARING ACCIDENT RECORD OF CAR MODELS*
THE
JEREMYBROUGHTON Transport Research Laboratory, Crowthome, Berkshire, RG116AU, U.K. (Received
6 April 1994;accepted 18 July 1995)
Abstract-The accident records of different models of car can be compared statistically, provided that accident data which allow the make and model of accident-involved cars to be identified are collected on a national scale: this has been done in Great Britain since 1989. This paper considers the theoretical basis for comparing safety and shows that, because of the lack of detailed exposure data, the most which can currently be achieved is to measure the level of secondary safety (also known as crashworthiness). Based on mathematical considerations, it is shown that the best measure of secondary safety of a particular model is the proportion of drivers who are injured when involved in a two-car accident where one or other driver is injured. In order to minimize bias, this proportion should be adjusted statistically to allow for the influence on the accident data of factors such as type of road and age of driver.
than 1 would imply a worse record. It would then be desirable to see whether differences could be explained by factors such as mass, to judge whether the record of a particular model was better or worse than would be expected for a car of that “size”. A sophisticated statistical model is required to achieve this ideal, but information about the accident exposure of different car models is not available at the required level of detail in Great Britain. Consequently, only a more limited goal is feasible. The most that can be achieved is to measure the level of secondary safety (“crashworthiness”) provided by each model, that is, the probability of an occupant being injured once a car has become involved in an accident. Even this is not straightforward to calculate, as there are various potential sources of bias. At present, two methods in regular use take some account of the complexities involved, one from the U.K. Department of Transport (DOT) and another from the Swedish company Folksam. The two methods lead to different safety indices, but it will be shown that they are directly linked. This paper introduces some of the factors that can influence the probability of an accident-involved driver being injured; these need to be taken into account to minimize bias when evaluating any safety index. The two existing methods for estimating such safety indices are then compared, using British data from 1989-92. A sequel to this paper (Broughton 1996) presents further developments of the modeling, and more detailed results.
INTRODUCTION In several countries, accident data are collected in sufficient detail and on a sufficient scale for it to be possible to assess the relative safety of different car models. Since 1989, the British Stats197 accident reports have included the Vehicle Registration Mark of accident-involved vehicles, which enables details such as vehicle make and model to be extracted from files at the Driver and Vehicle Licensing Agency. Thus, Great Britain is now also a member of this group. This paper considers the theoretical basis for assessing the relative safety of different models, with particular reference to the data available from the Stats19 database. The ideal basis for assessing the relative safety of any model can be stated simply. It is the ratio of the number of occupant casualties that occur in accidents involving that model relative to the number that would be expected if the accidents had instead involved cars of a “standard” model representing the current average. This could be extended to a range of ratios, to include casualties to other road users. Ratios less than 1 would imply an accident record that was better than average, while ratios greater
*Crown Copyright 1994. The views expressed in this publication are not necessarily those of the Department of Transport. tThe Stats19 form is used by all police forces in Great Britain to record details of any road accident involving human injury or death. 89
J. BROUGHTON
90
FACTORS
better than model C when severity <5 but worse when severity = 6, 7 or 8. It seems improbable that a method could be devised for estimating these injury probabilities without reliable reports of accident severity. Nonetheless, the following example demonstrates the importance of allowing for differences in accident severity as far as possible. Suppose that the accident records of car models D and E are to be compared, that both models actually offer the same level of protection and that model D is involved in relatively more high-speed impacts than model E because these cars travel more on rural roads. As model D is involved in accidents of higher severity, relatively more of its occupants will be injured, so a ndive comparison of the accident records will find that D offers a lower level of protection than E. Thus, failing to take account of accident severity can lead to misleading conclusions if different car models do actually experience different ranges of accident severity. While the change in velocity during an impact can be estimated with some success by trained investigators, it is not routinely available from police accident reports. The speed limit at the accident site is available from the Stats19 reports and can be used as a proxy for accident severity.
