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Crash injury loss: The effect of speed, weight and crash configuration
Acrid. Anal. .4 Pnt..
Vol 9, pp. 5.W.
Pcrgamon Press 1977
Primed tin Great Britam
CRASH INJURY LOSS: THE EFFECT OF SPEED, WEIGHT AND CRASH CONFIGURATION WILLIAML. CARLSON Department
of Economics,
St. Olaf College, Nortbtield,
(Received
MN 55057, U.S.A.
23 April 1976)
Abstract--Crash injury prediction models are developed for major crash configurations using data from in depth crash investigations. Analysis of these models demonstrates the positive contribution of higher impact velocity to crash injury, especially for head on and side impact crashes. An increase of the proportion of smaller cars is not predicted to substantially increase overall crash injury. The other injury predictor variables are crash configuration, occupant age, seating position, and restraint usage.
There are an increasing number of economic studies of the highway safety problem. Anderson [I9751 presented an argument for the application of marginal analysis to study the trade-off between input resources and the level of safety. Peltzman [I9751 raised the question of the relationship between specific regulations and consumer demand for risk. These and other studies emphasize the need for considering safety in a broader context. The recent implementation of a national speed limit and the trend toward smaller vehicles are two important changes that could have important influences. Recent studies have confirmed that the reduction of travel velocity and/or reduction of total travel has contributed to reduced highway injury. The ultimate result of changing vehicle sizes will have to wait until1 the present vehicle mix has been consumed. Economic analysis implies that rational consumers will choose a combination of goods such that the marginal utility divided by price is equal for all goods. If we treat reduced injury as a good desired by consumers, as was done by both the Anderson and Peltzman studies, then it is possible to analyze the question of optimal injury level. But specific solutions require a knowledge of the relationship between input decision variables and output injury variables. For example common wisdom suggests that reduced injury can be “purchased” through slower travel speed and increased vehicle size. But without knowledge of the relationship consumers are not able to perform the necessary analysis. Also policy makers do not have a rational basis for developing transportation policies. In this study, the velocity and vehicle size mix crash injury effects are examined. A number of empirical injury prediction models were developed for various crash configurations. Next these models were used to develop injury predictions as a function of changes in velocity and vehicle size mix. The importance of impact velocity as a predictor of injury was confirmed. Impact velocity interaction with crash configuration was also shown. That result helps to explain lower fatality levels on expressways in spite of higher speeds. Based upon this analysis we are optimistic concerning crash injury loss as vehicle size is reduced. Again, there is an important interaction between size and crash configuration. The study has also identified variables that should be controlled when analyzing crash injury loss. The work here draws on previous studies by O’Day er al. [ 19731,Carlson [ 19741,and Carlson and Kaplan [1975]. These studies identified relationships between occupant injury measured by the Abbreviated Injury Scale (A.I.S.) and various crash variables. Data for these studies have come from investigations by multidisciplinary teams [Marsh, 19731.Other important work has been done by Campbell and Reinfurt (19741 using police generated accident reports. A fundamental problem with all of these studies (including the study reported here) is the measurement of injury loss. Thus, continuing research is important to develop better dependent variables. METHODOLOGY
This paper presents the results of an empirical study of crash injury and a set of crash injury prediction models. The analysis began with a hypothesis concerning variables that predict injury severity. 55
56
WILLIAM L. CARLSON
(a) Impact velocity The basic transfer of momentum arguments from physics imply a strong direct effect, which has been confirmed by empirical studies, including those referenced above. One interesting exception is the struck vehicle in a rear end crash where higher velocity implies lower relative velocity. (b) Vehicle weight Here the relationship is hypothesized to be inverse but with some ambiguity. Basic physics and many studies confirm that the occupants of the larger vehicle have lower injury given a two vehicle crash. But, total injury loss requires combining occupant injuries from both the large and the small vehicle. For example total injury loss might be lower if two small vehicles collided compared to a large and a small vehicle. (c) Crash configuration
Since occupant dynamics differ for the various crash configurations we expect different injury patterns. Casual observation of crash data shows, for example, that head on crashes have more severe injuries compared to rear end crashes. Mixing data from various crash configurations will result in lower model precision. In addition interactions between crash configuration and other variables may lead to different models for different crash configurations. In addition to these major hypotheses several related but minor hypotheses were included, (i) Occupant age: Based upon previous work [Carlson 19741we hypothesized a positive linear relationship between occupant age and injury severity. (ii) Occupant seating position: Based upon previous work [O’Day et al., 19731we hypothesized that rear seat occupants would have lower injury levels. (iii) Restraint system utilization: Based upon numerous previous studies we hypothesized lower injury severity when restraint systems are used. These later variables were included in part as covariates to reduce unexplained variability and thus to increase the power of the tests on the main variables, impact velocity, vehicle size and crash configuration. Empirical analysis of highway injury has been limited by deficiencies in the available data. Data deficiency leads to problems of quantifying highway crash loss in a commonly accepted form. Development of relationships between policy variables and loss measurements is also difficult. The data commonly used comes from police generated routine accident reports or from intensive investigation of selected crashes by highly trained investigators [O’Day et al., 19731.The former lack measurement depth and precision while the later have problems of represenativeness. In some large cities certain crashes are not investigated by the law enforcement departments. Thus, the user of routinely collected accident data should understand the potential for bias in the data from some law enforcement jurisdictions. Methods to overcome the data problems are being developed at the Highway Safety Research Institute sponsored by the National Highway Traffic Safety Administration. The results of that work should substantially improve the application of the methodology presented in this paper [O’Day, 19751. Data for this study came from a large computerized file of in depth crash investigations maintained by the Highway Safety Research Institute (HSRI). The crash investigations were obtained from Crash Performance Injury Reports (CPIR) completed by investigators from locations across the United States and edited by HSRI [Marsh, 19731.Availability of this file and the associated statistical analysis software made this study possible. Using large samples allows us to assume the central limit theorem for the distribution of errors about the regression model. This avoids some of the distributional problems associated with using the AIS as the dependent variable. Crashes used in this study occurred during 1971-73. Only crashes involving automobiles produced by the four major U.S. manufacturers for model years 1970-73 were included. Thus the vehicles had most of the available injury reduction equipment. Also the analysis is not influenced by unusual foreign designs or by crashes involving trucks and cars. A generalized highway loss function can be defined as follows, L = L(X,, Xz, . . ., X,) + E,
(1)
Crash injury loss: the effect of speed, weight and crash configuration
57
where L is the loss from a particular crash type which is defined by a vector of measurable variables, X,((j = 1,. . ., k), is a vector of crash variable measurements and E is a disturbance or error term associated with the estimation of the loss, L. To obtain parameters for the loss function it is desirable to use data from well designed experiments which span the range of crash alternatives. Unfortunately it is not possible to conduct such experiments. But, previous work has developed the concept of retroactive experiments [Carlson and Kaplan, 19751 which provides a methodology for using data from intensive crash investigations conducted by well trained multidisciplinary teams. Essentially each case study is treated as an experiment defined as a point in a k + 1 dimensional experimental space by a set of k independent variables. The file of the available crash investigations is amenable to this approach because it contains a large number of crashes. In addition, high injury loss crashes are overrepresented. Thus the observations are distributed more uniformly over the range of the X’s as compared to using a random sample of all crashes. This reduces the variance of the regression model coefficients [Kmenta, 19713.The specific data used have occupant injury-measured on the AIS scale-as the dependent loss variable and various variables which define vehicle, environment and human characteristics associated with the injury as independent variables. Previous estimation work with this loss function has been directed toward the goal of finding a minimum variance estimator of the loss L. In contrast the emphasis of this study is the determination of the affect of specific independent variables on L. Therefore, model building must carefully consider the effect on multicolinearity-corelation between independent variables--on the cofficients for the independent variables of interest. If the true model includes correlated independent variables the coefficient estimators are unbiased. However, the correlation between independent variables increases the variance of coefficient estimates. The practical result of this increased variance is that the coefficient estimates for each of two correlated variables may vary substantially when regressions are run on different random samples. For example, the first random sample may yield a large coefficient for variable one and a small coefficient for variable two. Conversely the second random sample yields a small coefficient for variable one and a large coefficient for variable two. Results like that render the model highly suspect for policy analysis which uses the coefficient estimates directly. The problem is often solved by dropping one of the correlated variables. This approach is particularly useful if one of the correlated variables is not an important policy variable. But, if the dropped variable is contained in the true model then the coefficients for the remaining variables are biased. Thus, the model builder may be forced to deal with the choice between a biased small variance estimate and an unbiased large variance estimate. Statistical theory does not provide a global preference. From previous work the variables impact velocity and vehicle damage extent were shown to be strong predictors of injury [Carlson and Kaplan, 19751. However they are strongly correlated. Therefore, in the study reported here the vehicle damage extent was not used to avoid the problem of large coefficient variance. An additional justification for not using the vehicle damage variable is that vehicle damage and occupant injury are both caused by impact velocity but vehicle damage does not cause occupant injury. Another possible solution is to treat the vehicle damage variable as an endogenous variable in addition to the AIS variable and to develop a stimultaneous equation model. This approach was attempted using two stage least squares [Kmenta, 19711.The result was a model with a lower percent explained variability, R2. That approach also suffered from the ambiguity of the relationship between vehicle damage and injury. A large amount of vehicle damage may indicate severe crash dynamics and subsequent high injury. Alternatively high vehicle damage may indicate that substantial crash energy was absorbed in metal deformation and thus did not lead to higher occupant injury. Therefore, it was decided to develop the injury prediction models reported here by using ordinary least squares. The crash population was partitioned into the following subsets and models were fitted: (i) Single vehicle crashes. (ii) Head on crashes. (iii) Right angle crashes-striking vehicle. (iv) Right angle crashes-struck vehicle. (v) Rear end crashes-striking vehicle. (vi) Rear end crashesstruck vehicle. Justification for this partitioning is the difference in the relationship between vehicle dynamics and potential injury production associated with the different crash configurations. For
58
WILLIAML. CARLSON
example, high impact velocity is likely to increase occupant injury in a head on crash, but reduce injury for occupants in the struck vehicle in a rear end crash. Since differences exist between the crash dynamics and injury production mechanisms it was decided to treat each model separately. The overall loss function could be defined as a linear combination of the loss functions for each crash configuration as follows: L =
9 a,L,(X,,
x,, . . .,x62.
