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Accidents, mileage, and the exaggeration of risk
AC&. An& & Prev. Vet. 23. Nos. 213. pp. 183-188. 1991 Printed in Great Britain.
ACCIDENTS,
MILEAGE, AND THE EXAGGERATION OF RISK*
MARY IS. JANKE California Department of Motor Vehicles, Sacramento, CA 95818, U.S.A. (Received 29 March 1990; in revised form August 19, 1990) Abstract-The usual interpretation of accidents per mile as a measure of risk exaggerates the apparent risk of low-mileage groups-for example, teenagers and the elderly. The assumption of a linear proportionaf relationship between mileage and accidents is shown not to fit obtained data. Neither would it be expected to fit hypothetical data derived from a “standard driver” or a group of equally competent drivers driving different numbers of miles. People driving low mileages tend to accumulate much of their mileage on congested city streets with two-way traffic and no restriction of access, while high-mileage drivers typically accumulate most of those miles on freeways or other divided muItilane highways with limited access. Because the driving task is simpler, the accident rate per mile is much lower on freeways, and beyond a certain point, a person driving half as many miles as another would be expected to have considerably more than half as many accidents. This and other considerations lead to the suggestion that an induced exposure approach would be a more valid method of correcting accident rates for mileage.
Writers commonly state, and graphical displays imply, that some driver groups-for example, the elderly-are relatively hazardous drivers despite experiencing fewer accidents per driver than other groups, because they have much higher accident rates when the rate is adjusted for mileage (e.g. Transportation Research Board 1988; Mercer 1989). Using the average number of accidents per driver as a measure of group risk is an adequate measure of the group’s public safety impact, but plainly confounds innate accident liability-which may be the variable of interest- with exposure to accident risk. If two groups of drivers are equally competent and prudent, but differ in miles driven, the higher-mileage group will have more accidents on the average because of its greater exposure to risk. Therefore, it has seemed an obvious corrective measure to adjust group accident rates by group mileage in order to measure innate risk more accurately. This adjustment has commonly involved dividing a measure of group accidents by a measure of group mileage to get accidents per mile (or per 100,000 miles, etc.) as a criterion measure. If accidents per mile are accepted as being simply accidents per mile, this procedure is unobjectionable. But if the ratio is interpreted as being a quantitative measure of risk, extremely (and fallaciously) high apparent risk levels for low-mileage groups of drivers may be produced. For example, in Huston and Janke ~~986) a graph of total accidents per driver per mile as a function of age and sex revealed that the rate for women aged 75 or above (who drive only about 1,000 miies per year, on the average) was about six times as high as that of the “average driver.” The implication to a naive reader might well have been that elderly women drivers are six times as dangerous to society; the authors tried to temper this impression by discussing and qualifying the implications of their graph. Similar inferences have been made with respect to groups that also drive less, but have more accidents per driver, than the average. Their risk, already high, may be exaggerated when accidents per mile are used as a criterion. For example, in reference to young drivers, the C~~~~~~~~~ Driver Fact Book (California Department of Motor Vehicles 1981) noted that Teenage drivers average twice as many accidents as adult drivers, while driving only half as many miles; the young driver accident rate per mile is thus four times as great as that of adult drivers (p. 36). *Opinions and conclusions expressed in this paper are those of the author and not necessarily those of the California Department of Motor Vehicles or the State of California. 183
184
M. K. JANKE
Again, while the statement is true in itself, an uncritical reader might interpret it as indicating that teenage drivers are four times as hazardous as adults, The underlying assumption involved in using this type of statement as a basis for inferring quantitative relative risk is that, everything else being equal, there is a linear relationship between accidents per driver (A) and driver mileage (M), with the line going through the origin. Under this model (A = kM), if mileage is increased by a factor of 2, say, then accident involvements are also expected to increase by a factor of 2, for equally prudent drivers of equal competence. If a hypothetical “standard driver” were considered, such a person, when driving 10,~ miles per year, would be expected to be involved in twice as many accidents as when driving only 5,000 miles in the same period of time. Therefore, when it is found that a real-world 5,000-miles-per-year driver has more than half as many accidents as another driving 10,000 miles, a relative lack of either competence or prudence (i.e. increased accident risk) is assumed for the lowermileage driver. This reasoning ignores the consideration that the variable that should bear a linear relation to accidents, if any does, is not mileage but exposure to accident risk, of which mileage is only one component. Thus, not even for drivers equivalent in all relevant respects but miles driven would accident occurrence be expected to be proportional to mileage. For such to be true, one would have to posit the concept of a “standard mile” offering a mile’s worth of risk. Real-world driving does not consist of standard miles; different highway