R.
J.
V AUG H AN
B.Sc., M.Eng.Sc., M. E., F. S.S., Australia n Road Research Board Research Fellow, Universi ty of Adelaide
G. G 0 U L B ERG H
A.R.M .T.C. Cert.T. P.C., Research O fficer, Highways Department, South A ustralia
A TWO-STAGE M ODEL FOR TRAFF IC
A CCIDENT PREDICTION *
(Pa per No . 723)
The major street system of metropolitan Adelaide has been
divided into some 250 road sections for the purpose of accident
evaluation. The object of the study was to determine the worst
sections. For each section the million vehicle miles (m.v.m.) was
calculated and plotted against accident damage. A linear relationship was shown to exist between m.v.m. and accident damage;
from this a weight was assigned to each section to make a comparison between sections possible. Sections with major geometric
differences were not compared and the virtue of the system lies
in that it can detect minor differences in road geometries to which
the variation in accident damage is attributed.
INTRODUCTI ON
1.
To date little effective use has been made of the vast data on road accidents. The authors feel that the data can, and should be, used as a feedback
mechanism in traffic engineering, to reduce accidents. The method outlined in
this paper compares road sections (excluding intersections), to detect the minor
geometric differences between them, to which accidents (excluding accidents at
intersections) , may then be attributed. At present the only method to determine
the effectiveness of a traffic engineering measure is to conduct a before and after
study (Ref. 1).
2.
It is assumed that all road sections with major geometric differences can
be stratified (e.g. type 1 = hilly, type 2 = straight single carriageway, etc.) . In
thi s paper, which deals only with metropolitan roads, the road sections are more
or less homogeneous, so no stratification was made. After stratification, road
sections cannot be compared until allowance is made of their relative accident
exposure. As a measure of exposure the million vehicle miles (m.v.m.) travelled
on the road section was used in this study.
3.
For a given exposure, models for the expected distribution of fatal accidents, personal injury accidents, property damage accidents, total accidents and
dollar damage were tested.
4.
Tolerance limits can be constructed on any of these above factors, but here
they are tested on weighted combination described in the next section .
• ACKNOWL EDGEMENT - The authors would like to thank ti,e A .R.R.B. fo r their fin ancial assistance,
the Comm issio ner fo r H ighways for his enco uragement, and Mr. W. N. Vena bles of the Stat istics Depa rtment, Adelaide University for invaluable statistical advice.
Volume 5, Part 3, 1970
365
VAUGHAN, GOULBERGH -
MODEL FOR TRAFFIC ACCIDENT PREDICTION
DESCRIPTION OF DATA
5.
The data used are those from a report by Goulbergh (Ref. 2) Road sections (excluding intersections) are listed with their corresponding m.v.m., number
of fatal accidents, number of personal injury accidents, number of property damage
accidents, and the total dollar damage of property.
6.
Graphs of m.v.m. against number of personal injury accidents, and m.v.m.
against dollar damage are given in Fig. 1 and 2. Graphs similar to Fig. 1 and 2
were made of the means and variances (for sections with similar m.v.m.) against
m.v.m.
7.
It was fo und that the mean number of personal injury accidents increased
linearly with m.v.m. and the variance also increased linearly, or possibly quadratically, with m.v.m. The means and variances of dollar damage increased linearly
with m.v.m.
8.
To make a comparison between type of accidents possible, Goulbergh calculated the weighted damage for each section by allocating $A5000 to each fatal
accident, $A1500 to each personal injury accident (which is consistent with
English estimates of £ 2970 and £ 770 (Ref. 3)) and adding the dollar damage.
USE OF TOLERANCE LIMITS
9.
Tolerance limits are used to classify road sections into good, bad, and
intermediate. The upper and lower tolerance limits determine the three regions
of Fig. 2 in which I, II, III represent the above classifications. To classify a road
section, on dollar damage alone, a point is plotted on the graph, with co-ordinates
30
'" . ........ . .. ..
.. _........_"-'-"
.....
. ., .
