# Markov switching multinomial logit model: An application to accident-injury severities

**ABSTRACT** In this study, two-state Markov switching multinomial logit models are proposed for statistical modeling of accident-injury severities. These models assume Markov switching over time between two unobserved states of roadway safety as a means of accounting for potential unobserved heterogeneity. The states are distinct in the sense that in different states accident-severity outcomes are generated by separate multinomial logit processes. To demonstrate the applicability of the approach, two-state Markov switching multinomial logit models are estimated for severity outcomes of accidents occurring on Indiana roads over a four-year time period. Bayesian inference methods and Markov Chain Monte Carlo (MCMC) simulations are used for model estimation. The estimated Markov switching models result in a superior statistical fit relative to the standard (single-state) multinomial logit models for a number of roadway classes and accident types. It is found that the more frequent state of roadway safety is correlated with better weather conditions and that the less frequent state is correlated with adverse weather conditions.

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**ABSTRACT:**The analysis of highway-crash data has long been used as a basis for influencing highway and vehicle designs, as well as directing and implementing a wide variety of regulatory policies aimed at improving safety. And, over time there has been a steady improvement in statistical methodologies that have enabled safety researchers to extract more information from crash databases to guide a wide array of safety design and policy improvements. In spite of the progress made over the years, important methodological barriers remain in the statistical analysis of crash data and this, along with the availability of many new data sources, present safety researchers with formidable future challenges, but also exciting future opportunities. This paper provides guidance in defining these challenges and opportunities by first reviewing the evolution of methodological applications and available data in highway-accident research. Based on this review, fruitful directions for future methodological developments are identified and the role that new data sources will play in defining these directions is discussed. It is shown that new methodologies that address complex issues relating to unobserved heterogeneity, endogeneity, risk compensation, spatial and temporal correlations, and more, have the potential to significantly expand our understanding of the many factors that affect the likelihood and severity (in terms of personal injury) of highway crashes. This in turn can lead to more effective safety countermeasures that can substantially reduce highway-related injuries and fatalities.Analytic Methods in Accident Research. 01/2013; - SourceAvailable from: Baoshan Huang[Show abstract] [Hide abstract]

**ABSTRACT:**Objective: The severity of traffic-related injuries has been studied by many researchers in recent decades. However, the evaluation of many factors is still in dispute and, until this point, few studies have taken into account pavement management factors as points of interest. The objective of this article is to evaluate the combined influences of pavement management factors and traditional traffic engineering factors on the injury severity of 2-vehicle crashes. Methods: This study examines 2-vehicle rear-end, sideswipe, and angle collisions that occurred on Tennessee state routes from 2004 to 2008. Both the traditional ordered probit (OP) model and Bayesian ordered probit (BOP) model with weak informative prior were fitted for each collision type. The performances of these models were evaluated based on the parameter estimates and deviances. Results: The results indicated that pavement management factors played identical roles in all 3 collision types. Pavement serviceability produces significant positive effects on the severity of injuries. The pavement distress index (PDI), rutting depth (RD), and rutting depth difference between right and left wheels (RD_df) were not significant in any of these 3 collision types. The effects of traffic engineering factors varied across collision types, except that a few were consistently significant in all 3 collision types, such as annual average daily traffic (AADT), rural-urban location, speed limit, peaking hour, and light condition. Conclusions: The findings of this study indicated that improved pavement quality does not necessarily lessen the severity of injuries when a 2-vehicle crash occurs. The effects of traffic engineering factors are not universal but vary by the type of crash. The study also found that the BOP model with a weak informative prior can be used as an alternative but was not superior to the traditional OP model in terms of overall performance.Traffic Injury Prevention 07/2013; 14(5):544-553. · 1.04 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Referring to the 1 248 survey data of rural population in 14 provinces of China, the influencing factors of trip time choice were analyzed. Based on the basic theory of disaggregate model and its modelling method, nine grades were selected as the alternatives of trip time, the variables affecting time choice and the method getting their values were determined, and a multinomial logit (MNL) model was developed. Another 1 200 trip data of rural population were selected to testify the model’s validity. The result shows that the maximum absolute error of each period between calculated value and statistic is 3.6%, so MNL model has high calculation accuracy.Journal of Central South University. 20(1).

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arXiv:0811.3644v1 [stat.AP] 21 Nov 2008

Markov switching multinomial logit model: an

application to accident injury severities

Nataliya V. Malyshkina∗, Fred L. Mannering

School of Civil Engineering, 550 Stadium Mall Drive, Purdue University, West

Lafayette, IN 47907, United States

Abstract

In this study, two-state Markov switching multinomial logit models are proposed

for statistical modeling of accident injury severities. These models assume Markov

switching in time between two unobserved states of roadway safety. The states are

distinct, in the sense that in different states accident severity outcomes are generated

by separate multinomial logit processes. To demonstrate the applicability of the

approach presented herein, two-state Markov switching multinomial logit models

are estimated for severity outcomes of accidents occurring on Indiana roads over

a four-year time interval. Bayesian inference methods and Markov Chain Monte

Carlo (MCMC) simulations are used for model estimation. The estimated Markov

switching models result in a superior statistical fit relative to the standard (single-

state) multinomial logit models. It is found that the more frequent state of roadway

safety is correlated with better weather conditions. The less frequent state is found

to be correlated with adverse weather conditions.

Key words: Accident injury severity; multinomial logit; Markov switching;

Bayesian; MCMC

1Introduction

Vehicle accidents result in property damage, injuries and loss of people lives.

