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A bivariate normal model for intra-accident correlations of driver injury with application to the effect of mass ratio
A BIVARIATE NORMAL MODEL FOR INTRA-ACCIDENT CORRELATIONS OF DRIVER INJURY WITH APPLICATION TO THE EFFECT OF MASS RATIO
T. P. Department
of Statistics
HUTCHINSON
and Operational Research, Coventry CVI
(Rrcriwd
19 Febrwry
Coventry (Lanchester) SFB. England
1982; in revised form
16 September
Polytechnic,
Priory
Street.
1981)
Abstract-The data analysed consists of the joint distribution of severities of injury to vehicle drivers in head-on crashes. stratified according to the relative masses of the vehicles. On the basis of some fairly strong assumptions, a model is developed which results in the joint distribution being bivariate normal. The parameters are interpretable in terms of the effect of velocity change on injury severity. and the relative variability of velocity change and of injury severity at a particular velocity change. The predictions made by the model enjoy a considerable degree of success.
INTRODUCTION In
crashes between two vehicles, one of the most important factors affecting the severity of injury sustained by their occupants is the velocity change which the vehicles undergo at impact. This, in turn, is determined by the configuration of the impact, the relative speed of the vehicles, and their relative masses. Routinely-collected data enables us to determine the accident configuration (head-on, front-into-rear, etc.) and the masses of the vehicles (as make and model are known), but not their speedslas this is too lengthy and difficult a process to be done routinely). It is possible, therefore, to obtain a table of the frequency of occurrence of all combinations of injury severities to the two drivers; and to stratify this table according to mass ratio of the vehicles and according to accident configuration. Table I(a) is an example. Note that the number of accidents in which both drivers were uninjured is not known, since such accidents are not recorded unless another person was injured. The processing of British road accident data for 1969-72 in order to obtain tables like Table I(a) is described by Grime and Hutchinson [1979]. The analysis reported in that paper is confined to the marginal totals of such tables. One model for the joint distribution of severities is described by Hutchinson [ 1977, 19821; the present paper takes another approach, attempting to fit bivariate normal distributions to such data. A BIVARIATE It may happen that two observed random variables U, V, W thus:
NORMAL
MODEL
random variables
X and Y are created from three other
X=U+V)
Table
I. Numbers
of accidents in which each combination of injury severity occurred 1969-72. head-on accidents in urban areas. mass ratio = 0.60 - 0.69) (a)
Actual
Faral
DrlVer
(bl Predicted Driver
of heavier
Serlcus
Slight
None
I.5
28
Fatal
1
serious
4
125
148
Slight
4
56
None
3
68
10
vehicle
Fatal
Serious
Slight
None
3
15
9
11
269
4
141
141
251
271
793
2
61
310
791
267
?
2
60
269
11266
of
lighter vehicle
(Great
Britain.
II6
T. P. HCTCHIMON
If Lr. L’. W are mutually independent and normally distributed, X and Y jointly have a hivariate normal distribution. with correlation dependent on the variances of U. c’. and W. The symbol L(h. k. r) is used to denote the bivariate normal integral. that is the probability that S > h and Y > k given that X and Y jointly have a bivariate normal distribution with zero means. unit standard deviations. and correlation r. If two vehicles of equal mass collide head-on, there is a positive correlation between the severities of injury to their drivers. This arises because of the common factor of velocity at impact. which is the same for both drivers in the same accident, but differs between accidents. In terms of eqns (1). X and Y are the two severities of injury. U is a factor deriving from crash speed. and V and W are random variables influenced by how well the vehicle is designed. the driver’s susceptibility to injury. and so on. Since there is no means of quantitatively measuring injury severity, merely a grading into a few ordered categories, our problem is similar to that of estimating polychoric correlation. More generally, consider head-on collisions between vehicles, one of which is k times as massive as the other. (We take k s I). If their relative speed at impact was c‘, the velocity changes of the two vehicles are c/l + k and kr/l + k. The speed c is difficult to determine and is not generally available in routinely-collected data, but k is known from the makes and models of the vehicles. Assume that injury severity (X) is related to velocity change (1,) by the equation X=a+blog(A,)+e where E is a normally distributed random variable severities of injury X and Y are therefore given by
with mean 0 and variance
oe2. The two
so that
+ b log (u) + E:.
If impact speed has a log-normal distribution, log (c) will be normally distributed. Taking this to have mean ).Land variance (T’, X and Y jointly have a bivariate normal distribution with
Mean(X)
= a + b log
(
Mean(Y)=a+blog
(
i-:~
&
)
)
+ bp.
+ bp.
Variance (X) = Variance (Y) = b’o’+ Correlation
(XY) = blj:T’U
t
oc’.
I.
Therefore, if an injury is called “serious” if X is greater than a threshold less than T, the probability of both drivers being seriously injured is
Prob{X>TandY>T}=L
T - a - bp - b log(l/l d/(b%” + a,‘)
T, and “slight” if it is
+ k) T - a - bp - b log(k/l ’ d(b%‘+ a,‘)
+ k)
b’$ ’ b%‘t
cr;)
A bivariate normal model for intra-accident
correlations
21:
and other probabilities may be obtained similarly. When there are four levels of injury severity, as in Table I. there will be three thresholds T,, T?, and T3 to be fitted. Some of the parameters in this model are confounded with others. The following conventions will be adopted: a = 0, p = 0. b’a’+ a,‘= 1. Further, b’cT’/fb’a’+ uI’) will be referred to as the correlation r. Thus there will be five parameters to be determined: three thresholds, b, and cr. (When presenting results from this and related models (below). logarithms to base e will be used). In the above model. it has been assumed that the distribution of impact speed is the same for all masses of vehicles, and that the relation between injury severity and velocity change is the same for all masses of vehicles. British data examined by Grime and Hutchinson [I9791 shows that mass of vehicle does not affect severity of injury. neither in single-vehicle accidents nor in two-vehicle accidents in which the two vehicles were of almost equal weight. The simplest way to interpret this is to see it as confirmation of the aforementioned assumptions. (The possibility should be admitted that bigger cars, for instance, might be involved in higher-speed crashes, with injury at a given speed being lower in them than in smaller vehicles by just the right amount that the two effects cancel out). The contrary finding in North Carolina, that vehicle mass does affect occupant injury severity, the greater injury being in smaller cars, should be mentioned [Campbell and Reinfurt, 1973; O’Neill et al., 19741. The reason for the divergence between British and American data in this respect is an important unsolved question.