miles offer different amounts of exposure to risk. For example, in this society, high-mileage drivers typically accumulate most of their mileage on freeways or other divided multilane roadways with limited access. These generally have much lower accident rates per mile than do other types of roadways; recent data from the California Business, Transportation, and Housing Agency (1985) indicated that there were 2.75 times as many accidents per mile driven on non-freeways as on freeways. From this it can be predicted that with constant driver competence and prudence, accidents will tend to rise at a low and decreasing rate as mileage increases beyond a certain point. On the other end of the mileage scale, drivers with low mileage typically make short trips in congested city driving conditions. Inserting one’s vehicle into the traffic stream (as opposed to lowing with the traffic stream) offers considerable opportunity for accidents, and the need to do this occupies a much greater proportion of short stop-andgo trips than of long ones. Even the shortest trip affords this accident opportunity, and it would be expected that accidents as a function of mileage would rise steeply from the origin. More sophisticated models than the simple linear one have been introduced, for example by Quimby, Maycock, Carter, Dixon, and Wall (1986), who represented accident involvements by means of a multivariate model in which the mileage term appeared as a multiplicative factor raised to the .25 power. However, such complex models have apparently not been widely used. In a more general context, Mahalel (1986) opted for a broader definition of exposure. He noted that “there appears to be a vicious circle in which, on the one hand, the accepted definition of risk necessitates a linear relationship between accidents and exposure; on the other hand, it is difficult (or even impossible) to find exposure estimators that fulfill this limitation” (p. 86). In trying to determine the relationship between mileage and accidents, other difficulties arise as well. First, one would not expect to find a single “true” curve; in fact, there should be a whole family of mileage-accident curves, depending upon parameters representing different types of driving conditions. Second, mileage data are typically self-reported and therefore subject to considerable error. Third, and more critically, even if some more valid way of estimating mileage were used, it would still be the case that drivers are not randomly assigned to mileages. Since groups of people who customarily drive low or high mileages differ in age, sex, license class, and probably other characteristics like health status, the data points to which a hypothetical generalized mileage-accident curve is fitted represent different kinds of people at different ends of the mileage scale. It is highly likely that these groups have different accident expectancies
Accidents, mileage, and the exaggeration of risk
185
simply because of their different characteristics and the ways in which these affect driving competency, regardless of their mileage. In fact, it seems not unreasonable to hypothesize that drivers with a low level of competence tend to have low mileages. Circularity is introduced because the causality here might go in either direction. Because of the limitations noted above, one would not expect empirical accidentmileage data to be very meaningful. Nevertheless, some California data were examined to illustrate their failure to show propo~ionality of accidents to mileage. The Cu~~~o~~~u Driver Fact Book displays, on page 17, accident-per-person averages taken over a sixyear period by sex for seven annual mileage categories (under 1,000 through 30,000 + ). These data, derived in part from a survey, are due to Carpenter (1976). The first two columns of Table 1 show the mileage category medians (M) and corresponding average accident rates (A) for combined sexes. For analytical purposes, an arbitrary median of 500 was chosen for the under-l,000 category and one of 40,000 for the 30,000+ mileage category. These data are poor for illustrative purposes, because differences in mileage at the extremely low and the high ends of the scale are not discriminable. However, the gross shape and placement of the curve relating accident involvement to mileage are still of interest. A linear regression equation was fitted to the data by least squares, and the fit was excellent (r = .97). But this equation, A = 7.7 (1O-6 M) + .27, clearly does not go through the origin. Although it must necessarily be true that someone who drives zero miles will be involved (as a driver) in zero accidents, in order to pass through the origin the line fitting these points would have to assume a marked curvature as it closely approached M = 0. A linear and proportional relationship between accidents and mileage does not describe the data. Through trial and error, an equation was derived that roughly fit Carpenter’s data points, and for which A, the accident average, at least comes close to zero when mileage, M, equals zero. Again it is for illustrative purposes only. The equation is A = [ln (A4 + 1)]/21 - (.05 - [lo-‘O(M - 20,000)2]). The third column in Table 1 shows the value of A corresponding to each M, as calculated from the equation. This curve passes close to the origin, as noted, and also fits the Carpenter data points fairly well. If we were now using the model to compare two groups, one with an annual mileage of 3,000 and the other with an annual mileage of 9,000, the second group (A = .396) would be expected to have 1.10 times as many accidents as the first (A = .360), not 3 times as many. Proportionality similarly fails to apply for other hypothetical examples, and we would not expect it to. Other data, for example those of Burg (1973) support this general conclusion.
