---
category: literaturenote
citekey: tayageingdriversstormteacup2006
title: "Ageing drivers: Storm in a teacup?"
authors: "Tay, Richard"
year: 2006
date: 2006-00-00 2006
doi: 10.1016/j.aap.2005.07.009
publication: "Accident Analysis & Prevention"
url: "https://doi.org/10.1016/j.aap.2005.07.009"
zotero_key: PRZ9AANC
zotero_storage: 6ERQ53ZA
collections: transpordi_planeerimine
folder: 001_artiklid
firstAuthor: "Tay, Richard"
status: converted
---
Accident Analysis and Prevention 38 (2006) 112–121
# Ageing drivers: Storm in a teacup?
# Richard Tay ∗
*AMA/CTEP Chair in Road Safety, Department of Civil Engineering, University of Calgary, Calgary, Alta., Canada T2N 1N4* Received 3 May 2005; received in revised form 29 July 2005; accepted 29 July 2005
#### **Abstract**
The purpose of this paper is to address two important issues regarding ageing drivers. First, there is a presumption in the road safety arena that the expected increase in the number of ageing drivers on the roads will lead to an increase in crashes. Second, despite extensive research on ageing drivers, especially on their increased vulnerability and reduced driving abilities, the most widely recommended road safety strategy relates to the control of their driving licenses. This presumption and the associated policy recommendation are based mainly on the higher relative crash risks associated with ageing drivers compared to their younger counterparts. This study, however, argues that the average crash risks obtained in previous studies provide only indirect information concerning these issues and any decision based on these results may produce unexpected outcomes. An analytical framework to examine the marginal effect of changing the driver mix on the roads is proposed and a simple empirical model is estimated as an illustration. In contrast to previous studies, we found that increasing the number of licenses issued to ageing drivers had only an insignificant impact on the number of fatal crashes on the roads, suggesting that the concern over ageing drivers may be a storm in a teacup.
*Keywords:* Road safety; Ageing drivers; Licensing; Marginal effects; Australia; Econometrics
# **1. Introduction**
© 2005 Published by Elsevier Ltd.
The demographic profile and thus the profile of drivers in many countries are undergoing significant changes. In most developed countries, the population is ageing as the babyboomer generation matures. Furthermore, the new generation of seniors is expected to drive more frequently and continue to drive longer than their previous counterparts. The expected increase in the number of ageing drivers on the road has generated much concern among some road safety researchers and policy makers because of their anticipated decrease in driving ability and the disproportionate attention given to their age by the media when they are involved in a crash. These perceptions, supported by numerous studies on their driving performance, have generated frequent calls for policy makers to tighten the license renewal process of ageing drivers [\(Transportation Research Board, 1994;](#page-9-0) [Road Safety](#page-9-0) [Committee, 2003;](#page-9-0) [Dulisse, 1997; Rock, 1998; Lyman et al.,](#page-8-0) [2002; Stefano and MacDonald, 2003; Owsley et al., 2003;](#page-8-0) [Langford et al., 2004; Evans, 2004\).](#page-8-0)
However, studies on the driving performance of ageing drivers can at best provide an indirect link to the often recommended policy of license restriction, control and cessation. It is well accepted in the literature that driving performance and behavior are not the same ([Evans, 2004\)](#page-8-0) and there are many studies that suggest that ageing drivers self-regulate their driving including driving less at night, driving slower, driving during off-peak hours and driving only in familiar environment [\(Raitanen et al., 2003; Owsley et al., 2003; Newbold](#page-9-0) [et al., in press; Abdel-Aty et al., 1999](#page-9-0)). As a result, many studies have found that the simple crash involvement (number of crashes per unit time) of ageing drivers to be lower than drivers in other age groups. In fact, as shown in [Fig. 1,](#page-1-0) the driver fatalities per time period in the United States, for example, appear to decrease with age. This evidence suggests that ageing drivers are the safest group of drivers on the roads.
Proponents of license restrictions, control or cessation for ageing drivers, however, choose to focus on the driving risks per distance traveled and, with a few exceptions, the general finding in the literature is that ageing drivers have
∗ Tel.: +1 403 220 4725; fax: +1 403 282 7026. *E-mail address:* rtay@ucalgary.ca.