RESULTS
This model was fitted to five sets of data on injury in head-on accidents in rural areas (the data is given in Table 5 of Hutchinson, 197.7). The five sets were for mass ratios 0.60-0.69, 0.70-0.79, 0.80-0.89, 0.90-0.99, and equal masses. The values of k were assumed to be 0.65. 0.75, 0.85, 0.95, and 1.00 respectively. Certain preliminary points need to be mentioned: (I) Mass ratios less than 0.60 were omitkd because these will mostly involve a car and a commercial vehicle, where (a) incompatibility between vehicle structures may be of comparable importance to velocity change, and (b) impact speed may have a different distribution from that in car-car crashes. (2) For the case of equal masses, the two vehicles are indistinguishable, the entries in the data table below the diagonal are combined with those above, and there are 4 parameters to be fitted to 9 cells in the table. (3) The method of fitting was the minimisation of Pearson’s x2, Z(O- E)‘/E. (4) The average number of observations in the five data tables was over 2100. With such a large number, it is possible that a model that is largely realistic might give a significant ,$. (In addition, Hutchinson and Mayne [ 19771 have shown that there is rather more variability in road accident figures than would be expected on the Poisson assumption). The first model (model I) fitted’to the data was that described in the previous Section. The values of x’ obtained were as in column 12 of Table 2. As can be seen, in each case the value of x’ was highly significant. There was. moreover, a particular pattern to the difference between observed and fitted values in the data tables: the numbers of cases in which both drivers were seriously injured and both were slightly injured were greater in the data than the predicted numbers were and the numbers of cases in which one driver was seriously injured and one slightly injured were less. This occurred with each of the five datasets. A possible reason for this is the existence of variability in where the thresholds separating the four categories of injury are positioned. Although specific criteria are laid down for the categories, there is nevertheless some latitude in their interpretation. This is discussed further in the Appendix. Accordingly, a model was developed in which the position of the thresholds between the four categories of injury severity was the same for both drivers in the same accident, because they are both being classified by the same policeman at the same time, but was different for different accidents. It was assumed that there was no variability in the position of the fatality threshold, T,, but that thresholds TI and TJ jointly had a bivariate normal distribution with standard deviations (T: and g2, and correlation rz3. The effect of this is that there appears to be a greater correlation in injury severities when Tz and T, is used as a
dividing line than when T, is. (An ambiguity could arise from T, exceeding T,. But we assume this occurs so rarely as not to be a problem). The results for this model 2 are given in Columns 2-l 1 of Table 2. It may be seen that the model was a satisfactory fit in all five cases (none of the values of x2 given in Column IO are significant at the 5% level). Further, the estimates of the 8 parameters are quite similar in all five datasets. The modei was also fitted to five sets of data on injury in head-on accidents in urban areas (the data is given in Table 3 of Hutchinson [1982]). The average number of observations in the five data tables was over 1800. The results are given in Table 3. Of course, the parameter estimates in the five rows of Table 2 should be the same. Further, they should be the same in Table 3. except that (a) the variability of impact speed, g, might be different in urban areas from that in rural areas. and (b) because the average impact speed will probably be different in urban areas from that in rural areas,. and we are taking p = 0 in both cases, the thresholds T,, Tz, and Ti will appear to be different. (But the differences between the values for urban and rural areas will be the same for all three thresholds. This difference will equal bp,, where p,, is the mean of log(c) in urban areas where the mean of log(c) in rural areas is taken as zero).
Table 3. Results from fitting model to urban data
(11 L.CO
!6)
!7)
:31
!3)
(2)
(31
!4i
'51
2.21
1.61
(3.76
f
0.30
0.76
0.6:
0.84
2.L
0.63
1.30
0.45
0.57
3.79
(LO)
!1Ll
!12)
1
34.7
i3.2
6.0
6
66.2
16.3
.90 -
.99
2.75
1.98
1.51
.a0 -
.33
2.27
1.4L
0.7;
1.02
o.a3
0.54
0.61
0.70
14.0
6
ao.6
29.1
- .79
2.29
1.39
0.77
o.ao
1.01
0.45
0.60
3.75
3.1
6
33.3
LG.3
i.81
o.a9
'3.04 1.04
3.71
0.50
0.34
0.63
13.3
i
AL.7
51.3
.:O
.60 f
.63
NO: t?e
Cecarnrne:! sy a.ieraqe Value
.Abivariate normal
model
for intra-accident
correlations
119
Accordingly. in model 3.10 parameters were fitted simultaneously to all 10 datascts. Eight of the parameters had the same meanings as T,, TI. T+ b. u. u2. cr:, and rJ3 for the rural data. Another parameter was the amount by which the thresholds were increased for the urban data. The tenth parameter was the value of u for the urban data. The values of the parameters that gave the best fit are given in footnotes to Tables 7 and 3. The total value of x2 was 250.0. with 118 degrees of freedom, which is highly significant. The contribution to this x2 from each of the ten datasets is given in Column 13 of Tables 2 and 3. The difference between the thresholds in urban and rural areas was 0.44, meaning that the average impact speed in urban areas is estimated to be 64% of that in rural areas. x2 is useful for alerting US to the failures of our model. but is not so successful at telling us about its successes. The large number of observations has already been mentioned (preliminary point 4 above). One attempt to show the degree of success of the model is given in Tables I and 4. where the predicted distribution of casualties (model 3) is compared with the actual. Table I is the worst-fitting case (x2 = 51.3) and Table 4 is the best-fitting case (x’ = 14.0). Even in Table 1 the fit does not seem too bad to me. In another attempt to illustrate the success of this approach, model 4 was developed. Values of u. u?, uj, and rzz were taken as constants (the values found when fitting model 3). and six thresholds were fitted to each dataset-T,,. T,,. and TI1 for the driver of the lighter vehicle. and Tz,, T?,, and T2? for the driver of the heavier vehicle. For this model 4. results analogous to eqn (1) imply the following predictions: (i) When T,,. T,?, and T13 are plotted against log (I + k), the relationship in each case should be linear with slope b. (ii) When T2,. T,:, and T?, are plotted against log (1 + k/k) the relationship in each case should be linear with slope b. (iii) The three intercepts in (ii) above should be the same as the three intercepts in (i). (iv) When T:,-T,,, T?,-T,,, and Tz3-TII are each plotted against log(l/k), the relationship should be a straight line through the origin, with slope b. The results for the rural data are shown in Fig. I. Figure l(a) shows T,, plotted against log (I + k) and T?, plotted against log (I + k/k). Predictions (i)-(iii) mean that these points should lie on a single straight line of slope b. Figure I(b) shows T,? plotted against log (I + k) and ‘T??plotted against log (I C k/k). Predictions (i)-(iii) mean that these points should lie on a single straight line of slope b. Figure I(c) shows T,, plotted against log (1 + k) and Tz3 plotted against log (I + k/k). Predictions (i)-(iii) mean that these points should lie on a single straight line of slope b. Figure I(d) shows the average of T2,-T,,. Tz2-T,>. and T,,-T,, plotted against log (Ilk). Prediction (iv) says that these points should lie on a line of slope b through the origin. Though the degree of success of the predictions is a matter of opinion, they are clearly satisfied to some extent. The results for the urban data are shown in Fig. 1. Considered on its own, the same predictions should apply to the urban data as were made for the rural data. In addition, (c) the thresholds for the rural and urban datasets should differ by a constant (they will not be the same because I* will be different in the two cases), (vi) b should be the same in both cases. Figure 2 illustrates the extent to which the earlier predictions (i)-(iv) are fulfilled. As to (c), the quantity T,-T2 averaged 0.86 for the rural data and 0.80 for the urban data, and T,-T,
Table
1. Numbers
of accidents in which each combination of injury severity 1969-Z head-on accidents in urban areas. mass ratio = 0.70-
occurred 0.79)
(Great
Britain.