Table 1. Actual and calculated accident rates by annual mileage (6-year record) Mileage category/ Median
Average accidents
None Under l,OOOJ500 1,~-4,~9/3,~ 5,~-9,~/7,5~ 10,~14,999/12,5~ 15,oOQ-19,999/17,500
0 (definition) .30 .31 .34 .38 .39 .46 .50
2o,ooo-29,999/25,ooo
30,000+ /40.000
Caiculated accidents - .Ol .28 .36 .39 .40 .42 .43 .50
186
M. K.
JANKE
A method of taking account of exposure that holds promise of being more fruitful is that of induced exposure, originated by J. D. Thorpe in 1964 and elaborated on in many subsequent studies (e.g. Carlson 1970; Haight 1973; Cerrelli 1973; McKelvey, Maleck, Stamatiadis, and Hardy 1988). Here, instead of soliciting drivers’ self-report, the exposure of a category of drivers is measured by (or induced from) their involvement in accidents for which they were not responsible. Their risk is measured by (or induced from) their involvement in accidents for which they were responsible. Thus, of any two driver groups, the less competent is distinguished from the more competent one by being at fault in a greater proportion of their accidents, few or many as these may be. An example of the use of this technique is Cerrelli’s paper (1973). Restricting consideration to multiple-vehicle accidents, he used the induced exposure method to derive three measures. One, the relative exposure index, is the percentage of innocently accident-involved drivers in a category (e.g. an age-sex group), divided by the percentage of licensed drivers in that category within the driving population. The measure is intended to provide a measure of over- or under-exposure for a set of drivers. A second measure, the liability index, may be thought of as representing the group’s level of societal riski.e. their degree of threat to public safety-which is greater, everything else being equal, in a group that drives more. It is the percentage of accident-responsible drivers in the category divided by the percentage of licensed drivers in that category, and it is what an insurance company might be interested in when setting rates. Finally, the hazard index measures the risk, relative to exposure, of the categorywhat the present paper is concerned with. The hazard index is the ratio of the liability index to the exposure index. An index of 1.00 would be expected, over the long run, if members of a driver group were involved in only two-vehicle accidents, and in each accident culpability was attributed to one of the drivers by means of tossing a coin. It would represent the overall index for the entire driving population as well, if all accidents were two-vehicle with one driver culpable in each. Using Cerrelli’s method, the overall hazard index for men was 1.00, and for women 0.99. Cerrelli’s indices have been described at relative length in order to present and comment on some illustrative numbers he arrived at. For example, he found that women more than 64 years old drove very little, their overall exposure index being 0.42. (Indices vary for different types of roadways, times of day, and so forth.) Their liability index, measuring the degree of threat posed to society by women drivers in this age group, was also rather low at 0.63. This combination made for an overall hazard index of 1.5. (In contrast, the hazard index for both men and women aged 35-44 was 0.90.) The 1.5 figure for older women certainly indicates a degree of excessive risk but the excess is small when compared to that implied by the use of accidents per mile. As another example, the overall hazard index for males less than 20 was 1.2. For males over 64 it was 1.4 and, as seen above, it was 0.90 for males aged 35-44, 1.0 for males generally. This method, like the use of accidents per mile, leads to a concaveupward curve of the dependent variable as a function of age, with elderly men emerging as being at higher risk on the average than very young ones and both of these groups being at higher risk than men in the middle age ranges. But the “severalfold worse” quality of the relationship is lacking when the induced exposure method is used. Cerrelli treated only multiple-vehicle accidents in his 1973 paper. If he had included single-vehicle crashes (in which the driver is generally deemed to be responsible), older drivers would have tended to look relatively better, if anything. That is because they experience a much lower proportion of single-vehicle accidents than young drivers do (Cerelli 1989). Induced exposure may not be the optimal tool for measuring risk. One potential failing of the method as illustrated here is that a group that does not drive as defensively as another, and thus becomes “innocently” involved in more accidents, will tend to have a lower hazard index because its apparent exposure will be inflated. This might be corrected by assigning different degrees of responsibility to one or more crash-involved drivers, as appropriate. (See Wasielewski and Evans [ 19851, who formulated an “induced responsibility” model similar to that for induced exposure, in which they relaxed the
Accidents, mileage, and the exaggeration of risk
187