~
o
~~.:,. w
o
10
MILLION VEHICLE MI LES
Fig . 1 -
366
Each road section, Adelaide 1967, plotted with,
number of personal injury accidents and m.v.m.,
as co-ord inates
A.R .R.B.
PROCEEDINGS
VAUGHAN, GOUL BERGH -
MODEL FOR TRAFFIC ACCIDENT PREDICTION
4IJ
11
30
If)
a::
<
-J
-J
0
a
~
20
0
If)
a
z
<
If)
:::J
10
0
I
t-
10
20
MILLION VEHICLE MILES
30
Fig . 2 -
Ea ch road section, Adelaide 1967, plotted with, dollar damage and m.v. m., as co-o rdinates, together with the upper a nd lower 10 per cent tolera nce li mits
the dollar damage and the m.v.m. for that section. Then the region in which the
point lies determines its classification.
THE POISSON MODEL
10.
The first model investigated was the Poisson model of Norden et al (Ref.
4) used by Goulbergh in his report (Ref. 2). The above method compares each
road section with sections on the same road and assumes that the number of
accidents has a Poisson distribution . Approximate tolerance limits were given by
using the percentage points of the normal distribution (Ref. 4).
Am ± UeV Am
where m is the m.v.m. of the road section, A is the overall accident rate for the
road (total accidents/ total m.v.m.), and Ue is the lOOe per cent point of the normal distribution.
11.
Faults in this method are that tolerance limits are inaccurate at low m.v.m.
for here the normal approximation will not fit. Also the assumption of a Poisson
distribution is invalid for dollar damage as the mean was found to be twice the
variance. We tested the Poisson model to find it gave a very poor X2 value even
for the number of accidents.
12.
Goulbergh in using the Poisson model has obtained approximate tolerance
limits which are of the correct form but err slightly in that the percentage tolerance to which they correspond is incorrect.
Vo lu me 5, Pa rt 3, 197 0
367
VAUGHAN, GOULBERGH -
MODEL FOR TRAFFI C ACCIDENT PREDICTION
BASIC MODEL
13.
Let us consider a given road section which has exposure x m.v.m.
Let Y be a variate to describe the weighted damage,
F be a variate to describe the number of fatal accidents,
P be a variate to describe the number of personal injury accidents,
D be a variate to describe the total amount of property damage, and
C be a variate to describe the number of casualty accidents.
14.
Ideally Y is calculated by the formula
(1)
Y = 5000 . F + 1500 P + D
But since there were only 30 fatal accidents for the period, the fatal accidents were
combined with personal injury accidents and called casualty accidents. The adjusted form of (1) becomes
Y
=
5000 . Y • C
+
1500 P
+
(2)
D
where:
number of fatal accidents
number of casualty accidents
The fitted models for C, P, and D are described in the next section .
Y=
DISTRIBUTION OF PERSONAL INJURY ACCIDENTS
15.
The same basic model was used for P and C. It was assumed th at P, for
a given x, had a negative binomial distribution with density :
gp
( n)
=
PreP
=
n . x)
.
=
(h
+n-
n !(h - 1) !(l
1) !(f3x)n
+ f3x)h +n
(3)
11 = 0, 1,2, 3, .. .
with mean hf3x and variance hf3x(1 + f3x )
where h, f3 are parameters and x is the m.v.m. for the road section. This distribution has the required property of a mean which increases linearly with m.v.m .
and whose variance can increase linearly or qu adratically depending upon the
parameters h, f3.
16.
These parameters were estimated by the method of maximum likelihood
(see Appendix A) and their values were as follows .
2
h
X 17
f3
18.01
10.30
0.072963
Personal injury accidents
15.05
10.02
0.077697
Casualty accidents
46.56
5.870
All accidents
0.5089
TABLE I shows how, for personal injury accidents, the observed medians from
the records compare with the expected m~dians calculated by the model.
17.
The values for the X2 test of fit were very good for the number of personal
injury accidents, and for the number of casualty accidents, but for the total number of accidents a very poor value resulted. The most likely reason for these
values is that accidents with very light property damage are not reported , whilst
nearly all casualty accidents are reported.