INFLUENCING A MODEL’S ACCIDENT RECORD
The “Stats19 database” contains statistical details of all road accidents reported to the police in Great Britain in which one or more people are injured. A report is published annually to summarize the latest data (e.g. Department of Transport 1994). Injuries are classified as fatal, serious or slight, and all police forces supply data using the uniform Stats19 report form. The accident data analysed in this paper come from the Stats19 files for 1989-1992, so all empirical results relate to injury accidents. The accident record of a specific car model is likely to be affected by various factors whose presence is recorded in the Stats19 reports. These are examined below, and ways of using this information to achieve the index of safety with the lowest practicable level of bias will be discussed later. Accident severity
A sound conceptual basis from which to approach the definition of an index of safety is set out in various Folksam reports (e.g. HBgg et al. 1992). This uses the concept of “accident severity”, which is not exactly specified but seems to be best represented by the change in velocity during an impact. For any particular make of car, the probability of an occupant being injured increases with accident severity, but the rate of increase will differ between models. Figure 1 presents three hypothetical examples, using 10 levels of accident severity. No driver of any of the three models is injured at level 1, at level 10 all drivers of each of the three models are certain to be injured, but there are differences between the models at intermediate levels. Model B protects occupants better than models A and C; model A protects occupants
Diferences in accident distributions
Table 1 examines the extent to which models’ accident-involvement distributions varied in 1991, using a list of 74 models derived from a recent report (Department of Transport 1993a) which included all models with more than 20,000 vehicles registered. Older and newer cars (those first registered before and after 1 January 1989) are treated separately, and
1.0
1
2
3
4
5
6
Accident
severity
Fig. 1.
I
8
9
10
The theoretical
basis for comparing
Table 1. Distribution
the accident record of car models
of injury accident involvements,
91
1991
Size of car
Cars registered pre-1989 Number % with: male drivers drivers aged 17-24 drivers aged 25-34 drivers aged 35-54 drivers aged 55 and older % on nbu roads Cars registered 1989-91 Number % with: male drivers drivers aged 17-24 drivers aged 25-34 drivers aged 35-54 drivers aged 55 and older % on nbu roads
Small
Small/ medium
Medium
Large
All
44284
59615
42787
10189
156875
49 42 24 21 12 28
69 35 27 25 12 28
79 25 30 34 12 27
81 11 25 48 15 30
67 33 27 28 12 28
13938
19513
15177
3415
52043
53 26 29 29 15 30
71 20 32 34 14 34
81 12 31 46 12 40
86 6 21 60 13 44
70 18 30 38 14 35
the table includes the percentage of cars that were involved in accidents on non built-up (nbu) roads (‘non built-up’ is the term applied in Great Britain to any road with a speed limit exceeding 40 mph). Results are presented according to the size of car as defined in the DOT report; as an approximate guide, ‘small’ cars are 140-152 inches long, ‘small/medium’ cars are 153-167 inches long, ‘medium’ cars are 168-180 inches and ‘large’ cars are over 180 inches long. The table represents 67.8% of all cars involved in accidents in 1991, the others being excluded because the model could not be identified or was not on the list. The table shows clear variations between model groups in their accident circumstances. Among newer cars, larger cars are involved in relatively more accidents on nbu (high speed) roads, but the range of variation is much less among older cars: this presumably reflects the travel patterns of company-owned cars (which in Great Britain are rarely more than 3 years old and tend to be larger than privately-owned cars). Variations are even wider among individual models-one half of the newer Jaguars involved in accidents were travelling on nbu roads, compared with only one sixth of older Hyundai Stellars (the Stellar is a medium-sized model). These variations presumably indicate differences in exposure since, for example, it seems unlikely that Jaguars are involved in an especially high number of accidents per mile travelled on nbu roads, rather that a high proportion of their mileage is on these roads. A driver’s age and sex affect the likelihood of becoming involved in an accident, but they also affect the likelihood of being seriously injured and hence
the relationship between accident severity and injury probability. An elderly driver is more vulnerable in an accident than a younger driver, so models with a high proportion of elderly drivers will tend to have relatively many seriously injured drivers. One third of drivers of newer Toyota Carinas (a medium-sized model) are at least 55 years old, compared with one twelfth of drivers of older Fiat Pandas (a small model). Similarly, the injury distributions for men and women differ, and there are clear differences in the driver sex distributions: only one ninth of newer Ford Granadas (a large model) were driven by women, compared with three fifths of older Nissan Micras (a small model). Thus, age and sex of driver also need to be taken into account when comparing secondary safety. All cars involved in accidents (with the trivial exception of parked cars) have exactly one driver, so a consistent basis for comparing models is achieved by studying only driver casualties. The inclusion of passenger casualties appears attractive because it increases numbers and so raises the likelihood of identifying apparently significant differences. The comparison will, however, be biased against those models which carry relatively many passengers unless there is reliable evidence for the number of passengers carried per model. This does not exist in Great Britain and, moreover, the Stats19 data show wide variations in the ratio of passenger to driver casualties which cannot be explained simply by car size. Thus, for this country at least, it would not be safe to include passenger casualties when calculating a safety index.
