Discussion Papers In Economics
And Business

Smeed’s Law and the Role of Hospitals in Modeling Fatalities
and Traffic Accidents

Yueh-Tzu Lu and Mototsugu Fukushige

Discussion Paper 17-22

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANSmeed’s Law and the Role of Hospitals in Modeling Fatalities
and Traffic Accidents

Yueh-Tzu Lu and Mototsugu Fukushige

Discussion Paper 17-22

July 2017

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANSmeed’s Law and the Role of Hospitals in Modeling Fatalities and Traffic
Accidents
Yueh-Tzu Lu
Graduate School of Management
National Cheng Kung University
Tainan City, 701 Taiwan
e-mail: enjourney321@gmail.com
andMototsugu Fukushige*
Graduate School of Economics
Osaka University
Toyonaka, 560-0043, Japane-mail: mfuku@econ.osaka-u.ac.jp
Abstract
We applied Smeed’s Law to Japanese prefectural data from between 1988 and 2016. We
found that the coefficient for the number of vehicles was stable over the estimation period,
but that the constant term decreased gradually. We decomposed fatalities per capita into
fatalities per accidents and accidents per capita, and applied regression equations to thedata. We conclude the following from this study. First, the relationship between fatalities
per capita and the number of registered vehicles per capita was stable, which is consistent
with Smeed’s Law. Second, the effects of technological advances have changed theestimated coefficients for time dummies. The role of hospitals may be difficult toincorporate into Smeed’s Law because of the complicated relationship between the
distance to hospital and fatalities per capita.
JEL Classification Number: R41, I18, R42
Keywords: Smeed’s Law, Traffic Accidents, Fatalities, Hospitals*

Corresponding author:
Graduate School of Economics, Osaka University
1-7, Machikaneyama-cho, Toyonaka, 560-0043 Japan.
e-mail: mfuku@econ.osaka-u.ac.jp1Smeed’s Law and the Role of Hospitals in Modeling Fatalities and Traffic
Accidents

1. INTRODUCTION

After Smeed (1949) found a relationship between the numbers of fatalities andvehicles, researchers have continued to apply this law in many countries.2

For example,

Tamura (2013) has studied this law in Japan. However, this recent research by Tamura(2013) has indicated that the law is not stable over time in Japan.
One question about the stability of Smeed’s Law is that it does not incorporate
safety improvements in automobiles involved in traffic accidents or advances in medical
technologies. For example, one of the earliest studies, by Pantridge and Geddes (1967),
involved the cardiology unit and found that the time to hospital was one of the mostimportant factors affecting patient survival. Jones and Bentham (1995) also investigatedthe effect of time to hospital.

Technological advances in automobile safety andemergency rescue at traffic accidents can also affect survival. For example, since the law
requiring installation of airbags and provision of quicker transportation to hospitals, thesurvival of seriously injured people has increased.
This paper investigated the possible need for structural changes in Smeed’s Law
and the role of hospitals in Japan. We reformulated Smeed’s Law and used panel data inJapan to estimate whether the distance to hospital is an explanatory variable. We thendecomposed the fatalities per capita into the number of fatalities per accidents and traffic

2

There are several models to explain the number of fatalities in traffic accidents that do
not refer to Smeed’s Law. For example, Grimm and Treibich (2013) investigated thedeterminants of fatalities in different kinds of traffic accidents and found that income
could explain the differences.
2

accidents per capita. We also explain the determinants of these two variables using
Japanese prefectural data between 1988 and 2016.
In section 2 of this paper, we first reformulate and decompose Smeed’s Law andextend it by using a model that includes the distance to hospital. In section 3, we
decompose the fatalities per capita into two parts and estimate the relationships between
related variables. Finally, we conclude by discussing the empirical implications of therelationships between a stable Smeed’s Law and the role of hospitals.

2. SMEED’S LAW AND THE ROLE OF HOSPITALS

2.1 Traditional Approach to Smeed’s Law

Smeed’s original law is described as follows:1

Ndeath = 0.00030 ∗ (Nveh ∗ Pop2 )3 .
where Ndeathis the number of fatalities in traffic accidents, Nveh is the number ofregistered vehicles, and Pop is the size of the population, usually 100,000. This
relationship can be rewritten as number of fatalities per registered number of vehicles:2

NdeathNveh −3= 0.00030 ∗ (
)
Nveh
Pop
or fatalities per capita:
1

NdeathNveh 3= 0.00030 ∗ (
) .
Pop
Pop

3

We linearized this using logarithmic transformation and generalized to estimate data fromother countries as follows:
NdeathNveh

NdeathNvehln ( Nveh ) = α0 − α1 ln ( Pop ),(1)orln ( Pop ) = β0 + β1 ln ( Pop ) . 3(2)

Using Japanese prefectural data between 1988 and 2016, we applied the above equation(2).