(2)
I=.,
The a,_‘~are weighting factors which reflect the relative proportion of vehicles involved in each crash configuration. The models developed for each crash configuration have the following form: y = B,, + B,X: + &X:
+
$ B,Xj +AX,X,
i-3
(3)
where Y denotes the injury for a specific occupant in the primary vehicle measured on the AIS scale, and X, and Xt denote the impact velocity, in mph, for the primary? and for the secondary vehicle, and X, and X4 denote the weights in 1000Ibs. for the primary and secondary vehicles, and X5 denotes the occupant age in years, and X6 and X7 are dummy variables indicating seat belt and upper torso restraints usage by occupant, and X8 and X9 are dummy variables indicating occupant seated in the right front or the driver’s seat respectively, and AX,X, denotes the relative weight of the primary and secondary vehicles in the following categories, (1) under 3000 pounds, (2) 3oOO&OOpounds, and (3) over 4000 pounds. Table 1 presents the regression coefficients and their standard errors for each injury prediction model. The final models result from trying a large number of different variables and model forms. The following criteria were used to select the final forms of the models which are presented here: (i) The models should enable the initial hypotheses to be tested. (ii) The models should predict occupant injury with minimum possible error. (iii) The models should indicate the predictive effect of impact velocity and vehicle weight on occupant injury. (i) Prediction variables which are highly correlated should not be used to avoid multicollinearty effects. (v) The variables used should be consistent with the theory of crash injury development. The use of least squares to fit the injury prediction models requires a dependent variable, Y, measured on a continuous scale which is linearly related to a vector of prediction variables, X, plus a random disturbance. This assumption is not strictly met since the AIS dependent variable is reported only on discrete intervals over the range of O-6 as shown in Table 2. That problem was dealt with by introducing a rounding variable, 0, which represents the difference between a continuously measured injury variable and its nearest integer. An analysis of the effect of 6 on the estimation procedure is shown in Appendix A. From that analysis, the regression coefficient estimators obtained by using ordinary least squares are unbiased if the rounding error is not correlated with the X matrix of predictor variables. However, the estimated variance are all biased high implying that the significance level of ail tests.are overstated. In this way the results presented are conservative. ANALYSIS
OF REGRESSION
MODELS
Table 1, which contains the detailed regression results, will be used to test the original hypotheses. We will begin by testing the specific hypotheses concerning the predictor variables. Following that analysis we will discuss the overall implications for future crash injury loss. (a) Impact uelocity The results will be expressed in terms of an overall contribution, Yi(V), where i is the crash configuration; i = 1 for single vehicle, i = 2 for head on, i = 3 for right angle, and i = 4 for rear end crashes. In this way we present a loss function that relates impact velocity of all vehicles to tThe primary
vehicle
contains
the occupants
whose
injuries
have been measured.
Crash injury loss: the effec! of speed. weight and crash configuration Table I. Occupant crash injury prediction model coefficients
2.
3.
Head On Side Iwpact a. Striking
b.
4.
Rear
a.
b.
0.40
0.02
C.09)
C.21)
C.18)
(.004)
0.50 (.06>
O*lh
0.1a C.12)
0.25
C.06)
0.16 (.06)
0.58
0.17
0.10
(.16>
C.14)
0.69
Vehicle
.2a
495
.20
Struck Vehicle
-.a0
0‘77
589
1.13
.Oi
t.08)
(.lO)
f”d
.31
Striking Vehicle
Struck
I99
.23
Vehicle
.67
146
-.8?
0.16 C.05)
-.a0
.65
wsrxt*,s
monr
4
Driver
<3000
fast
3000 to 4000
-0.19
0.50
0.01
C.16)
C.11)
C.004)
-O.Bii
0.31
0.18
0.01
C.17)
C.09)
(.21)
0.29 C.19)
Privzy
cz.;,.;dG b? be Shouldar Right
-0.19
C.10)
Primary<3000
bb seat BO1t
-0.77
3000 to 4000
Primary+lOO Secrmder)l
beconuar):
r4000
.c3000
3000 to 4000
.02
.30
(.OW)
ZWOO
c3000
3000 to 4000
.38
.62
.75
,400O
-0.34 (.lZ)
-0.42 C.35)
0.22 C.35)
0.27 (.16)
-0.10 C.17)
-0.40 C.42)
O.b9 C.23)
0.59 C.22)
0.0
48
-.I8
-0.05 C.08)
-0.38 C.23)
0.38 C.13)
O.lb f.13)
0.0
-.25
m.09
-.55
-.53
-.?2
-.74
e.72
-1.21
0.46
0.13
0.09
0.0
-.oa
.a4
-.37
.*33
-.24
-.I+2
c.19
- .80
C.34)
C.15)
C.14)
0.34
-0.36
C.12)
0.10
-0.84
.17
-.25
-.09
-.55
-.53
e.72
-.74
-.72
-1.21
C.43)
(.18)
0.25 (.f7)
0.0
C.13) 0.05 C.17,
0.11 C.69,
0.30 f.19,
0.28 C.14)
0.0
-.OE
.w
-.37
-.33
-.24
-.42
W.19
* *m
Table 2. Abbreviated injury scale (ASS) Code -
Description
0
NO Injury
1
Minor Injury
2
Hodent
3
Severe (Not Life Threatening) Injury
4
Serious (Life Threatening Survival Probable)
5
Critical (Survival uncertain)
6
Fatal (Within 20 Hours)
Injury
Developed by the American Medical Association Committee on Medical Aspects of Automotive Safety Reference CPIR (1972)
60
WILLIAM L. CARLSON
average injury loss. If we assume that impact velocity is functionally related to travel velocity then, Yi(V) becomes a surrogate function of loss due to average highway travel velocity. For single vehicle crashes the regression coefficient is 0.48 x lo-‘, and the loss function is, Y,(v) = 0.48 x lo-3 v2,
(4)
where V is expressed in miles per hour. For head on crashes the coefficients for primary and secondary vehicle velocity squared, 0.48 x 10e3 and 0.69 x lo--‘, were added to obtain the total effect for velocity, Yz(v) = 1.17x lo+ v2.
(5)
For right angle crashes, coefficients for primary and secondary vehicle velocity squared were added within the striking and struck vehicle subsets. The resulting sums within each subset were then averaged to obtain the average effect of velocity, Y3(V) = [OS x (0.50 + 0.14) x lo-’ + 0.5 x (0.16 + 0.58) x lo-‘] Vz = 0.69 x lo-’ r.
(6)
For rear end crashes the velocity effect is more complex. A necessary condition for a crash is that the striking vehicle has a higher velocity. When the striking vehicle is the primary vehicle the injury loss function is, Y,(V),=O.l6x
lO-3 v-0.19x
10-3(V-AV)2,
where V is the primary vehicle impact velocity and AV is the positive difference between vehicle velocities. Similarly for the case with the struck vehicle as the primary vehicle the loss function is. Y*(
v)2
=
lO-3 v+O.31 x 1O-3(V-AVJ2.
-0.64x
The combined average contribution to injury per occupant is thus y,(v) =; [y4( v), + Ye*]
= [-0.18*+0.05VAV+0.06AV7
x lo-‘.
(7)
Define, 2 = A VI V, as the velocity ratio and the expression becomes, F
= 0.06T + 0.502 - 0.18.