92
J.BROUG~N
EXISTING
SAFETY
INDICES
Three series of publications have compared car safety on the basis of accident data: (a)The US Highway Loss Data Institute (HLDI) routinely publishes analyses of insurance claims data, principally from injury claims paid to the occupants of insured cars and collision claims paid to the owners of insured cars. The data are supplied by a group of 11 or 12 major motor insurers, with exposure defined as the number of “insured vehicle years” covered by policies with these companies (see, for example, Highway Loss Data Institute, 1993). Various types of claim are analysed, for example of injury claims, theft and collisions claims, also repair costs. Only basic allowances are made to compensate for the relatively high number of claims for cars with “youthful operators” and for policies with higher “deductible amounts”, however, so this method will not be considered further. (b)The Swedish insurance company Folksam has published comparisons for a decade, using a relatively sophisticated method to analyse its claims data. Its method will be considered in detail. (c) The U.K. Department of Transport has published three annual reports of analyses of the Stats19 data enhanced with vehicle details from the DVLA files. The analytical method has developed in the successive reports, and the method from the latest report (Department of Transport 1993a) will be considered. In addition, a recent German report (Engels and Lupsen 1993) has presented a safety ranking which is similar to that used by the DOT. It is described below. The Folksam method This discussion of the Folksam method is based upon a recent report (HBgg et al. 1992), although previous reports have contained the identical description. The method is applied to statistical data compiled from accident reports made by motorists insured with the company. Experience in Great Britain suggests that these may be completed rather less consistently than police accident reports, but it is the actual method rather than the reliability of the Swedish results that will be considered here. The method analysts only two-car accidents. It is claimed that this avoids bias due to variations in accident severity, on the grounds that in such accidents both cars experience the same potential risks and so the results are independent of any differences between accident severity distributions. The Folksam report deals with integrals over the range of accident
severities, but then returns to sums over discrete severity intervals. It seems simpler to adopt a discrete approach throughout. Let a, the range of accident severities, be divided into n intervals a,.~, of increasing severity, and let tti be the probability of the driver of a car of model c being injured in an accident of severity cli. The set of injury probabilities (t,-i) is central to studying the relative safety of model c, and Fig. 1 has illustrated the sets for three hypothetical cars. In addition, let fci be the proportion of the involvements of a car of model c in accidents of severity Ui$ so that f is the severity distribution for model c. If model c cars are involved in N accidents, the expected number of injured drivers of these cars is simply Z,=N* C t,i*fi I, is approximately equal to the actual number of injured drivers, but it is not clear how to identify (t,J To avoid this difficulty, consider only accidents of severity Cli where cars of model 1 and model 2 collide, then Drivers of in these N Drivers of in these
tli*fli
x
model 1 cars injured accidents model 2 cars injured accidents
The report claims that this ratio has the same value for all severity ranges and hence, summing over all ranges, that Z&z
+ = 2
Driver casualties in model 1 cars Driver casualties in model 2 cars
is independent of the severity distributions. Further, since for most model-pair., there will be rather few collisions on which to base the calculation, it is assumed that the distribution of “other” cars is the same for each model and the index actually calculated for model 1 is RI=
Driver casualties in model 1 cars Driver casualties in other cars
This is the “Folksam index”, which is also claimed to be unaffected by differences between severity distributions. These claims will now be reviewed. To see that R,, can in fact vary with the severity range, consider the following simple example. There are only two levels of accident severity, low and high; drivers of model 2 cars are injured in all accidents while model 1 provides better protection and its drivers are only injured in high severity accidents.