"0 __
T. P. HLTCHISSOS
ial i; or l,,
(bl T,, or T,, 2 0,
2.e
1
2 74 I 0
*.,I
0
.SO
-60 in [;*kl
:,9. ," [;I
.90
11 i
+
0
1.0 0
Fig. I. Testing the predictions for the rural data by plotting the thresholds against mass ratio. Key to (a). (b). (c): X thresholds for driver of lighter vehicle (T,,, TI:, T,I). + thresholds found for mass ratio = I. 0 thresholds for driver of heavier vehicle (7-21. 7~. T:x).
0.65 for the rural data and 0.67 for the urban data. As to (vi), this may be judged visually from Figs. 1 and 2. averaged
INTERPRETATION
OF
RESULTS
We are interested in describing the effect of velocity change on injury severity. However, we must appreciate the limitations we are working under. First, our dependent variable (injury severity) is inherently unquantifiable. Second, there is no information in our data about the absolute values of our independent variable (velocity change). One way of illustrating our results is Fig. 3. On the horizontal axis is log(v). The distribution of this is shown as being normal, with mean = 0 and standard deviation = 0.98. On the vertical axis is injury severity. X. The mean level of this is shown as 0.82 log(u). (We are illustrating the
A bivariate normal model for intra-accident correlations
33.
3.2.
3
l-
0
0
3.0.
+
0
0
+
2-9_
0
x
.50
1.9’
.70
-60
ln[1*4]
(dl 0.5
The
0
cf- In y
P
,90
I
average of T,,-T,,.T,,-5, and Tzl-J,
1 x
1.6 i
1.5 0
lI_. >. 1.3 I ,50
x 0
x .60
x .lO
.80
.90 ln $ [I
Fig. 2. Testing the predictions for the urban data by plotting the thresholds against mass ratio. Key: as for Fig. I.
case where velocity change = impact speed. This occurs when a vehicle crashes into a stationary object much more massive than itself, or when it crashes into a vehicle of the same mass travelling at the same speed as itself in the opposite direction). At two points, log (u) = - 1 and 1, the distribution of injury severity is shown as being normal, with mean = 0.82 log (u) and standard deviation a, = v/(1 - b*u2) = 0.60. Also shown are the average positions of the thresholds, at T, = 1.94, T2 = 1.12, and TJ = 0.47, and the distributions of r2 and TJ. Another illustration is given in Fig. 4. Again we are considering the case where velocity change is the same as impact speed. On the horizontal axis, the speed is shown on a percentile scale. On the vertical axis is shown the probability of sustaining a fatal injury at that speed (lowest line), a serious or fatal injury (middle line), or an injury of any severity (highest line). Thus, at the median speed (50th percentile), the probability of death is less than O.l%, the probability of a serious or fatal injury is 3%, and so on. At the impact speed such that 90% of impacts are slower than this, the probability of death is 6% and the probability of escaping without injury is 18%.
T. P.
NO INJURY
HCTCHINSOS
on
t Impact
Velocity
Fig. 3. The distributions
>
I logorithmrc of impact
scale
at in iv!
Severtty
i
of
Injury
Severity
a:
tntvl
= 1
= -1
1
velocity and injury severity Ghen velocity fuller explan3lion. see text.
change = impact velocity.
FOI
Having obtained estimates of the numbers of accidents at each mass ratio in which both drivers escaped injury, we can estimate the numbers of deaths and injuries per collision. The results are given in Table 5. This shows that within the range of mass ratios considered (i.e. 0.60 to 1.00). there is almost no effect of mass ratio. Thus it is a matter of indifference whether the population of vehicles is homogeneous or heterogeneous. This confirms the result of the simpler analysis of the same data reported earlier [Grime and I-Iutchinson. 19791. But. of course. if the population becomes very heterogeneous, then deaths in the heavier vehicle become negligible and decreases with increasing mass do not compensate for the increases in deaths in the lighter vehicle. The unexplained difference between British and American data mentioned earlier should also be remembered. CONCLUSIONS
In explaining the joint distrjbution of injury severities, the bivariate normal model has considerable success (witness the comparison between predictions and results). but not complete success (witness the significant x2). (2) Factors common to both drivers are shown to play an important. but not overwhelming. role in determining injury severity (the correlation b’a’ being approximately 0.65 in rural areas. 0.69 in urban areas). Relative velocity at impact is assumed to be the key common factor. (3) Variability in the location of the thresholds between serious and slight injury. and between slight and no injury, is of approximately the same size as the variability in injury severity at any given velocity change. The positions of these two thresholds are correlated. (4) The relationship between velocity change and severity of injury is estimated to be as in Fig. 4. (1)
A bivariate
normal
model
for intra-accident
correlations
11 3
The average impact speed in urban areas is estimated to be 64% of that in rural areas. (6) For the type of impact considered here, the ratio of the total number of accidents to the total number of injury accidents is estimated to be 3.8 in rural areas and 6.6 in urban areas. (9
99
/
98
2
95
5
90
IO
80
20
7C
30
60
40
50
50
40
63
30
70
20
80
10
90
4
95
4
cc
5OIOO4OlO2
I2
5
10
20
30 40 50 60 70
Percentile Fig. 1. Estimated
Table
relationship
5. Showing
Locale
between
IYa*S ratio NaA~ar of injuries'
l.M
63
657
.90 - .39
lL5
1148
.SO - .a9
I.33
1290
.70 - .79
39
.60 - .69
132
?-D
Slight Lcm
95
998999%%999
98 99
the numbers
TOtal
of deaths
of sustaining
and injuries
injury
of variuus
per accident
In3uries ?er LOO accident;
accidents' Fatal Serious SLl?Z 4733
1.3
13.7
22.1
2240
9782
L.2
13.3
22.0
2248
9695
1.1
13.7.