In practice, we added prefectural and time dummies to control the individual andtime effects. Additionally, to capture the possibility that time could alter the coefficient
Nveh

Nveh

for ln ( Pop ), we adopted the products of the trend term (Time) and ln ( Pop ) as
explanatory variables.
Before we explain the estimation results, we first describe the data source (TableⅠ) and variables that we used in this paper and their summary statistics (Table Ⅱ). The
estimated results of several specifications are shown in Table Ⅲ. From the viewpoint of
the maximum adjusted R-squared or minimum Akaike’s information criterion (AIC),
Model 1C is the best model. This model is a simple least-square with dummy variables(LSDV) model.4

The estimated coefficient of β1 was 0.962085, which means that α1

in equation (1) is 0.037013. We could not reject the hypothesis α1 = 0. 5 However, theHesse et al. (2016) and Koren and Borsos (2010) also investigate Smeed’s Law in theform of the same dependent variable (fatalities per capita).
4
Ponnaluri (2012) took a similar approach to Indian panel data and found regional
differences between states.
5
The estimated standard error for the coefficient was 0.0883.
3

4

estimated results indicated significant and stable relationships between the number offatalities in traffic accidents and registered vehicles.6
To investigate the effects of technological advances, we estimated thecoefficients for time dummies in Model 1C in Figure 1. This figure implies that theprobability of death in traffic accidents gradually decreases during this period as therelationship between number of deaths and vehicles remains stable. We consider thatthese changes reflect the technological advances applied to the rescue of victims in traffic
accidents. This result is consistent with Oppe’s (1991) findings for Japanese data.

2.2 Accessibility to Hospitals
When considering technological advances or transportation time to hospitals,
one should add some explanatory variables to equation (2). In this paper, we modified theconstant term as a linear function of the distance to hospital:
β0 = θ0 + θ2 ln(Hdist)where Hdist is the distance to hospital. In this paper, we estimated the distance tohospital using the following steps:
1. We estimated each hospital’s area of coverage.
2. We assumed that each area is circular.
3. Using the formula to estimate the area of the circle, we estimated the radius of thecircle.
Using these steps, we estimated Hdist as:

6Smith (1999) found a negative correlation between number of fatalities and vehicles
using OECD cross-sectional data. Our result is different from that of Smith (2010), but
our specification is expressed per capita.
5

Area√Nhospital⁄
Hdist =
π
where Nhospital is the number of hospitals in each prefecture, Area is the total area of
each prefecture, and π is the circular constant. We then modified equation (2) to:
Death

Nveh

ln ( Pop ) = θ0 + θ1 ln ( Pop ) + θ2 ln(Hdist).

(3)

In this empirical estimation, we estimated equation (3) in a similar way as for equation(2). The estimation results are reported in Table Ⅳ. Model 2B produced the minimum
Nveh

AIC. The estimated coefficient for ln ( Pop ) was 1.12921, which means that α1 inequation (1) was –0.12921. We also could not reject the hypothesis α1 = 0. 7

Theestimated coefficient for the additional variable Time ∗ ln(Hdist) was negative and wasstatistically significant. This result implies that the number of fatalities in traffic accidents
per capita decreases when the distance to hospital increases. This is not a plausible result.
In the following sections, we try to explain this result.

3. DECOMPOSITION APPROACH

To investigate why the distance and fatalities per capita correlated negatively
correlated, as shown in section 2.2, we decomposed the dependent variable in equation(3) as:NdeathNdeath Accident
=
∗
.
Pop
AccidentPop
To investigate this reason, we applied the following two equations:7

The estimated standard error for the coefficient was 0.1210.
6

NdeathNvehln (Accident) = 𝛾0 + 𝛾1 ln ( Pop ) + 𝛾2 ln(Hdist),

(4)

and
Accident

ln (

Pop

Nveh

) = 𝛿0 + 𝛿1 ln ( Pop ) + 𝛿2 ln(Hdist).