368
A.R.R.B.
P ROCEED INGS
VAUGHAN, GOULBERGH -
MODEL FOR TRAFFIC ACCIDENT PREDICTION
TABLE I
CO MPAR ISO N OF OBSERVE D AN D EXPECTED M EDIANS
Do llar Damage
Personal Injury
m.v. m.
Obse rved
Expected
Observed
Expected
1.0
1.0
1.0
167 1
492
3.0
2.0
2.0
1980
2038
5.0
I
7.0
3.0
3.0
3287
3568
5 .0
5.0
4750
5097
10.0
6.0
7.0
5780
7389
14.0
7 .0
10.0
10500
10445
INTERPRETATION
18.
The negative binomial distribution is a member of the family of compound Poisson distributions. The theory of this family and some interpretation
of the properties as they apply to traffic accidents, is given by Feller (Ref. 5).
19.
To obtain om model we have assumed firstly that a particular person has
an accident only as a result of a chance mishap; however some people are more
accident-prone than others. For a person with accident proneness, A accidents per
m.v.m. will have a mean number of accident AX in X m.v.m., and the probability
of just n accidents is
Pr(n accidents I AX)
=
e- J'"(,h)"
n!
n
=
0,1,2, . . .
(4)
20.
To describe the population exposed to accidents, we give A a distribution,
and we have assumed A has a gamma distribution with density
e -;·/p;..n- l
A> O
,Bhr(li) ,
with mean h,B (h, ,B are parameters to be estimated from the data).
(5)
21.
From equations 4, 5, we obtain the negative binomial distribution with
density (3) , for all the accidents on the road section for the year.
DISTRIBUTI O N O F DAMAG E
22.
A model for the amount of damage was built as follows. It was assumed,
on the basis of highway department experience, that the amount of damage per
accident had a negati.ve exponential distribution. This observation led to the model:
D = Dl
+ D2 + D3 + D4 +
D5
+ .. . + D k
where the D b i = 1.2.3 ... , k are independent damage 'components' each with
negative exponential density e-d/ aj ex, where ex is an unknown parameter and k
has a Po isson distribution with density
Volume 5, P a rt 3, 197 0
369'
VAUGHAN, GOULBERGH -
MODEL FOR TRAFFIC ACCIDENT PREDICTION
e -eX(8xt
k = 0, 1, 2, 3, .. .
k!
where 8 is a second unknown parameter and x is the m.v.m . for the road section.
Then D will have density:
go(d)
= e- ef~(8x) +
8xe:
d/a
of{2'
8: d) ]
(6)
where t. is the dirac delta function, and
<Xl
OF1(a, z) =
zn
L --
n=O(a)nn!
is the of! hypergeometric function , closely related to a Bessel Function. For a
description of these functions see Sneddon (Ref. 6). A sketch of some uses of
the density (6) is given by Fisher and Cornish (Ref. 7).
23.
D then has mean ex8x and variance 2ex 2 8x, so that the mean and variance
increase linearly with m.v.m .
e,
24.
The parameters
ex were estimated by the method of maximum likelihood
(see Appendix B) and the following values obtained
e=
r:t.
The value of
X 2! 7
1·540
= 504
was 26.4, which indicated that the model just fitted.
25 .
An explanation of the mediocrity of this fit may be the fact that the total
number of accidents does not follow the negative binomial distribution. Another
explanation is the apparent overvaluation of light property damage accidents indicated by TABLE I.
26.
One can roughly interpret a damage component as being due to a single
a(;cident. Alternatively, since exact correspondence does not hold, we can think
of it as an accident quantum, with a variable number of quanta to an accident.
CONSTRUCTION OF TOLERAN CE LIMITS
27 .
The upper 100 per cent tolerance limit for the weighted damage, y, (where
y, is the point which is exceeded by E of the total probability) may be determined
by the formula:
YE = 5000 . }' . CE + 1500PE + de
where c,' P., d, are the respective upper 100E per cent tolerance limits of the
casualty, personal injury, and dollar damage distributions. They are given by the
following equations :
= 1- e
Gp(p,) = 1 - e
Go(d e) = 1 - e
(7)
G C(c e)
(8)
(9)
where G is the relevant cumulative distribution given by:
370
A.R.R .B.
