Side 1

Today’s program1. Econometrics is better suited for accident analysis than for

economics. Why?

2. Accident modelling implications

3. Remedies against overfitting

4. Structure of the TRULS model

INRETS, Arcueil, 30-31 May 2007

Side 2

Aim of workshop

Explore the shared interest in monitoring, explaining and forecasting road safety developments at the national or regional level.

Approach

Multivariate, structural modelling, focusing on substantive, causal relationships distinguishing exposure from risk and accident frequency from severity, while acknowledging the importance of estimating

functional form as well as first derivatives.

Side 3

Today’s program10.00: The TRULS model for Norway10.30: The DRAG-3 model for Quebec11.00: The KILOM-2 and TAG-2 models for France11.30: The Intercity Traffic Model for France11.45: Coffee12.00: Developing a model for Spain12.45: The DRAG-Algeria model13.10: Lunch14.15: The national model for Belgium14.30: The regional model for Stockholm, Sweden 14.45: Prospects for a Danish model15.00: Modelling overall Dutch safety performance 15.30: On simultaneous traffic-accident structures15.55: Coffee16.10: Accounting for spatial correlation in classical regression16.30: The NERDS-RSVP Consortium position paper16.45: End

The TRULS model for Norway – and other issues in accident modelling

byLasse Fridstrøm

Managing Director Institute of Transport Economics (TØI)

Oslo, Norwaylef@toi.nowww.toi.no

Side 5

Outline1. Econometrics is better suited for accident analysis than for

economics. Why?

2. Accident modelling implications

3. Remedies against overfitting

4. Structure of the TRULS model

Side 6

1. “…although econometrics was originally developed as a toolbox for economic research, it may … be even better suited for accident analysis”. (Fridstrøm 1999a)

In most econometric applications, the “error” term is random only in the sense of being unknown to the analyst. It is epistemically (subjectively) random.

Accident counts, on the other, are ontologically (objectively) random. Their distribution in a perfectly specified model is known: Poisson.

Had the individual accident been anticipated, it would not have happened! It is thus logically unpredictable. We are dealing with the whitest noise in behavioural science.

Thus, accident counts lend themselves to a natural and clear-cut distinction between the causal and the casual: systematic vs. random variation.

Side 7

The linear probability model

),...,2,1(1

niuxby i

J

jijji

,

eller

uXby systematic (causal) part random part

Usually, u is random only in the sense of being unobservable to the analyst. It is epistemically random, like in random utility theory.

Side 8

1. “…although econometrics was originally developed as a toolbox for economic research, it may … be even better suited for accident analysis”. (Fridstrøm 1999a)

In most econometric applications, the “error” term is random only in the sense of being unknown to the analyst. It is epistemically (subjectively) random.

Accident counts, on the other, are ontologically (objectively) random. Their distribution in a perfectly specified model is known: Poisson.

Had the individual accident been anticipated, it would not have happened! It is thus logically unpredictable. We are dealing with the whitest noise in behavioural science.

Thus, accident counts lend themselves to a natural and clear-cut distinction between the causal and the casual: systematic vs. random variation.

Side 9

Random and systematic variation coexist

trtrtrtrtrtr uuyEy x

tr

x x

ie ei trii i tr i

While the u terms are probabilistically independent, the terms are functionally dependent on certain common

factors and hence empirically correlated.

In the accident modelling case, we know that if all systematic variation has been accounted for through the x terms, then the y terms are independent Poisson variates.

Side 10

1. “…although econometrics was originally developed as a toolbox for economic research, it may … be even better suited for accident analysis”. (Fridstrøm 1999a)

In most econometric applications, the “error” term is random only in the sense of being unknown to the analyst. It is epistemically (subjectively) random.

Accident counts, on the other, are ontologically (objectively) random. Their distribution in a perfectly specified model is known: Poisson.

Had the individual accident been anticipated, it would not have happened! It is thus logically unpredictable. We are dealing with the whitest noise in behavioural science.

Thus, accident counts lend themselves to a natural and clear-cut distinction between the causal and the casual: systematic vs. random variation.