assumption that only one driver is responsible in a two-car crash.) Another potential weakness of induced exposure is the difficulty of assigning accident responsibility; attribution of responsibility on an accident report is vulnerable to any preconceptions or biases held by the officer completing the report. Nevertheless, the induced exposure approach would seem to give a more balanced picture of risk than does the method of using accidents per mile. It also appears to circumvent effects of the probable circularity between mileage and driving competence that complicates interpretation of accidents-per-mile data. One might ask, however, what difference it makes if the perceived hazard of lowmileage groups is exaggerated. Ignoring the fact that it is always desirable to point out whatever erroneous assumptions may be underlying our conclusions, it makes a practical difference if the climate of,opinion is affected and unwarranted licensing actions result. In the specific case of the elderly driver, Ranney and Pulling (1990) have pointed out that more than 50 million persons aged 65 or older will be eligible to drive in 2020. While they noted that mileage-based rates have come into question on grounds that mileage estimates have not been validated (Yanik 1985) and that older drivers’ curtailment of mileage is greater than their increase in risk (Evans 1988), they warned that this may not be true in the future. “The average yearly miles driven by drivers 65 and older has increased with each major survey taken between 1969 and 1983 . . . and therefore, such a compensatory tendency [to drive less] may well be specific to the current generation of older drivers” (p. 1). Faced with these considerations (all of which are valid), licensing administrators influenced by the exaggerated implied risk of the elderly group when accidents are divided by miles may become more alarmed than is warranted about the safety hazard posed by increasing numbers of elderly drivers. Even though researchers in general would be the last to countenance such an outcome, this alarm could lead to the introduction of unduly repressive licensing measures. Although it is unlikely that any state would arbitrarily de-license drivers on the basis of age alone, an exaggerated notion of the risk older people pose could easily result, for example, in the licensing agency’s calling in all drivers beyond a certain age for special testineupon the results of which their driving privileges might depend. This would be an obvious inconvenience and source of anxiety to the elderly. It would also be a less than optimal use of public funds, if the relative public safety threat posed by the specially treated group was not as great as believed and not as great as the threat posed by other groups that do not get special agency attention. Therefore, it seems important to point out, from the standpoint of fairness, that quantitative estimation of the relative risk posed by the group, based on accidents per mile at their currently low mileage, is inflated us a consequence ofthat low mileagegiven the assumptions of linearity and proportionality discussed above. There is no reason to assume that the “actual” relationship between mileage and accidents, even for a hypothetical “standard driver,” is of the simple A = kM type. On the contrary, there is reason to assume that linearity and proportionality would not hold. And the empirical data derived from samples of real-world drivers apparently do not fit such a model. Thus there is no reason to expect that the number of accidents divided by the number of miles driven will remain constant for a “standard driver” or for groups having equal intrinsic risk. Regardless of whether an induced exposure approach or some other method is substituted, it seems that it is at least time to discard the assumption of accident-mileage proportionality that now implicitly forms the basis of many group risk comparisons.
REFERENCES Burg, A. The effects of exposure to risk on driving record. Los Angeles, CA: University of California; 1973. California Business, Transportation, and Housing Agency, Department of Transportation, Division of Traffic Engineering. 1984 accident data on California state highways (road miles, travel, accident, accident rates). Sacramento, CA: Author; 1985. California Department of Motor Vehicles. The California driver fact book. Sacramento, CA: Author; 1981. Carlson. W. L. Induced exposure revisited. HIT Lab Report; 1970.
188
M. K.
JANKE
Carpenter, D. W. An evaluation of the California driver knowledge test and the University of Michigan item pool. Sacramento. , CA: Department of Motor Vehicles; 1976. Cerrelli, E. C. Driver exposure: the indirect approach for obtaining relative measures. Accid. Anal. Prev. 5:147-156; 1973. Cerrelli, E. C. Older drivers, the age factor in traffic safety. Washington, DC: National Center for Statistics and Analysis; 1989. Evans, L. Older driver involvement in fatal and severe traffic crashes. J. Gerontology: Sot. Sciences 43:186193; 1988. Haight, F. A. Induced exposure. Accid. Anal. Prev. 5:111-126; 1973. Huston, R.; Janke, M. K. Senior driver facts. Sacramento, CA: California Department of Motor Vehicles; 1986. Mahalel, D. A note on accident risk. Trans. Res. Rec. 1068: 85-89; 1986. McKelvey, F. X.; Maleck, T. L.; Stamatiadis, N.; Hardy, D. K. Highway accidents and the older driver. Paper presented at Transportation Research Board 67th Annual Meeting, January 11-14, Washington, DC; 1988. Mercer, G. W. Traffic accidents and convictions: Group totals versus rate per kilometer driven. Risk Anal. 9:71-77, 1989. Quimby, A. R.; Maycock, G.; Carter, I. D.; Dixon, R.; Wall, J. G. Perceptual abilities of accident involved drivers. Crowthorne, Berkshire, U.K.: Transport and Road Research Laboratory; 1986. Ranney, T. A.; Pulling, N. H. Performance differences on driving and laboratory tasks between drivers of different ages. Trans. Res. Record, (in press). Thorpe, J. D. Calculating relative involvement rates in accidents without determining exposure. Aust. Road Res. 2125-36; 1964. Trans~rtation Research Board. Transportation in an aging society. Special Report 218, Vol. 1. Washington DC: Author, 1988. Yanik. A. J. What accident data reveal about elderly drivers. Paper 851688. Detroit, MI: Society; of Automotive Engineers; 1985. Wasielewski, P.; Evans, L. A statistical approach to estimating driver responsibility in two-car crashes. J. Safety Res. 1637-48, 1985.
