1

An Analysis of Transit Bus Driver Distraction Using Multinomial Logistic

Regression Models

Kelwyn A. D’Souza, Ph.D.

Professor of Management Science, and Director of Eastern Seaboard Intermodal Transportation

Applications Center (ESITAC), Hampton University, Hampton, VA 23668, U. S. A.

Email: kelwyn.dsouza@hamptonu.edu. Telephone: (757) 727-5037.

Sharad K. Maheshwari, Ph.D.

Associate Professor, Department of Business Administration, Hampton University, Hampton,

VA 23668, U. S. A. Email: sharad.maheshwari@hamptonu.edu. Telephone: (757) 727-5605.

ABSTRACT

This paper explores the problem of distracted driving at a regional bus transit agency to identify

the sources of distraction and provide an understanding of factors responsible for driver

distraction. A risk range system was developed to classify the distracting activities into four risk

zones. The high risk zone distracting activities were analyzed using multinomial logistic

regression models to determine the impact of various factors on the multiple categorical levels of

driver distraction. The models demonstrated that the highest source of driver distractions was

due to passenger-related activities and the level of distraction was influenced by driver

demographics, driving hours, and location. The model results were validated by simulation over

an entire range of random input variables. The model and results could assist in mitigating

distraction and improving transit performance.

BACKGROUND AND LITERATURE REVIEW

According to the National Highway Traffic Safety Administration (NHTSA), distracted driving

claimed over 3,000 lives across the U.S. in 2010 (Daily Press Newspaper, 2012). A study of

more than 20,000 drivers in the transit industry (all transit types) identified the statistically

significant distractions and found that transit drivers who have been involved in a collision are

twice as likely to regularly use a handheld cell phone compared to those drivers who have not

been involved in a collision (Metro Magazine 2011). Government regulators have responded by

proposing policies such as the recent National Transportation Safety Board’s (NTSB)

recommendation to ban all cell phone usage including hand-free devices while driving, except in

emergency situations (U.S. DOT 2011). Such policies are inconsistent because sufficient

information on the activities and factors that cause transit bus driver distraction is not available.

This study explores the problems of distracted driving for bus drivers at a regional transit

agency that provides service to seven cities and surrounding suburbs in the region. It is one of

only a few studies to examine the full range of distractions and associated risk in transit buses.

The reason for undertaking this study was to get a better understanding of distracting activities in

transit operations since most published research on distracted driving focus on personal and

commercial truck drivers. Research on transit bus driver distraction is very limited (D’Souza and

Maheshwari 2012 and Salmon et al. 2011), although injuries from transit vehicle accidents are

generally higher due to larger number of passengers. The accident reports filed by transit bus

drivers rarely document distraction as the cause of accidents. Such accidents are possibly recorded

2

in Virginia Traffic Crash Facts under the category of No Violation (Department of Motor Vehicles

2008). Yang (2007) analyzed trends in transit bus accidents and related factors such as road

design, weather, lighting condition, etc, recorded by the National Transit Database (NTD), but no

analysis was reported on driver distraction. Due to lack of reporting distractions by transit drivers,

the associated risks and impact on performance is difficult to study and hence, not been well-

understood.

Driver distraction represents a significant problem in the personal and public transport

sector, and has been studied by several researchers. A study funded by the AAA Foundation

(2003) identified the major sources of distraction for personal vehicles contributing to crashes,

developed taxonomy of driver distractions for the U.S. driving population, and examined the

potential consequences of these distractions on driving performance. The source of bus driver

distractions at a major Australian public transport company was investigated using ergonomics

methods through which, a taxonomy of the sources of bus driver distraction was developed,

along with countermeasures to remove/mitigate their effects on driver performance (Salmon et

al. 2011). In an earlier study, Salmon et al. (2006) developed taxonomy of distraction sources

and duration for bus drivers at the State Transit Authority New South Wales, Australia, but

provided insufficient inferential statistical analysis.

Factors such as location, driving hours/week; and driver age, gender, and experience have

an impact on transit bus driver distraction. The location of the driving route in a densely

populated area would have greater number of passengers and higher external sources of

distraction due to more frequent stops and more other road users or pedestrians (AAA

Foundation 2003). A driver less familiar with the driving routes is more likely to be involved in

rear-end accidents at signalized intersections (Yan et al. 2005). Studies on the impact of age,

gender, driving experience, and driving demands on driving performance suggests that younger

(below 25 years) and older (above 70 years) drivers tend to be more vulnerable to the effects of

distraction than middle-aged drivers (U. S. DOT 2009; Owens and Lehman 2001; Oxley et al.

2005). Blower et al. (2008) reported that age, sex, hours driving, trip type, method of

compensation, and previous driving records are related to driver error. The multivariate model

applied by Yan et al. (2009) to study accidents in trucks, identified driver age and gender among

the several other factors related to rear-end crashes.