The theoretical basis for comparing the accident record of car models
For the two accident severities, RI2 (low) = O/number of low severity accidents = 0
R,,(high)=
number of high severity accidents =l number of high severity accidents
and overall R,,=proportion of accidents of high severity. This example shows that the ratio can vary with injury distribution t, and the previous example involving models D and E can be developed to show that it can also vary with the severity distribution f and differences between the distributions of “other” cars. Although these examples are clearly artificial, they prove that the ratio R is not independent of the severity distributions. Perhaps in the real world the bias will not be as marked as in these examples, but it will undoubtedly exist to some degree. The restriction of the analysis to two-car accidents is probably effective in reducing bias, but it is still desirable to take account of accident severity whenever a suitable proxy exists. The DoTmethod
The latest DOT report dealing with car accident rates (Department of Transport 1993a) has two sets of tables, one relating to primary safety and the other to secondary safety. Comparisons by model are only made for secondary safety; the method used will be referred to as the DOT method and is now described. The DOT method is conceptually simpler than Folksam method. Again, only two-car accidents with at least one injured driver are included, and the index (to be denoted by D) which corresponds to the Folksam R is the proportion of drivers of cars of a particular model who are injured when involved in one of these accidents, i.e. numbers of drivers who are injured in these accidents D= number of drivers who are involved in these accidents A second index is the proportion of these accident-involved drivers who are killed or seriously injured (ksi). The DOT report makes no attempt to justify its approach theoretically, in contrast to the Folksam report. This and other differences in presentation tend to create the impression that the two indices differ markedly. It will be shown, however, that the two are directly related, and that analyses of a single data set by the two methods yield very similar rankings. The restricted set of accidents used to evaluate the index is intended to standardize the risks faced by the drivers, but the DOT method goes further by
93
introducing factors which are likely to influence the likelihood of a driver being injured. As described in detail in the DOT report, a logistic regression model is used to represent the independent influence of: speed limit of road sex of driver
first point of impact age of driver
Clearly, speed limit is only a proxy for the variable of interest, accident severity. In order to compare D and R in the following sections, only the basic definitions will be used and D will not be adjusted to allow for these influences. The accompanying paper (Broughton 1996) presents a statistical model which provides a theoretical justification for the DOT index. This approach also allows the index to be evaluated rather more rapidly than is possible with the logistic regression method currently used. The treatment of serious accidents
The Stats19 slight injury category covers a wide range of relatively minor types of injury, such as sprains, bruises and slight shocks. This might suggest that attention should be focused on serious and fatal injuries, which have greater personal and economic consequences than slight injuries. The Folksam and DOT indices can both be applied to this restricted range, but unfortunately the number of driver casualties is thereby reduced by approximately six-sevenths, making it less likely that statistically significant differences between models will be identified. The DOT report exemplifies the problem: with the index based on all driver casualties, 29 out of 90 models differ significantly from the appropriate size group means but only 2 differ significantly with the index based on drivers ksi. If it can be shown that the two DOT indices are sufficiently closely linked then the ‘all casualty’ index will indicate reliably the protection provided in fatal and serious accidents. The ksi index quoted in the DOT report for model m cars is: DA= Proportion
of drivers of model m cars who are ksi when involved in two-car accidents where at least one driver is injured.
There is an alternative definition, however: D,=
Proportion of drivers of model m cars who are ksi when involved in two-car accidents where at least one driver is ksi.
