Modeling of relative collision safety including driver characteristics

A.M VadebyAccident 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, γ

2).• 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.