PROCEEDINGS
VAUGHAN, GOULBERGH -
MODEL FOR TRAFFIC ACCIDENT PREDICTION
c
GC(C) =
L: gC(n)
n=O
=
L: gp(n)
n=O
(10)
p
Gp(p)
GD(d) =
28.
f
gD(t)dt
(11)
(12)
To solve (7) and (8) we cumulate the sum on the right of (10) and
(11) until l-E is attained.
29.
To solve (9) we use the expression for GD(d) given by Fisher and Cornish
(Ref. 7):
and Newton's method.
30.
TABLE II shows the values of y., c., P. and d. for each m.v.m . in the case
when E = 0 .1 and 0.9 . The d. curve obtained is that of Fig. 2 and each road section has been plotted to find approximately in which region it lies. Usually the
ovcrall tcst cmploying the y. curve would be applied.
Computer programmes have been developed to fit, test and plot these
31.
tolerance limits, for the successful models, and are available on request.
SUGGESTED ALLOCATION OF FUNDS
32.
After determining the bad road section how should funds be allocated to
each district and road type?
We would first form the road sections into groups according to their road
33.
type and district,
e.g. group 1 = district 1, road type 1
group 2 = district 1, road type 2
group 3 = district 2, road type 1
Now let
be the group number,
j be the road section number,
d ij be the vertical distance to the weighted damage upper tolerance limit
if the road is bad, if good set d ij to zero, and
(J' ij be the standard deviation determined by the type of road in group i
and the m.v.m. of section j.
Then we construct a measure for each group
c.I
=
I
d ..
---.!.1
j O'ij
Then the fraction of funds allocated to a group is given by
Volume 5, Part 3, 1970
371
V AUGH AN , GO ULBERGH -
MODE L FOR TRAFFIC ACCIDE NT PREDICTIO
TABLE II
TOLERANCE LIMITS
Upper 10% points
m .v.m.
Casualty
Person a l
Damage
372
Weighted
Damage
Injury
1.000
2 .000
2 .000
1950.245
5295.867
2 .000
3 .000
3 .000
3202.941
8221 .374
3.000
5 .000
4.000
4332 .851
11196.907
4 .000
6 .000
6 .000
5402.456
15439.322
5 .000
7 .000
7.000
6434.344
18144.021
6 .000
8 .000
8 .000
7439.765
20822.254
7.000
9 .000
9 .000
8425.289
23480.588
8.000
10.000
10.000
9395 .146
2 6 123.257
9.000
12.000
11 .000
10352 .253
28925 .986
10.000
13 .000
12.000
11298.721
31545.264
11 .000
14.000
13 .000
12236.133
34155.488
12.000
15.000
14.000
13165.717
36757.883
13 .000
16.000
16.000
14088.446
40853.423
14.000
17.000
17.000
i5005 .106
43442 .894
15.000
18 .000
18 .000
15916.343
46026 .943
16.000
19.000
19.000
16822.698
48606.108
17.000
21.000
20 .000
17724.626
51353.658
18.000
22.000
21.000
18622.515
53924.358
19.000
23.000
22 .000
19516.700
56491.355
20 .000
24 .000
23.000
20407.474
59054.939
21.000
25.000
24 .000
21295.089
61615.366
22 .000
26 .000
25 .000
22179.772
64172 .860
23.000
27 .000
26.000
23061.722
66727.620
24 .000
28 .000
27 .000
23941.115
69279.825
25 .000
30.000
28 .000
24818 .112
72002.443
26.000
31.000
30.000
25692.855
76049.998
27.000
32.000
31.000
26565.474
78595.428
28 .000
33 .000
32.000
27436.087
81138.852
29.000
34.000
33 .000
28304 .801
83680.377
30.000
35.000
34.000
29171.714
86220. 101
A.R.R.B. PROCEEDINGS
VAUGHAN, GOULBERGH -
MODEL FOR TRAFFIC ACCIDENT PREDICTION
TABLE II (continued)
lower 10% point s
m .v. m .