D, is the result of raising the “severity threshold” for D by excluding drivers with only slight injuries, so it is more closely related than DA to D. It has another simple advantage over DA-the values of D
J. BROUGHTON
94
and DB lie in the same range, so they can be compared directly and differences between models emerge more clearly. The relationship with the “all casualties” index is more important, however, and this can be checked statistically with accident data for 1989-92. Linear models have been fitted to see whether D, or D, is more closely related to D. The fitted line for D, is D,= l.l9D-0.14
@X2=0.87, N=91, t=21.6)
rather than a causal relation. There are devices which can protect in serious accidents but not in slight, in particular those which are triggered at a certain speed, and models equipped with these may not conform to the general relationship. This analysis has shown the current relationship to be reliable, but it will need to be rechecked in future. For the present, rankings based on data from all car accidents should also be valid for the more severe accidents. If, however, DB indicates a statistically significant difference between one model and other comparable models (only likely among the most popular models), this should be relied on rather than the comparison based on D.
(1)
while R2 =0.63 when the model is refitted for D,: Fig. 2 compares D, and D, and includes the fitted line. Hence, DB is also more closely related empirically than D, to D, and there is a simple explanation for this. Some models will be exposed to greater risk of fatal and serious accidents than the average because of the types of condition under which they are normally driven, so their drivers will be at above-average risk of being killed or seriously injured. The calculation of D, automatically takes this into account, as it is based solely on data from fatal and serious accidents; D,, however, includes all accidents and so is affected by atypical exposure to risk. The residuals from (1) are generally greatest for those car models which are involved in the fewest accidents, so deviations are probably statistical in origin rather than the result of systematic differences with accident severity in the protection provided by these models. Thus D, (the proportion of drivers involved in fatal or serious accidents who are killed or seriously injured) correlates well with D (the proportion of drivers involved in injury accidents who are injured). It is important to realize that this is a statistical
An alternative
treatment
If the relation between D, and D had not proved to be so strong, a method involving casualty costs could have been used to emphasize data from fatal and serious accidents. This is considered now because it has recently been applied by researchers at the University of Cologne (Engels and Ltipsen 1993) to compare secondary safety in cars on German roads. The Department of Transport estimates the cost of road accident casualties in order, for example, to assess proposed highway investments; an article (Department of Transport 1993b) has explained the recent revision of the methodology. At 1991 prices, the costs quoted are E683,2OO/fatal casualty, ~71,OOO/seriouscasualty and f5,8OO/slight casualty. A new safety index can be calculated using these figures: M = mean cost of driver casualties per model m car involved in an accident
0.8 -
G x E z z 'p %
O.l-
z ._ r z e a
0.5 -
0.6 -
0.4 -
0.3 -
0.2 ' 0.3
0.4
0.1
0.6
0.5 Proportion
of drivers Fig. 2.
injured
0.8
0.9
The theoretical basis for comparing the accident record of car models
This is, in effect, a version of the DOT index D that is heavily weighted towards fatal casualtieseach fatality is the equivalent of 118 slight casualties, each serious casualty is the equivalent of 12.3 slight casualties. Its chief advantage is to emphasize the more severe casualties, but the higher correlation found above means that rankings by the two indices will be very similar. This new index has some potential drawbacks:
(4 relatively few drivers are killed, but the ranking is heavily influenced by the distribution of these fatalities by model; hence, the ranking is influenced to some extent by chance; U-9if, as happened recently in Great Britain, casualty costs were to be revised, a ranking based on M would also have to be revised (although the high correlation between D and DB suggests that in practice the ranking would be little changed); (4 other countries have different approaches to casualty costing, so that rankings could not be compared internationally.
