navinroadsafetymodelsome1996.md 32 KB
category: literaturenote citekey: navinroadsafetymodelsome1996 title: "Road Safety Model: Some Fundamental Ideas" authors: "Navin, Francis P. D.; Bergan, Arthur; Zhang, Guanyu" year: 1996 date: 1996-00-00 1996 doi: 10.1177/0361198196156300107 publication: "Transportation Research Record: Journal of the Transportation Research Board" url: "https://doi.org/10.1177/0361198196156300107" zotero_key: VTREN8HH zotero_storage: 74R7V2SM collections: transpordi_planeerimine folder: 001_artiklid/Road Safety Indicators firstAuthor: "Navin, Francis P. D."
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Road Safety Model: Some Fundamental Ideas
FRANCIS P. D. NAVIN, ARTHUR BERGAN, AND GUANYU ZHANG
A fundamental relationship has been developed that explains road accident statistics in developed and developing countries. The model uses two variables, traffic hazard measured as deaths per vehicle and motorization measured as vehicles per person, to estimate personal hazard as deaths per person. Special cases of the model are those by Smeed, Trinca et al., and Koornstra. The model of fatalities has two extremes. Early motorization has high traffic hazard and personal safety is low and increasing. Full motorization is characterized by a moderate and falling traffic hazard and a low and decreasing personal safety. Between these extremes, there is a maximum number of fatalities per population. Models for personal injury and total road accidents in developed countries appear to follow a similar trend. Available world data fit the proposed relationships well. The models allow planners and engineers to estimate the future maximum road fatalities for developing countries. The model has been extended to incorporate an automobile ownership model that explains some of the growth in motorization. A traffic hazard model is also outlined, in part on the basis of the ideas developed by Koornstra. The extended models should allow a more detailed analysis of some of the social and engineering factors that contribute to road safety.
More than half a million road deaths occur annually in traffic crashes around the world. Stated in another way, approximately one person in 8000 in the world's population dies from injuries received in traffic crashes according to Trinca et al. (1). Traffic accidents constitute a major threat to public health and have been classified as a crisis by the Centers for Disease Control. Road accidents also have a high monetary loss. Since about 1970 developed countries have undergone a downward trend in traffic fatalities, while developing countries are facing significant increases.
The general downward trend of road fatalities is shown in Figure 1. Navin et al. (2) used only the fatality-vehicle plan of this figure to illustrate the downward tendency for Canada, the United Kingdom, and the United States. Using data from the beginning of this century, they observed at first a rapid increase in the number of deaths. This peaked in the late 1960s or early 1970s, followed by a steady decline. Available data also support the finding of Navin et al. (2) that not only developed countries but also developing countries will go through the same phases.
What insight may be found in such common trends? The study of fatal road accidents must consider major factors related to road safety. Factors such as the economic situation (3) or legislative intervention (4) play large roles in road safety. Other authorities ascribe change solely to driver behavior. Evans (5) has given a hierarchical ordering of road safety factors, ranked as human infrastructure and engineering infrastructure. Evans concluded that the
social norm, an element of human infrastructure, dominates the road safety trend.
More recent findings by Koornstra (6) demonstrate that the development of traffic and traffic accidents can be described as a semiclosed system or as a self-organizing social-economic system. The Koornstra system is a gradual learning process of society. In that system road safety benefits by changing the related environment to govern individual experiences.
An empirical model developed by Navin et al. (2) explains longrun road accidents in developed and developing countries. This paper extends the original model by incorporating some of Koornstra's ideas to provide additional explanation of the interaction of the variables.
LINKING POPULATION, VEHICLES, AND FATALITIES
Converting the fatality data for Canada, the United States, and the United Kingdom to the rate values shows that there is a relationship among traffic hazard (T)—death/vehicle; motorization (M)—vehicle/person; and personal hazard (H)—death/person as shown in Figure 2. The notation and the idea of using these variables come from the work of Trinca et al. (1). The T-M plane shows that the fatality rate continues to fall regularly as motorization increases.