(8)
A critical velocity ratio can be defined as the minimum velocity ratio at which the contribution to occupant injury is positive. This critical ratio, Z,, can be found by solving, 0.06zL + 0.502 - 0.18 > 0 for the minimum positive 2, which is, z, = 0.346. This implies that overall injury is not increased by impact velocity increases unless the velocity ratio exceeds 0.346. This analysis reflects the prediction equation result that relative velocity and not velocity alone is the important variable. (b) Vehicle weight Here the contribution per occupant will be expressed by, Y(W), where i again indicates the crash type. Except for single vehicle crashes the relationships are more complex since they must include parameters for the relative proportions of various size vehicles.
Crash injury loss: the effect of speed, weight and crash configuration
61
The injury contribution from vehicle weight for single vehicle crashes is unambiguous since it is given directly in the prediction equation, Y,(W) = -0.09 X lo-'w, where W is the vehicle weight in pounds. Vehicle weight has a very small effect in single vehicle crashes. Increasing vehicle weight by 1000 pounds reduces injury by 0.1. Also the coefficient is not statistically significant at the 0.05 level. This weak relationship in single vehicle crashes was also reported in the work by Campbell and Ream 119741,which used police generated accident reports. For two vehicle crashes the relationship is more complex. Occupants of larger vehicles have lower injury levels than do occupants of smaller vehicles, as indicated by the regression coefficients for vehicle weight. When vehicles of different size collide we would expect lower injuries for the occupants of the larger vehicle and vice versa. Thus to determine the influence of vehicle weight it is necessary to combine the injury levels for both vehicles. In addition interaction effects from combinations of vehicle weights are possible. In this study the linear and interaction components of vehicle weight were combined to yield the results shown in Tables 3-5. The number in each cell is the average occupant injury where the average includes occupants of both the striking and the struck vehicles. In this way it represents the average injury loss to occupants of the primary vehicle when vehicles of the indicated size groups collide. The cont~bution for each vehicle was found by multiplying the median weights for each cell by the respective primary and secondary vehicle weight coefficients and then adding the interaction effect. Finally the cases for which the primary vehicle is the striking vehicle and for which the primary vehicle is the struck vehicle are averaged to obtain the reported values. Table 3. Average occupant injury contribution
by vehicle size combination for head on crashes Secondary Vehicle
L3000 c 3000 #'8
- 0.72
I
> 4000 #'s
3000 to 4000 n's
#.s I
- 1.74
Table 4. Average occupant injury contribution
> uooo X's
- 0.46
- 0.12
- 1.17
- 1.31
by vehicle size combination for side impact crashes Secondary Vehicle
I
I
I
3000 to 4000 a's
c 3000 #'s L 3000 #'s
0.89
0.88
Table 5. Average occupant injury contribution
>4000
#'s 1.50
by vehicle size combination
for rear end crashes
Secondary Vehicle
/
Primary Vehicle
< 3000 #'s
I
3000 to 4000 U's
I
2.4000 #'S
4 3000 #'s
0.97
1.29
1.27
3000 to 4000 #'s
1.33
1.22
1.59
> 4000 R's
0.95
0.96
0.99
62
WILLIAM
L.
CARLSON
A relationship between vehicle size mix proportions and average occupant injury contribution was developed for head on, side impact, and rear end crashes using the values in Tables 3-5. The probability that a randomly selected vehicle comes from weight group j, was defined as, P,, j = 1.3. Assuming that the striking and struck vehicles are independent the probability of a particular vehicle combination in a crash is, Pip,_ where j is the primary vehicle weight group and k is the secondary vehicle weight group. In addition we define the injury contribution from each weight group combination as, X,. Then the expected contribution to injury is, E(Xi) =
yi( W) =
2i i
XiUPjPk. k
Equation (10) was simplified by substituting the identity, P3= I-P,-PZ,
for
P3.
The result was an equation for relative injury contribution as a function of the proportion of small and intermediate vehicles. These equations are, (i) Head on crashes y,(W)=P,(O.76-
1.17P,)-P2(4.13-0.66P2)+3.95P1Pz-l.31.
(11)
(ii) Rear end crashes Y3(W) = P,(O.24 -0.26P,) + Pz(O.57- 0.34PJ - 4.OgP1Pz+0.99.
(12)
(iii) Side impact crashes y4( w) = P,( 1.01-0.7oP,)+
Pt(0.73 -0.5OP,) +O.O7P,P* +0.57.
(13)
Since these expressions represent the average injury contribution as a function of vehicle size mix they may be used to forecast the potential injury loss resulting from changes such as introducing more small vehicles. These relationships are also shown graphically in Figs. 1-3.
P,=O20
Pr 2030
Pz:040
P,=O50
/
-2oL
005
Fig.
/ , 010 015 ProportIon
I. Average
injury
I
I
020
025 of
I
030
small
and vehicle
I
035 040 vehicles
I
045
050
P,
size mix: head on crashes.
Crash injury loss: the effect of speed, weight and crash configuration
63
1.5 -
s .-
p:“,:zg
5
P .-
$= 0.30 P2 = 0.20 P,=O.10
LO-
i ”
?
2
.c P
z 0.55
a
‘I
0.05
0.10
0.15
0.20
Proportion
of
0.25
0.30
small
vehicles
0.35
0.40
O.b5
0:50
q
Fig. 2. Average injury and vehicle size mix: side impact crash.
Pz = 0.10 P* 50.20
p2 = 0.30 p2 = 0.40
p,=o50
I
0.05
I
0.10
I
I
0.15 0.20 Proportion
I
I
I
I
I
,
0.25
0.30
0.35
0.40
0.45
0.50
of
small
vehicles
P,
Fig. 3. Average injury and vehicle size mix: rear end.
Study of these equations provides some interesting results. In Fig. 1 it can be seen that for head on crashes the substitution of small vehicles for large vehicles will increase average injury. But substituting intermediate vehicles for large vehicles, while holding small vehicles fixed actually reduces injury levels. The former result supports our original hypothesis while the latter does not. In Fig. 2 it can be seen that vehicle size mix has very little effect on relative average injury, given side impact crashes. Again the result does not support our original hypothesis. Finally in Fig. 3 the results for rear end crashes are seen to contradict the original hypothesis. In fact there is a strong interaction between the proportion of small cars, PI, and the proportion of intermediate cars, P2, This results in an inverse relationship between P, and average injury which increases absolutely as the proportion of intermediate cars, Pz, increases. Based on this analysis we see that the trends toward smaller vehicles will not substantially increase occupant crash injury. This is especially true for crashes other than head on. We do
64
WILLIAM L. CARLSON
however that large differences between vehicle size do contribute to increased injury for occupants of the smaller vehicle, as shown in Tables 3-5.
see
(c) Crash configuration
Comparison of injury severity for different crash configurations was performed by assigning fixed values to the various independent variables in the regression equations. These fixed values were chosen to be approximately the average observed values in the data. The fixed values used are as follows, Impact velocity 30 mph Impact velocity for the second vehicle in rear end crashes 15 mph Vehicle weight 3400 lbs Occupant seating position driver Average age 30 Restraint usage none Using these values the predicted injury for each was computed for each crash configuration as shown in Table 6. The standard error of the regression estimate and the sample size are also Table 6. Predicted average injury by crash configuration Crash Configuration
Predicted 1n'ury d
Sample Size N
Standard Error of Regression S(YlX)
Standard Error of Predicted Injury, S(YlX)
1.
Single Vehicle
1.52
989
1.36
.043
2.
Head On
1.86
355
1.08
.057
3.
Side Impact Striking
1.22
495
-77
.035
4.
Struck
1.68
589
1.13
.047
5.
Rear End Striking
.Sl
199
.67
.047
Struck
.98
146
.65
.054
6.
shown for each model. Since the predicted values are computed using the average of the independent variables the variance of the predicted value is approximately, S( r’lx,
= S( Y/X)/q(n)
where n is the sample size and, S(Y/X) is the standard error of the regression estimate. This approximate variance of the prediction can be easily computed and used to test for significant differences between predictions from the various models. Examination of the predicted values for each crash configuration reveals substantial differences between average injury. In particular head on, single vehicle, and side impact-struck vehicle configurations produce the highest injury level. In contrast rear end crashes produce substantially lower injury levels. The large standard error of the regression for single vehicle crashes suggests the possible need for additional analysis. One possible approach might involve partitioning the crashes into subgroups that are more homogeneous with respect to injury production mechanisms. For example roll over crashes might be separated from fixed object impact crashes. Hypothesis testing was performed to identify significant differences between predicted average injury for the various crash configurations. Determination of the significance level for these tests was complicated by the fact that multiple comparisons between predicted values are possible. Since we have six predicted values the number of possible comparisons are the combinations of six items taken two at a time which is fifteen. The joint significance level for
Crash injury loss: the effect of speed. weight and crash configuration
65
15 independent hypothesis tests is, Ly’ = l-(1
-*)I5
(15)
where Q is the significance level for a single test and LY’is the joint significance level. For example if P is 0.005 then cy’ is 0.035 and if P is 0.01 then ci’ is 0.068.The estimated standard deviation of the difference is, S(A 8) = #‘(I?) Ai’=&-
+ S’(c)) .