Thus, research efforts in predicting accident severity are clearly very impor-

tant. In the past there has been a large number of studies that focused on mod-

eling accident severity outcomes. Common modeling approaches of accident

∗Corresponding author.

Email addresses: nmalyshk@purdue.edu (Nataliya V. Malyshkina),

flm@ecn.purdue.edu (Fred L. Mannering).

Preprint submitted to Accident Analysis and Prevention

Page 2

severity include multinomial logit models, nested logit models, mixed logit

models and ordered probit models (O’Donnell and Connor, 1996; Shankar and Mannering,

1996; Shankar et al., 1996; Duncan et al., 1998; Chang and Mannering, 1999;

Carson and Mannering, 2001; Khattak, 2001; Khattak et al., 2002; Kockelman and Kweon,

2002; Lee and Mannering, 2002; Abdel-Aty, 2003; Kweon and Kockelman, 2003;

Ulfarsson and Mannering, 2004; Yamamoto and Shankar, 2004; Khorashadi et al.,

2005; Eluru and Bhat, 2007; Savolainen and Mannering, 2007; Milton et al.,

2008). All these models involve nonlinear regression of the observed accident

injury severity outcomes on various accident characteristics and related factors

(such as roadway and driver characteristics, environmental factors, etc).

In our earlier paper, Malyshkina et al. (2008), which we will refer to as Pa-

per I, we presented two-state Markov switching count data models of accident

frequencies. In this study, which is a continuation of our work on Markov

switching models, we present two-state Markov switching multinomial logit

models for predicting accident severity outcomes. These models assume that

there are two unobserved states of roadway safety, roadway entities (road-

way segments) can switch between these states over time, and the switching

process is Markovian. The two states intend to account for possible hetero-

geneity effects in roadway safety, which may be caused by various unpre-

dictable, unidentified, unobservable risk factors that influence roadway safety.

Because the risk factors can interact and change, roadway entities can switch

between the two states over time. Two-state Markov switching multinomial

logit models assume separate multinomial logit processes for accident severity

data generation in the two states and, therefore, allow a researcher to study

the heterogeneity effects in roadway safety.

2 Model specification

Markov switching models are parametric and can be fully specified by a like-

lihood function f(Y|Θ,M), which is the conditional probability distribution

of the vector of all observations Y, given the vector of all parameters Θ of

model M. First, let us consider Y. Let Ntbe the number of accidents ob-

served during time period t, where t = 1,2,...,T and T is the total number

of time periods. Let there be I discrete outcomes observed for accident sever-

ity (for example, I = 3 and these outcomes are fatality, injury and property

damage only). Let us introduce accident severity outcome dummies δ(i)

are equal to unity if the ithseverity outcome is observed in the nthaccident

that occurs during time period t, and to zero otherwise. Here i = 1,2,...,I,

n = 1,2,...,Ntand t = 1,2,...,T. Then, our observations are the accident

severity outcomes, and the vector of all observations Y = {δ(i)

outcomes observed in all accidents that occur during all time periods. Sec-

ond, let us consider model specification variable M. It is M = {M,Xt,n}

t,nthat

t,n} includes all

2

Page 3

and includes the model’s name M (for example, M = “multinomial logit”)

and the vector Xt,nof all accident characteristic variables (weather and envi-

ronment conditions, vehicle and driver characteristics, roadway and pavement

properties, and so on).

To define the likelihood function, we first introduce an unobserved (latent)

state variable st, which determines the state of all roadway entities during

time period t. At each t, the state variable st can assume only two values:

st= 0 corresponds to one state and st= 1 corresponds to the other state (t =

1,2,...,T). The state variable stis assumed to follow a stationary two-state

Markov chain process in time,1which can be specified by time-independent

transition probabilities as

P(st+1= 1|st= 0) = p0→1,P(st+1= 0|st= 1) = p1→0. (1)

Here, for example, P(st+1= 1|st= 0) is the conditional probability of st+1= 1

at time t + 1, given that st= 0 at time t. Transition probabilities p0→1and

p1→0 are unknown parameters to be estimated from accident severity data.

The stationary unconditional probabilities of states st = 0 and st = 1 are

¯ p0= p1→0/(p0→1+p1→0) and ¯ p1= p0→1/(p0→1+p1→0) respectively.2Without

loss of generality, we assume that (on average) state st = 0 occurs more

or equally frequently than state st = 1. Therefore, ¯ p0 ≥ ¯ p1, and we obtain

restriction3

p0→1≤ p1→0.(2)

We refer to states st= 0 and st= 1 as “more frequent” and “less frequent”

states respectively.

Next, a two-state Markov switching multinomial logit (MSML) model assumes

multinomial logit (ML) data-generating processes for accident severity in each

of the two states. With this, the probability of the ithseverity outcome ob-

served in the nthaccident during time period t is

1

Markov property means that the probability distribution of st+1depends only

on the value stat time t, but not on the previous history st−1,st−2,.... Stationarity

of {st} is in the statistical sense.

2These can be found from stationarity conditions ¯ p0 = (1 − p0→1)¯ p0+ p1→0¯ p1,

¯ p1= p0→1¯ p0+ (1 − p1→0)¯ p1and ¯ p0+ ¯ p1= 1.

3Without any loss of generality, restriction (2) is introduced for the purpose of

avoiding the problem of state label switching 0 ↔ 1. This problem would otherwise

arise because of the symmetry of Eqs. (1)–(4) under the label switching.