It is obvious that there is a common tendency of national traffic accidents and that the fatality model has two extremes (2). The first extreme occurs during early motorization, which has a high traffic hazard and low and increasing personal safety. The second extreme is at full motorization, which is characterized by a moderate and falling traffic hazard and a low decreasing personal safety. Between these extremes, there is a maximum value of fatalities per capita.
The current decreasing trend in road fatalities in the developed countries has attracted the attention of many researchers. Researchers used several approaches to construct models from the available statistical data to explain the decreasing trends. Smeed (7) used regression analysis, and a later work by Minter (8) used learning theory or risk theory as used by Koornstra (6).
Learning theory in terms of road safety was first introduced by Minter (8), who stresses that a collective individual learning is involved in improving road safety. That model suggests that the major feature of the traffic safety system is that drivers learn from accumulated driving experience. Later Koornstra (6) rejected Minter's interpretation and developed a risk-adaptation model on the basis of a community learning process.
In 1994, Navin et al. (2) investigated a fundamental relationship between traffic hazard, motorization, and personal hazard and built a model to link them. The model fits a wide range of data well and allows planners and engineers to estimate the future maximum road fatalities for developed and developing countries.
F. P. D. Navin, Department of Civil Engineering, University of British Columbia, 2324 Main Mall, Vancouver, B.C., Canada V6T 1Z4. A. Bergan and G. Zhang, Department of Civil Engineering, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 0W0.
FIGURE 1 General relationship between fatalities and vehicles over time.
In this paper, Navin's model has been extended to incorporate a traffic hazard model based in part on the ideas developed by Koornstra (6). The extended model should allow a more detailed analysis of some of the social and engineering factors that contribute to road safety.
The proposed revisions to Navin's model are simple empirical ways of summarizing relationships among the elements inside the traffic safety system. Plausible arguments also might be made to use registered vehicles as a measure of exposure. In the absence of accurate data on vehicle mileage over the long term, especially in developing countries where the national reporting system is not adequate, the number of vehicles is a more reliable statistic.
Because Navin's model is a generalization that allows the derivation of previous models, a review of the previous models is made first to help understand this new model.
PREVIOUS ROAD SAFETY MODELS
Smeed's Law
An early, well-known model was developed by Smeed in 1949 (7). It was found that traffic hazard was inversely related to motorization. Smeed used motorization as a measure of exposure to the risk of road accident and successfully fitted data to the law that bears his name. Smeed's law is
$$d = 0.0003(vp^2)^{1/3} \tag{1}$$
$$\frac{d}{v} = a \left(\frac{v}{p}\right)^b \tag{2}$$
where
d 5 number of deaths,
v 5 number of vehicles,
p 5 population, and
a, b 5 constants.
Many practical models have been established on the basis of Smeed's work. The models provide reasonable methods for predicting fatalities.
FIGURE 2 Fundamental relationship among deaths, vehicles, and population over time.
Smeed's law has critics, such as Andreassen (9) and Broughton (10). Andreassen pointed out that Smeed analyzed the data for 20 countries for only 1 year. The original work did not have any timeseries data. Andreassen and Broughton showed that each country has a unique set of constants a and b in Smeed's Law. Broughton went so far as to state, "This paper (his) has demonstrated unequivocally that the relationship detected by Smeed among data from 1938 has no general validity. . . . They are simply empirical relations that may indicate the existence of more fundamental relationships" (10). This paper explores some of those more fundamental relationships.
Adams (11) was more supportive of Smeed's Law and illustrated the robustness of the law by applying it to a number of countries. Adams considered the model as a national learning curve and was critical of any one group taking credit for the improvements in road safety. Adams cited economic studies by Cooper (12) that explained some of the variance in the main models. Adams also used U.S. data after 1966, the year the National Traffic and Motor Vehicle Safety Act and the Highway Safety Act were signed, to show that the same curve fit both data sets.
Fieldwick (13) developed an interesting application of Smeed's formula that included speed limits. The equations are
$$D = 0.00097215 (P^{1.02192}Ur^{1.786134}Ru^{1.064236})$$
$$C = 0.0358442 (V^{1.074341} Ur^{1.591740} Ru^{1.394894})$$
where
D 5 number of fatalities in 1984,
C 5 number of casualties in 1984,
P 5 population in 1984 in millions,
V 5 vehicles in 1984 in millions,
Ur 5 urban speed limit in kilometers per hour, and
Ru 5 rural speed limit in kilometers per hour.