(16)
pi, ?i, (i,i = 1,6; if j) are the average predicted injury levels for each crash configuration. Thus, for example in comparing side impact striking and rear end struck crashes we find, A?=
ps-- ?e= 1.22-0.98=0.24
S(A B) = ~((0.035)’ + (0.054)z)= 0.064. To perform a students t test we compute t I -Ap-0 = -0.24 = 3.75. S(Ap) 0.064
(17)
The critical value of t is 2.6 for LY= 0.005 which implies LY ’ = 0.035. Thus, a critical value of t = 2.6 is conservative for 15 independent comparisons. Following this procedure all differences between predicted injury are significant at Q’ = 0.055. Thus, we can with a small probability of error rank the crash aviations in order of severity using the predicted injury in Table 6. (d) Secondff~ hypotheses The remaining predictor variables in the regression models are mainly control variables which reduce model prediction variance. However, it should be observed that age has a significant positive coefficient in all modeis. Thus, as the average age of the population increases we would expect somewhat higher injury severity levels. In addition, the driver and front seat passenger has significantly higher injury levels compared to rear seat passengers. If we begin to increase occupancy levels then more people will occupy rear seats and average injury levels per occupant will be reduced. DISCUSSION
The analysis presented here provides strong evidence for injury reduction from reducing impact velocity. The Iogical extension is that impact velocity may be reduced by reducing travel speed. Thus, we have strong indirect evidence to support a continued Iower speed limit. Impact velocity is much more critical for head on and right angle crashes. Thus, one would expect that the greatest benefits from lower speed would occur on undivided highways which allow maximum access. This has some important implications for enforcement policy. The large injury differences between different crash configurations emphasizes again the contribution to injury reduction from traffic separation. Again the reduction of head on and right angle crashes would lead to substantial injury reductions. An important result of this study is the conclusion that a change in vehicle mix to a greater percentage of smaller cars will not substantially increase overall injury severity. The reduced injuries for occupants of smaller cars when struck by a smaller car compensates for the increased injuries to former large car occupants. To provide a worst case analysis of the vehicle weight versus input velocity trade off we examined the situation for head on crashes. The totaf impact velocity ~on~bution to average injury from eqn (5) is, Yz(v)= 1.17x lo-“V. MP
Vol. 9. No. 1-E
(5)
WILLIAM L. hS.SON
66
Similarly the total contribution to average injury from changes in vehicle size mix is, Y2(IV) = P,(O.76 - l.l7P, - Pz(4.13 -OX&P,) + 3.95P,P, - 1.31.
(11)
If we fix Pt at 0.20 and increase PI from 0.30 to 0.40 the increase in the average injury is, AYz(w) = 0.07 This increase could be compensated for by reducing impact velocity from 30 to 29 mph in eqn (5). Thus, even the potential average injury increase from vehicle weight reduction in heat on crashes can be easily compensated for by small impact velocity decreases. A study by O’Day, Minahan and Golumb [ 19751of Michigan accidents after the energy crisis induced speed limit change provides an opportunity to test the conclusions of this study. In that study they compared accident and travel speed data for comparable 5 month periods before and after the 55 mph speed limit was imposed. Consider the results extracted from their study as shown in Table 7. The effect of changes in travel speed for both total and fatal accidents on the Table 7. Road type Interstate/ Limited Access
U.S./state Trunklines
Couotyhocal Roads
Change in travel speed
-10 mph
-5 mph
-3 mph
Change in total accidents
-29.4%
-11.3%
-14.5%
Change in fatal accidents
-18.4%
-45.9%
-21.7%
Percent change in Eatals per mph of velocity reduction
-1.8%
-9.2%
-7.2%
different road types was examined. The much larger decrease in fatals per unit speed reduction on non Iimited access roads is consistent with our previous analysis. We would expect more head on and side impact crashes on these roads and hence velocity reductions would have a higher payoff. Thus, we see further evidence of the importance of considering interactions between road type and speed limit. These results might also be used to allocate enforcement resources in ways that lead to higher payoff. Acknowledgemenfs-Part of this research was conducted while the author was a visiting researcher at the Highway Safety Research Institute of the University of Michigan. The author wishes to recognize the support and encouragement of James O’Day in that effort.
REFERENCES Anderson R., The economics of highway safety, TM& Quarterly 29, 99-111, 1975. Campbell R. and Reinfurt D., Relationship Between Driver Injury and Passenger Car Weight. Highway Safety Research Center, The University of North Carolina, Chapel Hill. N.C., Nov. 1973. Carlson W. and Kaplan R., Case studies considered as retroactive experiments. Accid. Anal. & Preu. 7, 73-80, 1975. CPIR Revision 3 Codebook, The Highway Safety Research Institute, The University of Michigan, Ann Arbor, Michigan, June 1973. Kmenta Jan, Elements of Econometrics. MacMillan Co., New York. 1971. Marsh I., Review of MultidisciplinaryAccident Investigation (MDAI)Report Automation and Utilization Program. HITLAB Rep. 3(6), 1973. 0%~ J.. Carlson W., Kaplan R. and Douglass R.. Statistical Inference from Multidisciplinary Accident Investigation. Final Report to National Highway Tragic Safety Administration, Report N. Uhf-HSRI-SA73-4, 1973. 0%~ J., Minahan D. and Golomb D., The effects of the energy crisis and 55 mph speed limit in Michigan. HJ’FLAB Rep. s(ll), 1975. O’Day J., Accident data needs and a national accident sampling system. HSRI Res. 6(2), 1975. Peltsman S., The effects of automobile safety regulation, 1. Political &on. 83(4), 677-725, 1975. APPENDlX Effect
of dependent
variable
rounding
of coeficient
A estimates
Consider an experimental situation in which the value of a continuous stochastic dependent
Crash injury loss: the effect of speed. weight and crash configuration
67
Y is observed over the closed interval O-6. The value of Y is functionally related to a vector of n independent variables by the standard multiple regression model; variable
K=fB,Xi+E,
i=l,...,n
j-0
where; E(E,) = 0, Var (Ei) = a*, and E(Ei, EL)= 0 for i# L. In addition X0 = I is associated with an estimate of the equation constant. The observed value of Yi is rounded to the nearest 6). Therefore, the observed rounded values Y* must be used with the integer Y*(Y*=O,i,..., observed Xi’s to obtain estimates of the Bi’s. In summary the following structure exists,
where, Y* = reported rounded value, and Y = observed value, and 0 is the rounding effect added to the Y to obtain Y* (-0.5 < 6 < +OS). Using ordinary least squares to obtain the estimates of the Bi's,
B = (x=x)-'x=Y*
(2)
where B is K x 1, X is n x k and Y* is n x 1. The expected value of the B'sis,
E(B)=E[(X=X)-'X=Y*]=(X=X)-'X=E[XB+E+e]=B+(x=X)-'X=e
(3)
where 0 is n x 1. Estimated coefficients, B, are in general biased. This bias will approach zero as 8 approaches zero. The rounding effect is directly related to the product of the X matrix and the rounding effect vector 8. Thus, its effect is ambiguous. If the rounding effect is not related to X and if it is symmetricaHy distributed about 0 its effect is similar to that of randome error. In that case the rounding effect tends toward zero. Alternatively if the rounding effect were related to the value of X then serious bias could result in the estimates of the Bi's. Careful evaluation of the specific data generation process is thus important. The variance of B is, Var(B)=[(B
- E(B))']= E[(X=X)-'X=Y*-B-(X*X)-'X'e)']
= E[((X=X)-‘X=(XB + e + E) - B - (X=X)-‘X=0)‘] = E[(B-B+(XrX)-'XrE+(XrX)-'Xr(e-e)2]
= E[((X=x)-'X=E)']= (X=X)-'cr2.
(4)
Variances of the coefficients are not effected by 0 given that cr* is not effected. We will next consider the effect of rounding on the estimated variance. Since, ordinary least squares uses the residuals to estimate variance consider the residuals e (n x 1 vector) when rounding occurrs, e= Y*-P*=XB+E+e-Xj=XB+E+e-X(X=X)-'X'Y*
=XB+E+e-X(X=X)-'X=(XB+E+e) = E+ e-X(XTX)-'X=(E+e)=(I-X(X=X)-'X=)(E+ e) = [I-x(xTx)-'xT]E+[I-x(x7x)-'xT]e. The sum of the squared residuals are,
(5)
WILLIAM L. CARLSON
68
since I - X(X’X)-‘X’
is idempotent. Taking expected values,
2 (YT - YT)' =u2(n -
k) i- #-(I - x(x=x)-‘X’M
and thus,
G2
=
x tyt- w_t?‘(I - x(x=x)-‘X’yl n-k
n-k
(6)
From eqn (6) it can be seen that the variance estimator obtained using ordinary least squares is biased by a positive value. Therefore, the significance level for all tests of hypothesis are biased low.