3

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P(i)

t,n=

exp(β′

j=1exp(β′

exp(β′

?I

i=1,2,...,I,

(0),iXt,n)

?I

(0),jXt,n)

if st= 0,

(1),iXt,n)

j=1exp(β′

(1),jXt,n)

if st= 1,

(3)

n = 1,2,...,Nt,t = 1,2,...,T,

Here prime means transpose (so β′

vectors β(0),iand β(1),iare unknown estimable parameters of the two standard

multinomial logit probability mass functions (Washington et al., 2003) in the

two states, st= 0 and st= 1 respectively. We set the first component of Xt,n

to unity, and, therefore, the first components of vectors β(0),iand β(1),iare

the intercepts in the two states. In addition, without loss of generality, we set

all β-parameters for the last severity outcome to zero,4β(0),I= β(1),I= 0.

(0),iis the transpose of β(0),i). Parameter

If accident events are assumed to be independent, the likelihood function is

f(Y|Θ,M) =

T?

t=1

Nt

?

n=1

I?

i=1

?

P(i)

t,n

?δ(i)

t,n. (4)

Here, because the state variables st,nare unobservable, the vector of all es-

timable parameters Θ must include all states, in addition to model parameters

(β-s) and transition probabilities. Thus, Θ = [β′

vector S = [s1,s2,...,sT]′has length T and contains all state values. Eqs. (1)-

(4) define the two-state Markov switching multinomial logit (MSML) model

considered here.

(0),β′

(1),p0→1,p1→0,S′]′, where

3Model estimation methods

Statistical estimation of Markov switching models is complicated by unobserv-

ability of the state variables st.5As a result, the traditional maximum likeli-

hood estimation (MLE) procedure is of very limited use for Markov switching

models. Instead, a Bayesian inference approach is used. Given a model M

with likelihood function f(Y|Θ,M), the Bayes formula is

f(Θ|Y,M) =f(Y,Θ|M)

f(Y|M)

=f(Y|Θ,M)π(Θ|M)

?f(Y,Θ|M)dΘ

.(5)

4This can be done because Xt,nare assumed to be independent of the outcome i.

5Below we will have 208 time periods (T = 208). In this case, there are 2208possible

combinations for value of vector S = [s1,s2,...,sT]′.

4

Page 5

Here f(Θ|Y,M) is the posterior probability distribution of model parameters

Θ conditional on the observed data Y and model M. Function f(Y,Θ|M)

is the joint probability distribution of Y and Θ given model M. Function

f(Y|M) is the marginal likelihood function – the probability distribution of

data Y given model M. Function π(Θ|M) is the prior probability distribution

of parameters that reflects prior knowledge about Θ. The intuition behind

Eq. (5) is straightforward: given model M, the posterior distribution accounts

for both the observations Y and our prior knowledge of Θ.

In our study (and in most practical studies), the direct application of Eq. (5) is

not feasible because the parameter vector Θ contains too many components,

making integration over Θ in Eq. (5) extremely difficult. However, the poste-

rior distribution f(Θ|Y,M) in Eq. (5) is known up to its normalization con-

stant, f(Θ|Y,M) ∝ f(Y|Θ,M)π(Θ|M). As a result, we use Markov Chain

Monte Carlo (MCMC) simulations, which provide a convenient and practi-

cal computational methodology for sampling from a probability distribution

known up to a constant (the posterior distribution in our case). Given a large

enough posterior sample of parameter vector Θ, any posterior expectation and

variance can be found and Bayesian inference can be readily applied. A reader

interested in details is referred to our Paper I or to Malyshkina (2008), where

we describe our choice of the prior distribution π(Θ|M) and the MCMC sim-

ulation algorithm.6Although, in this study we estimate a two-state Markov

switching multinomial logit model for accident severity outcomes and in Pa-

per I we estimated a two-state Markov switching negative binomial model for

accident frequencies, this difference is not essential for the Bayesian-MCMC

model estimation methods. In fact, the main difference is in the likelihood

function (multinomial logit as opposed to negative binomial). So we used the

same our own numerical MCMC code, written in the MATLAB programming

language, for model estimation in both studies. We tested our code on arti-

ficial data sets of accident severity outcomes. The test procedure included a

generation of artificial data with a known model. Then these data were used

to estimate the underlying model by means of our simulation code. With this

procedure we found that the MSML models, used to generate the artificial

data, were reproduced successfully with our estimation code.

For comparison of different models we use a formal Bayesian approach. Let

there be two models M1and M2with parameter vectors Θ1and Θ2respec-

tively. Assuming that we have equal preferences of these models, their prior

probabilities are π(M1) = π(M2) = 1/2. In this case, the ratio of the models’

posterior probabilities, P(M1|Y) and P(M2|Y), is equal to the Bayes fac-

tor. The later is defined as the ratio of the models’ marginal likelihoods (see

Kass and Raftery, 1995). Thus, we have

6Our priors for β-s, p0→1and p1→0are flat or nearly flat, while the prior for the

states S reflects the Markov process property, specified by Eq. (1).