Fig. 1. Driver fatalities by age. Source: [Evans \(2004\).](#page-8-0)
higher crash risks per distance traveled compared to drivers between 25 and 65 years old [\(Dulisse, 1997; Rock, 1998;](#page-8-0) [Hakamies-Blomqvist and Peters, 2000; Hakamies-Blomqvist](#page-8-0) [et al., 2003; Evans, 2004; Dellinger et al., 2004\)](#page-8-0). This finding can be summarized in the widely accepted U-curve as shown in Fig. 2. However, the relationship shown is of direct relevance only if the policy decision concerns restricting the length of trips or number of trips taken by ageing drivers but provides only indirect information regarding the policy decision to tighten the control of licenses issued to ageing drivers in general.
The most appropriate method to answer the question regarding licensing is to directly examine the risks associated with licensing ageing drivers. Several studies have examined this issue using crash risks per licensed driver for various agegroups [\(Rock, 1998; Evans, 2004\) a](#page-9-0)nd the results can best be summarized by Fig. 3. Again, these results show that the fatality risks exhibit the U-shaped relationship with age. These studies, however, examine how the average fatality risks per licensed driver vary with age but not how the fatality risks vary with the number of licenses issued for the various age groups.
It is well accepted in the field of economics that decisions regarding resource allocation should be made using the
Fig. 2. Driver fatalities per distance traveled by age groups. Source: [Evans](#page-8-0) [\(2004\).](#page-8-0)
Fig. 3. Driver fatalities per licensed driver by age. Source: [Evans \(2004\).](#page-8-0)
marginal effects (e.g., change in the number of fatalities due to a unit change in the number of licenses issued) and not average effects (e.g., fatalities per license issued which is total fatalities divided by total number of licenses issued). Since, except in rare instances, the average effects and the marginal effects are not the same, using the results of these studies to decide on the optimal number of licenses among the different age groups may result in unexpected outcomes ([Tay, 2003b\).](#page-9-0)
### **2. Analytical framework for policies to change the mix of drivers on the roads**
In practice, however, many decisions are made based on the relative or average effects, which may lead to an inefficient allocation of resources. For example, in an effort to minimise the social costs of road crashes, especially fatal crashes, policy makers may implement regulatory, administrative or fiscal policies to encourage the licensing of drivers who are deemed to be safer and discourage the licensing of drivers who are deemed to be less safe. Differences in the requirements for knowledge tests, on-road driving tests, medical examinations, insurance premiums and driving restrictions are, however, usually based on either the average fatality risks given that a crash has occurred or the average involvement rates in fatal crashes. The expected effects of such policies on the total number of fatal crashes may be ambiguous because of two important considerations that have yet to be adequately addressed in the literature.
First, the average risk associated with each group of drivers is assumed in most literature to be constant with respect to the number of drivers in each group. Although widely accepted among many road safety researchers ([Evans, 2004\),](#page-8-0) this assumption is very restrictive [\(Peltzman, 1975; Harvey](#page-9-0) [and Durbin, 1986\).](#page-9-0) As an illustration, consider an increase or decrease in the number of licenses issued to a particular segment of the driving population. This change will increase or decrease the total number of licensed drivers as well as increase or decrease the relative share of drivers on the road who belong to that segment, all else held constant.
In general, a change in the total number of drivers (size effect) on the roads will increase or decrease the number of fatal crashes depending on the current level traffic flow relative to the saturation point. If the current level of traffic flow is well below the saturation point and experiencing free flow condition, then an increase in the number of vehicles on the roads is expected to increase the number of fatal crashes, all else held constant. On the other hand, if the current level of traffic flow is near or above the saturation point, then increasing the number of drivers on the roads is expected to have little impact or even reduce the number of fatal crashes as traffic congestion sets in. With respect to ageing drivers, since a large proportion of their travel is undertaken during off peak hours, they are not likely to contribute significantly to congestion or a reduction in crash risks.
A change in the relative share of drivers on the roads also has important implications for the road safety because the mix of drivers has changed, thereby changing other traffic characteristics and relative risks associated with the different groups. All else held constant, as the one group becomes more dominant, their relative risk as a group may decrease because their variance from the mean traffic stream decreases (own effect) but the relative risks associated with other drivers may increase (interactive effect) as their variance increases. For example, the average speed of the traffic in general is expected to fall with an increase in the relative share of ageing drivers and moves closer to the average speed of ageing drivers and further from the average speed of younger drivers, especially young male drivers. This change in speed is expected to decrease the relative crash risks of ageing drivers but increase the relative crash risks of younger drivers ([Dee and Sela,](#page-8-0) [2003\).](#page-8-0)
More importantly, the change in the relative risks may be significant enough to overcome the poorer driving performance or ability of ageing drivers, thus resulting in an improved overall effect rather than the adverse effect predicted by using only the relative risks and assuming that the risks are constant. Conversely, increasing the number of younger drivers on the roads may change the dynamics of the traffic to their advantage thereby decreasing their crash risks (own effect) but may increase the crash risks associated with ageing drivers (interactive effect).