23.2
LL32
1540
7965
1.2
13.3
23.1
1074
1398
S-i29
1.2
L2.3
22.3
373
388
6:Li
0.4
5.3
L3.3
39
a:3
2094
14234
3.3
5.3
LG.'
.30 - .39
37
363
zz53
I.5125
3.3
5.7
11.3
.;o - .7?
:I
720
i:33
i2339
0 .3
5.3
LA.5
.i3 - .G3
G6
a:0
:3:3
L3323
5.5
6.1
13.3
.?O - .Y9
KNo.
90
2.!
lJr~a.nL.Ca
?\AP Vd
Se=iou+
80
speed
speed and the probability severities.
that mass ratio does not affect
‘stat: ihI
impact
of Impact
“1 __
T. P. HLTCHINSO~
Campbell B. J and Reinfurt Highway Safety ResearLh Chapman D. P. and Neilson
REFERENCES D W.. Rrf~irionship Brruern Dricrr Crush [njurv und Pusrengrr Car W’eiehr. Report from the Center. Cniversity of North Carolina at Chapel Hill. 1973. I. D.. .S\;i~trs jar Lsrrs of Natiuntil .4ccidrnr Duiu (Sr[~fs 191. Unpublished note. Transport &
Road Research Laboratory. Crowthorns. 19-l. Grime G. and Hutchinson T. P Vehicle mass and drivjer injury. Ergonomics 22. 93-104. 1979. (A fuller version, “Some implications of vehicle-weight for the risk of injury to drivers”. was issued as a report from the Transport Studies Group. University College London). Hutchinson T. P. Intro-accident correlations of driver injury and their application to the effect of mass ratio on injury severity. Accid. ;\md. 5: Prrr. 9. 117-27. 1977. Hutchinson T. P.. Statistical aspects of injury severity. Part IV: Matched data. Transp.Sci. 16. 83-105. 1982. Hutchinson T. P. and Slavne .A. I.. The year-to-year variability in the numbers of road accidents. Truflc Engineerinu and Conirol. 18. 13?-43!. I+. Lai P. W’.. \fodel of injury severity allowing for different gradings of severity: some applications using the British road accident data. Accitl. .4nd. & Prrr. 12. 221-139. 1980. Newby R. F.. Proposed Clrcssificc~tiorr of Ncrrionul Rod Accident Starisrics by Sewrity of Injury. Presented at a Seminar organised by Planning and Transport Research and Computation. London. 1969. O’Neill B.. Joksch H. and Haddon W., Relationship between car size. car weight, and crash injuries in car-to-car crashes. Proc. 3rd Int. Congrers on Automo~ice Safefv. held in San Francisco. Washington. D.C.: National Motor V’ehicle Safety ;\dvisory Council. 19:1. Satterthwaite S. P., A note on the classification of accidents by severity. Report from the Traffic Studies Group. University College
London.
1975.
APPESDIS.
VARIATIONS
IN
THE
GRADING
OF
INJURY
SEVERITY
The definitions of the three degrees of injury are as follows: FATAL: death within 30 days as a result of the accident. SERIOUS: injury for which a person is detained in hospital as an “inpatient” or any of the following injuries. whether or not he is detained in hospital: fractures. concussion, internal injuries. crushings. severe cuts and lacerations. severe general shock requiring medical treatment. _ SLIGHT: injury of a minor character such as a sprain. bruise, or a cut or laceration not judged to be severe. Chapman and Neilson (19711 have this to say about these definitions: “Injury categories are a considerable source of variation. The four categories. fatally. seriously. slightly or not injured are barely sufficient for many purposes and the two borderlines between the last three categoriezsare very uncertain. A fatality is precisely classified by the 30 days rule which includes only those dying vvithin 30 days of the accident. But many injured are detained overnight in hospital because they cannot be examined until morning, because they may have suffered concussion or even because they find themselves at a hospital and there is no alternative accommodation for the night, The result is that many of those falling within the definition of seriously injured do not necessarily have any of the injuries listed for serious injury. Again in some Police Forces it is customary for the policeman at the scene to assess injury severity, while in others enquiry is made afterwards at the hospitals. The distinction between slight and no injury determines whether or not a Stats 19 entry is made. Many accidents of a minor nature are not seen by the police but are reported to them afterwards to comply with the legal requirement to do so by those involved, The police opinion then determines the severity rating.” (Stats I9 is the accident report form filled in by the police.) Newsby [I9691 has examined the proportions of casualties classified as having fatal. serious, or slight injuries in the (then) I.(? British police forces. He shows that those forces with a high ratio of fatalities to fatal plus serious injuries tend to have a low ratio of fatal plus serious casualties to the total, The reason for this is the varying definition of a “serious” injury between police forces: if the boundary between slight and serious injury is at a relatively low level of severity. fatals/(fatals - serious) will be too low and (fatals - serious)/total will be too high. New by also gives an example from one particular English town where the numbers of fatal and serious casualties remained consistent over the period (195(L67) examined. but the number of slight casualties suddenly increased in I959 by 80% and continued to increase from the ne* level. Thus apparently some accidents that previously would have been classified as non-injury in that town are now regarded as slight injury. A similar instance is given by Satterthwaite [I9751 in which it was apparently the boundary between serious and slight injuries that moved. In one particular region of England between 1968 and 1969. there was a fall from 30 to 217~ in the proportion of injury accidents classified as serious in this region. whereas in the rest of Great Britain there w’as a much smaller chance in the oooosite direction. At the same time the total number of accidents in that region remained much the same, as did the proportion of fatalities. Finally. Lai [I9801 has used data in which casualties were classified according to both injury severity and the police force area where the accident occurred to establish that consistent differences exist between police forces in their placement of the boundary between “serious” and “slight” severities. We mav therefore conclude that some of the variability in T? and T, is attributable to rules of interpretation that are followed throughout a police force. The remainder is presumably attributable to the judgements of individual police officers. 1
1
T. P. Department
of Statistics
HUTCHINSON
and Operational Research, Coventry CVI
(Rrcriwd
19 Febrwry
Coventry (Lanchester) SFB. England
1982; in revised form
16 September
Polytechnic,
Priory
Street.