5

Page 6

P(M2|Y)

P(M1|Y)=f(M2,Y)/f(Y)

f(M1,Y)/f(Y)=f(Y|M2)π(M2)

f(Y|M1)π(M1)=f(Y|M2)

f(Y|M1),(6)

where f(M1,Y) and f(M2,Y) are the joint distributions of the models and

the data, f(Y) is the unconditional distribution of the data. As in Paper I,

to calculate the marginal likelihoods f(Y|M1) and f(Y|M2), we use the

harmonic mean formula f(Y|M)−1= E [f(Y|Θ,M)−1|Y], where E(...|Y)

means posterior expectation calculated by using the posterior distribution. If

the ratio in Eq. (6) is larger than one, then model M2is favored, if the ratio

is less than one, then model M1is favored. An advantage of the use of Bayes

factors is that it has an inherent penalty for including too many parameters

in the model and guards against overfitting.

To evaluate the performance of model {M,Θ} in fitting the observed data Y,

we carry out the Pearson’s χ2goodness-of-fit test (Maher and Summersgill,

1996; Cowan, 1998; Wood, 2002; Press et al., 2007). We perform this test by

Monte Carlo simulations to find the distribution of the Pearson’s χ2quan-

tity, which measures the discrepancy between the observations and the model

predictions (Cowan, 1998). This distribution is then used to find the goodness-

of-fit p-value, which is the probability that χ2exceeds the observed value of

χ2under the hypothesis that the model is true (the observed value of χ2is

calculated by using the observed data Y). For additional details, please see

Malyshkina (2008).

4Empirical results

The severity outcome of an accident is determined by the injury level sustained

by the most injured individual (if any) involved into the accident. In this study

we consider three accident severity outcomes: “fatality”, “injury” and “PDO

(property damage only)”, which we number as i = 1,2,3 respectively (I = 3).

We use data from 811720 accidents that were observed in Indiana in 2003-2006.

As in Paper I, we use weekly time periods, t = 1,2,3,...,T = 208 in total.7

Thus, the state stcan change every week. To increase the predictive power

of our models, we consider accidents separately for each combination of acci-

dent type (1-vehicle and 2-vehicle) and roadway class (interstate highways, US

routes, state routes, county roads, streets). We do not consider accidents with

more than two vehicles involved.8Thus, in total, there are ten roadway-class-

accident-type combinations that we consider. For each roadway-class-accident-

7A week is from Sunday to Saturday, there are 208 full weeks in the 2003-2006

time interval.

8Among 811720 accidents 241011 (29.7%) are 1-vehicle, 525035 (64.7%) are 2-

vehicle, and only 45674 (5.6%) are accidents with more than two vehicles involved.

6

Page 7

type combination the following three types of accident frequency models are

estimated:

• First, we estimate a standard multinomial logit (ML) model without Markov

switching by maximum likelihood estimation (MLE).9We refer to this

model as “ML-by-MLE”.

• Second, we estimate the same standard multinomial logit model by the

Bayesian inference approach and the MCMC simulations. We refer to this

model as “ML-by-MCMC”. As one expects, the estimated ML-by-MCMC

model turned out to be very similar to the corresponding ML-by-MLE model

(estimated for the same roadway-class-accident-type combination).

• Third, we estimate a two-state Markov switching multinomial logit (MSML)

model by the Bayesian-MCMC methods. In order to make comparison of ex-

planatory variable effects in different models straightforward, in the MSML

model we use only those explanatory variables that enter the corresponding

standard ML model.10To obtain the final MSML model reported here, we

also consecutively construct and use 60%, 85% and 95% Bayesian credible

intervals for evaluation of the statistical significance of each β-parameter.

As a result, in the final model some components of β(0)and β(1)are re-

stricted to zero or restricted to be the same in the two states.11We refer

to this final model as “MSML”.

Note that the two states, and thus the MSML models, do not have to exist

for every roadway-class-accident-type combination. For example, they will not

exist if all estimated model parameters turn out to be statistically the same

in the two states, β(0)= β(1), (which suggests the two states are identical and

the MSML models reduce to the corresponding standard ML models). Also,

the two states will not exist if all estimated state variables stturn out to be

close to zero, resulting in p0→1 ≪ p1→0 [compare to Eq. (2)], then the less

9

To obtain parsimonious standard models, estimated by MLE, we choose the

explanatory variables and their dummies by using the Akaike Information Criterion

(AIC) and the 5% statistical significance level for the two-tailed t-test. Minimization

of AIC = 2K − 2LL, were K is the number of free continuous model parameters

and LL is the log-likelihood, ensures an optimal choice of explanatory variables in

a model and avoids overfitting (Tsay, 2002; Washington et al., 2003). For details on

variable selection, see Malyshkina (2006).

10A formal Bayesian approach to model variable selection is based on evaluation

of model’s marginal likelihood and the Bayes factor (6). Unfortunately, because

MCMC simulations are computationally expensive, evaluation of marginal likeli-

hoods for a large number of trial models is not feasible in our study.

11A β-parameter is restricted to zero if it is statistically insignificant. A β-parameter

is restricted to be the same in the two states if the difference of its values in the

two states is statistically insignificant. A (1 − a) credible interval is chosen in such

way that the posterior probabilities of being below and above it are both equal to

a/2 (we use significance levels a = 40%,15%,5%).

7

Page 8

frequent state st= 1 is not realized and the process stays in state st= 0.

Turning to the estimation results, the findings show that two states of roadway

safety and the appropriate MSML models exist for severity outcomes of 1-

vehicle accidents occurring on all roadway classes (interstate highways, US

routes, state routes, county roads, streets), and for severity outcomes of 2-

vehicle accidents occurring on streets. We did not find two states in the cases of

2-vehicle accidents on interstate highways, US routes, state routes and county

roads (in these cases all estimated state variables stwere found to be close to

zero). The model estimation results for severity outcomes of 1-vehicle accidents

occurring on interstate highways, US routes and state routes are given in

Tables 1–3. All continuous model parameters (β-s, p0→1and p1→0) are given

together with their 95% confidence intervals (if MLE) or 95% credible intervals

(if Bayesian-MCMC), refer to the superscript and subscript numbers adjacent

to parameter estimates in Tables 1–3.12Table 4 gives summary statistics of

all roadway accident characteristic variables Xt,n(except the intercept).