ACCIDENTS,
MILEAGE, AND THE EXAGGERATION OF RISK*
MARY IS. JANKE California Department of Motor Vehicles, Sacramento, CA 95818, U.S.A. (Received 29 March 1990; in revised form August 19, 1990) Abstract-The usual interpretation of accidents per mile as a measure of risk exaggerates the apparent risk of low-mileage groups-for example, teenagers and the elderly. The assumption of a linear proportionaf relationship between mileage and accidents is shown not to fit obtained data. Neither would it be expected to fit hypothetical data derived from a “standard driver” or a group of equally competent drivers driving different numbers of miles. People driving low mileages tend to accumulate much of their mileage on congested city streets with two-way traffic and no restriction of access, while high-mileage drivers typically accumulate most of those miles on freeways or other divided muItilane highways with limited access. Because the driving task is simpler, the accident rate per mile is much lower on freeways, and beyond a certain point, a person driving half as many miles as another would be expected to have considerably more than half as many accidents. This and other considerations lead to the suggestion that an induced exposure approach would be a more valid method of correcting accident rates for mileage.
Writers commonly state, and graphical displays imply, that some driver groups-for example, the elderly-are relatively hazardous drivers despite experiencing fewer accidents per driver than other groups, because they have much higher accident rates when the rate is adjusted for mileage (e.g. Transportation Research Board 1988; Mercer 1989). Using the average number of accidents per driver as a measure of group risk is an adequate measure of the group’s public safety impact, but plainly confounds innate accident liability-which may be the variable of interest- with exposure to accident risk. If two groups of drivers are equally competent and prudent, but differ in miles driven, the higher-mileage group will have more accidents on the average because of its greater exposure to risk. Therefore, it has seemed an obvious corrective measure to adjust group accident rates by group mileage in order to measure innate risk more accurately. This adjustment has commonly involved dividing a measure of group accidents by a measure of group mileage to get accidents per mile (or per 100,000 miles, etc.) as a criterion measure. If accidents per mile are accepted as being simply accidents per mile, this procedure is unobjectionable. But if the ratio is interpreted as being a quantitative measure of risk, extremely (and fallaciously) high apparent risk levels for low-mileage groups of drivers may be produced. For example, in Huston and Janke ~~986) a graph of total accidents per driver per mile as a function of age and sex revealed that the rate for women aged 75 or above (who drive only about 1,000 miies per year, on the average) was about six times as high as that of the “average driver.” The implication to a naive reader might well have been that elderly women drivers are six times as dangerous to society; the authors tried to temper this impression by discussing and qualifying the implications of their graph. Similar inferences have been made with respect to groups that also drive less, but have more accidents per driver, than the average. Their risk, already high, may be exaggerated when accidents per mile are used as a criterion. For example, in reference to young drivers, the C~~~~~~~~~ Driver Fact Book (California Department of Motor Vehicles 1981) noted that Teenage drivers average twice as many accidents as adult drivers, while driving only half as many miles; the young driver accident rate per mile is thus four times as great as that of adult drivers (p. 36). *Opinions and conclusions expressed in this paper are those of the author and not necessarily those of the California Department of Motor Vehicles or the State of California. 183
184
M. K. JANKE
Again, while the statement is true in itself, an uncritical reader might interpret it as indicating that teenage drivers are four times as hazardous as adults, The underlying assumption involved in using this type of statement as a basis for inferring quantitative relative risk is that, everything else being equal, there is a linear relationship between accidents per driver (A) and driver mileage (M), with the line going through the origin. Under this model (A = kM), if mileage is increased by a factor of 2, say, then accident involvements are also expected to increase by a factor of 2, for equally prudent drivers of equal competence. If a hypothetical “standard driver” were considered, such a person, when driving 10,~ miles per year, would be expected to be involved in twice as many accidents as when driving only 5,000 miles in the same period of time. Therefore, when it is found that a real-world 5,000-miles-per-year driver has more than half as many accidents as another driving 10,000 miles, a relative lack of either competence or prudence (i.e. increased accident risk) is assumed for the lowermileage driver. This reasoning ignores the consideration that the variable that should bear a linear relation to accidents, if any does, is not mileage but exposure to accident risk, of which mileage is only one component. Thus, not even for drivers equivalent in all relevant respects but miles driven would accident occurrence be expected to be proportional to mileage. For such to be true, one would have to posit the concept of a “standard mile” offering a mile’s worth of risk. Real-world driving does not consist of standard miles; different highway miles offer different amounts of exposure to risk. For example, in this society, high-mileage