Cas ua lty
Personal
Damage
Injury
1.000
0 .000
0 .000
Weighted
Damage
0.000
0 .00
2 .000
0.000
0.000
0 .000
3 .000
0.000
0.000
574.386
574.386
4 .000
1.000
1.000
1031.712
2704 .523
5.000
1.000
1.000
1526.938
3199.7 49
0 .00
6 .000
2 .000
2.000
2048 .840
5394.462
7 .000
2.000
2 .000
2590.779
5936.401
8.000
2 .000
2.000
3148.477
6494 .099
9 .000
3 .000
3 .000
3718 .989
8737.422
10.000
3 .000
3 .000
4300.187
9318.620
11 .000
4 .000
4.0 00
48 90.473
11581.718
12.000
4.000
4 .000
5488. 613
12179.858
13.000
5 .000
5.000
6093 .628
14457.684
14.000
5 .000
5 .000
6704.728
15068.783
15.000
6.000
5 .000
7321.262
15858.129
16.000
6.000
6.000
7942 .689
17979.556
17.000
7.000
6.000
8568.552
18778.230
18.000
7 .000
7.000
9198 .461
20908 .138
19.000
8 .000
7 .000
9832.078
21714 .567
20 .000
8 .000
8 .000
10469.113
23851.602
21.000
9 .000
8.000
11109.310
24664. 6 10
22 .000
9.000
9.000
11752.444
26807.743
23 .000
9.000
9 .000
12398.314
27453 .613
24.000
10.000
10.000
13046.743
29774.853
25 .000
10.000
10.000
13697.571
30425.682
26.000
11 .000
11 .000
14350.655
32751.577
27.000
11 .000
11 .000
15005 .865
33406.786
28.000
12.000
11 .000
15663 .083
34236.815
29.000
12.000
12.000
16322.201
36395.934
30.000
13.000
12 .000
16983 .122
37229.666
Volume 5, Part 3, 1970
373
VAUGHAN , GOULBERGH -
MODEL FOR TRAFFIC ACCIDENT PREDICTION
CONCLUSION
34.
The methods contained in this paper will enable a valid comparison of
road sections over a wider range of exposure than was possible with the Poisson
model ; particularly with respect to rural roads and intersection accidents to which
the authors will next turn attention.
35.
In comparing the two methods, it was found that they did not differ greatly
indicating the robustness of use of the Poisson and normal distributions.
The fact that the variance of the number of accidents increases almost
36.
linearly with m.v.m . indicates that the increase is due mostly to cross factors (e.g.
people crossing the road, side distractions, etc.) and not between vehicles themselves. This confirms a study by Raff (Ref. 8), in which he could find no relation
between accidents and traffic volume on freeways, yet there was one on other
roads.
The method of this paper has shown that the original models for the ex37.
planation of general accident behaviour of Greenwood and Yule (Ref. 9) is valid
for road traffic accidents. The interpretation of these models is that for a given
person, accidents will occur to him in a chance (Poisson) manner. For a given
exposure, as each individual has a different accident proneness, the number of
accidents occurring to a group of people will form a negative binomial distribution. Therefore each road section will have a different distribution of accidents
according to its exposure and the type of drivers using the road section.
APPENDIX A
MAXIMUM LIKELIHOOD (M.l.) ESTIMATION OF hAND
f3
38.
Let (nb Xi) be the number of accidents and the m.v.m. for a road section i.
From (3) the likelihood function is
h
L(h, f3:n j , x) =
(h + n - l)!p Q"'
j = l
(h-l)!n j !
fI
1
p.= - -
where
I
1
+
f3Xj
Then the log likelihood is
InL = c +
m
L {Iogrch + n j) -logrch) + h .10gP j + njlog QJ
i= 1
and
374
A.R .R.B.
