TRAFFIC CRASH EXPOSURES BY INJURY SEVERITY
Young-Jun Kweon, Ph.D.
Associate Research Scientist
Virginia Transportation Research Council
530 Edgemont Road
Charlottesville, VA 22903
Phone: 434-293-1949
Fax: 434-293-1990
Email: Young-Jun.Kweon@VDOT.Virginia.gov
TRAFFIC CRASH EXPOSURES BY INJURY SEVERITY
ABSTRACT
A crash exposure has been used to normalize different levels of an exposure to traffic crashes so
that a comparison of safety performance over time can be made using a crash rate, a crash count
divided by an exposure. However, a crash exposure is thought to be valid only when a crash
count is a linear function of an exposure going through the origin. To this end, this study
attempts to empirically verify the linear relationship using autoregressive error models. The
study uses the annual data of three crash/victim counts (fatalities, injuries, and crashes) and four
exposure measures (VMT, vehicles, drivers, and population) of Virginia. Population was found
to be a better exposure measure than the other three for all three counts. Limitations include
population is usable as an exposure for fatalities for the years after 1995, for injuries for the years
except for the period from late 1980s through early 1990s, and for total crashes for the years after
1995. In addition, a qualitative comparison using the crash/victim rates per population rather
than a quantitative comparison is recommended.
ACKNOWLEDGMENT
The author thanks the Virginia Department of Transportation for sponsoring this study and two
anonymous reviewers for their valuable comments. A fully revised version of this paper will be
available soon upon request.
Kweon
1
INTRODUCTION
Traffic crash/victim rates (e.g., fatality rate per million vehicle miles traveled [VMT]) are
frequently used as a safety performance measure for the purpose of comparison across
geographical areas and/or temporal periods (e.g., Subramanian 2006, Washington State
Department of Transportation, 2006, National Center for Statistics and Analysis 2006). For
example, the U.S. Federal Highway Administration (2006) has been using the traffic fatality rate
per 100 million VMT as a safety performance measure of the nation.
Researchers questioned about validity of using crash exposures in a crash rate for
measuring a safety performance. For example, Khan et al. (1999) explored to the relationship
between a set of fatal, injurious, and property damage only (PDO) crash counts, and a set of
traffic volume, segment length and VMT in Colorado. They found that VMT is significant for
fatal crashes whereas traffic volume and segment length are significant for injury and PDO
crashes. For pedestrian safety, Raford et al. (2006) confirmed the finding of Leden (2002) and
Jacobsen (2003) that the pedestrian injury risk decreases with increasing pedestrian volume.
Using detailed data collected in Oakland, California, they found decreasing collision risk
between pedestrians and vehicles with increasing pedestrian volume and decreasing vehicle
volume.
In 1995, Hauer raised an interesting question about qualifications for a good exposure.
An exposure is mainly used to calculate a crash rate, crash frequency divided by a crash
exposure, so that a fair comparison of traffic safety can be made among cases with different
crash exposure levels. The numerator and the denominator in a crash rate are related, and a crash
frequency can be a function of a crash exposure, which is called a safe performance function.
Hauer illustrated that a comparison using a typical crash rate results in an erroneous
conclusion unless the relationship between a crash frequency and an exposure is a straight line
going through the origin. This origin requirement guarantees that no occurrence of traffic
crashes should be expected when there is no crash exposure. Hauer’s arguments were examined
by later studies using empirical data. For example, Qin et al. (2004) explored relationships of
exposure measures (e.g., AADT, segment length, and VMT) with crash occurrences by four
crash types. Using data from highways in Michigan, they found those relationships are nonlinear and vary by the crash types and exposure measures.
However, all the relevant studies focused on cross-sectional perspectives of the
relationship, not temporal perspective. To this end, this paper attempts to empirically examine
the linear relationship passing through the origin between each of three crash/victim counts
(fatalities, injuries, and total crashes) and each of four crash exposures (vehicle miles traveled,
vehicles, drivers, and population). The study examines such relationship over time in Virginia
using annual time-series data to determine whether a crash/victim rate based on exposures can be
used for measuring a safety performance of Virginia over years.
METHOD
Autoregressive Error Model
The linear relationship between a crash/victim count and an exposure over time is
examined using a linear regression model. When an error term of the model is found to be
serially correlated, autoregressive (AR) error terms are used to correct standard errors of the
coefficients of the model. This type of a regression model is called as an autoregressive error