drivers typically accumulate most of their mileage on freeways or other divided multilane roadways with limited access. These generally have much lower accident rates per mile than do other types of roadways; recent data from the California Business, Transportation, and Housing Agency (1985) indicated that there were 2.75 times as many accidents per mile driven on non-freeways as on freeways. From this it can be predicted that with constant driver competence and prudence, accidents will tend to rise at a low and decreasing rate as mileage increases beyond a certain point. On the other end of the mileage scale, drivers with low mileage typically make short trips in congested city driving conditions. Inserting one’s vehicle into the traffic stream (as opposed to lowing with the traffic stream) offers considerable opportunity for accidents, and the need to do this occupies a much greater proportion of short stop-andgo trips than of long ones. Even the shortest trip affords this accident opportunity, and it would be expected that accidents as a function of mileage would rise steeply from the origin. More sophisticated models than the simple linear one have been introduced, for example by Quimby, Maycock, Carter, Dixon, and Wall (1986), who represented accident involvements by means of a multivariate model in which the mileage term appeared as a multiplicative factor raised to the .25 power. However, such complex models have apparently not been widely used. In a more general context, Mahalel (1986) opted for a broader definition of exposure. He noted that “there appears to be a vicious circle in which, on the one hand, the accepted definition of risk necessitates a linear relationship between accidents and exposure; on the other hand, it is difficult (or even impossible) to find exposure estimators that fulfill this limitation” (p. 86). In trying to determine the relationship between mileage and accidents, other difficulties arise as well. First, one would not expect to find a single “true” curve; in fact, there should be a whole family of mileage-accident curves, depending upon parameters representing different types of driving conditions. Second, mileage data are typically self-reported and therefore subject to considerable error. Third, and more critically, even if some more valid way of estimating mileage were used, it would still be the case that drivers are not randomly assigned to mileages. Since groups of people who customarily drive low or high mileages differ in age, sex, license class, and probably other characteristics like health status, the data points to which a hypothetical generalized mileage-accident curve is fitted represent different kinds of people at different ends of the mileage scale. It is highly likely that these groups have different accident expectancies
Accidents, mileage, and the exaggeration of risk
185
simply because of their different characteristics and the ways in which these affect driving competency, regardless of their mileage. In fact, it seems not unreasonable to hypothesize that drivers with a low level of competence tend to have low mileages. Circularity is introduced because the causality here might go in either direction. Because of the limitations noted above, one would not expect empirical accidentmileage data to be very meaningful. Nevertheless, some California data were examined to illustrate their failure to show propo~ionality of accidents to mileage. The Cu~~~o~~~u Driver Fact Book displays, on page 17, accident-per-person averages taken over a sixyear period by sex for seven annual mileage categories (under 1,000 through 30,000 + ). These data, derived in part from a survey, are due to Carpenter (1976). The first two columns of Table 1 show the mileage category medians (M) and corresponding average accident rates (A) for combined sexes. For analytical purposes, an arbitrary median of 500 was chosen for the under-l,000 category and one of 40,000 for the 30,000+ mileage category. These data are poor for illustrative purposes, because differences in mileage at the extremely low and the high ends of the scale are not discriminable. However, the gross shape and placement of the curve relating accident involvement to mileage are still of interest. A linear regression equation was fitted to the data by least squares, and the fit was excellent (r = .97). But this equation, A = 7.7 (1O-6 M) + .27, clearly does not go through the origin. Although it must necessarily be true that someone who drives zero miles will be involved (as a driver) in zero accidents, in order to pass through the origin the line fitting these points would have to assume a marked curvature as it closely approached M = 0. A linear and proportional relationship between accidents and mileage does not describe the data. Through trial and error, an equation was derived that roughly fit Carpenter’s data points, and for which A, the accident average, at least comes close to zero when mileage, M, equals zero. Again it is for illustrative purposes only. The equation is A = [ln (A4 + 1)]/21 - (.05 - [lo-‘O(M - 20,000)2]). The third column in Table 1 shows the value of A corresponding to each M, as calculated from the equation. This curve passes close to the origin, as noted, and also fits the Carpenter data points fairly well. If we were now using the model to compare two groups, one with an annual mileage of 3,000 and the other with an annual mileage of 9,000, the second group (A = .396) would be expected to have 1.10 times as many accidents as the first (A = .360), not 3 times as many. Proportionality similarly fails to apply for other hypothetical examples, and we would not expect it to. Other data, for example those of Burg (1973) support this general conclusion.