Nonetheless, the German researchers have ranked the secondary safety of 125 car models based on police accident reports from several German Ltinder for 1991 and (in part) 1992. These reports include the manufacturer and model of car, also an assessment of which driver was responsible for the accident. The analysis is restricted to the cars in twocar accidents whose drivers were judged to be not responsible for the accident. For each model, the mean casualty cost is calculated (including injured passengers), an adjustment is made for the mean number of passengers per model (the standard German accident report form includes the number of vehicle occupants, including those not injured) and the models are ranked accordingly. No allowance is made for factors such as age of driver or type of road, as with the Folksam reports, and this may introduce bias. The introduction of the question of which driver was responsible is a curious feature of the German approach. Even if responsibility can be attributed reliably (and such factors are excluded from the Stats19 report form in part because of doubts about the reliability of police reporting of such subjective data), there is little reason to expect that the “responsible” and “not responsible” drivers face different risks in the typical accident. The exclusion of the “responsible” drivers simply halves the number of cars included in the analysis and reduces the precision of the results.
COMPARISON An aggressivity
95
OF FOLKSAM INDICES
AND DOT
index
The question of whether certain models are relatively “aggressive” in accidents, in the sense that they are associated with unexpectedly many casualties in the “other” car, will be examined as a prelude to comparing the DOT and Folksam indices in more detail. Ideally, an index of secondary safety should be independent of aggressivity and measure purely the occupant protection provided by the model being evaluated, but it will be shown that neither D nor R achieves this ideal. Suppose that models 1 and 2 are identical except that the latter possesses a feature which leads to greater damage to the ‘other’ car in two-car accidents, and that there are two sets of two-car accidents which are identical except that the first involves model 1 cars and the second involves model 2 cars. The following table shows the hypothetical numbers in the two sets of accidents: a and b represent those accidents in which drivers of the other cars are injured by the particular feature of model 2 (non-injury accidents are included): Driver in model
Driver in other car
Injured Not injured
1
Injured
Not injured
n2
n3
n1
n4 Driver in model 2
Driver in other car
By definition,
Injured Not injured
Injured
Not injured
n,+a n,-a
n,+b n,-b
the DOT indices for the two models are D, = (nl+ nz)/(n, +
n2 +
~1
D2 = (nl + n2)/(n1 + n2 + n3 + b)
so D,/D2 = 1 + b/(nt + n2 + n3)
The Folksam indices for the two models are R, = (~1 +nz)hz
+
~1
R, = (nl + n2)/(n2 + a + n3 4 b)
so RI/R2 = 1 +(a + b)/(n, + n3)
Thus, with either the DOT or Folksam index, if both models are involved in identical sets of accidents, the index will be less for model 2 than for model 1. Although, by assumption, both models protect their
J BROUGHTON
96
which is also shown in the figure. The fact that there are no clear outliers suggests that no models are markedly more or less aggressive than other models which provide a similar level of occupant protection. When the calculation is repeated for drivers ksi, a similarly close relation is found between A, and 0;
drivers to the same extent, the model which is more aggressive (i.e. it “inflicts more injuries”) has the lower index. The DOT index comes closer than the Folksam index to the ideal of being independent of the casualties in the “other” car. An aggressivity Index will now be defined to complement the DOT and Folksam “Protection” indices. The natural definition is, by analogy with the DOT index, A=
AB = 0.92 -0.040,
- 0.860;
(R2=0.91, N=91,
Proportion of drivers of cars in collision with model m cars who are injured.
t=0.2, 4.5)
Thus, the relationship found for drivers ksi differs slightly from (2), having rather more curvature and, predictably, the data are slightly more scattered. There are no clear outliers.
This proportion is calculated over the same category of two-car accidents that was used before, so it can be written
The Folksam and DoTindices compared
Using the notation introduced in the previous section, the Folksam index can be written
As with the DOT index, A is influenced by differences between accident distributions; these could be allowed for using a statistical model but for the present the index will be calculated directly from the accident data. The index can also be calculated for more severe accidents by restricting n,, nz and n3 to drivers who were killed or seriously injured: this index will be written A, as its definition parallels that of D,. Figure 3 compares A and D. The data lie in a narrow pencil, with the aggressivity index A falling as the ‘protection index’ D rises. The relationship is not quite linear, however, and is well represented by the quadratic A=0.95-0.180-0.66D2
R=(nl+n2Mn2+n3)
while the DOT index is D = (nr + Mnl+
n2 + n3)
so D/R=(n2+n3)/(n,+n2+n3)=A.