Minter (8) showed that Smeed's Law is a consequence of Wright's learning curve model. Wright's learning curve model is a double logarithmic that measures performance and is given as
$$t_n = t_1 n^{-b}$$
where
tn 5 time for *n*th item to be made on *n*th repetition,
t1 5 time for first repetition,
n 5 cumulative number of repetitions, a measure of experience, and
b 5 measure of rate of reduction of time per occasion, about 0.4 for United Kingdom.
Minter showed that Smeed's formula is a double-logarithmic model that is analogous to Wright's learning curve model. Minter argues that Smeed's measure of performance is deaths per vehicle and the measure of accumulated experience is vehicles per unit of population. This last measure is close to a measure of accumulated experience, because the number of vehicles grows annually and adds to the travel experience. Also vehicles per head indicates a country's accumulated experience. Minter suggests that because of the underlying learning process road safety will improve in spite of other interventions.
Trinca et al. Model (1988)
Trinca et al. (1) showed that there is a relationship among T, M, and P, and this relationship can reflect a country's motorizing stage. The relationship is
$$P = T \times M \tag{3}$$
where
P 5 personal safety measured as number of deaths per 100,000 people,
T 5 traffic safety measured as number of deaths per 10,000 motor vehicles, and
M 5 motorization measured as number of registered motor vehicles per 1,000 people.
The model is descriptive and graphically showed a relationship among the three variables using data from 33 developed and developing countries. The single data point for each country used information between 1980 and 1986. In general the model showed that at low levels of traffic safety, that is, few deaths per vehicle,
motorization was high and personal safety, or deaths per person, was also high. This situation is representative of developed countries. Developing countries have a high value of traffic safety (deaths/vehicle), low motorization, and low personal safety (deaths/person).
They looked at the time trends from 1970 to 1985. This particular graph provided the inspiration and key ideas used to develop this paper. The time trend shows that countries at the start of motorization will have low values of motorization and personal safety and a high value of traffic safety. The country then, as motorization increases, moves toward a peak value of personal safety, that is, a maximum rate of deaths per population. As motorization increases beyond this maximum value personal safety and traffic safety values decrease.
The research of this paper took this as a starting point and developed mathematical relationships based on time-series data for Canada, the United States, and the United Kingdom.
Towill Learning Curves
This is an exponential learning curve model that assumes that the number of vehicles in a country grows annually and each year adds to the driving experience (8). The performance function is
$$y_x = y_s[1 - \exp(1 - x/t)] + c$$ (4)
where
yx 5 rate of production (measure of performance) at time x from start,
ys , c 5 constants, and
t 5 time constant of exponential curve.
The performance measure is the casualty rate defined as casualties per unit distance traveled. Minter (8) showed that U.K. road death data starting from 1945 could be fitted successfully to the Towill model. The model has a ceiling in the value of yx as x tends to infinity. The value of x that measures experience is the cumulative distance driven. The equation developed by Minter was
$$y_x = a(1 - e^{-x/c}) + c$$
$y_x = 2.27 + 119.832(1 - e^{-x/14011.5})$ (5)
where yx is millions of kilometers per death, and x is cumulative distance traveled by year, in kilometers.
The estimate of x used an exponential model to estimate first the vehicle population by year and then a linear model to estimate the average annual distance traveled per vehicle. Good results were obtained with an R2 value of 0.75. It was also found that for the United Kingdom an upper limit to yx is about 156 million km per death and a motor vehicle saturation of about 17.5 million automobiles.
Minter showed that although the models to forecast road casualties are robust there are other variables that overlay the exponential learning curve models. He suggests that these controls arise in social and general state of mind of the road user, and he further states, "The road safety industry would say that the improvement in casualty rate is due only to its efforts over the years. This study suggests otherwise—that things will improve anyway, and left to themselves they might even improve better than with legislative intervention" (8).