Clearly, a change in the number of drivers in any segment of the population may change the average crash risks within that segment as well as the average crash risks associated with other segments of the driving population. Although we are unable to delineate the separate effects empirically, the above discussion serves to illustrate the point that assuming a constant risk for each segment of the population is very restrictive and needs to be reconsidered. Moreover, changing traffic characteristics is not the only influence that may change the average crash risk. Other relevant influences include the change in the social and political climate with respect to acceptable driving behaviours, legislation, enforcement and prioritization of public policies.
Moreover, since the average crash risk at any severity level for any group age cannot be negative; assuming that it is constant automatically precludes the possibility of a reduction in crashes when the exposure increases in that segment of the driving population—an assumption that is clearly violated in many situations (e.g., congested roads).
#### **3. Marginal versus average effects**
The above discussion also leads us to another important consideration that has yet to receive sufficient attention in road safety, except among transport economists: the difference between average effect and marginal effect. As discussed above, the marginal effect of changing the number of drivers in any segment consists of the sum of three separate and uncertain effects due to changes in the dynamics of the traffic flow alone. The marginal effect may therefore be very different from the average effect and using the average effect as a basis for making policy decisions may well produce an opposite effect from the intended outcome [\(Tay,](#page-9-0) [2003b\).](#page-9-0)
In optimising the allocation of scarce resources, economic theory mandates that decisions have to be made at the margin: decision makers should consider the marginal effects of changing the consumption or investments in any transport policies rather than the corresponding average effects. Except in rare instances, the average effect associated with a change in the number of drivers in a particular segment, for example, is expected to be different from the corresponding marginal effect and the difference may be significant enough to change the decision or outcome.
As an illustration, consider the fatality risks associated with two subgroups of drivers, A and B, shown in [Fig. 4.](#page-3-0) Suppose the number of fatalities associated with the current number of type-A drivers, *Q*A, is *F*A and the corresponding number of fatalities associated with type-B drivers at the current level, *Q*B, is *F*B. It is clear from the figure that the average fatality risk of type-A driver (*F*A/*Q*A) is higher than the average fatality risk of type-B driver (*F*B/*Q*B). If decisions are made based on these statistics, then policy makers may implement regulatory, administrative or fiscal policies that discourage the licensing of type-A drivers and encourage the licensing of type-B drivers, resulting in an increase in the number of fatalities associated with both types of drivers and a resultant increase in the total fatalities.
To better assist policy makers in designing prudent regulatory and administrative policies regarding licensing, we need to estimate the unconditional marginal effects of changing the number of licensed drivers for various age-groups on the number of fatal crashes. It is clear from [Fig. 4](#page-3-0) that the marginal effect of increasing the number of type-A drivers (the slope of the line on the left graph) is negative whereas the marginal effect of increasing the number of type-B drivers
Fig. 4. Average and marginal effects of changing driver mix.
(slope of the line on the right graph) is positive, implying that increasing the number of type-A driver will reduce the number of fatalities whereas increasing the number of type-B driver will increase the number of fatalities. The recommendation based on marginal effects will instead suggest that policy makers enact regulations or administrative policies, or set prices, premiums and other costs that encourage the licensing of type-A drivers and discourage the licensing of type-B drivers. In contrast to the recommendations based on average effects, this recommendation will result in an overall decrease in the number of fatalities.
#### **4. Data**
Since the policy variable of interest is the number of licenses issued to ageing drivers, sufficient variation in this variable is required to perform any statistical analysis. It should be noted that this type of data can only be obtained from historical time series data or cross sectional data from different jurisdictions because it is not feasible for researchers to conduct experiments that manipulate this policy variable. The primary data used in this study are the number of licenses issued to different groups (under 25 years old, 25–59 years old, 60–69 years old and 70 years old and above) between October 1997 and July 2001 (46 observations) for an Australian state and was provided by the state's Department of Transport.