Vol 9, pp. 5.W.
Pcrgamon Press 1977
Primed tin Great Britam
CRASH INJURY LOSS: THE EFFECT OF SPEED, WEIGHT AND CRASH CONFIGURATION WILLIAML. CARLSON Department
of Economics,
St. Olaf College, Nortbtield,
(Received
MN 55057, U.S.A.
23 April 1976)
Abstract--Crash injury prediction models are developed for major crash configurations using data from in depth crash investigations. Analysis of these models demonstrates the positive contribution of higher impact velocity to crash injury, especially for head on and side impact crashes. An increase of the proportion of smaller cars is not predicted to substantially increase overall crash injury. The other injury predictor variables are crash configuration, occupant age, seating position, and restraint usage.
There are an increasing number of economic studies of the highway safety problem. Anderson [I9751 presented an argument for the application of marginal analysis to study the trade-off between input resources and the level of safety. Peltzman [I9751 raised the question of the relationship between specific regulations and consumer demand for risk. These and other studies emphasize the need for considering safety in a broader context. The recent implementation of a national speed limit and the trend toward smaller vehicles are two important changes that could have important influences. Recent studies have confirmed that the reduction of travel velocity and/or reduction of total travel has contributed to reduced highway injury. The ultimate result of changing vehicle sizes will have to wait until1 the present vehicle mix has been consumed. Economic analysis implies that rational consumers will choose a combination of goods such that the marginal utility divided by price is equal for all goods. If we treat reduced injury as a good desired by consumers, as was done by both the Anderson and Peltzman studies, then it is possible to analyze the question of optimal injury level. But specific solutions require a knowledge of the relationship between input decision variables and output injury variables. For example common wisdom suggests that reduced injury can be “purchased” through slower travel speed and increased vehicle size. But without knowledge of the relationship consumers are not able to perform the necessary analysis. Also policy makers do not have a rational basis for developing transportation policies. In this study, the velocity and vehicle size mix crash injury effects are examined. A number of empirical injury prediction models were developed for various crash configurations. Next these models were used to develop injury predictions as a function of changes in velocity and vehicle size mix. The importance of impact velocity as a predictor of injury was confirmed. Impact velocity interaction with crash configuration was also shown. That result helps to explain lower fatality levels on expressways in spite of higher speeds. Based upon this analysis we are optimistic concerning crash injury loss as vehicle size is reduced. Again, there is an important interaction between size and crash configuration. The study has also identified variables that should be controlled when analyzing crash injury loss. The work here draws on previous studies by O’Day er al. [ 19731,Carlson [ 19741,and Carlson and Kaplan [1975]. These studies identified relationships between occupant injury measured by the Abbreviated Injury Scale (A.I.S.) and various crash variables. Data for these studies have come from investigations by multidisciplinary teams [Marsh, 19731.Other important work has been done by Campbell and Reinfurt (19741 using police generated accident reports. A fundamental problem with all of these studies (including the study reported here) is the measurement of injury loss. Thus, continuing research is important to develop better dependent variables. METHODOLOGY
This paper presents the results of an empirical study of crash injury and a set of crash injury prediction models. The analysis began with a hypothesis concerning variables that predict injury severity. 55
56
WILLIAM L. CARLSON
(a) Impact velocity The basic transfer of momentum arguments from physics imply a strong direct effect, which has been confirmed by empirical studies, including those referenced above. One interesting exception is the struck vehicle in a rear end crash where higher velocity implies lower relative velocity. (b) Vehicle weight Here the relationship is hypothesized to be inverse but with some ambiguity. Basic physics and many studies confirm that the occupants of the larger vehicle have lower injury given a two vehicle crash. But, total injury loss requires combining occupant injuries from both the large and the small vehicle. For example total injury loss might be lower if two small vehicles collided compared to a large and a small vehicle. (c) Crash configuration
Since occupant dynamics differ for the various crash configurations we expect different injury patterns. Casual observation of crash data shows, for example, that head on crashes have more severe injuries compared to rear end crashes. Mixing data from various crash configurations will result in lower model precision. In addition interactions between crash configuration and other variables may lead to different models for different crash configurations. In addition to these major hypotheses several related but minor hypotheses were included, (i) Occupant age: Based upon previous work [Carlson 19741we hypothesized a positive linear relationship between occupant age and injury severity. (ii) Occupant seating position: Based upon previous work [O’Day et al., 19731we hypothesized that rear seat occupants would have lower injury levels. (iii) Restraint system utilization: Based upon numerous previous studies we hypothesized lower injury severity when restraint systems are used. These later variables were included in part as covariates to reduce unexplained variability and thus to increase the power of the tests on the main variables, impact velocity, vehicle size and crash configuration. Empirical analysis of highway injury has been limited by deficiencies in the available data. Data deficiency leads to problems of quantifying highway crash loss in a commonly accepted form. Development of relationships between policy variables and loss measurements is also difficult. The data commonly used comes from police generated routine accident reports or from intensive investigation of selected crashes by highly trained investigators [O’Day et al., 19731.The former lack measurement depth and precision while the later have problems of represenativeness. In some large cities certain crashes are not investigated by the law enforcement departments. Thus, the user of routinely collected accident data should understand the potential for bias in the data from some law enforcement jurisdictions. Methods to overcome the data problems are being developed at the Highway Safety Research Institute sponsored by the National Highway Traffic Safety Administration. The results of that work should substantially improve the application of the methodology presented in this paper [O’Day, 19751. Data for this study came from a large computerized file of in depth crash investigations maintained by the Highway Safety Research Institute (HSRI). The crash investigations were obtained from Crash Performance Injury Reports (CPIR) completed by investigators from locations across the United States and edited by HSRI [Marsh, 19731.Availability of this file and the associated statistical analysis software made this study possible. Using large samples allows us to assume the central limit theorem for the distribution of errors about the regression model. This avoids some of the distributional problems associated with using the AIS as the dependent variable. Crashes used in this study occurred during 1971-73. Only crashes involving automobiles produced by the four major U.S. manufacturers for model years 1970-73 were included. Thus the vehicles had most of the available injury reduction equipment. Also the analysis is not influenced by unusual foreign designs or by crashes involving trucks and cars. A generalized highway loss function can be defined as follows, L = L(X,, Xz, . . ., X,) + E,
(1)
Crash injury loss: the effect of speed, weight and crash configuration
57
where L is the loss from a particular crash type which is defined by a vector of measurable variables, X,((j = 1,. . ., k), is a vector of crash variable measurements and E is a disturbance or error term associated with the estimation of the loss, L. To obtain parameters for the loss function it is desirable to use data from well designed experiments which span the range of crash alternatives. Unfortunately it is not possible to conduct such experiments. But, previous work has developed the concept of retroactive experiments [Carlson and Kaplan, 19751 which provides a methodology for using data from intensive crash investigations conducted by well trained multidisciplinary teams. Essentially each case study is treated as an experiment defined as a point in a k + 1 dimensional experimental space by a set of k independent variables. The file of the available crash investigations is amenable to this approach because it contains a large number of crashes. In addition, high injury loss crashes are overrepresented. Thus the observations are distributed more uniformly over the range of the X’s as compared to using a random sample of all crashes. This reduces the variance of the regression model coefficients [Kmenta, 19713.The specific data used have occupant injury-measured on the AIS scale-as the dependent loss variable and various variables which define vehicle, environment and human characteristics associated with the injury as independent variables. Previous estimation work with this loss function has been directed toward the goal of finding a minimum variance estimator of the loss L. In contrast the emphasis of this study is the determination of the affect of specific independent variables on L. Therefore, model building must carefully consider the effect on multicolinearity-corelation between independent variables--on the cofficients for the independent variables of interest. If the true model includes correlated independent variables the coefficient estimators are unbiased. However, the correlation between independent variables increases the variance of coefficient estimates. The practical result of this increased variance is that the coefficient estimates for each of two correlated variables may vary substantially when regressions are run on different random samples. For example, the first random sample may yield a large coefficient for variable one and a small coefficient for variable two. Conversely the second random sample yields a small coefficient for variable one and a large coefficient for variable two. Results like that render the model highly suspect for policy analysis which uses the coefficient estimates directly. The problem is often solved by dropping one of the correlated variables. This approach is particularly useful if one of the correlated variables is not an important policy variable. But, if the dropped variable is contained in the true model then the coefficients for the remaining variables are biased. Thus, the model builder may be forced to deal with the choice between a biased small variance estimate and an unbiased large variance estimate. Statistical theory does not provide a global preference. From previous work the variables impact velocity and vehicle damage extent were shown to be strong predictors of injury [Carlson and Kaplan, 19751. However they are strongly correlated. Therefore, in the study reported here the vehicle damage extent was not used to avoid the problem of large coefficient variance. An additional justification for not using the vehicle damage variable is that vehicle damage and occupant injury are both caused by impact velocity but vehicle damage does not cause occupant injury. Another possible solution is to treat the vehicle damage variable as an endogenous variable in addition to the AIS variable and to develop a stimultaneous equation model. This approach was attempted using two stage least squares [Kmenta, 19711.The result was a model with a lower percent explained variability, R2. That approach also suffered from the ambiguity of the relationship between vehicle damage and injury. A large amount of vehicle damage may indicate severe crash dynamics and subsequent high injury. Alternatively high vehicle damage may indicate that substantial crash energy was absorbed in metal deformation and thus did not lead to higher occupant injury. Therefore, it was decided to develop the injury prediction models reported here by using ordinary least squares. The crash population was partitioned into the following subsets and models were fitted: (i) Single vehicle crashes. (ii) Head on crashes. (iii) Right angle crashes-striking vehicle. (iv) Right angle crashes-struck vehicle. (v) Rear end crashes-striking vehicle. (vi) Rear end crashesstruck vehicle. Justification for this partitioning is the difference in the relationship between vehicle dynamics and potential injury production associated with the different crash configurations. For
58
WILLIAML. CARLSON
example, high impact velocity is likely to increase occupant injury in a head on crash, but reduce injury for occupants in the struck vehicle in a rear end crash. Since differences exist between the crash dynamics and injury production mechanisms it was decided to treat each model separately. The overall loss function could be defined as a linear combination of the loss functions for each crash configuration as follows: L =
9 a,L,(X,,
x,, . . .,x62.