model and can be written as follows:
yt  xt β  t and  t   t  1 t 1    m t m
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2
where t is a time index (e.g., year); yt is a dependent variable (e.g., the number of fatalities); xt
is a set of independent variables (e.g., exposure measure);  t ,, t m are m+1 serially correlated
errors;  t is a normal and independent error term; β is a set of structural coefficients; and
1 ,, m are m autoregressive coefficients. It should be noted that no constant term is included
in the set of structure coefficients to satisfy the origin requirement (i.e., the straight line going
through the origin).
A Durbin-Watson test is used to identify autocorrelation among the residuals from the
ordinary least square (OLS) regression. If autocorrelation is found, the order of AR error terms
included in the regression needs to be determined. To suggest an appropriate AR order, the
stepwise autoregression method is used, which is similar to the stepwise regression method for
cross-sectional data. The model starts with a high AR order and removes statistically
insignificant AR error terms in sequence. The Durbin-Watson test is used again to check the
presence of any remaining autocorrelation in the residuals from a final autoregressive error
model.
2
To evaluate goodness of fit models, a regression R-squared measure ( Rreg
) is computed.
This measure gauges the fit of the structural part of the model after the model is transformed for
the identified autocorrelation. Autoregressive error models are estimated using the maximum
likelihood (ML) technique, and all calculations and estimations are performed using SAS 9.1.
Data
Annual counts of traffic fatalities, injuries, and crashes from 1951 through 2004 in
Virginia were compiled and formed as a historical crash dataset. However, the number of
drivers and population were obtainable only after around 1970. The merged annual crash and
exposure data are displayed in Figures 1 and 2. It should be noted that the VMT estimation
method has been changed in Virginia since 2002, explaining a sudden drop of the VMT estimate
in 2002 shown in Figure 2.
[Figures 1 and 2]
All three counts show overall increasing trends over time. However, they seem quite
fluctuated in certain periods of time. On the other hands, the four exposures show stable
increasing trends with much fluctuation. This implies that the relationship between a
crash/victim count and an exposure might be linear overall, but it may not be well fitted for
certain time periods. In such a case, a crash rate (a crash count  an exposure) might be usable
for a long-term comparison (e.g., 20 years) but not for a short-term comparison.
RESULTS AND DISCUSSIONS
The ARIMA(1,1,1) model was fitted to the VMT data to adjust the effect of the change in
the VMT estimation method. Using the parameter estimates of the model, VMT data in the years
of 2002 through 2004 were adjusted and used for further analysis.
One regression model for each of a combination between one from the three crash/victim
counts and one from the four exposures was estimated. The estimated models were presented
and discussed below separately by injury severity. All the estimates presented here are
statistically significant at the 0.05 significance level unless indicated otherwise.
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3
Fatality
VMT, vehicles, and drivers were found to be statistically insignificant even at the 0.1
significance level in their fatality models. With an intercept term in the model, which violates
the origin requirement, the number of drivers was significant, but its coefficient estimate was
negative, which is counterintuitive in terms of the definition of a crash exposure.
Only population turned out to be statistically significant at the 0.1 significance level in
the final fatality model. Thus, among the four exposures, population is the only candidate for an
exposure measure to traffic fatalities. The estimated model with an AR order 1 is written as
follows:
2
=0.177)
fatalityt  1.383   populationt   0.979  t 1 ( Rreg
The coefficient of population (1.383) is statistically significant at the 0.084 significance level.
For a visual comparison of the model fit, Figure 3 was provided. For population to be
usable as a fatality exposure, observation points should be in line with the dotted line, which
represents the structural part of the model (i.e., a straight line going through the origin). The
figure suggests that population might be usable as a fatality exposure for the population values
greater than about 6.5 millions, corresponding to the years after around 1995.
In summary, none of the four exposures well satisfied the straight-line relationship going
through the origin. However, for the high population values (accordingly, recent years),
population might be usable as an exposure for fatalities in Virginia.
[Figure 3]
Injury
As for traffic injuries, all four exposure measures were statistically significant in their
injury models. The final AR error models and regression R-squared values are shown as follow:
VMT t