(5)

The estimated results for equation (4) are shown in Table Ⅴ. Model 3C produced theminimum AIC. This result shows that the number of fatalities per accident correlated
positively with the products of the trend term and the distance to hospital: Time ∗ln(Hdist) This variable is also included in Table Ⅳ, which shows the estimation resultsof equation (3). This result seems to be natural. If the distance to hospital increases, theaccess time to hospital for an injured person in a traffic accident also increases, and thepotential of a fatality becomes high.
The estimated results for equation (5) are shown in Table Ⅵ. Model 4B produced
the minimum AIC. This result shows that the number of accidents per capita correlated
negatively with the products of the trend term and the distance to hospital: Time ∗ln(Hdist) This variable is also included in Table Ⅳ, which shows the estimation resultsof equation (3). This relationship explains the implication that the number of fatalities intraffic accidents per capita decreases when the distance to hospital increases whenestimated using equation (3). However, we could not identify any reason why the numberof accidents per capita increased when the number of hospitals increased and the distanceto hospital decreased.
Next, we investigated the possibility that there is reverse causality between thetwo variables.8

We estimated the following regression equation:

8

Some researchers have investigated the number of traffic accidents. For example,
Hakim et al. (1991) reviewed models for traffic accidents and proposed some additionalvariables as explanatory variable in macro models. In this paper, we did not search forthe best model to explain the number of traffic accidents. For example, Bishai et al. (2006)
7

Nveh

ln(Hdist) = ϕ0 + ϕ1 ln ( Pop ) + ϕ2 ln (Accident
Pop

).

(6)
Nveh

In practice, to capture the possibility that the time coefficient varied for ln ( Pop ) and
Accident

ln (

Pop

Accident

ln (

Pop

Nveh

), we adopted the products of the trend term (Time) and both ln ( Pop ) and) as explanatory variables.

Additionally, we approximated the nonlinear
2
Accident

relationship between ln(Hdist) and ln (Pop

Accident), and we added (ln (Pop)) as anadditional explanatory variable. The estimation results with several specifications areshown in Table Ⅶ. Model 5C produced the minimum AIC model. In this result, theAccident

relationship between ln(Hdist) and ln (Pop

) was unclear because the estimatedrelationship was quadratic. According to Table Ⅱ, we estimated the relationship between
Hdist andAccidentPop

when

AccidentPop

was from 1.822 (=exp(0.6)) to 13.4637 (=exp(2.6)),

which were the minimum and maximum values, respectively. Figure 3 shows that theestimated relationship between these two variables was negative.
We considered that hospitals were constructed in areas where traffic accidents
occurred frequently during these years. In other words, people made the distance tohospitals shorter in the place where traffic accidents occurred frequently.

This is anatural policy to cope with injured people in traffic accidents.

investigated the determinants of the number of traffic accidents and found that GDP,
population, number of vehicles, and length of roads were significant factors. However,their results are not useful for investigating traffic accidents per capita in prefectures inJapan because the time series data are relatively short and road lengths are almost fixedduring the sample period. García-Ferrer, De Juan, and Poncela (2007) also investigated
the relationship between number of traffic accidents and economic activity by
constructing time series models with monthly data. We could not obtain monthly data,
and therefore, we could not construct such a complicated model to investigate the numberof traffic accidents. These are questions for future investigations of the determinants oftraffic accidents per capita.
8

Following the estimation result described above, we next estimated equation (2)2by adding (ln (AccidentPop)) as an additional explanatory variable. We obtained thefollowing result:

Death
ln (
)
Pop
2

Nveh
Accident= 1.050 ∗ ln () − 0.00178 ∗ (Time ∗ ln(Hdist)) + 0.0144 ∗ (ln ()) + (others)PopPop(8.286)(−1.375)(2.075).

where t values are in parentheses. In this result, the estimated coefficient for ln(Hdist)remained negative, but it was not statistically significant. We conclude that this reflects areverse causality between the two variables.

4. CONCLUSION

In this paper, we estimated Smeed’s Law using Japanese prefectural data from
NdeathNveh

between 1988 and 2016 in the form of ln ( Pop ) = β0 + β1 ln ( Pop ) . We thenNvehconfirmed that the coefficient of ln ( Pop ) was stable over the estimation period, but theconstant term decreased gradually. The latter phenomenon might be related to thetechnological advances in the rescue of victims of traffic accidents. Including the distanceto hospital as an additional explanatory variable to explain the role of hospital in Smeed’s

9Law did not produce plausible results because of the complicated relationships between
Deathln ( Pop ) and ln(Hdist) . We decomposed the number of fatalities per capita into
Ndeathln (Accident) and ln (AccidentPop), and we used different regression equations with the same

explanatory variables. Finally, we reached the following conclusions. First, therelationship between fatalities per capita and number of registered vehicles per capita is
stable, which is also implied by Smeed’s Law. However, estimation using the traditional
linearized form:ln (NdeathNveh) = α0 − α1 ln (
),
Nveh
Pop

which is used by most researchers, could not produce statistically significant relationships.
Second, the effect of technological advances is reflected in changes in the estimatedcoefficients for time dummies. However, the role of hospitals is difficult to incorporate
into Smeed’s Law because of the complicated relationship between the distance tohospital and fatalities per capita.
Finally, we note some remaining problems. We found it difficult to model therole of hospitals when modeling traffic accidents, but we have not proposed any solutions.
Our analysis was limited to Japanese data, and it may not be applicable to use ourdecomposition approach in other countries to check the robustness of our findings. These
are remaining problems for future research.