(2)
I=.,
The a,_‘~are weighting factors which reflect the relative proportion of vehicles involved in each crash configuration. The models developed for each crash configuration have the following form: y = B,, + B,X: + &X:
+
$ B,Xj +AX,X,
i-3
(3)
where Y denotes the injury for a specific occupant in the primary vehicle measured on the AIS scale, and X, and Xt denote the impact velocity, in mph, for the primary? and for the secondary vehicle, and X, and X4 denote the weights in 1000Ibs. for the primary and secondary vehicles, and X5 denotes the occupant age in years, and X6 and X7 are dummy variables indicating seat belt and upper torso restraints usage by occupant, and X8 and X9 are dummy variables indicating occupant seated in the right front or the driver’s seat respectively, and AX,X, denotes the relative weight of the primary and secondary vehicles in the following categories, (1) under 3000 pounds, (2) 3oOO&OOpounds, and (3) over 4000 pounds. Table 1 presents the regression coefficients and their standard errors for each injury prediction model. The final models result from trying a large number of different variables and model forms. The following criteria were used to select the final forms of the models which are presented here: (i) The models should enable the initial hypotheses to be tested. (ii) The models should predict occupant injury with minimum possible error. (iii) The models should indicate the predictive effect of impact velocity and vehicle weight on occupant injury. (i) Prediction variables which are highly correlated should not be used to avoid multicollinearty effects. (v) The variables used should be consistent with the theory of crash injury development. The use of least squares to fit the injury prediction models requires a dependent variable, Y, measured on a continuous scale which is linearly related to a vector of prediction variables, X, plus a random disturbance. This assumption is not strictly met since the AIS dependent variable is reported only on discrete intervals over the range of O-6 as shown in Table 2. That problem was dealt with by introducing a rounding variable, 0, which represents the difference between a continuously measured injury variable and its nearest integer. An analysis of the effect of 6 on the estimation procedure is shown in Appendix A. From that analysis, the regression coefficient estimators obtained by using ordinary least squares are unbiased if the rounding error is not correlated with the X matrix of predictor variables. However, the estimated variance are all biased high implying that the significance level of ail tests.are overstated. In this way the results presented are conservative. ANALYSIS
OF REGRESSION
MODELS
Table 1, which contains the detailed regression results, will be used to test the original hypotheses. We will begin by testing the specific hypotheses concerning the predictor variables. Following that analysis we will discuss the overall implications for future crash injury loss. (a) Impact uelocity The results will be expressed in terms of an overall contribution, Yi(V), where i is the crash configuration; i = 1 for single vehicle, i = 2 for head on, i = 3 for right angle, and i = 4 for rear end crashes. In this way we present a loss function that relates impact velocity of all vehicles to tThe primary
vehicle
contains
the occupants
whose
injuries
have been measured.
Crash injury loss: the effec! of speed. weight and crash configuration Table I. Occupant crash injury prediction model coefficients
2.
3.
Head On Side Iwpact a. Striking
b.
4.
Rear
a.
b.
0.40
0.02
C.09)
C.21)
C.18)
(.004)
0.50 (.06>
O*lh
0.1a C.12)
0.25
C.06)
0.16 (.06)
0.58
0.17
0.10
(.16>
C.14)
0.69
Vehicle
.2a
495
.20
Struck Vehicle
-.a0
0‘77
589
1.13
.Oi
t.08)
(.lO)
f”d
.31
Striking Vehicle
Struck
I99
.23
Vehicle
.67
146
-.8?
0.16 C.05)
-.a0
.65
wsrxt*,s
monr
4
Driver
<3000
fast
3000 to 4000
-0.19
0.50
0.01
C.16)
C.11)
C.004)
-O.Bii
0.31
0.18
0.01
C.17)
C.09)
(.21)
0.29 C.19)
Privzy
cz.;,.;dG b? be Shouldar Right
-0.19
C.10)
Primary<3000
bb seat BO1t
-0.77
3000 to 4000
Primary+lOO Secrmder)l
beconuar):
r4000
.c3000
3000 to 4000
.02
.30
(.OW)
ZWOO
c3000
3000 to 4000
.38
.62
.75
,400O
-0.34 (.lZ)
-0.42 C.35)
0.22 C.35)
0.27 (.16)
-0.10 C.17)
-0.40 C.42)
O.b9 C.23)
0.59 C.22)
0.0
48
-.I8
-0.05 C.08)
-0.38 C.23)
0.38 C.13)
O.lb f.13)
0.0
-.25
m.09
-.55
-.53
-.?2
-.74
e.72
-1.21
0.46
0.13
0.09
0.0
-.oa
.a4
-.37
.*33
-.24
-.I+2
c.19
- .80
C.34)
C.15)
C.14)
0.34
-0.36
C.12)
0.10
-0.84
.17
-.25
-.09
-.55
-.53
e.72
-.74
-.72
-1.21
C.43)
(.18)
0.25 (.f7)
0.0
C.13) 0.05 C.17,
0.11 C.69,
0.30 f.19,
0.28 C.14)
0.0
-.OE
.w
-.37
-.33
-.24
-.42
W.19
* *m
Table 2. Abbreviated injury scale (ASS) Code -
Description
0
NO Injury
1
Minor Injury
2
Hodent
3
Severe (Not Life Threatening) Injury
4
Serious (Life Threatening Survival Probable)
5
Critical (Survival uncertain)
6
Fatal (Within 20 Hours)
Injury
Developed by the American Medical Association Committee on Medical Aspects of Automotive Safety Reference CPIR (1972)
60
WILLIAM L. CARLSON
average injury loss. If we assume that impact velocity is functionally related to travel velocity then, Yi(V) becomes a surrogate function of loss due to average highway travel velocity. For single vehicle crashes the regression coefficient is 0.48 x lo-‘, and the loss function is, Y,(v) = 0.48 x lo-3 v2,
(4)
where V is expressed in miles per hour. For head on crashes the coefficients for primary and secondary vehicle velocity squared, 0.48 x 10e3 and 0.69 x lo--‘, were added to obtain the total effect for velocity, Yz(v) = 1.17x lo+ v2.
(5)
For right angle crashes, coefficients for primary and secondary vehicle velocity squared were added within the striking and struck vehicle subsets. The resulting sums within each subset were then averaged to obtain the average effect of velocity, Y3(V) = [OS x (0.50 + 0.14) x lo-’ + 0.5 x (0.16 + 0.58) x lo-‘] Vz = 0.69 x lo-’ r.
(6)
For rear end crashes the velocity effect is more complex. A necessary condition for a crash is that the striking vehicle has a higher velocity. When the striking vehicle is the primary vehicle the injury loss function is, Y,(V),=O.l6x
lO-3 v-0.19x
10-3(V-AV)2,
where V is the primary vehicle impact velocity and AV is the positive difference between vehicle velocities. Similarly for the case with the struck vehicle as the primary vehicle the loss function is. Y*(
v)2
=
lO-3 v+O.31 x 1O-3(V-AVJ2.
-0.64x
The combined average contribution to injury per occupant is thus y,(v) =; [y4( v), + Ye*]
= [-0.18*+0.05VAV+0.06AV7
x lo-‘.
(7)
Define, 2 = A VI V, as the velocity ratio and the expression becomes, F
= 0.06T + 0.502 - 0.18.