Thus, R and D are directly linked by the aggressivity index, and if the DOT and aggressivity indices are known for a particular model, the Folksam index can be calculated directly (the same applies with the ksi indices provided DB is used). Figure 4 illustrates the close relationship between the two indices, and
(2)
(R2=0.96, N=91, t= 1.4, 5.1)
0.8
0.2
0.3
0.4
0.5
0.6 DOT index
Fig. 3.
0.7
0.8
The theoretical
basis for comparing
the accident
0.5
record
of car models
0.7
0.6
97
0.8
0.9
DOT index Fig. 4.
includes the fitted line implied by (2)
Driver in model I
R=D/[0.95-O.lSD-0.66D2]
The figure relating the Folksam ksi index and D, is similar, but the fitted line is more curved and the maximum value of R is 5. As Fig. 4 suggests, the safety ranking based on R is very similar to the ranking based on D. The rankings differ for a few models, all of which have relatively high or low aggressivity indices. It has been seen that D is less sensitive than R to differences in aggressivity, so the ranking provided by the DOT index should give the more reliable guide to the relative protection provided by the various models.
The e#ect of safety improvements The effect on the indices D and R of improving
the safety of a particular model of car is now investigated with the new notation. First, suppose that model I is an improved version of model 1, in the sense that the probability of drivers of model I cars being injured in a collision is 0 times the probability for drivers of model 1 cars; 0 is less than 1 but in practice only a little less, unless there is a major breakthrough in secondary safety. If all other models are unchanged, and model I cars are involved in identical collisions to those in the previous example, the expected distribution of accidents involving model I cars will be:
Driver in other models
Injured Not injured
Injured
Not injured
en, On,
(i-e)n,+n, (i-e)n,+n4
Applying the definitions of D and R with these data gives: D(1) = B(n, + n2)/(Onl + n2 + n3) =8(n,+n2)/(n,+n2+n3-(l-8)n,)z0D(l)
since (1 - 0) is small. R(1) = 0(n, + n2)/(n2 + n3) = BR( 1) (Collisions between two model I cars can be ignored as they will be relatively infrequent, given the low proportions of the car fleet accounted for by any single model.) Thus, D(1) is almost exactly equal to O.D( 1) and R(1) is exactly equal to O.R( 1). This is an important result: when the safety of one model is improved, all other models being unchanged, D reflects the improvement almost exactly and R reflects the improvement exactly. This has assumed that only model I is improved, but consider now the situation where all models are improved to the same degree. In collisions involving model I cars, the drivers of the other cars will also have a reduced risk of injury, so the expected distribution of accidents will now be:
J. BROUGHTON
98
Driver in model I
Driver in other models
Injured Not injured
Injured
Not injured
@n, O(l -@z~+&,
8(1-B)n,+Bn3 (1 -8)Sz2+ (l-WQ+~,)+~,
Hence, in this new scenario
R(I)=R(l) Thus, if in future the secondary safety of each model is uniformly better than its current analogue, as represented by 0, the indices will be effectively unchanged. This apparent paradox is easily explained. A number of collisions that currently involve driver casualties will in future involve no casualties and the numerators and denominators will decline to leave D virtually unchanged and R exactly unchanged. Thus, the indices measure the relative secondary safety of current car models, but cannot evaluate the general progress in improving secondary safety. They compare models with the average for a particular year but other measures are needed to determine whether this average changes over the years. The ideal measure would incorporate information about damage-only accidents to see whether the proportion of accidents involving driver casualties has varied, but such information is not collected systematically at present in Great Britain and the position is unlikely to change. CONCLUSIONS This paper has considered the ways in which national accident data can be analysed to compare the safety of different car models. It has been written in the context of the data available in Great Britain, but the conclusions should be generally applicable. In addition to vehicle design, various factors influence the number of accidents and casualties involving a particular model, the principal one being exposure to accident risk. As there is virtually no information about the relative levels of exposure of different models, only simple comparisons of accident involvement rates are possible. The main index of model safety which does not require exposure data, and so can be calculated with the data available, is the proportion of car occupants involved in accidents who are injured. This index of secondary or passive safety (also known as “crashworthiness”) has been the main subject of this paper. The number of accidents and casualties is also influenced by accident severity. There is no consensus