The main outcome of interest in this study is the total number of fatal crashes per month and data on this variable was also provided by the Department of Transport for the same period stated above. It should be noted that the total number of crashes per month across all age groups rather than the number of fatal crashes per month by driver age is selected as the dependent variable in this analysis because of the potential of interactive effects among groups and also to examine the system wide effect of changing the driver mix. Finally, data on the retail index, which serves as a measure of economic activities, was provided by the Office of Economics and Statistical Research for the period of analysis.
#### **5. Preliminary analysis of trends and correlations**
The number of fatal crashes per month for the study period is shown in Fig. 5. The trend line fitted using the least squares technique showed a slightly increasing trend during this period. However, the standard *t*-test indicated that the estimate of the coefficient for the slope (β = 0.064) is not statistically significant (*p* = 0.31). The main independent variables of interest are the numbers of licensed drivers in each age group, especially the number of licenses issued to ageing drivers. As shown in [Fig. 6,](#page-4-0) except for younger drivers under the age of 25, the number of licensed drivers per month have been increasing over the period analyzed. It should be noted that the estimated coefficients of the slope of all the trend lines for the number of licenses issued, fitted using the least squares technique, are statistically significant at α = 0.05 level, using the standard *t*-test.
It is not surprising therefore that these trends result in a negative correlation between the number of fatal crashes per month and the number of licensed drivers under the age of
Fig. 5. Number of fatal crashes per month.
Fig. 6. Trends in number of licensed drivers by age.
25 but a positive correlation between the number of fatal crashes per month and the number of licensed drivers in all other age groups (see Fig. 7), even though the estimates of these correlation coefficients are not statistically significant, using the standard *t*-test and α = 0.05. Also shown in Fig. 7 are the least square estimates of the linear relationships between the total number of fatal crashes per month and the number of licenses issued in each of the age group.
### **6. Preliminary empirical model with least squares estimators**
All else held constant, the number of fatal crashes per month, *y*, is assumed to be dependent on the number of drivers, measured in units of 10,000 drivers, in each of the following age categories: AGE1 (under 25), AGE2 (25–59 years old), AGE3 (60–69 years old) and AGE4 (70 years old and above). Although a more detailed breakdown, especially for the ageing drivers, would improve the analysis, these categories (used by the Department of Transport when providing the data used in this study) are sufficient for the purpose of this paper which is to examine whether the marginal effect is consistent with the average effects found in previous studies, specifically the monotonically decreasing relationship found between the simple fatality rates (fatalities per time period) and age as well as the U-shaped relationship found between age and fatalities per distance traveled or fatalities per licensed driver per time period.
One of the main determinants of the number of motor vehicle crashes is exposure, which is often captured by proxy variables such as vehicle kilometres travelled, driving population, vehicle population or gasoline consumption. Unfortunately, no data on either vehicle kilometres traveled or gasoline consumed are available while the number of drivers and the number of vehicles are highly correlated with the main independent variables. Therefore, an alternate measure, *Retail Index*, is included in this study to capture the effects of both traffic and economic activities. The latter effect has also been found to be a significant determinant of road safety [\(Joksch,](#page-8-0) [1984; Tay, 2003a,b, 2005b\).](#page-8-0) The amount of retail activity is
Fig. 7. Relationship between fatal crashes and licensed drivers.
chosen over other common economic indicators such as the unemployment rate because of its availability and its closer association with monthly changes in traffic and economic activities [\(Tay, 2003b, 2005b\).](#page-9-0)
Finally, a *Trend* term is included in the model to capture any longer term trend that may arise due to gradual changes in road environment or vehicle improvement over time ([Joksch, 1984; Tay, 2003a,b, 2005a,b,c\).](#page-8-0) For simplicity, a linear-in-parameters model will be assumed in the preliminary analysis.
$$y = \beta_0 + \beta_1 \text{ Trend} + \beta_2 \text{ Retail Index} + \beta_3 \text{ AGE1}$$
$+ \beta_4 \text{ AGE2} + \beta_5 \text{ AGE3} + \beta_6 \text{ AGE4} + \varepsilon$ (1)
where ε is the normal error term.