1981)
Abstract-The data analysed consists of the joint distribution of severities of injury to vehicle drivers in head-on crashes. stratified according to the relative masses of the vehicles. On the basis of some fairly strong assumptions, a model is developed which results in the joint distribution being bivariate normal. The parameters are interpretable in terms of the effect of velocity change on injury severity. and the relative variability of velocity change and of injury severity at a particular velocity change. The predictions made by the model enjoy a considerable degree of success.
INTRODUCTION In
crashes between two vehicles, one of the most important factors affecting the severity of injury sustained by their occupants is the velocity change which the vehicles undergo at impact. This, in turn, is determined by the configuration of the impact, the relative speed of the vehicles, and their relative masses. Routinely-collected data enables us to determine the accident configuration (head-on, front-into-rear, etc.) and the masses of the vehicles (as make and model are known), but not their speedslas this is too lengthy and difficult a process to be done routinely). It is possible, therefore, to obtain a table of the frequency of occurrence of all combinations of injury severities to the two drivers; and to stratify this table according to mass ratio of the vehicles and according to accident configuration. Table I(a) is an example. Note that the number of accidents in which both drivers were uninjured is not known, since such accidents are not recorded unless another person was injured. The processing of British road accident data for 1969-72 in order to obtain tables like Table I(a) is described by Grime and Hutchinson [1979]. The analysis reported in that paper is confined to the marginal totals of such tables. One model for the joint distribution of severities is described by Hutchinson [ 1977, 19821; the present paper takes another approach, attempting to fit bivariate normal distributions to such data. A BIVARIATE It may happen that two observed random variables U, V, W thus:
NORMAL
MODEL
random variables
X and Y are created from three other
X=U+V)
Table
I. Numbers
of accidents in which each combination of injury severity occurred 1969-72. head-on accidents in urban areas. mass ratio = 0.60 - 0.69) (a)
Actual
Faral
DrlVer
(bl Predicted Driver
of heavier
Serlcus
Slight
None
I.5
28
Fatal
1
serious
4
125
148
Slight
4
56
None
3
68
10
vehicle
Fatal
Serious
Slight
None
3
15
9
11
269
4
141
141
251
271
793
2
61
310
791
267
?
2
60
269
11266
of
lighter vehicle
(Great
Britain.
II6
T. P. HCTCHIMON
If Lr. L’. W are mutually independent and normally distributed, X and Y jointly have a hivariate normal distribution. with correlation dependent on the variances of U. c’. and W. The symbol L(h. k. r) is used to denote the bivariate normal integral. that is the probability that S > h and Y > k given that X and Y jointly have a bivariate normal distribution with zero means. unit standard deviations. and correlation r. If two vehicles of equal mass collide head-on, there is a positive correlation between the severities of injury to their drivers. This arises because of the common factor of velocity at impact. which is the same for both drivers in the same accident, but differs between accidents. In terms of eqns (1). X and Y are the two severities of injury. U is a factor deriving from crash speed. and V and W are random variables influenced by how well the vehicle is designed. the driver’s susceptibility to injury. and so on. Since there is no means of quantitatively measuring injury severity, merely a grading into a few ordered categories, our problem is similar to that of estimating polychoric correlation. More generally, consider head-on collisions between vehicles, one of which is k times as massive as the other. (We take k s I). If their relative speed at impact was c‘, the velocity changes of the two vehicles are c/l + k and kr/l + k. The speed c is difficult to determine and is not generally available in routinely-collected data, but k is known from the makes and models of the vehicles. Assume that injury severity (X) is related to velocity change (1,) by the equation X=a+blog(A,)+e where E is a normally distributed random variable severities of injury X and Y are therefore given by
with mean 0 and variance
oe2. The two
so that
+ b log (u) + E:.
If impact speed has a log-normal distribution, log (c) will be normally distributed. Taking this to have mean ).Land variance (T’, X and Y jointly have a bivariate normal distribution with
Mean(X)
= a + b log
(
Mean(Y)=a+blog
(
i-:~
&
)
)
+ bp.
+ bp.
Variance (X) = Variance (Y) = b’o’+ Correlation
(XY) = blj:T’U
t
oc’.
I.
Therefore, if an injury is called “serious” if X is greater than a threshold less than T, the probability of both drivers being seriously injured is
Prob{X>TandY>T}=L
T - a - bp - b log(l/l d/(b%” + a,‘)
T, and “slight” if it is
+ k) T - a - bp - b log(k/l ’ d(b%‘+ a,‘)
+ k)
b’$ ’ b%‘t
cr;)
A bivariate normal model for intra-accident
correlations
21:
and other probabilities may be obtained similarly. When there are four levels of injury severity, as in Table I. there will be three thresholds T,, T?, and T3 to be fitted. Some of the parameters in this model are confounded with others. The following conventions will be adopted: a = 0, p = 0. b’a’+ a,‘= 1. Further, b’cT’/fb’a’+ uI’) will be referred to as the correlation r. Thus there will be five parameters to be determined: three thresholds, b, and cr. (When presenting results from this and related models (below). logarithms to base e will be used). In the above model. it has been assumed that the distribution of impact speed is the same for all masses of vehicles, and that the relation between injury severity and velocity change is the same for all masses of vehicles. British data examined by Grime and Hutchinson [I9791 shows that mass of vehicle does not affect severity of injury. neither in single-vehicle accidents nor in two-vehicle accidents in which the two vehicles were of almost equal weight. The simplest way to interpret this is to see it as confirmation of the aforementioned assumptions. (The possibility should be admitted that bigger cars, for instance, might be involved in higher-speed crashes, with injury at a given speed being lower in them than in smaller vehicles by just the right amount that the two effects cancel out). The contrary finding in North Carolina, that vehicle mass does affect occupant injury severity, the greater injury being in smaller cars, should be mentioned [Campbell and Reinfurt, 1973; O’Neill et al., 19741. The reason for the divergence between British and American data in this respect is an important unsolved question.
