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: Taking into consideration the increasing availability of real-time traffic data and stimulated by the importance of proactive safety management, this paper attempts to provide a review of the effect of traffic and weather characteristics on road safety, identify the gaps and discuss the needs for further research. Despite the existence of generally mixed evidence on the effect of traffic parameters, a few patterns can be observed. For instance, traffic flow seems to have a non-linear relationship with accident rates, even though some studies suggest linear relationship with accidents. On the other hand, increased speed limits have found to have a straightforward positive relationship with accident occurrence. Regarding weather effects, the effect of precipitation is quite consistent and leads generally to increased accident frequency but does not seem to have a consistent effect on severity. The impact of other weather parameters on safety, such as visibility, wind speed and temperature is not found straightforward so far. The increasing use of real-time data not only makes easier to identify the safety impact of traffic and weather characteristics, but most importantly makes possible the identification of their combined effect. The more systematic use of these real-time data may address several of the research gaps identified in this research.Accident Analysis & Prevention 07/2014; 72C:244-256. · 1.87 Impact Factor - SourceAvailable from: Juan De OñaTransport 10/2013; 166(5):255-270. · 0.32 Impact Factor
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ABSTRACT: Multi-vehicle motorcycle crashes combine elements of design, behavior, and traffic. One challenge with working with motorcycle data are the inherit difficulties associated with missing data – such as motorcycle-specific: vehicle miles traveled (VMT) and average daily traffic (ADT). To address the challenges of the missing data, a random effects Bayesian negative binomial model is developed for the state of Ohio. In this study, the random effect terms improve the general model by describing the spatial correlation with fixed effects, the neighborhood criteria, and the uncorrelated heterogeneity for all the multi-vehicle motorcycle crashes that occurred on the 32,289 state-maintained roadway segments in Ohio. Some key findings from this study include regional data improves the goodness-of-fit, and further improvement of the models may be gained through a distance-based neighborhood specification of conditional autoregressive (CAR). In addition to the model improvement using the random effect terms, key variables such as smaller lane and shoulder widths, increases in the horizontal degree of curvature and increases in the maximum vertical grade will increase the prediction of a crash.Safety Science 07/2014; 66:47–53. · 1.67 Impact Factor
Page 1
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
Page 4
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
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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
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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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