This work will show that thoughtful intervention into the technology of driving produces beneficial results above those that could be obtained by simple learning curves.
Koornstra's Function (1992)
Koornstra (6) showed that the long-term trends in mobility and safety can be related by an evolutionary learning model of growth and risk adaptation.
Growth of mobility can be described by sigmoid curves with a saturation level (taken from biology or econometrics) and risk in road traffic as the fatality rate which follows a decreasing adaptation curve (taken from mathematical learning theory). A single peaked curve for the road fatalities per years, is a necessary result from saturating growth and steadily decreasing risk adaptation. Such growth and risk adaptation curves are related.
Koornstra used a simple model of the following form:
$$F = \frac{F}{D} \times D \tag{6}$$
where F is fatalities per year, and D is vehicle-kilometers per year. The mathematical function is
$$F{t} = zV{t}^{x}V{t-k}^{w} \left[ \left( \frac{V{\text{max}}}{V_{t-k}} \right)^{c} - 1 \right]^{y}$$ (7)
where
Ft 5 number of fatalities in year t,
Vt 5 vehicle-kilometers in year t,
*V*max 5 saturation level of vehicle-kilometers,
k 5 time lag in years, and
w, x, y, z, and c 5 constants.
He showed that this followed Hauer's (14) definition of safety:
safety 5 (risk)(exposure)
His model is broader in concept than those proposed by either Broughton (10) or Minter (8). Minter concentrated on collective individual learning; Koornstra uses a gradual learning process of society that enhances safety through changes in the road, vehicle, operating rules, and individual behavior. He stresses the concept of learning as a function of time, which is better described as adaptation. He also points out that there is a time lag in realizing safety benefits from any improvement.
He went on to successfully apply the model to selected developed countries for both fatalities and casualties and outlined a combined open- and closed-loop system to explain road safety.
Navin Model (1995)
The fundamental model of road safety developed by Navin et al. (2) is shown in Figure 3. It is easily seen that Smeed's Law is on the T-M plane, but the hyperbola function has been changed to an exponential function of the form
$$T = T_f e^{\frac{-M}{M_o}} \tag{8}$$
where
Mo 5 value of motorization at maximum personal hazard, *P*max, Tf 5 point where exponential curve pierces T-axis,
FIGURE 3 Four-dimensional view of fundamental relationship over time.
T 5 traffic hazard, deaths per vehicle, and M 5 motorization, vehicles per population.
In applying Equation 3, the personal hazard is derived as
$$P = MT_f e^{\frac{-M}{M_o}} \tag{9}$$
Note that this model defines hazard rather than safety. The coordinates of maximum personal hazard are given by
$$P_{\text{max}} = M_o \frac{T_f}{e} \tag{10}$$
$$T_o = \frac{T_f}{e} \tag{11}$$
Equation 8 is analogous to the linear-operational model by Broughton (10). Both are shown to correspond with Towill's learning curve in an inverse way and categorized as two special cases. The model is also found in traffic flow theory as proposed by Underwood (15).
Navin's model can be built up from basic time-series data as shown in Figure 4. The time-series data of fatalities and vehicles (or any other measure of exposure) as given in Figure 4a were used by Koornstra (3) as the basis of his model. Smeed (7) used a single year time slice from the F-V and V-P planes of Figures 4 a and b to obtain the model shown in Figure 4c. Adams (11) and others have shown that Smeed's formulation may also be applied over time as implied in the diagrams. Navin et al. (2) extended the two-dimensional Smeed model to three dimensions.
The basic element of the theory is found, as pointed out by Koornstra, by study of fatalities over time. Fatalities may be considered failures in the system. The system, in this case the technicalsocial road system, is made up of the following components: road, vehicle, driver, operating rules, and devices. The engineering failures of the road vehicle and operating devices may be studied by traditional reliability analysis (16). The driver may also be studied with
FIGURE 4 Relationship among road fatality models.
reliability analysis or with some learning curve. Because engineers study failures to learn how to avoid them, the reliability analysis also needs a learning component to reflect the new devices introduced into an existing system. Ideally one would have a set of evolving equations to learn from past failures to forecast future successes. The human-social elements of the system tend to frustrate such an ideal situation, as suggested by Wilde (17).