Table 1. Actual and calculated accident rates by annual mileage (6-year record) Mileage category/ Median
Average accidents
None Under l,OOOJ500 1,~-4,~9/3,~ 5,~-9,~/7,5~ 10,~14,999/12,5~ 15,oOQ-19,999/17,500
0 (definition) .30 .31 .34 .38 .39 .46 .50
2o,ooo-29,999/25,ooo
30,000+ /40.000
Caiculated accidents - .Ol .28 .36 .39 .40 .42 .43 .50
186
M. K.
JANKE
A method of taking account of exposure that holds promise of being more fruitful is that of induced exposure, originated by J. D. Thorpe in 1964 and elaborated on in many subsequent studies (e.g. Carlson 1970; Haight 1973; Cerrelli 1973; McKelvey, Maleck, Stamatiadis, and Hardy 1988). Here, instead of soliciting drivers’ self-report, the exposure of a category of drivers is measured by (or induced from) their involvement in accidents for which they were not responsible. Their risk is measured by (or induced from) their involvement in accidents for which they were responsible. Thus, of any two driver groups, the less competent is distinguished from the more competent one by being at fault in a greater proportion of their accidents, few or many as these may be. An example of the use of this technique is Cerrelli’s paper (1973). Restricting consideration to multiple-vehicle accidents, he used the induced exposure method to derive three measures. One, the relative exposure index, is the percentage of innocently accident-involved drivers in a category (e.g. an age-sex group), divided by the percentage of licensed drivers in that category within the driving population. The measure is intended to provide a measure of over- or under-exposure for a set of drivers. A second measure, the liability index, may be thought of as representing the group’s level of societal riski.e. their degree of threat to public safety-which is greater, everything else being equal, in a group that drives more. It is the percentage of accident-responsible drivers in the category divided by the percentage of licensed drivers in that category, and it is what an insurance company might be interested in when setting rates. Finally, the hazard index measures the risk, relative to exposure, of the categorywhat the present paper is concerned with. The hazard index is the ratio of the liability index to the exposure index. An index of 1.00 would be expected, over the long run, if members of a driver group were involved in only two-vehicle accidents, and in each accident culpability was attributed to one of the drivers by means of tossing a coin. It would represent the overall index for the entire driving population as well, if all accidents were two-vehicle with one driver culpable in each. Using Cerrelli’s method, the overall hazard index for men was 1.00, and for women 0.99. Cerrelli’s indices have been described at relative length in order to present and comment on some illustrative numbers he arrived at. For example, he found that women more than 64 years old drove very little, their overall exposure index being 0.42. (Indices vary for different types of roadways, times of day, and so forth.) Their liability index, measuring the degree of threat posed to society by women drivers in this age group, was also rather low at 0.63. This combination made for an overall hazard index of 1.5. (In contrast, the hazard index for both men and women aged 35-44 was 0.90.) The 1.5 figure for older women certainly indicates a degree of excessive risk but the excess is small when compared to that implied by the use of accidents per mile. As another example, the overall hazard index for males less than 20 was 1.2. For males over 64 it was 1.4 and, as seen above, it was 0.90 for males aged 35-44, 1.0 for males generally. This method, like the use of accidents per mile, leads to a concaveupward curve of the dependent variable as a function of age, with elderly men emerging as being at higher risk on the average than very young ones and both of these groups being at higher risk than men in the middle age ranges. But the “severalfold worse” quality of the relationship is lacking when the induced exposure method is used. Cerrelli treated only multiple-vehicle accidents in his 1973 paper. If he had included single-vehicle crashes (in which the driver is generally deemed to be responsible), older drivers would have tended to look relatively better, if anything. That is because they experience a much lower proportion of single-vehicle accidents than young drivers do (Cerelli 1989). Induced exposure may not be the optimal tool for measuring risk. One potential failing of the method as illustrated here is that a group that does not drive as defensively as another, and thus becomes “innocently” involved in more accidents, will tend to have a lower hazard index because its apparent exposure will be inflated. This might be corrected by assigning different degrees of responsibility to one or more crash-involved drivers, as appropriate. (See Wasielewski and Evans [ 19851, who formulated an “induced responsibility” model similar to that for induced exposure, in which they relaxed the