(8)
A critical velocity ratio can be defined as the minimum velocity ratio at which the contribution to occupant injury is positive. This critical ratio, Z,, can be found by solving, 0.06zL + 0.502 - 0.18 > 0 for the minimum positive 2, which is, z, = 0.346. This implies that overall injury is not increased by impact velocity increases unless the velocity ratio exceeds 0.346. This analysis reflects the prediction equation result that relative velocity and not velocity alone is the important variable. (b) Vehicle weight Here the contribution per occupant will be expressed by, Y(W), where i again indicates the crash type. Except for single vehicle crashes the relationships are more complex since they must include parameters for the relative proportions of various size vehicles.
Crash injury loss: the effect of speed, weight and crash configuration
61
The injury contribution from vehicle weight for single vehicle crashes is unambiguous since it is given directly in the prediction equation, Y,(W) = -0.09 X lo-'w, where W is the vehicle weight in pounds. Vehicle weight has a very small effect in single vehicle crashes. Increasing vehicle weight by 1000 pounds reduces injury by 0.1. Also the coefficient is not statistically significant at the 0.05 level. This weak relationship in single vehicle crashes was also reported in the work by Campbell and Ream 119741,which used police generated accident reports. For two vehicle crashes the relationship is more complex. Occupants of larger vehicles have lower injury levels than do occupants of smaller vehicles, as indicated by the regression coefficients for vehicle weight. When vehicles of different size collide we would expect lower injuries for the occupants of the larger vehicle and vice versa. Thus to determine the influence of vehicle weight it is necessary to combine the injury levels for both vehicles. In addition interaction effects from combinations of vehicle weights are possible. In this study the linear and interaction components of vehicle weight were combined to yield the results shown in Tables 3-5. The number in each cell is the average occupant injury where the average includes occupants of both the striking and the struck vehicles. In this way it represents the average injury loss to occupants of the primary vehicle when vehicles of the indicated size groups collide. The cont~bution for each vehicle was found by multiplying the median weights for each cell by the respective primary and secondary vehicle weight coefficients and then adding the interaction effect. Finally the cases for which the primary vehicle is the striking vehicle and for which the primary vehicle is the struck vehicle are averaged to obtain the reported values. Table 3. Average occupant injury contribution
by vehicle size combination for head on crashes Secondary Vehicle
L3000 c 3000 #'8
- 0.72
I
> 4000 #'s
3000 to 4000 n's
#.s I
- 1.74
Table 4. Average occupant injury contribution
> uooo X's
- 0.46
- 0.12
- 1.17
- 1.31
by vehicle size combination for side impact crashes Secondary Vehicle
I
I
I
3000 to 4000 a's
c 3000 #'s L 3000 #'s
0.89
0.88
Table 5. Average occupant injury contribution
>4000
#'s 1.50
by vehicle size combination
for rear end crashes
Secondary Vehicle
/
Primary Vehicle
< 3000 #'s
I
3000 to 4000 U's
I
2.4000 #'S
4 3000 #'s
0.97
1.29
1.27
3000 to 4000 #'s
1.33
1.22
1.59
> 4000 R's
0.95
0.96
0.99
62
WILLIAM
L.
CARLSON
A relationship between vehicle size mix proportions and average occupant injury contribution was developed for head on, side impact, and rear end crashes using the values in Tables 3-5. The probability that a randomly selected vehicle comes from weight group j, was defined as, P,, j = 1.3. Assuming that the striking and struck vehicles are independent the probability of a particular vehicle combination in a crash is, Pip,_ where j is the primary vehicle weight group and k is the secondary vehicle weight group. In addition we define the injury contribution from each weight group combination as, X,. Then the expected contribution to injury is, E(Xi) =
yi( W) =
2i i
XiUPjPk. k
Equation (10) was simplified by substituting the identity, P3= I-P,-PZ,
for
P3.
The result was an equation for relative injury contribution as a function of the proportion of small and intermediate vehicles. These equations are, (i) Head on crashes y,(W)=P,(O.76-
1.17P,)-P2(4.13-0.66P2)+3.95P1Pz-l.31.
(11)
(ii) Rear end crashes Y3(W) = P,(O.24 -0.26P,) + Pz(O.57- 0.34PJ - 4.OgP1Pz+0.99.
(12)
(iii) Side impact crashes y4( w) = P,( 1.01-0.7oP,)+
Pt(0.73 -0.5OP,) +O.O7P,P* +0.57.
(13)
Since these expressions represent the average injury contribution as a function of vehicle size mix they may be used to forecast the potential injury loss resulting from changes such as introducing more small vehicles. These relationships are also shown graphically in Figs. 1-3.
P,=O20
Pr 2030
Pz:040
P,=O50
/
-2oL
005
Fig.
/ , 010 015 ProportIon
I. Average
injury
I
I
020
025 of
I
030
small
and vehicle
I
035 040 vehicles
I
045
050
P,
size mix: head on crashes.
Crash injury loss: the effect of speed, weight and crash configuration
63
1.5 -
s .-
p:“,:zg
5
P .-
$= 0.30 P2 = 0.20 P,=O.10
LO-
i ”
?
2
.c P
z 0.55
a
‘I
0.05
0.10
0.15
0.20
Proportion
of
0.25
0.30
small
vehicles
0.35
0.40
O.b5
0:50
q
Fig. 2. Average injury and vehicle size mix: side impact crash.
Pz = 0.10 P* 50.20
p2 = 0.30 p2 = 0.40
p,=o50
I
0.05
I
0.10
I
I
0.15 0.20 Proportion
I
I
I
I
I
,
0.25
0.30
0.35
0.40
0.45
0.50
of
small
vehicles
P,
Fig. 3. Average injury and vehicle size mix: rear end.
Study of these equations provides some interesting results. In Fig. 1 it can be seen that for head on crashes the substitution of small vehicles for large vehicles will increase average injury. But substituting intermediate vehicles for large vehicles, while holding small vehicles fixed actually reduces injury levels. The former result supports our original hypothesis while the latter does not. In Fig. 2 it can be seen that vehicle size mix has very little effect on relative average injury, given side impact crashes. Again the result does not support our original hypothesis. Finally in Fig. 3 the results for rear end crashes are seen to contradict the original hypothesis. In fact there is a strong interaction between the proportion of small cars, PI, and the proportion of intermediate cars, P2, This results in an inverse relationship between P, and average injury which increases absolutely as the proportion of intermediate cars, Pz, increases. Based on this analysis we see that the trends toward smaller vehicles will not substantially increase occupant crash injury. This is especially true for crashes other than head on. We do
64
WILLIAM L. CARLSON
however that large differences between vehicle size do contribute to increased injury for occupants of the smaller vehicle, as shown in Tables 3-5.
see
(c) Crash configuration
Comparison of injury severity for different crash configurations was performed by assigning fixed values to the various independent variables in the regression equations. These fixed values were chosen to be approximately the average observed values in the data. The fixed values used are as follows, Impact velocity 30 mph Impact velocity for the second vehicle in rear end crashes 15 mph Vehicle weight 3400 lbs Occupant seating position driver Average age 30 Restraint usage none Using these values the predicted injury for each was computed for each crash configuration as shown in Table 6. The standard error of the regression estimate and the sample size are also Table 6. Predicted average injury by crash configuration Crash Configuration
Predicted 1n'ury d
Sample Size N
Standard Error of Regression S(YlX)
Standard Error of Predicted Injury, S(YlX)
1.
Single Vehicle
1.52
989
1.36
.043
2.
Head On
1.86
355
1.08
.057
3.
Side Impact Striking
1.22
495
-77
.035
4.
Struck
1.68
589
1.13
.047
5.
Rear End Striking
.Sl
199
.67
.047
Struck
.98
146
.65
.054
6.
shown for each model. Since the predicted values are computed using the average of the independent variables the variance of the predicted value is approximately, S( r’lx,
= S( Y/X)/q(n)
where n is the sample size and, S(Y/X) is the standard error of the regression estimate. This approximate variance of the prediction can be easily computed and used to test for significant differences between predictions from the various models. Examination of the predicted values for each crash configuration reveals substantial differences between average injury. In particular head on, single vehicle, and side impact-struck vehicle configurations produce the highest injury level. In contrast rear end crashes produce substantially lower injury levels. The large standard error of the regression for single vehicle crashes suggests the possible need for additional analysis. One possible approach might involve partitioning the crashes into subgroups that are more homogeneous with respect to injury production mechanisms. For example roll over crashes might be separated from fixed object impact crashes. Hypothesis testing was performed to identify significant differences between predicted average injury for the various crash configurations. Determination of the significance level for these tests was complicated by the fact that multiple comparisons between predicted values are possible. Since we have six predicted values the number of possible comparisons are the combinations of six items taken two at a time which is fifteen. The joint significance level for
Crash injury loss: the effect of speed. weight and crash configuration
65
15 independent hypothesis tests is, Ly’ = l-(1
-*)I5
(15)
where Q is the significance level for a single test and LY’is the joint significance level. For example if P is 0.005 then cy’ is 0.035 and if P is 0.01 then ci’ is 0.068.The estimated standard deviation of the difference is, S(A 8) = #‘(I?) Ai’=&-
+ S’(c)) .