Side 11

Eeyore is right

”I’m not saying there won’t be an Accident now, mind

you.They’re funny things,

Accidents. You never have them till

you’re having them.” (A.A. Milne: The House at

Pooh Corner)

Side 12

The (generalized) Poisson distribution In the Poisson distribution, the variance equals the mean. Once

we have estimated the mean, we also know the amount of objectively random variation.

If the model is not perfectly specified, or if the individual events counted are not probabilistically independent, overdispersion most be expected, i. e. the variance exceeds the mean.

Thus, victim counts will typically be overdispersed.

It may be preferable to work with accident counts in combination with severity measures capturing victims per accident.

To allow - or test - for overdispersion, use the generalized Poisson model, i. e. the negative binomial regression model.

Side 13

Poisson distribution

95 per cent probability bounds around expected

value Expected number

Observed number

Side 14

The negative binomial distribution

tr tr tr2 1 ,

Suppose the Poisson parameter is itself random, and drawn from a gamma distribution with shape parameter (say). In this case the observed number of accidents can be shown to follow a negative binomial distribution with expected value (say) and variance

1

E tr tr

being the overdispersion parameter.

Two interpretations:

a. Unobserved heterogeneity (Greenwood & Yule 1920)

b. True contagion (Eggenberger and Pólya 1923)

Side 15

Misspecification may show up as overdispersion Suppose one relevant variable has been left out.

In this case some systematic variation is indeed contained in the error term:

trtrtrtrtrtrtr uuy

trIItri

I

iitri

I

iitr xxx

1

11

ln

Side 16

Is (generalized) Poisson regression the only way to go about?No. The limiting distribution of the Poisson is the normal. Approximation is good already for mean 10 and above.

But dependent variable should be log-transformed.

Since the variance of a Poisson variable equals its mean, “objective” heteroskedasticity can be accounted for through appropriate weighting. This requires iteration and sometimes cumbersome transformations.

Box-Cox regression models are useful, since for many partial relationships, curvature is not known a priori.

Side 17

For large Poisson counts y, the variance of ln(y) is inversely proportional to the expected value ω.

2)a(

)ayln(var

1

The Box-Tukey constant is needed, since the log of a Poisson variate has infinite variance.

a

Side 18

The variance of ln(y+a), where y is Poisson distributed with parameter ω.

var[ln(y+a)]

0.001

0.01

0.1

1

10

0.001 0.01 0.1 1 10 100 1000

a=0.01

a=0.1

a=0.5

a=1

Source: TØI report 457

Side 19

The IRPOSKML method of estimation: improved error variance approximation for small accident counts

TRIO PROJECT: TRULS - an econometric model of road use, accidents and their severityDATE: 99 04 13USER: toi

ω values ranging from 0.000248

to 692

Source: TØI report 457

Side 20

2. Implications for accident modelling

Concentrate on substance Concentrate on exposure - estimate exposure elasticities Use multiplicative models Use cumulative severity categories Avoid autoregressive models

Side 21

Concentrate on substance Accident models are useful in estimating a variety of policy relevant

parameters, including the marginal external accident cost, the (marginal) contribution of various road user categories to risk, the effect of accident countermeasures, and the importance of behavioral response (risk compensation).

The policy relevant, explanatory factors are in the systematic part, not in the random term.

There is no need to further explore the random term, whose properties are already better known than in any other econometric application!

Advanced filtering and transformation will drain the juice out of the pattern of co-variation, without adding substantive knowledge. Avoid differencing! Let the levels speak!

Spend your resources on identifying systematic, causal factors, and on estimating their curvature and elasticities!

Side 22

Concentrate on exposure

The most important explanatory factor in any accident model is going to be exposure.

Give priority to its measurement and estimation!

Note that exposure is multidimensional! There are cars, trucks, buses, tramways, motorcycles, bicycles and pedestrians. Their interaction in producing accidents is of prime interest.

The elasticity of accident frequency w. r. t. exposure is not necessarily 1. Estimate it! The marginal (external) cost of accidents depends on it.

The relationships are not necessarily (log-)linear. Estimate their curvatures!

Side 23

Risk and accident frequency are non-negative magnitudesA minimum logical requirement is that models do not predict

negative accident frequencies or risk.

Additive linear regression models are therefore ruled out.

Multiplicative models are the canonical form. Risk factors compound. The systematic part of the regression should be decomposable as the product of various factors.