In the simple linear model given above, the estimated coefficients of the age related variables represent the linear association between monthly fatalities and the number of drivers in that age group. As the number of drivers in any age group changes, its impact on the number of fatalities can be estimated using its estimated coefficient. The marginal effect of changing the number of licenses issued to drivers under 25 years old (AGE1), for example, is given by the estimate of the corresponding coefficient β3 because a one unit change in AGE1 will change *y* by β3 fatal crashes per month. Note that the marginal effect can either be negative or positive whereas the average effect can only be non-negative.
The choice of estimation method in any statistical analysis is a combination of both art and science because the statistical properties of different models may vary depending on the validity of their underlying assumptions ([Gujarati, 1995;](#page-8-0) [Pindyck and Rubenfeld, 1986; Kennedy, 2001; Noland and](#page-8-0) [Karlaftis, 2005\).](#page-8-0) The approach taken in this paper is to start with the simplest and most widely used model in order to focus the attention on the policy analysis and recommendations. The underlying assumptions made are then tested and if the assumptions are not valid, then alternative methods to address these shortcomings will then be used and their results presented as a comparison.
Therefore, the above model will first be estimated using the ordinary least squares (OLS) technique, which is the most widely used method in crash data analysis [\(Hakim et al.,](#page-8-0) [1991; Tay, 2005a,b,c\).](#page-8-0) Although some researchers prefer to estimate only count data models such as Poisson or Negative Binomial, the OLS model will be estimated first in this study because of several reasons. First, OLS estimators are still the best linear unbiased estimator (BLUE) of the coefficients even if the Normality assumption is violated. Second, OLS estimates appear to be quite robust and very similar to those obtained in either the Poisson or Negative Binomial models even if the Normality assumption is violated [\(Tay,](#page-9-0) [2005a,b,c\).](#page-9-0) Third, many of the underlying assumptions in the model can be easily tested in the OLS model (e.g. Normality, serial correlation and misspecification tests) but not in the Poisson model.
Fourth, testing and correction for the presence of autocorrelation or serial correlation is relatively simple in OLS model but not so in Poisson or Negative Binomial models. Last but more importantly, when serial correlation is present, estimates using the Poisson model will be biased and inconsistent ([Johansson, 1996; McCarthy and Tay, 2005](#page-8-0)). Since time series data are used in this analysis, the presence of autocorrelation is a distinct possibility and will be tested using the standard Durbin–Watson statistics. Although the Durbin–Watson statistics is designed specifically to test for first-order autocorrelation, [Gujarati \(1995\)](#page-8-0) suggested that it can also be used to test for the omission of any significant time dependent explanatory variable from the model.
Obviously, the number of fatal crashes per time period depends on numerous factors, many of which cannot be adequately captured in any empirical model and as a result, most models are developed with specific foci. To focus this analysis on the impact of different types of drivers on the number of fatal crashes, only these variables and a few general variables to account for other major confounding factors are included in the model. The omission of other important variables may result in some specification bias. Consequently, the Regression Specification Test or the RESET test proposed by [Ramsey \(1969\)](#page-9-0) will be conducted to check for specification errors arising from the omission of significant independent variables and non-linearity in the functional forms ([Gujarati,](#page-8-0) [1995\).](#page-8-0)
As discussed above, one concern in using the standard regression model is that the number of fatal crashes per time period may not have a Normal distribution but instead follows a Poisson distribution [\(Jovanis and Chang, 1986; Michener](#page-8-0) [and Tighe, 1992; Tay, 2001, 2003b, 2005a,b,c](#page-8-0)). Therefore, the Jarque–Bera Test of Normality [\(Jarque and Bera, 1987;](#page-8-0) [Gujarati, 1995\)](#page-8-0) will be conducted. It should be noted that although the OLS estimators for the coefficients are still BLUE even if the assumption of normality is violated, the estimate of the variance may be biased. In such cases, caution should be exercised in interpreting the test results because they may no longer be valid.
# **7. Poisson regression model**
As noted above, the number of fatal crashes per month is a count data and thus is more likely to have a Poisson distribution. Therefore, a Poisson regression model shown below will also be estimated for comparison.
$$P(Y = y) = \frac{e^{-\lambda} \lambda^y}{y!}$$
$y = 0, 1, 2, 3, ...$ (2)
$$ln(\lambda) = \beta_0 + \beta_1 \operatorname{Trend} + \beta_2 \operatorname{RetailIndex} + \beta_3 \operatorname{AGE1}$$
$$+ \beta_4 \text{ AGE2} + \beta_5 \text{ AGE3} + \beta_6 \text{ AGE4} \tag{3}$$
where λ is the mean of the distribution.