12Note that MLE assumes asymptotic normality of the estimates, resulting in con-

fidence intervals being symmetric around the means (a 95% confidence interval is

±1.96 standard deviations around the mean). In contrast, Bayesian estimation does

not require this assumption, and posterior distributions of parameters and Bayesian

credible intervals are usually non-symmetric.

8

Page 9

Table 1

Estimation results for multinomial logit models of severity outcomes of one-vehicle accidents on Indiana interstate highways

(the superscript and subscript numbers to the right of individual parameter estimates are 95% confidence/credible intervals)

MSMLc

VariableML-by-MLEa

ML-by-MCMCb

state s = 0state s = 1

fatalityinjuryfatalityinjuryfatalityinjuryfatalityinjury

Intercept (constant term)−11.9−10.1

−13.7

−3.69−3.53

−3.84

−12.4−10.6

−14.5

−3.72−3.56

−3.88

−12.2−10.5

−14.4

−3.98−3.79

−4.17

−12.2−10.5

−14.4

−3.22−2.98

−3.45

Summer season (dummy)

.235.329

.142

.235.329

.142

.237.329

.143

.237.329

.143

.176.293

.0551

.176.293

.0551

.176.293

.0551

.615.959

.282

Thursday (dummy)−.798−.115

−1.48

–−.853−.206

−1.59

–−.872−.225

−1.61

–−.872−.225

−1.61

–

Construction at the accident location (dummy)−.418−.213

−.623

−.418−.213

−.623

−.425−.224

−.632

−.425−.224

−.632

−.566−.319

−.822

−.566−.319

−.822

−.566−.319

−.822

–

Daylight or street lights are lit up if dark (dummy)−.392−.0368

−.748

−1.38−.830

.137.224

.0501

−.387−.0301

−.740

−1.41−.884

.143.230

.0568

−.378−.0236

−.729

−1.54−1.03

.139.226

.0522

−.378−.0236

−.729

−1.54−1.03

.139.226

.0522

Precipitation: rain/freezing rain/snow/sleet/hail (dummy)

−1.92

−.361−.264

−.457

−1.99

−.363−.267

−.460

−2.10

−.671−.0515−.361

.566.930

.211

−.563−.404

−.729

−.671−.0515−.361

–

−2.10

−.671−.0515−.361

.566.930

.211

–

Roadway surface is covered by snow/slush (dummy)−1.28−.0917

−2.46

.571.929

−.432−.280

−.583

–

−1.43−.328

−2.84

−.438−.288

−.590

–

−.0515−.361

−.671

Roadway median is drivable (dummy)

.213

.577.939

.223

–

Roadway is at curve (dummy)

.114.212

.0165

.114.212

.0165

.116.213

.0186

.116.213

.0186

––––

Primary cause of the accident is driver-related (dummy)4.245.30

3.18

1.531.64

1.43

4.395.64

3.39

1.541.64

1.43

4.485.73

3.48

2.002.18

1.84

4.485.73

3.48

.715.946

.468

Help arrived in 20 minutes or less after the crash (dummy)

.790.887

.693

.790.887

.693

.790.891

.691

.790.891

.691

.785.886

.684

.785.886

.684

.785.886

.684

.785.886

.684

The vehicle at fault is a motorcycle (dummy)3.884.59

3.17

2.743.12

2.36

3.874.57

3.13

2.753.15

2.37

4.615.49

3.74

–

3.233.83

2.70

–1.392.49

.326

Age of the vehicle at fault (in years)

.0285.0370

.0201

.0285.0370

.0201

.0286.0370

.0201

.0286.0370

.0201

.0286.0371

.0200

–

.0286.0371

.0200

Number of occupants in the vehicle at fault

.366.463

.269

.123.159

.0859

.367.465

.264

.123.159

.0861

.366.464

.263

.124.161

.0874

.366.464

.263

.124.161

.0874

Roadway traveled by the vehicle at fault is multi-lane and

divided two-way (dummy)2.604.00

1.20

–2.864.63

1.56

–2.864.66

1.56

–2.864.66

1.56

–

At least one of the vehicles involved was on fire (dummy)1.242.12

−.345−.0257

−.665

.328.410

1.182.02

.206

−.345−.0335

−.669

.331.413

1.662.56

.621

−.332−.0198

−.659

.224.338

–−.332−.0198

−.659

.479.637

Gender of the driver at fault (dummy)–

.246

–

.248

–

.107

–

.328

9

Page 10

Table 1

(Continued)

MSMLc

VariableML-by-MLEa

ML-by-MCMCb

state s = 0state s = 1

fatalityinjuryfatalityinjuryfatalityinjuryfatalityinjury

Probability of severity outcome [P(i)

t,ngiven by Eq. (3)], averaged

over all values of explanatory variables Xt,n

––

.00724

.176

.00733

.174

.00672

.192

Markov transition probability of jump 0 → 1 (p0→1)––

.151.254

.0704

Markov transition probability of jump 1 → 0 (p1→0)––

.330.532

.164

Unconditional probabilities of states 0 and 1 (¯ p0 and ¯ p1)––

.683.814

.540

and

.317.460

.186

Total number of free model parameters (β-s)252528

Posterior average of the log-likelihood (LL)–−8486.78−8480.82

−8494.61

−8396.78−8379.21

−8416.57

Max(LL):estimated max. log-likelihood (LL) for MLE;

maximum observed value of LL for Bayesian-MCMC−8465.79(MLE)−8476.37(observed)−8358.97(observed)

