Modeling of relative
collision safety including
driver characteristics
A.M Vadeby
Accident Analysis & Prevention 36
(2004) 909-917
Introduction
• The aim of this study was
mainly to develop a new
mathematical model for relative
collision safety between
different car makes and
methods to study the statistical
uncertainty for the estimates
obtained.
Physical background
• 兩車頭部相撞時，會受到change of
speed 和vehicle mass的影響。
• we used the law of
conservation of linear
momentum, which states that
linear momentum before crash
equals linear momentum after
crash.
where △vj is the change of speed of vehicle i in a crash and
mj is the mass.
Description of our
database
• A database from Statistics
Sweden (SCB) containing traffic
accidents reported to the police
during the years 1992–1993.
• From 706 different head-on
crashes in this database, we
have extracted information
about car model, car mass,
injury, age and sex of the driver.
• The injuries are classified from
0 to 3, where 0: the driver is
unhurt, 1: the driver is mildly
hurt, 2:the driver is seriously
hurt and 3: the driver is dead
within 30 days.
•這資料庫並沒有收集到1.是否使用安全
帶或安全氣囊；2. 是否使用手機等。
Model formulation
• We have data from n head-on
crashes, l = 1, 2, . . . , n. In every
crash there are two cars involved, r
= 1, 2 and we study m different car
makes, k = 1,. . . , m.
• θrl to be a nuisance parameter
substituting the lack of information
about the change of speed car
number r undergoes in crash number
l.
• Let α be our so called design
parameter. It is our parameter
of interest and in this study it is
connected to the different car
makes. Each car make
corresponds to one particular αk,
k = relative risk of car make k.
• Let t be a measure on how
much violence the driver is
exposed to in the crash, i.e. t is
a function of car mass, change
of speed and car make: t = t(m,
△ v, α). The risk of being injured
increases when t increases.
• Define Pj(t) = pr (the driver is in
injury class j when exposed to
violence t), j = 0, 1, 2, 3.
• P 0(t) should be close to one when t
is small and then decrease to zero
when t increases. On the other hand
P 3(t) should be close to zero when t
is small and then increase to one
when t increases.
• Further we consider that when a person ends
up in, for example, injury class 2, he has in
some sense passed through the states 0 and 1.
• Thus we use the birth process as a model tool
and replace the time parameter with the
violence t.
• Formally we model this as a pure birth-process
{X(t), t > 0}, starting in state 0, with time
replaced by the violence t and the states
corresponding to the injury classes. When we
introduce a birth process we have to introduce
further nuisance parameters, namely the birth
rates: λ0, λ1 and λ2.
• So, according to Kolmogorov’s forward equations
for a pure birth process with X(0) = 0,given in, for
example, Ross (1993, p. 271) states that:
and krl is the car make of car number r in crash number l,
r = 1, 2, and l = 1, 2, . . . , n.
The effect of driver
characteristics
• Evans (1991) has created a so called
“double pair comparison method”
with which he has been able to
determine the differences in age and
sex in accidents of the same type.
• 所以在模式中將加入age and sex的影響。
•圖2和圖3分別為資料庫中的整理
• In Evans (1991), relations of fatality
risk relative to 20-year old males for
both male and female are derived.
• Since death risk may behave
differently from injury risk, we
introduce a new parameter γ
such that γ = 0 gives no effect
of driver characteristics, γ = 1
gives the effect proportional to
Evans’ death risks and γ > 1
gives stronger effect.
• This means that we instead of t =
• trl in Eq. (2) have t = trls where
Let us call the model with the effect of driver
characteristicsincluded in the risks for Model 1
(i.e. in Model 1,γ1 = γ2 = 0). In Model 2, we let
γ1 = γ2 = γ and Model3 is the most flexible
model with γ1 and γ2
Estimation method
• The parameters we are most
interested in are α =(α1, . . ., αm)
and γ = (γ1, γ2).
• The parameters will be
estimated by the maximum
likelihood method.
• Using (2) and (3) the likelihood for this crash
becomes
The likelihood for the entire problem contains 2n factor
and the log-likelihood is a sum:
• Before proceeding with an uncertainty
analysis we compare Models 1, 2 and 3 by
a likelihood ratio test.
• This is confirmed when we compare Model
3 with Model 2. With the likelihood ratio
test described above we have: W =
2(−922.5 − (−935.1)) = 25.2, ma = 1 and
critical value 6.64 which also is significant
at the 1% level.
• Consequently, we choose to go further
with Model 3.
Estimation results
• Our main purpose with this study is still to
estimate the relative risks of the different
car makes, namely α = (α1, . . ., αm) and of
course to study the influences of age and
sex through the parameter γ = (γ1, γ2).
• The estimated relative risks in Models 1
and 3, the number of crashes each car
make has been involved in and average
driver age and sex for each car mmodel are
stated in Table 2.
Uncertainty analysis
• One important issue when dealing
with statistical inference is the
uncertainty of the estimated
parameter values.
• In order to come to a conclusion of
the uncertainty of the estimates we
perform a bootstrap analysis.
Bootstrap analysis
•建立statistical model時，我們常常將可用
的資料分為training data and testing
data，先以訓練組的資料來建立模型，然後
再用測試組的資料來評估所建立模型的準確度。
•在同一組資料庫裡，隨機抽取一定的比例（通
常是可重複抽樣）當成訓練組，剩下的當作測
試組，如此反覆多次，以建立較為可靠的模型。
• We made nb = 500 bootstrap
simulations to study the
uncertainties of the estimates
of γ and α.
• The bootstrap results
concerning γ1 and γ2 are stated
in Table 3.
• Since they give very similar
results, we choose to show only
the percentile intervals for
Models 1 and 3, respectively.
The results can be studied in
Table 6. The average driver age
and sex are already given in
Table 2 and account for some of
the differences.
Summary
• We have found a useful model and have
shown that it is possible to consider the
driver population by including parameters
related to the driver’s age and sex into our
relative-risk-model.
• The model improvements of Model 3 are
clearly significant according to a likelihood
ratio test. But if we compare the relative
risks between the two models we can see
that there are no major differences.