Note that the Poisson regression model is a non-linear model and care should be exercised in interpreting the results, especially the estimates of the coefficients. Unlike the OLS model, the estimates of the coefficients for the age related variables do not represent the linear association between the number of fatal crashes per month and the number of licenses issued to each of the age group. It can be easily shown that the expected number of fatal crashes per month and the marginal effect of any independent variable *x*i are given by
$$E(y|x_i) = e^{X\beta} \tag{4}$$
$$\frac{\partial E(y|x_i)}{\partial x_i} = \lambda \beta_i \tag{5}$$
The above model will be estimated using the statistical software LIMDEP version 8. It should be noted that a common concern of using the Poisson regression model to analyse crash data is the possibility of over-dispersion in the data. This concern will be checked using the *t*-tests proposed by [Cameron and Trivedi \(1990\).](#page-8-0)
#### **8. Results**
The results of preliminary analysis using the standard regression model, estimated using the ordinary least squares technique, are reported in Table 1. The model fitted the data moderately well, with an *R*-square value of 0.25, given the small sample size and relatively disaggregated data used. The Durbin–Watson statistic was 2.2, indicating that there is no conclusive evidence to suggest that there is a problem associated with the presence of autocorrelation or the omission of a time-dependent variable [\(Gujarati, 1995\).](#page-8-0) As an additional check for possibility of a more general misspecification error, the Ramsay's Reset Test [\(Gujarati, 1995\) w](#page-8-0)as also performed and the test provided additional support that the model was well specified. This result implies that the estimated coefficients are not likely to be biased due to the omission of any important variables or incorrect function form ([Gujarati,](#page-8-0) [1995\).](#page-8-0) However, the Jarque–Bera Test of Normality ([Jarque](#page-8-0) [and Bera, 1987; Gujarati, 1995\) r](#page-8-0)ejected the assumption that the errors obtained came from a Normal distribution.
Wither respect to the estimated coefficients, both the control variables, *Trend* and *Retail Index*, had the expected sign and were statistically significant at α = 0.05. In terms of the age related variables, the estimated coefficients for AGE1
Table 1 Preliminary estimation results (OLS)
| Dependent variable | Number of fatal crashes per month |
|--------------------|-----------------------------------|
| No. of observation | 46 |
| R-square | 0.250 |
| F-statistics | 2.163 |
| P-value | <0.06 |
| Independent variables | Coefficients | t-Statistics | |
|-------------------------------|--------------|--------------|--|
| Constant | −729.35 | −1.95 | |
| Trend** | −3.606 | −2.06 | |
| Retail Index*** | 9.648 | 2.56 | |
| AGE1 (Under 25 years old)* | −33.95 | −1.77 | |
| AGE2 (25–59 years old)* | 5.99 | 1.85 | |
| AGE3 (60–69 years old) | 16.24 | 1.04 | |
| AGE4 (70 years old and above) | 2.27 | 0.56 | |
\* Denotes statistically significant at α = 0.10.
was negative whereas the estimated coefficient for AGE2 was positive. Since they were marginally significant (α = 0.10), these results imply that an increase in the number of licenses issued to young adult drivers (under 25 years old) is moderately likely to reduce the number of fatal crashes per month whereas increasing the number of middle-aged drivers (25–59 years old) is moderately likely to increase the number of fatal crashes. More importantly, the estimated coefficients for AGE3 (60–69 years old) and AGE4 (70 years old and above) are statistically insignificant even at the conservative level of α = 0.10. These results suggest that increasing the number of licenses issued to ageing drivers is expected to have little or no impact on the number of fatal crashes per month.
However, it should be noted that the Normality assumption was violated and thus the OLS estimator used is no longer efficient and the estimate of the variance may be biased. Therefore, caution should be exercised in interpreting the results of any hypothesis test. To overcome this shortcoming, the model was re-estimated using the Poisson regression model and the results are shown in Table 2 for comparison. To check the concern of over-dispersion in the Poisson model, the *t*-tests proposed by [Cameron and Trivedi \(1990\)](#page-8-0) was performed (see Table 2) and no significant over-dispersion was found.