Logarithm of marginal likelihood of data (ln[f(Y|M)])–−8498.46−8494.22

−8499.21

0.255

−8437.07−8424.77

−8440.02

0.222Goodness-of-fit p-value–

Maximum of the potential scale reduction factors (PSRF)d

–1.003021.00060

Multivariate potential scale reduction factor (MPSRF)d

–1.003251.00067

Number of available observationsaccidents = fatalities + injuries + PDOs:19094 = 143 + 3369 + 15582

aStandard (conventional) multinomial logit (ML) model estimated by maximum likelihood estimation (MLE).

bStandard multinomial logit (ML) model estimated by Markov Chain Monte Carlo (MCMC) simulations.

cTwo-state Markov switching multinomial logit (MSML) model estimated by Markov Chain Monte Carlo (MCMC) simulations.

dPSRF/MPSRF are calculated separately/jointly for all continuous model parameters. PSRF and MPSRF are close to 1 for converged MCMC chains.

10

Page 11

Table 2

Estimation results for multinomial logit models of severity outcomes of one-vehicle accidents on Indiana US routes

(the superscript and subscript numbers to the right of individual parameter estimates are 95% confidence/credible intervals)

MSMLc

Variable ML-by-MLEa

ML-by-MCMCb

state s = 0state s = 1

fatalityinjuryfatalityinjuryfatalityinjuryfatalityinjury

Intercept (constant term)−6.51−5.00

−8.03−2.13−1.79

.514.894

.134

−.498−.142

−.855

−1.17−.170

−2.18

.7011.25

.149

−.741−.383

−2.47−6.62−5.16

.200.305

.0947

.194.287

.101

−8.14−2.12−1.78

.509.883

.124

−.492−.136

−.848

−1.30−.357

−2.47

.7271.31

.199

−.399−.739−.377

−4.18−1.89−1.78

.594.681

.507

.594.681

.507

2.623.47

1.78

3.203.55

2.86

.0363.0444

.0283

.0363.0444

.0283

.0121.0178

.00640

−2.47

−5.72−4.69

−6.92

−2.05−1.71

−2.40−5.72−4.69

.190.300

.0789

.197.290

.105

.165.317

.0115

.214.332

.0965

−.294−.189

−6.92−2.79−2.37

.190.300

.0789

−3.23

Summer season (dummy)

.200.305

.0951

.190.300

.0789

–

Daylight or street lights are lit up if dark (dummy)

.203.296

.110

−.493−.136

−.857

–

.197.290

.105

Snowing weather (dummy)––−1.10−.151

−2.27

−1.10−.151

−2.27

.165.317

.0115

No roadway junction at the accident location (dummy)

.217.335

.0994

.213.331

.0968

.7871.36

.259

.7871.36

.259

−1.09−.294−.189

−4.40−.701−.263

.560.648

.472

3.223.58

2.88

.214.332

.0965

Roadway is straight (dummy)

−1.10−.295−.191

−3.45−2.72

−1.09−.296−.192

−4.32−1.89−1.79

.562.650

.475

2.573.38

1.65

.0367.0448

.0287

.0373.0643

.0117

−.399

−7.37−.372

−1.09

−.398−7.37−.372

−2.09−1.96

−.398

Primary cause of the accident is environment-related (dummy)

−1.99−3.51−2.81

−2.00

−3.59−2.89

−4.40

−2.24−3.59−2.89

.560.648

.472

3.223.58

2.88

.0366.0447

.0285

.0102.0178

.00635

−1.16

Help arrived in 10 minutes or less after the crash (dummy)

.562.650

.475

.560.648

.472

.560.648

.472

The vehicle at fault is a motorcycle (dummy)3.213.56

2.87

3.223.58

2.88

3.223.58

2.88

Age of the vehicle at fault (in years)

.0367.0448

.0287

––

.0366.0447

.0285

Speed limit (used if known and the same for all vehicles involved) .0363.0631

.00950

.0118.0176

.00616

.0285.0495

.0104

–

.0120.0178

.00635

Roadway traveled by the vehicle at fault is two-lane and

one-way (dummy)−.216.0417

−.391−.216.0417

1.191.94

.439

.0114.0213

.00150

−.391−.223.0517

1.131.85

−.398−.223.0517

−.398

−.224.0504

−.401

−.224.0504

−.401−.224.0504

1.271.98

−.401−.224.0504

−.401

At least one of the vehicles involved was on fire (dummy)–

.315

–1.271.98

.452

–

.452

–

Age of the driver at fault (in years)–

.0113.0211

.00137

–

.0101.0200

.0000542

–––

Weekday (Monday through Friday) (dummy)–−.104.0116

−.196

–−.104.0124

−.196

–−.125.0242

−.227

––

Gender of the driver at fault (dummy)–

.272.362

.183

–

.276.365

.186

–

.280.369

.190

–

.280.369

.190

11

Page 12

Table 2

(Continued)

