Highway accident severities and the mixed

logit model: An exploratory analysis

John Milton, Venky Shankar, Fred Mannering

Background

State agencies generally look only at accident frequencies when programming safety highway improvement.

Example: Washington State uses negative

binomial and zero-inflated models to forecast accident frequencies.

Problems with frequency-dominated approaches

Some do not consider severity which may be the critical element.

Some only simplistically consider severity

leading to problematic assumptions.

Frequency-dominated approaches tend to overlly favor urban areas.

How to forecast injury severity?

Detailed severity models based on individual accidents. Too complex for forecasting purposes (require

information on age and gender of driver, type of car, restraint usage, alcohol consumption, etc.).

Separate frequency models for different severity types. Ignores correlation among severity outcomes. Can lead to very complex modeling structures.

Past methodological approaches

Logistic regression and bivariate models.

Ordered probability models.

Multinomial and nested logit models.

Proposed approach

Assume the frequency of accidents is known (well developed methods exist for determining these).

Divide highways into segments.

Develop a model to forecast the proportion of accidents by severity levels on highway segments.

Differences relative to existing approaches:

More aggregate – cannot include specific accident characteristics (driver characteristics, vehicle characteristics, restraint usage, alcohol consumption, etc.).

Has advantage of easy application (does not require forecasting of many accident-specific variables).

Methodological approach

Without detailed accident information, our approach potentially introduces a heterogeneity problem.

Heterogeneity could result in varying effects of X that could be captured with random parameters.

Mixed logit may be appropriate.

Define:

where Sin is a severity function determining the

injury-severity category i proportion on roadway segment n;

Xin is a vector of explanatory variables (weather, geometric, pavement, roadside and traffic variables);

βi is a vector of estimable parameters; and

εin is error term.

in i in inS X

If εin’s are assumed to be generalized extreme value distributed,

where

Pn(i) is the proportion of injury-severity category (from the set of all injury-severity

categories I) on roadway segment n .

i in

n

i InI

EXPP i

EXP

X

X

The mixed logit is:

where f (β | φ) is the density function of β with φ referring to a vector of parameters of the density function (mean and variance).

With this, β can now account for segment-specific variations of the effect of X on injury-severity proportions, with the density function f (β | φ) used to determine β .

i inin

i InI

EXP XP f | d

EXP X

Mixed logit

Relaxes possible IIA problems with a more general error-term structure.

Can test a variety of distribution options for β .

Estimated with simulation based maximum likelihood.

Empirical setting Seek to model the annual proportion of

accidents by injury severity on roadway segments.

Injury-severities: property damage only; possible injury; injury.

Multilane divided highways in Washington State.

274 roadway segments defined (average length 2.7 miles).

Empirical setting

Accident data from 1990-94 (22,568 accidents; 56% property damage only; 22% possible injury; 22% injury).

Accident data linked with weather, geometric, pavement, roadside and traffic data.

Descriptive statisticsVariable Mean Std. Dev. Minimum Maximum

Average daily traffic 37,354 3,696 3,347 172,557

Average annual precipitation in inches 29.90 21.84 4.56 131.76

Average annual snowfall in inches 15.12 42.6 0 652

Percentage of trucks 14.16 6.68 3.20 32.00

Number of interchanges per mile 0.85 0.83 0 4

Speed limit in miles per hour 59.67 5.50 20.00 65

Friction number of pavement surface 46.82 5.63 20.00 61.5

Number of horizontal curves per mile 1.44 0.95 0 5

Number of changes in vertical profile per mile

1.88 1.69 0 20

Average daily truck traffic 4,165 3070 549 14,032

Estimation results: Variable

Value

Standard Error

t-statistic

Property damage only

Constant (Standard error of parameter distribution)

-0.6847 (1.7184)

0.2822 (0.8152)

-2.43 (2.11)

Average daily traffic per lane in thousands (Standard error of parameter distribution)

0.0792 (0.7143)

0.0365 (0.2438)

2.17 (2.93)

Average annual snowfall in inches (Standard error of parameter distribution)

0.0116 (0.0475)

0.0051 (0.0220)

2.29 (2.16)

Possible injury

Constant (Standard error of distribution)

-0.6205 (0.4338)

0.1995 (0.5507)

-3.11 (0.79)

Percentage of trucks (Standard error of parameter distribution)

-0.1617 (0.1350)

0.0506 (0.0453)

-3.20 (2.99)

Injury

Average daily truck traffic in thousands (Standard error of parameter distribution)

-0. 4669 (0. 6771)

0.1085 (0.1932)

-4.30 (3.51)

Number of horizontal curves per mile (Fixed parameter)

-0.3274 0.0761 -4.30

Number of changes in vertical profile per mile (Fixed parameter)

-0.0947 0.0257 -3.69

Findings: Average Daily Traffic

Defined for Property damage only

Parameter normally distributed; mean = 0.0792; s.d. = 0.7143

For roadway segments, 45.6% less than zero, 54.4% greater than zero.

Possible differences in driver behavior across the state changes this effect.

Findings: Average Annual Snowfall

Defined for Property damage only

Parameter normally distributed; mean = 0.1390; s.d. = 0.5703

For roadway segments, 37.9% less than zero, 62.1% greater than zero.

Most sections reduce severity but not all. Again, driver behavior differences.

Findings: Percentage of trucks

Defined for Possible Injury

Parameter normally distributed; mean = -0.1617; s.d. = 0.1350

For roadway segments, 88.1% less than zero, 11.9% greater than zero.

For most sections increasing percentages push severities to high/low extremes.

Findings: Average daily number of trucks

Defined for Injury

Parameter normally distributed; mean = -0.4669; s.d. = 0.6771

For roadway segments, 76% less than zero, 24% greater than zero.

For most sections increasing number of trucks reduces “injury” proportions.

Findings: Number of horizontal curves

Defined for Injury

Fixed Parameter; mean = -0.3274

Drivers compensating by driving more cautiously?

Findings: Number of changes in vertical profile

Defined for Injury

Fixed Parameter; mean = -0.0947

Drivers compensating by driving more cautiously?

Summary

Mixed logit has the potential to provide highway agencies with a new way of estimating injury severities.

Method needs to be applied to more diverse road classes, such as two-lane highways.