RESULTS
This model was fitted to five sets of data on injury in head-on accidents in rural areas (the data is given in Table 5 of Hutchinson, 197.7). The five sets were for mass ratios 0.60-0.69, 0.70-0.79, 0.80-0.89, 0.90-0.99, and equal masses. The values of k were assumed to be 0.65. 0.75, 0.85, 0.95, and 1.00 respectively. Certain preliminary points need to be mentioned: (I) Mass ratios less than 0.60 were omitkd because these will mostly involve a car and a commercial vehicle, where (a) incompatibility between vehicle structures may be of comparable importance to velocity change, and (b) impact speed may have a different distribution from that in car-car crashes. (2) For the case of equal masses, the two vehicles are indistinguishable, the entries in the data table below the diagonal are combined with those above, and there are 4 parameters to be fitted to 9 cells in the table. (3) The method of fitting was the minimisation of Pearson’s x2, Z(O- E)‘/E. (4) The average number of observations in the five data tables was over 2100. With such a large number, it is possible that a model that is largely realistic might give a significant ,$. (In addition, Hutchinson and Mayne [ 19771 have shown that there is rather more variability in road accident figures than would be expected on the Poisson assumption). The first model (model I) fitted’to the data was that described in the previous Section. The values of x’ obtained were as in column 12 of Table 2. As can be seen, in each case the value of x’ was highly significant. There was. moreover, a particular pattern to the difference between observed and fitted values in the data tables: the numbers of cases in which both drivers were seriously injured and both were slightly injured were greater in the data than the predicted numbers were and the numbers of cases in which one driver was seriously injured and one slightly injured were less. This occurred with each of the five datasets. A possible reason for this is the existence of variability in where the thresholds separating the four categories of injury are positioned. Although specific criteria are laid down for the categories, there is nevertheless some latitude in their interpretation. This is discussed further in the Appendix. Accordingly, a model was developed in which the position of the thresholds between the four categories of injury severity was the same for both drivers in the same accident, because they are both being classified by the same policeman at the same time, but was different for different accidents. It was assumed that there was no variability in the position of the fatality threshold, T,, but that thresholds TI and TJ jointly had a bivariate normal distribution with standard deviations (T: and g2, and correlation rz3. The effect of this is that there appears to be a greater correlation in injury severities when Tz and T, is used as a
dividing line than when T, is. (An ambiguity could arise from T, exceeding T,. But we assume this occurs so rarely as not to be a problem). The results for this model 2 are given in Columns 2-l 1 of Table 2. It may be seen that the model was a satisfactory fit in all five cases (none of the values of x2 given in Column IO are significant at the 5% level). Further, the estimates of the 8 parameters are quite similar in all five datasets. The modei was also fitted to five sets of data on injury in head-on accidents in urban areas (the data is given in Table 3 of Hutchinson [1982]). The average number of observations in the five data tables was over 1800. The results are given in Table 3. Of course, the parameter estimates in the five rows of Table 2 should be the same. Further, they should be the same in Table 3. except that (a) the variability of impact speed, g, might be different in urban areas from that in rural areas. and (b) because the average impact speed will probably be different in urban areas from that in rural areas,. and we are taking p = 0 in both cases, the thresholds T,, Tz, and Ti will appear to be different. (But the differences between the values for urban and rural areas will be the same for all three thresholds. This difference will equal bp,, where p,, is the mean of log(c) in urban areas where the mean of log(c) in rural areas is taken as zero).
Table 3. Results from fitting model to urban data
(11 L.CO
!6)
!7)
:31
!3)
(2)
(31
!4i
'51
2.21
1.61
(3.76
f
0.30
0.76
0.6:
0.84
2.L
0.63
1.30
0.45
0.57
3.79
(LO)
!1Ll
!12)
1
34.7
i3.2
6.0
6
66.2
16.3
.90 -
.99
2.75
1.98
1.51
.a0 -
.33
2.27
1.4L
0.7;
1.02
o.a3
0.54
0.61
0.70
14.0
6
ao.6
29.1
- .79
2.29
1.39
0.77
o.ao
1.01
0.45
0.60
3.75
3.1
6
33.3
LG.3
i.81
o.a9
'3.04 1.04
3.71
0.50
0.34
0.63
13.3
i
AL.7
51.3
.:O
.60 f
.63
NO: t?e
Cecarnrne:! sy a.ieraqe Value
.Abivariate normal
model
for intra-accident
correlations
119
Accordingly. in model 3.10 parameters were fitted simultaneously to all 10 datascts. Eight of the parameters had the same meanings as T,, TI. T+ b. u. u2. cr:, and rJ3 for the rural data. Another parameter was the amount by which the thresholds were increased for the urban data. The tenth parameter was the value of u for the urban data. The values of the parameters that gave the best fit are given in footnotes to Tables 7 and 3. The total value of x2 was 250.0. with 118 degrees of freedom, which is highly significant. The contribution to this x2 from each of the ten datasets is given in Column 13 of Tables 2 and 3. The difference between the thresholds in urban and rural areas was 0.44, meaning that the average impact speed in urban areas is estimated to be 64% of that in rural areas. x2 is useful for alerting US to the failures of our model. but is not so successful at telling us about its successes. The large number of observations has already been mentioned (preliminary point 4 above). One attempt to show the degree of success of the model is given in Tables I and 4. where the predicted distribution of casualties (model 3) is compared with the actual. Table I is the worst-fitting case (x2 = 51.3) and Table 4 is the best-fitting case (x’ = 14.0). Even in Table 1 the fit does not seem too bad to me. In another attempt to illustrate the success of this approach, model 4 was developed. Values of u. u?, uj, and rzz were taken as constants (the values found when fitting model 3). and six thresholds were fitted to each dataset-T,,. T,,. and TI1 for the driver of the lighter vehicle. and Tz,, T?,, and T2? for the driver of the heavier vehicle. For this model 4. results analogous to eqn (1) imply the following predictions: (i) When T,,. T,?, and T13 are plotted against log (I + k), the relationship in each case should be linear with slope b. (ii) When T2,. T,:, and T?, are plotted against log (1 + k/k) the relationship in each case should be linear with slope b. (iii) The three intercepts in (ii) above should be the same as the three intercepts in (i). (iv) When T:,-T,,, T?,-T,,, and Tz3-TII are each plotted against log(l/k), the relationship should be a straight line through the origin, with slope b. The results for the rural data are shown in Fig. I. Figure l(a) shows T,, plotted against log (I + k) and T?, plotted against log (I + k/k). Predictions (i)-(iii) mean that these points should lie on a single straight line of slope b. Figure I(b) shows T,? plotted against log (I + k) and ‘T??plotted against log (I C k/k). Predictions (i)-(iii) mean that these points should lie on a single straight line of slope b. Figure I(c) shows T,, plotted against log (1 + k) and Tz3 plotted against log (I + k/k). Predictions (i)-(iii) mean that these points should lie on a single straight line of slope b. Figure I(d) shows the average of T2,-T,,. Tz2-T,>. and T,,-T,, plotted against log (Ilk). Prediction (iv) says that these points should lie on a line of slope b through the origin. Though the degree of success of the predictions is a matter of opinion, they are clearly satisfied to some extent. The results for the urban data are shown in Fig. 1. Considered on its own, the same predictions should apply to the urban data as were made for the rural data. In addition, (c) the thresholds for the rural and urban datasets should differ by a constant (they will not be the same because I* will be different in the two cases), (vi) b should be the same in both cases. Figure 2 illustrates the extent to which the earlier predictions (i)-(iv) are fulfilled. As to (c), the quantity T,-T2 averaged 0.86 for the rural data and 0.80 for the urban data, and T,-T,
Table
1. Numbers
of accidents in which each combination of injury severity 1969-Z head-on accidents in urban areas. mass ratio = 0.70-
occurred 0.79)
(Great
Britain.