MSMLc

VariableML-by-MLEa

ML-by-MCMCb

state s = 0state s = 1

fatalityinjuryfatalityinjuryfatalityinjuryfatalityinjury

Probability of severity outcome [P(i)

t,ngiven by Eq. (3)], averaged

over all values of explanatory variables Xt,n

––

.00747

.179

.00823

.183

.00218

.158

Markov transition probability of jump 0 → 1 (p0→1)––

.0767.157

.0269

Markov transition probability of jump 1 → 0 (p1→0)––

.613.864

.337

Unconditional probabilities of states 0 and 1 (¯ p0 and ¯ p1)––

.887.959

.770

and

.113.230

.0409

Total number of free model parameters (β-s)242425

Posterior average of the log-likelihood (LL)–−7406.39−7400.61

−7414.03

−7349.06−7335.46

−7364.47

Max(LL):estimated max. log-likelihood (LL) for MLE;

maximum observed value of LL for Bayesian-MCMC−7384.05(MLE)−7396.37(observed) −7318.21(observed)

Logarithm of marginal likelihood of data (ln[f(Y|M)])–−7417.98−7413.72

−7420.23

−7377.49−7369.62

−7380.00

Goodness-of-fit p-value–0.3370.255

Maximum of the potential scale reduction factors (PSRF)d

–1.003191.00073

Multivariate potential scale reduction factor (MPSRF)d

–1.003761.00085

Number of available observationsaccidents = fatalities + injuries + PDOs:17797 = 138 + 3184 + 14485

aStandard (conventional) multinomial logit (ML) model estimated by maximum likelihood estimation (MLE).

bStandard multinomial logit (ML) model estimated by Markov Chain Monte Carlo (MCMC) simulations.

cTwo-state Markov switching multinomial logit (MSML) model estimated by Markov Chain Monte Carlo (MCMC) simulations.

dPSRF/MPSRF are calculated separately/jointly for all continuous model parameters. PSRF and MPSRF are close to 1 for converged MCMC chains.

12

Page 13

Table 3

Estimation results for multinomial logit models of severity outcomes of one-vehicle accidents on Indiana state routes

(the superscript and subscript numbers to the right of individual parameter estimates are 95% confidence/credible intervals)

MSMLc

VariableML-by-MLEa

ML-by-MCMCb

state s = 0state s = 1

fatality injuryfatalityinjuryfatalityinjuryfatalityinjury

Intercept (constant term)−3.98−3.66

−4.30

−1.67−1.53

−1.80

−4.03−3.71

−4.36

−1.71−1.58

−1.85

−3.44−3.10

−3.79

−1.68−1.54

−1.81

−4.96−4.15

−5.96

−1.68−1.54

−1.81

Summer season (dummy)

.232.307

.156

.232.307

.156

.232.307

.157

.232.307

.157

.238.314

.163

–

.238.314

.163

.238.314

.163

.238.314

.163

Roadway type (dummy: 1 if urban, 0 if rural)−.390−.302

−.478

−.390−.302

−.478

−.395−.306

−.483

−.395−.306

−.483

−.385−.296

−.474

–

−2.05−.954

−3.62

−3.85−.296

−.474

Daylight or street lights are lit up if dark (dummy)−.646−.408

−.884

.193.261

.125

−.641−.404

−.879

.199.267

.132

−.689−.448

−.931

−.689−.448

−.931

.277.378

.177

Precipitation: rain/freezing rain/snow/sleet/hail (dummy)−.854.466

−1.24

–−.868−.494

−1.27

–−.829−.448

−1.24

–−.829−.448

−1.24

–

Roadway median is drivable (dummy)−.583−.225

−.940

–−.596−.250

−.964

–−.589−.241

−.960

–−.589−.241

−.960

–

Roadway is straight (dummy)−.284−.214

−.353

−.284−.214

−.353

−.283−.214

−.352

−.283−.214

−.352

−.117−.0184

−.214

−4.40−3.79

−.117−.0184

−.214

−2.30−2.16

−.117−.0184

−.214

−4.40−3.79

−.465−.360

−.573

Primary cause of the accident is environment-related (dummy)−4.23−3.59

−4.86

−1.83−1.76

−1.91

−4.28−3.67

−4.97

−1.84−1.76

−1.91

−5.10

–

−2.44

−5.10

−1.41−1.26

−1.55

Help arrived in 20 minutes or less after the crash (dummy)

.840.917

.762

.840.917

.762

.863.945

.781

.863.945

.781

.861.944

.778

1.642.64

.856

.861.944

.778

The vehicle at fault is a motorcycle (dummy)3.103.31

2.89

3.103.31

2.89

3.103.31

2.89

3.103.31

2.89

3.373.66

3.09

3.373.66

3.09

3.373.66

3.09

2.823.19

2.47

Number of occupants in the vehicle at fault

.0557.0850

.0265

.0557.0850

.0265

.0565.0858

.0276

.0565.0858

.0276

.0942.138

.0528

.0942.138

.0528

.0942.138

.0528

–

At least one of the vehicles involved was on fire (dummy)1.902.45

14.621.4

× 10−3

−.496−.211

1.33

.456.780

−2.80−.800

× 10−3

.279.344

.133

1.872.42

14.521.3

× 10−3

−.505−.225

1.28

.447.768

−2.71−.723

× 10−3

.278.343

.124

1.872.43

14.521.4

× 10−3

−.473−.192

1.28

.461.782

−2.46−.469

× 10−3

.283.348

.137

1.872.43

14.521.4

× 10−3

−.473−.192

1.28

.461.782

−2.46−.469

× 10−3

.283.348

.137

Age of the driver at fault (in years)