PROCEEDINGS
VAUGHAN, GOULBERGH -
MODEL FOR TRAFFIC ACCIDENT PREDICTION
Also
E
- a21nL) -
.( afJ2
m hp2 x2
'\'
II_t
- 1=
.L...1 -' Q i 2
E(- a2!n~) =
ah
-2
=
m
11
f: PjX j = t12 = t21
afJoh
- a21nL
-
j=1
1
" ,- I
I I --- =
j=1 k=1 (n + k)2
t22
We define the information matrix
T = (til' t12)
t 21, t22
and the score vector
(:~)
s =
and the initial estimate vector
where {30 and ho are initial estimates for {3 and h, in our least squares estimates.
Then an iterative procedure,
A
A
CX 1
=
CXo
+
T-1s
(13)
is used to obtain a better approximation ~l to the m.l. estimates of {3, h .
APPENDIX B
MAXIMUM LIKELIHOOD ESTIMATION OF a AND
0
39.
Let (Yb Xj) be the dollar damage and the m .v.m. for a road section i
(i = 1, ... , n). From (6) and for convenience letting (3 = l / a, the likelihood
function becomes
L(O, fJ: Xj , Y;) = " fJex je- OX,- PY 'oF 1(2, (Jex jy;)
IT
i :::; 1
The log likelihood is
InL = n log fJ
+ n log e + I"
j
[log Xj
+ log of 1(2, fJexjy;)]
- ne x
-
nfJy
= 1
Noting that
Volume 5, Part 3, 1970
375
VAUGHAN , GOULBERGH -
MODEL FOR TR AFFIC ACCIDE T PREDICTION
we have
n
olnL
I
op
ceri XiYi - Yi
+ I jP)
= Sl
i= l
where
and
where
or
- i =
ap
l j I-'R - 2r. j PR - er·2 x.y.
or.
---1 =
oe
1j e - 2r.1 j e - I>Rr 1.- x 1.y.1
I
I
I
I
?
40.
Forming the information matrix T, the score and initial estimate vectors
in Appendix A, we may use the same procedure as equation (13) to obtain
m.l. estimates of () and f3 = 1/ 0:.
a3
REFERENCES
l. R esearch on road traffic, H.M.S. O. (London 1965).
2. Goulbergh, G ., Analysis of accident rates for metropolitan roads (for 1967) ,
S.A. Highw. Dept (Jan. 1969).
3. R esearch on road safety, H .M .S.O. (London 1963).
4. Norden, M., Orlansky, J . and Jacobs, H ., Application of statistical qualitycontrol techniques to analysis of highway accident data, H .R .B. Bull. 117
(1955).
5. Feller, W. , An introduction to probability theory and its applications, 1, 2nd
ed. (1957) .
6. Sneddon, I. N., Special junctions of mathematical physics and chemistry, 2nd
ed. , Oliver and Boyd (1961).
7. Fisher, R. A. and Cornish, E . A. , The percentile points of distributions having known cumulants, Techometrics, 2: 2, 209 (May 1960) .
376
A.R.R.B .
PROCEE DINGS
VAUGHAN , GOULBERGH -
MODEL FOR TRAFFIC ACCIDENT PREDICTION
DISCUSSIONS
8. Raff, M. S., Interstate highway-accident study, H.R.B. Bull. 74, 18 (1954).
9. Greenwood, M. and Yule, G. U. , An enquiry into the nature of frequency
distributions with reference to the occurrence of accidents, J. Roy. Stat. Soc.,
83,255 (1920).
DISCUSSIONS
D.
J.
B U C K LE Y,
Ph.D., M.Eng .Sc., B.E., School of Traffic Engineerin g, University of New So uth Wales.
41.
It is not immediately clear whether para. 33 refers to personal injury
accidents or dollar-damage, but this discussion applies in both cases.
42.
Am I correct in assuming that para. 33 defines a criterion which allocates
funds so that (roughly speaking and subject to some mathematical reservations)
roads having a high dollar-damage per million vehicle-miles will be improved
first? This might be called a democratic criterion because it tends to produce an
approximate equality of hazard to the individual driver. Opposed to this is the
idea of using a minimization of dollar-damage (or accidents) criterion. This involves
no mathematics and is equivalent to improving roads having large ordinates on
Fig. 2 . Instead of reducing the variability of hazard it seeks to spend public money
so as to reduce dollar-damage (or accidents).