Acknowledgments: This research stared when Lu visited Graduate School of Economics
in Osaka University as an exchange student. This research was financially supported by10Grants-in-Aid for Scientific Research of the Japan Society for the Promotion of Science (JSPS)(KAKENHI Grant Numbers JP15H03336 and JP15K12461).

11

References
Bishai, D., Quresh, A., James, P., & Ghaffar, A. (2006). National road casualties andeconomic development. Health Economics, 15(1), 65–81.
García-Ferrer, A., De Juan, A., & Poncela, P. (2007) The relationship between road traffic
accidents and real economic activity in Spain: common cycles and health issues.
Health Economics, 16(6), 603–626.
Grimm, M., & Treibich, C. (2013) Determinants of road traffic crash fatalities acrossIndian states. Health Economics, 22(8): 915–930.
Hakim, S., Shefer, D., Hakkert, A. S., & Hocherman, I. (1991). A critical review of macromodels for road accidents. Accident Analysis & Prevention, 23, 379–400.
Hesse, C.A., Oduro, F.T., Ofosu, J.B., & Kpeglo, E.D. (2016) Assessing the risk of roadtraffic fatalities across sub-populations of a given geographical zone, using amodified Smeed’s Model. International Journal of Statistics and Probability, 5(6),121–132.
Jones, A.P., & Bentham, G. (1995) Emergency medical service accessibility and outcome
from road traffic accidents, Public Health, 109(3), 169–177.
Koren, C., & Borsos, A. (2010) Is Smeed’s law still valid? A world-wide analysis of thetrends in fatality rates. Journal of Society for Transportation and Traffic Studies,
1(1), 64–76.
Oppe, S. (1991) The development of traffic and traffic safety in six developed countries.
Accident Analysis & Prevention, 23(5), 401–412.
Pantridge, J.F., & Geddes, J.S. (1967) A mobile intensive-care unit in the management of
myocardial infarction. The Lancet, 290(7510), 271–273.
Ponnaluri, R.V. (2012) Modeling road traffic fatalities in India: Smeed’s law, timeinvariance and regional specificity. IATSS Research, 36(1), 75–82.

12Smeed, R.J. (1949) Some statistical aspects of road safety research. Journal of the Royal
Statistical Society. Series A (General), 112(1), 1–34.
Smith, I. (1999) Road fatalities, modal split and Smeed's Law, Applied Economics Letters
6(4), 215–217.
Tamura, K. (2013) Mathematical study on trend of traffic accident deaths in Japan:
Analysis of macro trend using Smeed’s Law, AGI Working Paper Series 2013, 1–
23. [in Japanese]

13

Table Ⅰ. Sources of Data
Variable

Data

Survey Date

Year

SourceAccidents

Number of Traffic Accidents

Calendar Year

1988–2015

National Police Agency

NdeathNumber of Fatalities in Traffic Calendar Year

1988–2015

National Police Agency

1988–2015

Car ownership number:

Accidents
Nveh

Number of Registered Vehicles

End of March

Automobile Inspection & Registration Information Association
Pop

Prefecture Population

October 1

1987–2014

Basic resident register

Nhospital Number of Hospitals

October 1

1988–2015

Medical Facilities Survey: Ministry of Health, Labor and Welfare

October 1

1985, 1990, Census:

Area

Prefecture Area

1995, 2000, Statistical Survey Department, Statistics Bureau, Ministry of
2005, 2010, Internal Affairs and Communications
2015
Note: We fixed the area data in 1985–1989 to be equal to the surveyed data in 1985 and the data after 1990 are fixed in the same manner.

14

Table Ⅱ. Summary Statistics
Variable

MeanStandard Deviation

Minimum

Maximum

Ndeathln ()
Pop–2.69361

0.45435–4.43015–1.73229

Ndeathln ()
Accident–4.43517

0.50995–5.70787–3.31624

Accident
ln (
)
Pop

1.74157

0.32360

0.60851

2.58690

Nveh
ln (
)
Pop

6.43783

0.20993

5.788536.80430ln(Hdist)

1.224070.45435–4.43015–1.73229

15

Table Ⅲ. Estimation Results of Traditional Models

Ndeathln ()
Pop

Dependent Variables

ln (

Nveh
)
pop

Model 1A

Model 1B

Model 1C

Model 1D

Model 1E

0.098858–1.94222

0.962085

0.959081

1.53061(1.657)(–37.624)(10.894)(6.093)(25.371)―――0.00007541–0.0095714(0.023)(–63.4000)

Nveh
Time ∗ ln ()
popTime Dummies

No

No

Yes

Yes

No

Prefectural Dummies

No

Yes

Yes

Yes

Yes

Constant

Yes

No

No

No

No

Adjusted R-squared

0.00130.69570.93440.93430.9270AIC

829.277

69.7879–926.534–925.534–869.179

Note: t values are in parentheses.