(16)
pi, ?i, (i,i = 1,6; if j) are the average predicted injury levels for each crash configuration. Thus, for example in comparing side impact striking and rear end struck crashes we find, A?=
ps-- ?e= 1.22-0.98=0.24
S(A B) = ~((0.035)’ + (0.054)z)= 0.064. To perform a students t test we compute t I -Ap-0 = -0.24 = 3.75. S(Ap) 0.064
(17)
The critical value of t is 2.6 for LY= 0.005 which implies LY ’ = 0.035. Thus, a critical value of t = 2.6 is conservative for 15 independent comparisons. Following this procedure all differences between predicted injury are significant at Q’ = 0.055. Thus, we can with a small probability of error rank the crash aviations in order of severity using the predicted injury in Table 6. (d) Secondff~ hypotheses The remaining predictor variables in the regression models are mainly control variables which reduce model prediction variance. However, it should be observed that age has a significant positive coefficient in all modeis. Thus, as the average age of the population increases we would expect somewhat higher injury severity levels. In addition, the driver and front seat passenger has significantly higher injury levels compared to rear seat passengers. If we begin to increase occupancy levels then more people will occupy rear seats and average injury levels per occupant will be reduced. DISCUSSION
The analysis presented here provides strong evidence for injury reduction from reducing impact velocity. The Iogical extension is that impact velocity may be reduced by reducing travel speed. Thus, we have strong indirect evidence to support a continued Iower speed limit. Impact velocity is much more critical for head on and right angle crashes. Thus, one would expect that the greatest benefits from lower speed would occur on undivided highways which allow maximum access. This has some important implications for enforcement policy. The large injury differences between different crash configurations emphasizes again the contribution to injury reduction from traffic separation. Again the reduction of head on and right angle crashes would lead to substantial injury reductions. An important result of this study is the conclusion that a change in vehicle mix to a greater percentage of smaller cars will not substantially increase overall injury severity. The reduced injuries for occupants of smaller cars when struck by a smaller car compensates for the increased injuries to former large car occupants. To provide a worst case analysis of the vehicle weight versus input velocity trade off we examined the situation for head on crashes. The totaf impact velocity ~on~bution to average injury from eqn (5) is, Yz(v)= 1.17x lo-“V. MP
Vol. 9. No. 1-E
(5)
WILLIAM L. hS.SON
66
Similarly the total contribution to average injury from changes in vehicle size mix is, Y2(IV) = P,(O.76 - l.l7P, - Pz(4.13 -OX&P,) + 3.95P,P, - 1.31.
(11)
If we fix Pt at 0.20 and increase PI from 0.30 to 0.40 the increase in the average injury is, AYz(w) = 0.07 This increase could be compensated for by reducing impact velocity from 30 to 29 mph in eqn (5). Thus, even the potential average injury increase from vehicle weight reduction in heat on crashes can be easily compensated for by small impact velocity decreases. A study by O’Day, Minahan and Golumb [ 19751of Michigan accidents after the energy crisis induced speed limit change provides an opportunity to test the conclusions of this study. In that study they compared accident and travel speed data for comparable 5 month periods before and after the 55 mph speed limit was imposed. Consider the results extracted from their study as shown in Table 7. The effect of changes in travel speed for both total and fatal accidents on the Table 7. Road type Interstate/ Limited Access
U.S./state Trunklines
Couotyhocal Roads
Change in travel speed
-10 mph
-5 mph
-3 mph
Change in total accidents
-29.4%
-11.3%
-14.5%
Change in fatal accidents
-18.4%
-45.9%
-21.7%
Percent change in Eatals per mph of velocity reduction
-1.8%
-9.2%
-7.2%
different road types was examined. The much larger decrease in fatals per unit speed reduction on non Iimited access roads is consistent with our previous analysis. We would expect more head on and side impact crashes on these roads and hence velocity reductions would have a higher payoff. Thus, we see further evidence of the importance of considering interactions between road type and speed limit. These results might also be used to allocate enforcement resources in ways that lead to higher payoff. Acknowledgemenfs-Part of this research was conducted while the author was a visiting researcher at the Highway Safety Research Institute of the University of Michigan. The author wishes to recognize the support and encouragement of James O’Day in that effort.
REFERENCES Anderson R., The economics of highway safety, TM& Quarterly 29, 99-111, 1975. Campbell R. and Reinfurt D., Relationship Between Driver Injury and Passenger Car Weight. Highway Safety Research Center, The University of North Carolina, Chapel Hill. N.C., Nov. 1973. Carlson W. and Kaplan R., Case studies considered as retroactive experiments. Accid. Anal. & Preu. 7, 73-80, 1975. CPIR Revision 3 Codebook, The Highway Safety Research Institute, The University of Michigan, Ann Arbor, Michigan, June 1973. Kmenta Jan, Elements of Econometrics. MacMillan Co., New York. 1971. Marsh I., Review of MultidisciplinaryAccident Investigation (MDAI)Report Automation and Utilization Program. HITLAB Rep. 3(6), 1973. 0%~ J.. Carlson W., Kaplan R. and Douglass R.. Statistical Inference from Multidisciplinary Accident Investigation. Final Report to National Highway Tragic Safety Administration, Report N. Uhf-HSRI-SA73-4, 1973. 0%~ J., Minahan D. and Golomb D., The effects of the energy crisis and 55 mph speed limit in Michigan. HJ’FLAB Rep. s(ll), 1975. O’Day J., Accident data needs and a national accident sampling system. HSRI Res. 6(2), 1975. Peltsman S., The effects of automobile safety regulation, 1. Political &on. 83(4), 677-725, 1975. APPENDlX Effect
of dependent
variable
rounding
of coeficient
A estimates
Consider an experimental situation in which the value of a continuous stochastic dependent
Crash injury loss: the effect of speed. weight and crash configuration
67
Y is observed over the closed interval O-6. The value of Y is functionally related to a vector of n independent variables by the standard multiple regression model; variable
K=fB,Xi+E,
i=l,...,n
j-0
where; E(E,) = 0, Var (Ei) = a*, and E(Ei, EL)= 0 for i# L. In addition X0 = I is associated with an estimate of the equation constant. The observed value of Yi is rounded to the nearest 6). Therefore, the observed rounded values Y* must be used with the integer Y*(Y*=O,i,..., observed Xi’s to obtain estimates of the Bi’s. In summary the following structure exists,
where, Y* = reported rounded value, and Y = observed value, and 0 is the rounding effect added to the Y to obtain Y* (-0.5 < 6 < +OS). Using ordinary least squares to obtain the estimates of the Bi's,
B = (x=x)-'x=Y*
(2)
where B is K x 1, X is n x k and Y* is n x 1. The expected value of the B'sis,
E(B)=E[(X=X)-'X=Y*]=(X=X)-'X=E[XB+E+e]=B+(x=X)-'X=e
(3)
where 0 is n x 1. Estimated coefficients, B, are in general biased. This bias will approach zero as 8 approaches zero. The rounding effect is directly related to the product of the X matrix and the rounding effect vector 8. Thus, its effect is ambiguous. If the rounding effect is not related to X and if it is symmetricaHy distributed about 0 its effect is similar to that of randome error. In that case the rounding effect tends toward zero. Alternatively if the rounding effect were related to the value of X then serious bias could result in the estimates of the Bi's. Careful evaluation of the specific data generation process is thus important. The variance of B is, Var(B)=[(B
- E(B))']= E[(X=X)-'X=Y*-B-(X*X)-'X'e)']
= E[((X=X)-‘X=(XB + e + E) - B - (X=X)-‘X=0)‘] = E[(B-B+(XrX)-'XrE+(XrX)-'Xr(e-e)2]
= E[((X=x)-'X=E)']= (X=X)-'cr2.
(4)
Variances of the coefficients are not effected by 0 given that cr* is not effected. We will next consider the effect of rounding on the estimated variance. Since, ordinary least squares uses the residuals to estimate variance consider the residuals e (n x 1 vector) when rounding occurrs, e= Y*-P*=XB+E+e-Xj=XB+E+e-X(X=X)-'X'Y*
=XB+E+e-X(X=X)-'X=(XB+E+e) = E+ e-X(XTX)-'X=(E+e)=(I-X(X=X)-'X=)(E+ e) = [I-x(xTx)-'xT]E+[I-x(x7x)-'xT]e. The sum of the squared residuals are,
(5)
WILLIAM L. CARLSON
68
since I - X(X’X)-‘X’
is idempotent. Taking expected values,
2 (YT - YT)' =u2(n -
k) i- #-(I - x(x=x)-‘X’M
and thus,
G2
=
x tyt- w_t?‘(I - x(x=x)-‘X’yl n-k
n-k
(6)
From eqn (6) it can be seen that the variance estimator obtained using ordinary least squares is biased by a positive value. Therefore, the significance level for all tests of hypothesis are biased low.