43.
I am sure that the authors could provide a better non-mathematical description of their criterion. Perhaps they might care to do so.
L.
A.
FOLDVARY ,
B.E., B.S c. (Econ.), Ph .D., F.S.S .( Hun gary), Rese arch Officer, A.R.R.B.
44.
The model for traffic accident prediction prese nted by the authors will be
a useful addition to the repertoire of methods at hand , and the authors are commended for their contribution.
45.
A little note about a misquotation of Raff in para. 36, i.e. " . .. he could
find no relation between accidents and traffic volume on freeways ... " What Raff
stated in his quoted paper is the opposite: that there is a direct relationship between
the number of accidents and traffic volume on freeways , making accident rates per
m.v.m. uniform throughout the volume range.
D.
C.
AND REA SSE N D ,
B.Sc. , M.Tech ., M.H.F.S., Chief Engin eer, Traffic Commission of Victori a .
46.
Have the authors considered investigating the use of traffic density
(vehs/ ml) against the accidents per mile on the road section, i.e. using vehs/ ml
instead of veh. miles? Work done by D. J. Delaney* concluded in one section
that "the occurrence of accidents on a length of street, in a given period might be
expressed as a function of the average density of vehicles on that street" and I
wondered if the Adelaide data might confirm this?
" Del aney, D . J., A pilot stlldy 0/ M elbo llrne's traffic problem , A.R.R.B . Proc. I : 1,388 (1962).
Volume 5, Part 3, 1970
377
VA UGHAN, GOULBERGH -
MODEL FOR TRAFFIC ACCID ENT PREDICTION
A UTHOR'S CLOSURE
47 .
I was also pleased to se that the authors have mad a positive point of
omitting the accidents at intersections along the road sections they studied; whereas
a lot of other 'practitioners' seem to still, incorrectly I feel, cheerfully ignore this
aspect. Would the authors clarify which types of accidents remained in their
road sections and whether any of these could be attributed to the nearby intersections? That is, has any arbitrary cut off point been used or has some other
criterion been used to determine 'intersection accidents '?
AU TH O RS'
To
J.
CLOS URE
F. M. B R Y ANT
(See Introductory Remarks )
48 .
The models of Ashton referred to in Mr. Bryant's comments describe the
idiosyncrasies of individual drivers and are only of interest in interpretation of
the model. The extension of th e model presented to compare road sections over
areas where the weather pattern is non-homogeneous would indeed be important.
However, the authors would try to use weather as a measure of exposure together
with m.v.m. as used in this paper.
49 .
Mr. Goulbergh's paper (Ref. 2) is not difficult to obtain and has been
distributed to the A.R.R.B ., State road authorities and libraries. Moreover, all the
data has been plotted in Fig. 1 and 2 of our paper to avoid inclusion of very lengthy
tables.
50.
Mr. Bryant highlights the main point of the paper, namely, to determine
those sections of the road system which require a detailed examination by a
traffic engineer, who is informed automatically if the change is significant.
To
D.
J.
B U C K LEY
51.
The authors propose that funds should be spent where they will be the
most effective. The further the dollar damage ordinate is above the expected
ordinate the more likely the section will respond to treatment. Funds could be
allocated in proportion to this difference, if the expected variability about the
expected number of accidents was constant. However, the variability increases
with m.v.m . and to account for it we divide difference by the standard deviation
and allocate funds in proportion.
To
L.
A.
F0 L D V A RY
52.
The authors thank Dr. Foldvary for his comments. The authors have
erred in para. 36 of pre-print, last sentence, 'accidents' should be 'accident rate'.
To
D.
C.
AND REA SSE N D
53 .
Our paper would confirm that the accidents on a street would be proportion al to the vehs/ mile on that street.
54.
All accidents within 30 ft of an intersection were considered to be intersection accidents.
378
A.R.R .B.
PROCEEDINGS