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assumption that only one driver is responsible in a two-car crash.) Another potential weakness of induced exposure is the difficulty of assigning accident responsibility; attribution of responsibility on an accident report is vulnerable to any preconceptions or biases held by the officer completing the report. Nevertheless, the induced exposure approach would seem to give a more balanced picture of risk than does the method of using accidents per mile. It also appears to circumvent effects of the probable circularity between mileage and driving competence that complicates interpretation of accidents-per-mile data. One might ask, however, what difference it makes if the perceived hazard of lowmileage groups is exaggerated. Ignoring the fact that it is always desirable to point out whatever erroneous assumptions may be underlying our conclusions, it makes a practical difference if the climate of,opinion is affected and unwarranted licensing actions result. In the specific case of the elderly driver, Ranney and Pulling (1990) have pointed out that more than 50 million persons aged 65 or older will be eligible to drive in 2020. While they noted that mileage-based rates have come into question on grounds that mileage estimates have not been validated (Yanik 1985) and that older drivers’ curtailment of mileage is greater than their increase in risk (Evans 1988), they warned that this may not be true in the future. “The average yearly miles driven by drivers 65 and older has increased with each major survey taken between 1969 and 1983 . . . and therefore, such a compensatory tendency [to drive less] may well be specific to the current generation of older drivers” (p. 1). Faced with these considerations (all of which are valid), licensing administrators influenced by the exaggerated implied risk of the elderly group when accidents are divided by miles may become more alarmed than is warranted about the safety hazard posed by increasing numbers of elderly drivers. Even though researchers in general would be the last to countenance such an outcome, this alarm could lead to the introduction of unduly repressive licensing measures. Although it is unlikely that any state would arbitrarily de-license drivers on the basis of age alone, an exaggerated notion of the risk older people pose could easily result, for example, in the licensing agency’s calling in all drivers beyond a certain age for special testineupon the results of which their driving privileges might depend. This would be an obvious inconvenience and source of anxiety to the elderly. It would also be a less than optimal use of public funds, if the relative public safety threat posed by the specially treated group was not as great as believed and not as great as the threat posed by other groups that do not get special agency attention. Therefore, it seems important to point out, from the standpoint of fairness, that quantitative estimation of the relative risk posed by the group, based on accidents per mile at their currently low mileage, is inflated us a consequence ofthat low mileagegiven the assumptions of linearity and proportionality discussed above. There is no reason to assume that the “actual” relationship between mileage and accidents, even for a hypothetical “standard driver,” is of the simple A = kM type. On the contrary, there is reason to assume that linearity and proportionality would not hold. And the empirical data derived from samples of real-world drivers apparently do not fit such a model. Thus there is no reason to expect that the number of accidents divided by the number of miles driven will remain constant for a “standard driver” or for groups having equal intrinsic risk. Regardless of whether an induced exposure approach or some other method is substituted, it seems that it is at least time to discard the assumption of accident-mileage proportionality that now implicitly forms the basis of many group risk comparisons.
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Carpenter, D. W. An evaluation of the California driver knowledge test and the University of Michigan item pool. Sacramento. , CA: Department of Motor Vehicles; 1976. Cerrelli, E. C. Driver exposure: the indirect approach for obtaining relative measures. Accid. Anal. Prev. 5:147-156; 1973. Cerrelli, E. C. Older drivers, the age factor in traffic safety. Washington, DC: National Center for Statistics and Analysis; 1989. Evans, L. Older driver involvement in fatal and severe traffic crashes. J. Gerontology: Sot. Sciences 43:186193; 1988. Haight, F. A. Induced exposure. Accid. Anal. Prev. 5:111-126; 1973. Huston, R.; Janke, M. K. Senior driver facts. Sacramento, CA: California Department of Motor Vehicles; 1986. Mahalel, D. A note on accident risk. Trans. Res. Rec. 1068: 85-89; 1986. McKelvey, F. X.; Maleck, T. L.; Stamatiadis, N.; Hardy, D. K. Highway accidents and the older driver. Paper presented at Transportation Research Board 67th Annual Meeting, January 11-14, Washington, DC; 1988. Mercer, G. W. Traffic accidents and convictions: Group totals versus rate per kilometer driven. Risk Anal. 9:71-77, 1989. Quimby, A. R.; Maycock, G.; Carter, I. D.; Dixon, R.; Wall, J. G. Perceptual abilities of accident involved drivers. Crowthorne, Berkshire, U.K.: Transport and Road Research Laboratory; 1986. Ranney, T. A.; Pulling, N. H. Performance differences on driving and laboratory tasks between drivers of different ages. Trans. Res. Record, (in press). Thorpe, J. D. Calculating relative involvement rates in accidents without determining exposure. Aust. Road Res. 2125-36; 1964. Trans~rtation Research Board. Transportation in an aging society. Special Report 218, Vol. 1. Washington DC: Author, 1988. Yanik. A. J. What accident data reveal about elderly drivers. Paper 851688. Detroit, MI: Society; of Automotive Engineers; 1985. Wasielewski, P.; Evans, L. A statistical approach to estimating driver responsibility in two-car crashes. J. Safety Res. 1637-48, 1985.