The marginal effects of changes in the independent variables were also estimated and reported in Table 2. Again, an increase in the number of licenses issued to ageing drivers was found to have an insignificant effect on the expected number of fatal crashes whereas an increase in the number of licensed young drivers was expected to decrease the number of fatal crashes and an increase in the number of "middleaged" drivers was expected to increase the number of fatal
Table 2 Estimation results using Poisson model
| Dependent variable | Fatal |
|--------------------------------------|---------|
| Number of observations | 46 |
| Log likelihood function | −136.87 |
| Restricted log likelihood | −144.39 |
| Chi squared | 15.02 |
| Degrees of freedom | 7 |
| Prob[Chi squared > value] | 0.020 |
| Overdispersion tests: g = mu(i) | −0.024 |
| 2
Overdispersion tests: g = mu(i) | −0.070 |
| | Coefficient
Standard error | | Marginal effect | |
|----------------------------|-------------------------------|-------|-----------------|--|
| Variable | | | | |
| Constant** | −30.945 | 15.06 | −709.71 | |
| Trend** | −0.166 | 0.07 | −3.80 | |
| Retail*** | 0.404 | 0.14 | 9.26 | |
| Number of licensed drivers | | | | |
| Under 25** | −1.626 | 0.08 | −37.29 | |
| 25–59 years old** | 0.286 | 0.13 | 6.56 | |
| 60–69 years old | 0.697 | 0.62 | 15.98 | |
| 70 and above | 0.100 | 0.16 | 2.29 | |
\*\* Denotes statistically significant at α = 0.05 levels.
\*\* Denotes statistically significant at α = 0.05.
\*\*\* Denotes statistically significant at α = 0.01.
\*\*\* Denotes statistically significant at α = 0.01 levels.
crashes. It is not surprising that the estimated marginal effects are very similar to those obtained in the OLS model albeit with a stronger statistical significance since the OLS estimators of the coefficients are BLUE although they are not efficient.
## **9. Discussion and conclusion**
Extant research has been conducted on measuring the relative fatality risks associated with different segments of the driving population given that a crash has occurred and the average involvement rates of the different driver groups on the road. Although these probabilities are useful for making a variety of decisions such as understanding the relative needs of the different driver groups, they are not appropriate for use in designing regulatory or administrative policies that attempt to change the driver mix on the roads because utilising these results as the basis for decision making regarding the number of licenses issued to each driver segment requires the restrictive assumption that the fatality risk associated with each segment of driver is constant with respect to the number of drivers in each group. This assumption is likely to be violated because a change in the drive mix is likely to change the dynamics of the traffic stream, as well as other factors, thereby changing the relative risks of different segments of the driving population.
This paper presented a conceptual framework for analysing the potential effects of a change in the driver mix on roads due to a change in the dynamics of the traffic alone. The marginal effect of changing the number of drivers in any segment consists of three effects: size, own and interactive effects. As a result, the marginal effect of changing the number of drivers in any segment of the population is uncertain and may not even have the same qualitative effect (direction of change) as predicted by assuming a constant average risk for each segment. Therefore, using the conditional or unconditional average probabilities as a guide to change the mix of drivers on the road may not result in the desired outcomes. As an illustration, we also estimated a simple model of the marginal effect of changing the driver mix on the number of fatal crashes in an Australian state and compared the results obtained with those prescribed by using the average probabilities. The expected effects on the number of fatalities and the associated ranking of safety are summarized in Table 3.
It is well accepted in the literature that the simple average fatality risks associated with increasing age exhibit a monotonically decreasing relationship, which suggests that policy makers should reduce the number of younger drivers the most, followed by the number of "middle-aged" drivers and perhaps increase the number of ageing drivers on the road. On the hand, using either the relative fatality risks per licensed driver or per distance traveled yields a U-shaped relationship; which suggests that policy makers should decrease the number of younger and ageing drivers on the roads. This study however, found that the marginal effect of changing the driver mix was very different from those predicted by using these average crash risks associated with the different driver groups.
Our study found that increasing the number of ageing drivers licensed had no significant effect on the number of fatal crashes whereas increasing the number of "middleaged" drivers had an adverse effect but increasing the number of younger drivers had a beneficial effect. In contrast to previous recommendations, these results suggest that policy makers should decrease the number of "middle-aged" drivers and increase the number of younger drivers.
These results are somewhat surprising but not entirely unexpected when we consider the impact of changing driver mix on the dynamics of the traffic stream as discussed earlier. First, increasing the number of "middle-aged" drivers licensed will increase the total number of drivers on the roads, all else held constant, and this increase in exposure is expected to increase the number of crashes in general. Also, it will not change the dominance of this group but may increase the variance of the young and ageing drivers, resulting in an increase in the risks associated with the two non target groups. The expected overall effect is therefore to increase the number of fatal crashes on the roads.