16

Table Ⅳ Estimation Results of Accessibility to Hospital Models

Ndeathln ()
Pop

Dependent Variable
Model 2A

Model 2B

Model 2C

1.002481.03195

1.12921(6.109)(6.427)(9.332)0.00363244

0.00329325―(1.011)(0.922)–0.106817――–0.00287974–0.00302830–0.00251735(–2.093)(–2.218)(–2.002)

Time Dummies

Yes

Yes

Yes

Prefectural Dummies

Yes

Yes

Yes

Constant

No

No

No

Adjusted R-squared

0.93450.93450.9345AIC–926.546–927.142–927.690

ln (

Nveh
)
pop

Nveh
Time ∗ ln ()
pop
ln(Hdist)(–0.872)
Time ∗ ln(Hdist)Note: t values are in parentheses.

17

Table Ⅴ. Estimation Results of Fatalities per Accidents Models

Ndeathln ()
Accident

Dependent Variable
Model 3A

Model 3B

Model 3C

0.209680――–0.032053–0.029014–0.028817(–5.959)(–7.195)(–7.150)–0.196897–0.229130―(–1.074)(–1.278)0.020719

0.021113

0.020905(10.062)(10.523)(10.451)

Time Dummies

Yes

Yes

Yes

Prefectural Dummies

Yes

Yes

Yes

Constant

No

No

No

Adjusted R-squared

0.8836

0.8836

0.8835AIC–396.123–396.736–396.869

ln (

Nveh
)
pop

(0.854)
Nveh
Time ∗ ln ()
pop
ln(Hdist)
Time ∗ ln(Hdist)Note: t values are in parentheses.

18

Table Ⅵ. Estimation Results of Per Capita Accidents Models

Accident
ln (
)
Pop

Dependent Variables

ln (

Nveh
)
pop

Nveh
Time ∗ ln ()
pop
ln(Hdist)

Model 4A

Model 4B

0.792801

0.0767950(4.133)(4.092)0.035686

0.035972(8.493)(8.613)0.090081―(0.629)–0.023599–0.023474(–14.671)(–14.710)

Time Dummies

Yes

Yes

Prefectural Dummies

Yes

Yes

Constant

No

No

Adjusted R-squared

0.82350.8236AIC–721.073–721.862

Time ∗ ln(Hdist)Note: t values are in parentheses.

19

Table Ⅶ Estimation Results of Checking Reverse Causalityln(Hdist)

Dependent Variable
Model 5A

Model 5B

Model 5C

–0.107059–0.223446–0.220667(–0.238)(–5.698)(–6.054)0.0046643

0.00422610

0.00417486(3.640)(5.238)(5.483)–0.00977686――–0.119793–0.119768–0.121706(–5.159)(–5.238)(–5.819)–0.0000904455–0.0000710097―(–0.241)(–0.193)0.035709

0.035602

0.035762(5.646)(5.643)(5.720)Time Dummies

Yes

Yes

Yes

Prefectural Dummies

Yes

Yes

Yes

Constant

No

No

No

0.99620.99620.9962–2854.769–2855.733–2856.713

ln (

Nveh
)
pop

Nveh
Time ∗ ln (
)
pop

2

Nveh
(ln (
))
pop

(–0.260)ln (Accident)
Pop

Accident
Time ∗ ln (
)
Pop

2

Accident
(ln (
))
Pop

Adjusted R-squared
AIC
Note: t values are in parentheses.

20

1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015

-8.2

-8.4

-8.6

-8.8

-9.0

-9.2

-9.4

-9.6

Figure 1 Estimated Coefficients of Time Dummies in Model 1C
21

1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015

3.3

3.2

3.1

3.0

2.9

2.8

2.7

2.6

Figure 2 Estimated Coefficients of Time Dummies in Model 5C
22

0.95

ln(Hdist)

0.90

0.85

0.80

0.75

0.70
0

2

4

6

8

ln(Accident/Pop)
Figure 3 Relationship between Distance to Hospital and Accidents Per Capita
23

10

12

14