Models for non-negative integer-valued variables are a natural choice.

Side 24

Use cumulative severity categories Unless severity is defined

cumulatively, models may provide counterintuitive effect parameters that are hard to interpret.

Road safety measures may inflate any category except the uppermost. Hence always include the more serious accidents than the ones considered.

Fatal injuries

Critical injuries

Serious injuries

Slight injuries

Property damage only

Side 25

Avoid autoregressive models

Trying to explain the causal part in period t by means of the

white noise in period t-1, t-2, etc.

This is obfuscation, not explanation.

But: Do not confuse autoregression with autocorrelation. Autocorrelated models are OK.

trrtrtrt

trrttr

uuyE

uyy

,1,1,1

,1

....

....

x

Side 26

3. Remedies against overfitting Use specialized goodness-of-fit measures Use casualty subset tests Splitting the sample – and out-of-sample prediction

Side 27

The upper bound on explanatory power is computable

• On account of the Poisson assumption, it is possible, for a given accident data set, to calculate the normal amount of random variation and hence also the maximal amount of explainable, systematic variation.

Using this information, one may calculate goodness-of-fit measures for the systematic variation of interest, thus comparing the explained to the explainable.See AA&P vol 27, pp 1-20 (1995)

Side 28

Specialized goodness-of-fit Consider the well-known coefficient of determination

An observable upper bound on the coefficient of determination is given by

Compute the coefficient of determination for systematic variation

Side 29

Randomness accounts for a large part of variation in smaller accident counts

Source: AA&P 27 (1):1-20 (1995)

Side 30

Victim counts are overdispersed

Source: AA&P 27 (1):1-20 (1995)

Side 31

The casualty subset testThe affirmative casualty subset test: For any explanatory

factor operating through its presumed effect on a particular subset of casualties, the effect should be extra strong as applied to this subset. Ex.: seat belts and car occupants.

The complement casualty subset test: For any explanatory factor not affecting a particular subset of casualties, the effect should be zero as applied to this subset. Ex.: seat belts and pedestrians.

The converse casualty subset test: For any explanatory having an opposite effect on a particular subset of casualties, the effect should be sign-reversed as applied to this subset. Ex.: seat belts and seat belt users killed or injured.

Side 32

Out-of-sample prediction

A model can provide a perfect fit inside its own sample, and yet quite bad out-of-sample predictions.

The proof of the pudding is in the eating!

Source: Partyka (1991) (AA&P 23:423-430), quoted by Elvik (2007)

Side 33

4. The TRULS model for Norway – a member of the DRAG familyRecursive system of equations at the county and month level: 19

counties x 264 months (22 years) = 5016 observations. Observations cover 1973-94. The model has not been updated.Equations:1. Car ownership2. Exposure: light and heavy vehicle road use, MCs, and public

transport 3. Seat belt use 4. Injury accident frequency5. Severity: fatalities, dangerously injured, severely injured6. Various casualty subset equations: • single vs multiple vehicle crashes; • heavy vehicle crashes; • car occupant, bicyclist, and pedestrian victims; • (non-)seat belt users injured

Side 34

The TRULS modelInjury accident frequency:

Severity:

tri

triitr

trA

trA

trHtrAtr ux

lv

vv

lnvlnayln i

3321

3

tri

)(trii

tr

tr uxay

ahxi

Side 35

The TRULS model for Norway

Estimated elasticities w r t exposure, by severity.

0.000

0.000

0.000

0.000

-0.062

-0.253

0.911

-0.414

0.149

0.026

0.212

0.240

1.033

-0.238

0.165

0.034

0.216

-0.144

0.680

0.175

-0.043

0.001

0.168

-1.339

0.761

0.228

0.110

-0.007

-0.014

-1.143

-1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Motor vehiclekilometers

Traffic density (vehkms pr km road)

Heavy vehicle trafficshare

MC exposure (proxy)

Bus service density

Streetcar/subwaydensity

Elasticity

Fatalities

Dangerously injured

Severely injured

Injury accidents

Traffic volume

Source: TØI report 457

Side 36

The TRULS model for Norway

Estimated elasticities w r t exposure, by road user category.