"0 __
T. P. HLTCHISSOS
ial i; or l,,
(bl T,, or T,, 2 0,
2.e
1
2 74 I 0
*.,I
0
.SO
-60 in [;*kl
:,9. ," [;I
.90
11 i
+
0
1.0 0
Fig. I. Testing the predictions for the rural data by plotting the thresholds against mass ratio. Key to (a). (b). (c): X thresholds for driver of lighter vehicle (T,,, TI:, T,I). + thresholds found for mass ratio = I. 0 thresholds for driver of heavier vehicle (7-21. 7~. T:x).
0.65 for the rural data and 0.67 for the urban data. As to (vi), this may be judged visually from Figs. 1 and 2. averaged
INTERPRETATION
OF
RESULTS
We are interested in describing the effect of velocity change on injury severity. However, we must appreciate the limitations we are working under. First, our dependent variable (injury severity) is inherently unquantifiable. Second, there is no information in our data about the absolute values of our independent variable (velocity change). One way of illustrating our results is Fig. 3. On the horizontal axis is log(v). The distribution of this is shown as being normal, with mean = 0 and standard deviation = 0.98. On the vertical axis is injury severity. X. The mean level of this is shown as 0.82 log(u). (We are illustrating the
A bivariate normal model for intra-accident correlations
33.
3.2.
3
l-
0
0
3.0.
+
0
0
+
2-9_
0
x
.50
1.9’
.70
-60
ln[1*4]
(dl 0.5
The
0
cf- In y
P
,90
I
average of T,,-T,,.T,,-5, and Tzl-J,
1 x
1.6 i
1.5 0
lI_. >. 1.3 I ,50
x 0
x .60
x .lO
.80
.90 ln $ [I
Fig. 2. Testing the predictions for the urban data by plotting the thresholds against mass ratio. Key: as for Fig. I.
case where velocity change = impact speed. This occurs when a vehicle crashes into a stationary object much more massive than itself, or when it crashes into a vehicle of the same mass travelling at the same speed as itself in the opposite direction). At two points, log (u) = - 1 and 1, the distribution of injury severity is shown as being normal, with mean = 0.82 log (u) and standard deviation a, = v/(1 - b*u2) = 0.60. Also shown are the average positions of the thresholds, at T, = 1.94, T2 = 1.12, and TJ = 0.47, and the distributions of r2 and TJ. Another illustration is given in Fig. 4. Again we are considering the case where velocity change is the same as impact speed. On the horizontal axis, the speed is shown on a percentile scale. On the vertical axis is shown the probability of sustaining a fatal injury at that speed (lowest line), a serious or fatal injury (middle line), or an injury of any severity (highest line). Thus, at the median speed (50th percentile), the probability of death is less than O.l%, the probability of a serious or fatal injury is 3%, and so on. At the impact speed such that 90% of impacts are slower than this, the probability of death is 6% and the probability of escaping without injury is 18%.
T. P.
NO INJURY
HCTCHINSOS
on
t Impact
Velocity
Fig. 3. The distributions
>
I logorithmrc of impact
scale
at in iv!
Severtty
i
of
Injury
Severity
a:
tntvl
= 1
= -1
1
velocity and injury severity Ghen velocity fuller explan3lion. see text.
change = impact velocity.
FOI
Having obtained estimates of the numbers of accidents at each mass ratio in which both drivers escaped injury, we can estimate the numbers of deaths and injuries per collision. The results are given in Table 5. This shows that within the range of mass ratios considered (i.e. 0.60 to 1.00). there is almost no effect of mass ratio. Thus it is a matter of indifference whether the population of vehicles is homogeneous or heterogeneous. This confirms the result of the simpler analysis of the same data reported earlier [Grime and I-Iutchinson. 19791. But. of course. if the population becomes very heterogeneous, then deaths in the heavier vehicle become negligible and decreases with increasing mass do not compensate for the increases in deaths in the lighter vehicle. The unexplained difference between British and American data mentioned earlier should also be remembered. CONCLUSIONS
In explaining the joint distrjbution of injury severities, the bivariate normal model has considerable success (witness the comparison between predictions and results). but not complete success (witness the significant x2). (2) Factors common to both drivers are shown to play an important. but not overwhelming. role in determining injury severity (the correlation b’a’ being approximately 0.65 in rural areas. 0.69 in urban areas). Relative velocity at impact is assumed to be the key common factor. (3) Variability in the location of the thresholds between serious and slight injury. and between slight and no injury, is of approximately the same size as the variability in injury severity at any given velocity change. The positions of these two thresholds are correlated. (4) The relationship between velocity change and severity of injury is estimated to be as in Fig. 4. (1)
A bivariate
normal
model
for intra-accident
correlations
11 3
The average impact speed in urban areas is estimated to be 64% of that in rural areas. (6) For the type of impact considered here, the ratio of the total number of accidents to the total number of injury accidents is estimated to be 3.8 in rural areas and 6.6 in urban areas. (9
99
/
98
2
95
5
90
IO
80
20
7C
30
60
40
50
50
40
63
30
70
20
80
10
90
4
95
4
cc
5OIOO4OlO2
I2
5
10
20
30 40 50 60 70
Percentile Fig. 1. Estimated
Table
relationship
5. Showing
Locale
between
IYa*S ratio NaA~ar of injuries'
l.M
63
657
.90 - .39
lL5
1148
.SO - .a9
I.33
1290
.70 - .79
39
.60 - .69
132
?-D
Slight Lcm
95
998999%%999
98 99
the numbers
TOtal
of deaths
of sustaining
and injuries
injury
of variuus
per accident
In3uries ?er LOO accident;
accidents' Fatal Serious SLl?Z 4733
1.3
13.7
22.1
2240
9782
L.2
13.3
22.0
2248
9695
1.1
13.7.