2
injuryt  0.272  
  0.736  t 1  0.261  t 2 ( Rreg =0.078)
1
,
000
,
000


vehiclet
2
  0.968 
injuryt  111.990  

t 1 ( Rreg =0.331)
10
,
000


driverst
2
  0.790 
injuryt  167.000  

t 1 ( Rreg =0.925)
10
,
000


2
injuryt  114.2014  populationt  0.774  t 1 ( Rreg
=0.937)
The coefficient of  t  2 in the model with VMT (0.261) is statistically significant at the 0.066
significance level.
According to the regression R-squared values measuring the fit of the structural part of
the models, drivers and population appear to be much better than the other two measures, and
population is slightly better than drivers. Thus, population appears to be the best candidate for
an injury exposure measure.
For a visual assessment of the model fit, Figure 4 was created for the injury model with
population. The dotted line represents the structural part of the model predictions. Although the
regression R-squared value is very high, 0.937, observations are not in line with the structural
part, especially in the population values around 6 millions corresponding to late 1980s through
early 1990s.
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[Figure 4]
However, except for those population values, population seems to be capable of serving
as an exposure to traffic injuries. When population is used as an injury exposure, comparison of
a injury rate per population for a long-term (e.g., 10 years) might be possible because year-byyear variations are somewhat large relative to the slope of the structural part. Even for a longterm comparison, a qualitative rather than quantitative comparison appears to be proper. This
means that an indication of being safer or more dangerous can be suggested using a change in the
injury rate per population, but using the change to quantify the amount of safety improvement or
deterioration is not appropriate.
It seems a moving average technique over a short time period (e.g., 3-5 years) might
reduce the year-by-year variation in traffic injuries so that an averaged injury rate per population
(e.g., average number of injuries over 5 years divided by average population over the same 5
years) might serve as a stable safety performance measure for temporal comparison. However,
this technique still may not produce an appropriate injury rate for comparison during the period
when population is highly fluctuated (e.g., population of around 6 millions).
Crash
Vehicles and population were found to be statistically significant in their crash models
whereas VMT and drivers were not statistically significant at the 0.1 level even with an intercept.
The final models and the R-squared values are shown as follow:
vehiclet
2
  1.385   0.401
( Rreg
=0.197)
crasht  201.625  