0.000

0.000

0.000

0.000

-0.062

-0.253

0.911

-0.414

0.149

0.026

0.212

0.240

0.962

-0.319

-0.146

0.001

0.149

-0.245

0.749

0.012

0.476

0.208

0.362

-0.505

1.079

-0.604

0.529

0.254

0.110

1.104

1.109

-0.972

0.105

0.034

0.756

1.196

-1.0 -0.5 0.0 0.5 1.0 1.5

Motor vehiclekilometers

Traffic density(veh kms pr km

road)

Heavy vehicletraffic share

MC exposure(proxy)

Bus servicedensity

Streetcar/subwaydensity

Elasticity

Pedestrians injured

Bicyclists injured

MC occupants injured

Car occupants injured

Injury accidents

Traffic volume

Source: TØI report 457

Side 37

TRIO PROJECT: TRULS - an econometric model of road use, accidents and their severityDATE: 99 01 10USER: toi

The TRULS model for Norway:

relative injury accident risk as a function

of traffic density.

5016 sample points (19 counties x

264 months).

Side 38

TRIO PROJECT: TRULS - an econometric model of road use, accidents and their severityDATE: 99 01 11USER: toi

The TRULS model for Norway: relative accident

elasticities with respect to road use, as a function

of traffic density.

5016 sample points (19 counties x

264 months).

Side 39

The TRULS model for Norway:

relative injury accident

frequency as a function of aggregate

seat belt use.

5016 sample points (19 counties x

264 months).

TRIO PROJECT: TRULS - an econometric model of road use, accidents and their severityDATE: 99 02 09USER: toi

Side 40

According to TRULS, heavy vehicles are 3.8 times (=1.321/0.345) more dangerous than light ones.

Table 2: Estimated measures of partial association between injury accidents and overall, light vehicle and heavy vehicle road use. Minimal, mean and maximal sample point values.

Traffic category Elasticity Inverse traffic share times elasticity

Minimum Mean Maximum Minimum Mean Maximum

Total vehicle kilometres 0.484 0.494 0.506 0.484 0.494 0.506

Light vehicle kilometres 0.248 0.291 0.361 0.335 0.345 0.357

Heavy vehicle kilometres 0.181 0.202 0.236 0.909 1.321 1.974

Light vehicle road users generate a positive external accident costs only if their own share of the accident cost is less than 34 %.

Side 41

Thank you for listening!

Read more: TØI report 457/1999 Acc. Anal. & Prev. 27 (1):1-20 (1995)

Side 42

The Poisson distributionThere are compelling theoretical and empirical reasons to

assume that accident counts are Poisson distributed. The Poisson is a one parameter distribution:

trtrtr yvaryE

When we know the mean, we also know how much variance to expect around it!

The coefficient of variation decreases with the mean:

trtr

tr

tr

tr

yE

y

1var

Side 43

Generalized Poisson variates Integer valued: 0, 1, 2, … Zero occurrences OK. Poisson invariance under summation Non-negative outcome and positive expected value.

Suggests multiplicative structure of cofactors/independent variables.

Estimable through maximum likelihood (ML) methods. ML implicitly takes account of heteroskedasticity

Side 44

Probabilistic theories are complete Einstein: “He [God] does not play dice.”

Salmon (1984): Certain laws are ”irreducibly statistical”,

i. e. they include an inevitable, objectively random component. Single events may occur at random intervals, but with an almost constant overall frequency in the long run. Such laws are common in particle physics, but rare in behavioral science.

Although the single event is all but impossible to predict, the collection of such events may very well behave in a perfectly predictable way, amenable to description by means of precise mathematical-statistical relationships.

Ex.: radioactive decay (C14 method), die tossing, road accidents.

Side 45

95 per cent, overdispersed probability interval around trend-fitted annual road fatalities in Norway.

0

100

200

300

400

500

600

700

1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

Yearly road fatalities in Norway

Source: Elvik (2005), TØI report 792

Side 46

The law of rare eventsConsider a time-varying random variable Y(t) such that

Then ,,,m,

!met

msYstYPtm

210

.

i. e., the number of events occurring during any interval of length t (say) has a Poisson distribution with mean

00 Y ,

0t,tY has stationary independent increments,

totYP 2 , and

tottYP 1 ,

where a function f is said to be to if

00

t

tflimt

.

t