23.2
LL32
1540
7965
1.2
13.3
23.1
1074
1398
S-i29
1.2
L2.3
22.3
373
388
6:Li
0.4
5.3
L3.3
39
a:3
2094
14234
3.3
5.3
LG.'
.30 - .39
37
363
zz53
I.5125
3.3
5.7
11.3
.;o - .7?
:I
720
i:33
i2339
0 .3
5.3
LA.5
.i3 - .G3
G6
a:0
:3:3
L3323
5.5
6.1
13.3
.?O - .Y9
KNo.
90
2.!
lJr~a.nL.Ca
?\AP Vd
Se=iou+
80
speed
speed and the probability severities.
that mass ratio does not affect
‘stat: ihI
impact
of Impact
“1 __
T. P. HLTCHINSO~
Campbell B. J and Reinfurt Highway Safety ResearLh Chapman D. P. and Neilson
REFERENCES D W.. Rrf~irionship Brruern Dricrr Crush [njurv und Pusrengrr Car W’eiehr. Report from the Center. Cniversity of North Carolina at Chapel Hill. 1973. I. D.. .S\;i~trs jar Lsrrs of Natiuntil .4ccidrnr Duiu (Sr[~fs 191. Unpublished note. Transport &
Road Research Laboratory. Crowthorns. 19-l. Grime G. and Hutchinson T. P Vehicle mass and drivjer injury. Ergonomics 22. 93-104. 1979. (A fuller version, “Some implications of vehicle-weight for the risk of injury to drivers”. was issued as a report from the Transport Studies Group. University College London). Hutchinson T. P. Intro-accident correlations of driver injury and their application to the effect of mass ratio on injury severity. Accid. ;\md. 5: Prrr. 9. 117-27. 1977. Hutchinson T. P.. Statistical aspects of injury severity. Part IV: Matched data. Transp.Sci. 16. 83-105. 1982. Hutchinson T. P. and Slavne .A. I.. The year-to-year variability in the numbers of road accidents. Truflc Engineerinu and Conirol. 18. 13?-43!. I+. Lai P. W’.. \fodel of injury severity allowing for different gradings of severity: some applications using the British road accident data. Accitl. .4nd. & Prrr. 12. 221-139. 1980. Newby R. F.. Proposed Clrcssificc~tiorr of Ncrrionul Rod Accident Starisrics by Sewrity of Injury. Presented at a Seminar organised by Planning and Transport Research and Computation. London. 1969. O’Neill B.. Joksch H. and Haddon W., Relationship between car size. car weight, and crash injuries in car-to-car crashes. Proc. 3rd Int. Congrers on Automo~ice Safefv. held in San Francisco. Washington. D.C.: National Motor V’ehicle Safety ;\dvisory Council. 19:1. Satterthwaite S. P., A note on the classification of accidents by severity. Report from the Traffic Studies Group. University College
London.
1975.
APPESDIS.
VARIATIONS
IN
THE
GRADING
OF
INJURY
SEVERITY
The definitions of the three degrees of injury are as follows: FATAL: death within 30 days as a result of the accident. SERIOUS: injury for which a person is detained in hospital as an “inpatient” or any of the following injuries. whether or not he is detained in hospital: fractures. concussion, internal injuries. crushings. severe cuts and lacerations. severe general shock requiring medical treatment. _ SLIGHT: injury of a minor character such as a sprain. bruise, or a cut or laceration not judged to be severe. Chapman and Neilson (19711 have this to say about these definitions: “Injury categories are a considerable source of variation. The four categories. fatally. seriously. slightly or not injured are barely sufficient for many purposes and the two borderlines between the last three categoriezsare very uncertain. A fatality is precisely classified by the 30 days rule which includes only those dying vvithin 30 days of the accident. But many injured are detained overnight in hospital because they cannot be examined until morning, because they may have suffered concussion or even because they find themselves at a hospital and there is no alternative accommodation for the night, The result is that many of those falling within the definition of seriously injured do not necessarily have any of the injuries listed for serious injury. Again in some Police Forces it is customary for the policeman at the scene to assess injury severity, while in others enquiry is made afterwards at the hospitals. The distinction between slight and no injury determines whether or not a Stats 19 entry is made. Many accidents of a minor nature are not seen by the police but are reported to them afterwards to comply with the legal requirement to do so by those involved, The police opinion then determines the severity rating.” (Stats I9 is the accident report form filled in by the police.) Newsby [I9691 has examined the proportions of casualties classified as having fatal. serious, or slight injuries in the (then) I.(? British police forces. He shows that those forces with a high ratio of fatalities to fatal plus serious injuries tend to have a low ratio of fatal plus serious casualties to the total, The reason for this is the varying definition of a “serious” injury between police forces: if the boundary between slight and serious injury is at a relatively low level of severity. fatals/(fatals - serious) will be too low and (fatals - serious)/total will be too high. New by also gives an example from one particular English town where the numbers of fatal and serious casualties remained consistent over the period (195(L67) examined. but the number of slight casualties suddenly increased in I959 by 80% and continued to increase from the ne* level. Thus apparently some accidents that previously would have been classified as non-injury in that town are now regarded as slight injury. A similar instance is given by Satterthwaite [I9751 in which it was apparently the boundary between serious and slight injuries that moved. In one particular region of England between 1968 and 1969. there was a fall from 30 to 217~ in the proportion of injury accidents classified as serious in this region. whereas in the rest of Great Britain there w’as a much smaller chance in the oooosite direction. At the same time the total number of accidents in that region remained much the same, as did the proportion of fatalities. Finally. Lai [I9801 has used data in which casualties were classified according to both injury severity and the police force area where the accident occurred to establish that consistent differences exist between police forces in their placement of the boundary between “serious” and “slight” severities. We mav therefore conclude that some of the variability in T? and T, is attributable to rules of interpretation that are followed throughout a police force. The remainder is presumably attributable to the judgements of individual police officers. 1
1