as to how this should be measured, although change in velocity during an impact is a reasonable proxy. Different car models are likely to have different distributions of accident severity, but ideally a safety index should be free of bias arising from differences between the distributions. As no measure of accident severity is routinely available in accident reports, this ideal is unlikely to be achieved, but bias can be reduced by using speed limit as a proxy and by restricting the analysis to two-car accidents. Three indices of model safety are in regular use, in Sweden, Great Britain and U.S.A., and in addition an analysis has recently appeared in Germany using an index whose basic formulation is essentially similar to that used in Great Britain. The American. index makes no serious attempt to represent accident severity, so it has not been studied in detail. This paper has concentrated on the approaches of the Swedish Folksam insurance company and the British DOT, both of which measure the protection provided to a car driver in an accident. It has been seen that their basic formulations are directly linked by an aggressivity index which measures the infhience of each model on the risk of injury to the drivers of the cars with which it collides. As a result of this direct link, the two indices give very similar rankings, with differences occurring only where models are more or less aggressive than usual. The DOT ranking is preferable. The Folksam index is claimed to be unaffected by differences between models in their accident severity distributions, but this has been seen to be incorrect. No such claim is made for the DOT index, and instead a’statistical procedure is used to minimise bias by taking account explicitly of speed limit and driver characteristics. In this respect as well, the DOT index is preferable. Both indices are relative rather than absolute measures. They will compare the safety of different models at one time but cannot assess the changes in safety over a number of years, as new models with improved safety features enter the car fleet. Two DOT indices are published annually, one based on all driver casualties in two-car accidents and the other on drivers killed or seriously injured (ksi). The latter may seem the more pertinent because it deals with more severe accidents, but the former is more discriminating because of the much larger number of accidents used in its calculation. The two indices have been shown to be highly correlated, once the ksi index has been redefined slightly, so they give model rankings that are virtually identical. Thus, with most models it is sensible to concentrate on the more discriminating “all casualties” index in the knowledge that it also represents the ranking of secondary safety in the more severe accidents. For the most popular
The theoretical basis for comparing the accident record of car models
models, however, the ksi index will be sufficiently precise for its ranking to be preferred.
An
accompanying
paper
(Broughton
1996)
develops certain aspects of the analysis, examines the stability of the DOT index and demonstrates a method for adjusting it to take account of the greater occupant protection provided by heavier cars. work described in this paper was carried out in the Safety and Environment Resource Centre of TRL.
Acknowledgement-The
REFERENCES Broughton, J. The British index for comparing the accident record of car models. Accid. Anal. and Prev. 28: 101-109; 1996.
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Department of Transport. Cars: make and model: injury accident and casualty rates: Great Britain: 1991. London: HMSO; 1993a. Department of Transport. Road Accidents Great Britain 1992. London: HMSO; 1993b. Department of Transport. Road Accidents Great Britain 1993. London: HMSO; 1994. Engels, K.; Ltipsen, H. Rangreihenfolge von Pkw-Typen hinsichtlich ihrer passiven bzw. inneren Sicherheit (Ranking of passenger car models with respect to their passive and interior safety). University of Cologne; 1993. Hlgg, A.; KamrCn, B.; v Koch, M; Kullgren, A, Lie, A;
Malmstedt, B; Nygren, A; Tingvall, C. Folksam car model safetv rating 1991-92. Stockholm: Folksam Research; 1992. Highway Loss Data Institute. Insurance collision report R93-2; 1993.