The key to the successful implementation of any of the models to forecast future accident rates is understanding how accidents occurred. It is known that the causes of fatal accidents are 62 percent driver, 5 percent driver-vehicle, and 1 percent driver-road-vehicle (18). This implies that the traffic hazard function is
T 5 f (driver, road, vehicle, rules, society)
The remainder of this paper deals with this equation and how it may be applied.
IMPLICATION AND DEVELOPMENT OF NAVIN ET AL. MODEL
The first approach is to review the results of data in the postwar period of Canada, the United Kingdom, and the United States on T-M planes. Attempts were made to fit data to the models starting in 1900. The disruptions caused by the Depression and World War II introduced difficulties that did nothing for the analysis, so only data from after 1949 were considered. These data cover half a century that had an enormous range of changes. Tables 1 and 2 show that Navin's model as given in Equation 8 is more suitable for the data because it has much higher R2 values and fewer standard errors (see Figure 5).
Navin et al. (2) indicate that the proposed model can have as the dependent variable the stage of development, or time, or both. This is because the curve is to a large degree linked to the timedependent forces and economic efficiency that somehow make the
TABLE 1 Regression Result for Smeed's Law
resulting accident rate change. Partykas (3) and Cooper (12) have written on the economic aspects of accident rate change.
What roles do those factors play in reducing the traffic hazard? Are there any other factors? Minter (8) and Koornstra (6) argue that the reduction of fatality rate is the consequence of a community learning process reflected by improved driver attitude to traffic safety. In other words, the key factor is the human learning process in terms of time and other factors that have their own proportional contribution to traffic safety. They provide no guidance for how the other factors may be introduced.
Second, some fluctuation along the curve is found to correspond to economic activity, such as the bump on the curve during the early 1970s and the oil crisis. This finding implies that the curve can be adapted by introducing economic factors as shown by Partykas (3) and Cooper (12).
The third finding is that erratic deviations along the curve are associated with legislative intervention. This intervention is defined as teaching elements in the learning process. Evans (5) considers that such a teaching process is achieved by providing drivers with more frequent feedback to unsafe driving actions.
On the basis of those assumptions, a function of the time-based interventions is imposed on Navin's model in an exponential declining form. The traffic hazard becomes
$$T = e^{a+bx+cy+dz} (12)$$
where
x 5 year 2 1949,
y 5 economic proxy in terms of personal disposable income,
z 5 safety improvements introduced by various legislative interventions, and
a, b, c, d 5 constants.
Using the Canadian data set from 1949 to 1991 gives the following relationship:
$$T = e^{-6.97 - 0.02x + 0.01y + (-0.24z_1 - 0.45z_2 - 0.0004z_3 - 0.08z_4)}$$ (13)
where
x 5 year 2 1949,
y 5 personal disposable income (year 1949 5 1),
- z1 5 55 mph dummy variable (1949 to 1973 5 0, 1974 to 1991 5 1),
- z2 5 seat-belt percentage at each year,
- z3 5 per capita consumption of alcohol at each year, and
- z4 5 dummy variable for vehicles not equipped with regulated safety equipment (1949 to 1965 5 1, otherwise 0).
The result is plotted in Figure 6 from the Trinca et al. (1) model. The result is found to be satisfactory with an R2 of 0.985 for the regression function and is shown in Figure 7. Similar curves can be found for the United Kingdom and the United States. A general fatality trend on a time series is developed in Figure 8.
The characteristic feature of this model is to evaluate the efficiency of the three main factors related to reduction of fatalities. Figure 8 demonstrates a relationship among the learning process, economic situation, and legislative intervention. It indicates the following:
- 1. Improvement of fatality rate is the outcome of the interaction of these three factors;
- 2. Learning process dominates the broad trend, which is a curve starting from the original surge line and gradually settling down to the social minimum acceptable level when time tends to infinity;
- 3. Coefficient of economic activity is significant and positively related to the fatality rate; and
- 4. Introduction of significant legislative intervention is associated with substantial reduction of fatalities.