Next, increasing either the number of ageing drivers or younger drivers licensed will also increase the overall exposure. This increase is expected to be smaller when compared to "middle-aged" drivers because a larger percentage of these license holders may not drive. Regardless, these increases in crash risks due to increase in exposure have to be considered together with the potential beneficial own effect due to their increase in the share of drivers on the roads and the interactive effects these increases may have on other drivers on the roads. In the case of younger drivers, the benefit arising from the own and interactive effects may be large enough to overcome the increased risk due to an increase in exposure to produce a net benefit while the corresponding benefit arising from the own and interactive effects due to an increase in the number of ageing drivers is only sufficient
Table 3 Policy recommendations based on average and marginal effects
| Evidence | Under 25 | 25–59 | 60–69 | 70 and above |
|-----------------------------------------------------------------------|-----------|---------|-----------|--------------|
| Average fatalities per time period (see Fig. 1) | (↑) (4) | (↑) (3) | (↑) (2) | (↑) (1) |
| Average fatalities per time period per distance traveled (see Fig. 2) | (↑) (3/4) | (↑) (1) | (↑) (2) | (↑) (3/4) |
| Average fatalities per time period per license issued (see Fig. 3) | (↑) (3/4) | (↑) (1) | (↑) (2) | (↑) (3/4) |
| Marginal fatalities per time period (see Tables 1 and 2) | (↓) (1) | (↑) (4) | (–) (2/3) | (–) (2/3) |
*Note*: Direction of change and ranking of safety shown in the first and second parentheses respectively.
to negate the potential adverse effect due to an increase in exposure.
However, caution should be exercised in generalizing the results of this study, especially those associated with younger drivers because of the moderately small sample size and peculiarity of the data used. During the period examined, the number of fatal crashes per month experienced a slightly upward trend, albeit statistically insignificant, while the number of licensed drivers under the age of 25 experienced a downward trend whereas the number of licensed drives in all other age groups experienced an increasing trend. Although several statistical tests were conducted to check for potential specification bias in the regression model, these tests were not comprehensive enough to exclude the influence of all possible confounding factors that may be driving these unusual trends. Nevertheless, the results obtained serve to further illustrate the need to reconsider the current basis for making policy decisions regarding the licensing of ageing drivers which are based on the higher average fatality risks per distance travelled instead of the marginal fatality risks due to a change in the number of licenses issued.
In summary, this study found that increasing the number of licensed drivers among senior citizens, aged 60 and above, would not result in an increase in the number of fatal crashes as posited by many researchers who based their predictions on the average fatality rates (per distance traveled or per license issued) of drivers in different age groups or the anticipated decrease in the driving performance of ageing drivers. This finding further reinforces the adage that driving ability and driving behavior are not the same and policy makers should base their decisions more on the expected behavioral outcome of the countermeasures. Moreover, the presumption in road safety arena that the expected increase in the number of ageing drivers on the roads is a significant safety concern is not supported by this preliminary study.
More importantly, any regulatory, administrative or fiscal policy based on age that attempts to assess, restrict, control or add to the cost of ageing driver obtaining a license is not recommended. This recommendation is consistent with the finding in Australia that the various mandatory assessment procedures for managing older driver safety cannot be associated with road safety benefits and thus their continued use cannot be justified (Langford et al., 2004).
Our results also questions the restrictive assumption of a constant fatality risk within a particular segment of the driving population with respect to the number of drivers in each group because of the different size, own and interactive effects of changing the driver mix on the traffic characteristics. In addition to changing the traffic characteristics, any substantial change in driver mix will also have other social and political effects that may change the relative crash risks associated with different driver segments. Furthermore, since average risks cannot be negative, it is assumed that an increase in any group of drivers will automatically increase the number of fatalities on the road—an assumption that also needs to be reviewed. Finally, this study showed that the marginal effect of increasing the number of licensed ageing drivers was different from the average effects and policy makers should rely more on the marginal effect to guide their regulatory, administrative or fiscal policies on licensing.
#### **Acknowledgement**
Financial Support from the Alberta Motor Association is gratefully acknowledged but the views expressed by the author do not necessarily reflect those of the association.
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