t 1
t 2
10
,
000


2
=0.743)
crasht  220.800  populationt  0.932  t 1 ( Rreg
Population is much better than the number of vehicles for a crash exposure according the
regression R-squared values. Figure 5 shows how well the structural part of the crash AR error
model fits with observations. To be a good exposure, observation points should be in line with
the dotted line, which is the structural part. Population appears to be usable as a crash exposure
for the range of the population values greater than about 6.5 millions, corresponding to the years
from 1995.
[Figure 5]
CONCLUSIONS
The four crash exposures (VMT, vehicles, drivers, and population) frequently used to
measure a safety performance were examined to determine whether they are appropriate as an
exposure for each of three traffic crash/victim counts (fatalities, injuries, and total crashes).
Hauer (1995) suggested that the relationship between a crash count and an exposure should be
linear and go through the origin. This study attempted to empirically verify that such
relationship exists and if so, to determine which exposure is best for each of the three counts.
Using autoregressive error models and annual crash and exposure data of Virginia, the
crash/victim counts were linearly related to the four exposures, and all linear models were forced
to pass through the origin. According to the statistical significance t-tests and the regression Rsquared values, the most appropriate exposure measure among the four was selected for each of
the three counts.
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Interestingly, population turned out to be better than the other three exposures for all the
crash/victim counts in Virginia. However, limitations were identified when population is used as
an exposure. For fatalities, population appears to be usable for the population values greater than
6.5 millions corresponding to the years after 1995, but its use may not be valid for a short-term
comparison of safety performance. Population might be used as an injury exposure for a longterm comparison except for the population values around 6 millions corresponding to the period
from late 1980s through early 1990s. For total crashes, population might be usable for the
population values greater than 6.5 millions corresponding to the years from 1995.
Even when population was selected as an exposure carefully considering the above
limitations, a qualitative comparison rather than a quantitative one is recommended. This
implies that a decrease/increase in a crash/victim rate per population should not be interpreted as
an absolute measure of safety improvement/deterioration.
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REFERENCES
Federal Highway Administration. (2006) Safety: A FHWA Vital Few: Fact Sheet. Road Safety
Fact Sheet. U.S. Department of Transportation, safety.fhwa.dot.gov/facts
/road_factsheet.htm. Accessed March 18, 2007.
Hauer, E. (1995) ‘On exposure and accident rate’, Traffic Engineering & Control, 36(3),
pp. 134-138.
Jacobsen, P.L. (2003) ‘Safety in numbers: more walkers and bicyclists, safer walking and
bicycling’, Injury Prevention, 9, pp. 205-209.
Khan, S., Shanmugam, R., and Hoeschen, B. (1999) ‘Iinjury, fatal, and property damage accident
models for highway corridors’, Transportation Research Record, 1665, pp. 84-92.
Leden, L. (2002) ‘Pedestrian risk decrease with pedestrian flow: a case study based on data from
signalized intersections in Hamilton, Ontario’, Accident Analysis and Prevention, 34, pp.
457-464.
National Center for Statistics and Analysis. (2006) State Traffic Safety Information For Year
2005. National Highway Traffic Safety Administration, U.S. Department of
Transportation, www-nrd.nhtsa.dot.gov/ departments/nrd-30/ncsa/STSI/USA
%20WEB% 20REPORT.HTM. Accessed April 17, 2007.
Qin, X., Ivan, J.N., and Ravishanker, N. (2004) ‘Selecting exposure measures in crash rate
prediction for two-lane highway segments’, Accident Analysis and Prevention, 36(2), pp.
183-191.
Raford, N., Ragland, D.R., Geyer, J.A., and Pham, T. (2006) ‘The continuing debate about safety
in numbers—data from Oakland, CA’, Proceeding of the Transportation Research Board
85th Annual Meeting, Washington, DC, January 2006, Paper No. 06-2616.
Subramanian, R. (2006) Traffic Safety Facts, Research Note: Total and Alcohol-Related Fatality
Rates by State, 2003-2004. Publication DOT-HS-810-556. National Center for Statistics
and Analysis, National Highway Traffic Safety Administration, U.S. Department of
Transportation.
Washington State Department of Transportation. (2006) Motor Vehicle Fatalities.
www.wsdot.wa.gov/planning/wtp/datalibrary/Safety/MVFatalities.htm. Accessed July
20, 2006.
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7
1,400
100,000
90,000
1,200
Number of injuries
Number of fatalities
80,000
1,000
800
600
400
70,000
60,000
50,000
40,000
30,000
20,000
200
10,000
0
1950
0
1960
1970
1980
1990
2000
1950
Year
(b) injury
180,000
Number of total crashes.
160,000
140,000
120,000
100,000
80,000
60,000
40,000
20,000
0
1960
1970
1980
1970
1980
Year
(a) fatality
1950
1960
1990
2000
Year
(c) total crash
Figure 1. Trends of traffic fatalities, injuries, and crashes in Virginia.
1990
2000
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8
100,000
8,000,000
90,000
7,000,000
Number of vehicles
VMT in million
80,000
70,000
60,000
50,000
40,000
30,000
6,000,000
5,000,000
4,000,000
3,000,000
2,000,000
20,000
1,000,000
10,000
0
0
1950
1960
1970
1980
1990
1950
2000
1960
1970
Year
(a) VMT
(b) vehicle
6,000,000
8,000,000
1990
2000
1990
2000
7,000,000
5,000,000
6,000,000
4,000,000
Population
Number of drivers
1980
Year
3,000,000
2,000,000
5,000,000
4,000,000
3,000,000
2,000,000
1,000,000
1,000,000
0
1950
0
1960
1970
1980
1990
2000
1950
1960
1970
Year
(c) driver
(d) population
Figure 2. Trends of VMT, numbers of vehicles, drivers, and population in Virginia.
1980
Year
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9
Number of fatalities .
2,000
prediction
95% limits
prediction by population
observation
1,500
1,000
500
0
450
500
550
600
Population in 10,000
Figure 3. Fatality model prediction.
650
700
750
Kweon
10
100,000
Number of injuries .
90,000
80,000
70,000
60,000
50,000
prediction
95% limits
prediction by population
observation
40,000
30,000
450
500
550
600
Population in 10,000
Figure 4. Injury model prediction.
650
700
750
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11
190,000
Number of crashes .
170,000
150,000
130,000
110,000
90,000
prediction
95% limits
prediction by population
observation
70,000
50,000
450
500
550
600
Population in 10,000
Figure 5. Crash model prediction.
650
700
750