These findings appear to violate some previous investigations about the relative weights of the three factors. Among previous authors, one focused only on the learning process (8) and another on factors of economic activity or legislative intervention (4). However, one key point is missed by these researchers; that is, road transport is performed in a complicated and dynamic way where people, road, environment, and legislative intervention interact. This model considers such interaction.
DISCUSSION OF EXTENDED MODEL
Similarities can be found with the Navin et al. extended model and Leob (4) and Garbacz (19) cross-sectional models. Garbacz analyzes
TABLE 2 Regression Result for Navin's Model
FIGURE 5 Fitting Navin's model to data (*2*).
FIGURE 6 Comparison of various fatality models with Canadian data.
FIGURE 7 Fatalities over time forecast by various models.
FIGURE 8 General relationship among fatalities and vehicles over time and various forecasting methods.
the impact of various safety inspections on the reduction of fatalities by using time-series data. Garbacz (19) comments on the coefficient of economic activity using a time-series model.
It is expected that as income rises the value of time rises, and drivers may increase speed in order to cut the cost of travel. Higher speed leads to a higher risk of a traffic accident. Greater speed may lead to more deaths, but this is offset as the value of life rises with rising income. It is expected that drivers with higher income exercise more caution. The predicted sign on the coefficient can therefore only be decided empirically by the net effect of off-setting factors.
However, from analysis of the United States, Japan, Taiwan, and Canada (results of the Navin et al. extended model), the coefficients associated with economic activity tend to be generally positive.
The negative coefficients associated with legislative intervention indicate that intervention makes drivers aware of traffic safety. A great deal of evidence is obtained from the large drop after 1970 in developed counties since the enormous legislative interventions of the late 1960s had taken place. Before this phase, the development of mechanical and electrical engineering on the vehicle fleet and civil engineering in the form of roadway construction contributed to a large degree to accident reduction.
This highlights the point of interaction of three main factors only; details on other factors such as roadway, vehicle, medical care, and population change have been ignored and lumped in the equations constant. These factors might affect the model coefficients but not the broad downward trend.
As increasing accidents and traffic hazards in the world are more closely monitored, the implication of the Navin et al. extended model is available to developing countries; that is, greater care should be placed on road safety as economies undergo extreme change. Effective legislative intervention should be implemented to shape the public attitude to road safety.
CONCLUSION
Smeed's law (7), developed in 1949, has been successfully applied to multicountry and country time-series data. The formulation is not without its critics and defenders. Minter (8) has shown that it is a
variation of Wright's learning curve. Koornstra (6) took a slightly different approach to forecast the change of total road deaths or safety over time as a function of risk and exposure. Risk is defined as deaths per vehicle kilometers per year and exposure is vehicle kilometers per year.
Navin et al. (2) extended Smeed's two-dimensional traffic hazard and motorization model to a third dimension, personal hazard. It is shown in this paper how the models are conceptually related and that Koornstra's model is in fact an underlying component in Smeed's model.
By using the start dates of various road programs and indicators of economic activity such as gross national product and alcohol consumption, a time-based model exists for traffic hazard for Canada between 1949 and 1991. Data before 1949 were not used because they made model correlation difficult and were not as well-behaved as after 1949.
The model presented in this paper permits replication of past trends and future forecasts of fatalities. The difficulties fundamental to many regression models are present in this model, that is, how to forecast the results of innovations not represented in the equation variables.
What is needed is a more fundamental mechanistic model of road safety and the effects of new intervention on road safety. The model will require an understanding of driver behavior, vehicle dynamics, consequences of a crash, and response of the emergency support services. To do this probably requires a computer simulation that allows drivers and others to make mistakes as an ideal or look up tables of expected consequences as a minimum. The next major thrust in road safety forecasting model development is to look for an even more fundamental relationship.
ACKNOWLEDGMENTS
This work has been supported by operating grants from the Natural Science and Engineering Research Council of Canada. The graphics and most of the accident data for Canada, the United States, and the United Kingdom were organized by Connie Leung of the University of British Columbia Department of Civil Engineering.
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