7.80

−4.70

7.67

−4.69

7.63

−4.44

7.63

−4.44

Gender of the driver at fault (dummy)

−.780

–

.214

−.794

–

.213

−.764

–

.218

−.764

–

.218

Age of the vehicle at fault (in years)

.0334.0392

.0276

.0335.0393

.0277

.0332.0390

.0274

.0332.0390

.0274

license state of the vehicle at fault is a U.S. state except Indiana

and its neighboring states (IL, KY, OH, MI)” indicator variable–−.449−.217

−.681

–−.444−.217

−.679

–−.436−.208

−.671

–−.436−.208

−.671

13

Page 14

Table 3

(Continued)

MSMLc

VariableML-by-MLEa

ML-by-MCMCb

state s = 0state s = 1

fatalityinjuryfatalityinjuryfatalityinjuryfatalityinjury

Probability of severity outcome [P(i)

t,ngiven by Eq. (3)], averaged

over all values of explanatory variables Xt,n

––

.0089

.179

.00951

.180

.00804

.179

Markov transition probability of jump 0 → 1 (p0→1)––

.335.465

.216

Markov transition probability of jump 1 → 0 (p1→0)––

.450.610

.313

Unconditional probabilities of states 0 and 1 (¯ p0 and ¯ p1)––

.574.681

.504

and

.426.496

.319

Total number of free model parameters (β-s)222228

Posterior average of the log-likelihood (LL)–−13867.40−13861.92

−13874.73

−13781.76−13765.02

−13800.89

Max(LL):estimated max. log-likelihood (LL) for MLE;

maximum observed value of LL for Bayesian-MCMC−13846.60(MLE)−13858.00(observed)−13745.61(observed)

Logarithm of marginal likelihood of data (ln[f(Y|M)])–−13877.89−13874.24

−13880.38

0.515

−13820.20−13808.85

−13821.73

0.445 Goodness-of-fit p-value–

Maximum of the potential scale reduction factors (PSRF)d

–1.000271.00029

Multivariate potential scale reduction factor (MPSRF)d

–1.000411.00045

Number of available observationsaccidents = fatalities + injuries + PDOs:33528 = 302 + 6018 + 27208

aStandard (conventional) multinomial logit (ML) model estimated by maximum likelihood estimation (MLE).

bStandard multinomial logit (ML) model estimated by Markov Chain Monte Carlo (MCMC) simulations.

cTwo-state Markov switching multinomial logit (MSML) model estimated by Markov Chain Monte Carlo (MCMC) simulations.

dPSRF/MPSRF are calculated separately/jointly for all continuous model parameters. PSRF and MPSRF are close to 1 for converged MCMC chains.

14

Page 15

The top, middle and bottom plots in Figure 1 show weekly posterior proba-

bilities P(st= 1|Y) of the less frequent state st= 1 for the MSML models

estimated for severity of 1-vehicle accidents occurring on interstate highways,

US routes and state routes respectively.13Because of space limitations, in this

paper we do not report estimation results for severity of 1-vehicle accidents on

county roads and streets, and for severity of 2-vehicle accidents. However, be-

low we discuss our findings for all roadway-class-accident-type combinations.

For unreported model estimation results see Malyshkina (2008).

We find that in all cases when the two states and Markov switching multi-

nomial logit (MSML) models exist, these models are strongly favored by the

empirical data over the corresponding standard multinomial logit (ML) mod-

els. Indeed, from lines “marginal LL” in Tables 1–3 we see that the MSML

models provide considerable, ranging from 40.5 to 61.4, improvements of the

logarithm of the marginal likelihood of the data as compared to the corre-

sponding ML models.14Thus, from Eq. (6) we find that, given the accident

severity data, the posterior probabilities of the MSML models are larger than

the probabilities of the corresponding ML models by factors ranging from e40.5

to e61.4. In the cases of 1-vehicle accidents on county roads, streets and the

case of 2-vehicle accidents on streets, MSML models (not reported here) are

also strongly favored by the empirical data over the corresponding ML models

(Malyshkina, 2008).

Let us now consider the maximum likelihood estimation (MLE) of the standard

ML models and an imaginary MLE estimation of the MSML models. We

find that, in this imaginary case, a classical statistics approach for model

comparison, based on the MLE, would also favors the MSML models over the

standard ML models. For example, refer to line “max(LL)” in Table 1 given

for the case of 1-vehicle accidents on interstate highways. The MLE gave

the maximum log-likelihood value −8465.79 for the standard ML model. The

maximum log-likelihood value observed during our MCMC simulations for the

MSML model is equal to −8358.97. An imaginary MLE, at its convergence,

would give a MSML log-likelihood value that would be even larger than this

observed value. Therefore, if estimated by the MLE, the MSML model would

provide large, at least 106.82 improvement in the maximum log-likelihood

value over the corresponding ML model. This improvement would come with

only modest increase in the number of free continuous model parameters (β-

s) that enter the likelihood function (refer to Table 1 under “# free par.”).

13Note that these posterior probabilities are equal to the posterior expectations of

st, P(st= 1|Y) = 1 × P(st= 1|Y) + 0 × P(st= 0|Y) = E(st|Y).

14We use the harmonic mean formula to calculate the values and the 95% confidence

intervals of the log-marginal-likelihoods given in lines “marginal LL” of Tables 1–3.

The confidence intervals are calculated by bootstrap simulations. For details, see

Paper I or Malyshkina (2008).

15

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