Multinomial logistic regression (MLR) models are widely used in transportation to study

the relationship between the categorical dependent variable and a set of continuous and

categorical independent predictor variables collected through surveys. Washington et al. (2011)

developed a MLR model consisting of 18 independent variables covering driver factors, traffic

flow, distance, and number of signals etc. to study factors that influence driver’s selection of

route on their morning commute to work. Yan et al. (2009) utilized MLR to study the impact of

potential factors such as driver factors, road layout, and environmental conditions on rear-end

truck to car, car to truck, and car to car crashes. A MLR model was developed by Morfoulaki et

al. (2007) to identify the factors contributing to service quality and customer satisfaction (very

satisfied, satisfied, somewhat dissatisfied, and very dissatisfied) with a public transit service in

Greece. Gkritza et al. (2006) conducted an empirical study using multinomial logit models to

investigate the socio-economic and demographic factors that significantly affect passenger

satisfaction with airport security screening process. Bhadra (2005) applied multinomial logistic

regression to determine the choice of aircraft in the U. S, National Airspace System and could

predict exact choices 51% of the time. Petrucci (2009) and Hickman et al. (2010) computed the

odds ratios for the tasks/variables, along with 95 percent confidence intervals (CI) to identify the

3

high risk tasks/variables. An odds ratio for an event is the probability of the event occurring

divided by the probability of the event not occurring (Yan et al. 2005) thus indicating the

strength of association between the categorical dependent variable and independent variables.

The Monte Carlo simulation method (Bonate 2001) is often applied to validate empirical

results obtained from conceptual models. Carlson (1972) demonstrated the application of Monte

Carlo simulation to evaluate proposed component changes on highway crash reduction. The

impact of age and cognitive functions on driving performance has been studied extensively to

predict cognitive distraction with a computational cognitive model and validating the results

through simulation (Salvucci et al. 2004). A simulation approach was developed by Smith et al.

(2005) to evaluate the safety impacts of driver getting distracted by secondary tasks.

METHODOLOGY

A self-administered survey instrument developed by Salmon et al., (2006) was redesigned to

collect distraction data from a sample of drivers employed by the regional transit agency. The

driver location was recorded according to the geographical districts covered while driving the

bus: Northside and Southside. Data was collected on 18 sources of driver distraction and

perceived risks associated with a particular distracting activity along with independent factors

including location, driving hours/week, age, gender, and driving experience. The ratings and

durations for each activity were averaged and then each activity was ranked based on average

rating as well as average duration (Salmon et al. 2006 and Cross et al. 2010). The activities

involving perceived visual, manual, and cognitive effects risk to drivers were ranked based on

the aggregate count. The activities belonging to the top five average distraction rating, average

distraction duration, and driver’s perception of risk were formulated into a risk range system that

was used to classify distracting activities into respective risk zone.

Multinomial logistic regression (MLR) was utilized to model the seven risk zone

distracting activity using levels of distraction as the dependent variable and correlating it with the

factors as independent/predictor variables. The categorical dependent variable (driver

distraction) had four levels: Not Distracted, Slightly Distracted, Distracted, and Very Distracted.

This is similar to the categorical dependent variable used for studying customer satisfaction in

public transport systems (Morfoulaki et al. 2007). The independent variables included

categorical variables such as gender and location, and continuous variables such as age, driving

experience, and driving hours per week. The MLR was modeled as an extension of the binary

logistic regression (Moutinho and Hutcheson 2011 and Field 2009) by comparing each level of

distraction (Slightly Distracted, Distracted, Very Distracted) with a reference level of distraction

(Not Distracted), thus producing three binary logistic regression outputs. A stepwise procedure

included all the selected factors in the model initially; non-significant factors were eliminated

until a good fit was achieved with significant factors. The model’s goodness of fit was

statistically tested and verified graphically (Landwehr et al. 1984).

Furthermore, the model’s output were validated by Monte Carlo simulation with the

random input variables assuming values expected under actual driving environment. A

spreadsheet simulation model (Albright and Winston 2005) was constructed with cumulative

probability distribution for each input variable, formulas and logic for computing the distraction

probabilities, and the outputs containing the average probability of 1,000 random drivers. The

RAND function was embedded into each input variable cell providing a set of random numbers

4

equally likely to be between 0 and 1 for replication. Each input variable was replicated for 1,000

drivers keeping the remaining input variables fixed.

The levels of each dependent variable appear to be ordered (ranging from Not Distracted

to Very Distracted); hence one might consider using the ordinal logistic regression model. An

ordinal logistic regression test model developed for Passenger Using Mobile Phone activity

exhibited a poor fit (p = 0.381) with no significant independent variables. In addition, the

ordinal logistic regression models place a restriction on how the variables affect outcome

probabilities (Washington et al. 2011). Due to the poor fit, the reported problems with the

ordinal logistic regression model restriction (Gkritza et al. 2006 and Washington et al. 2011), and

the MLR for multiple types of distraction was modeled as a series of binary logistic regression

models, the widely used MLR was adopted in this study although the dependent variable appears

to be ordered.

ANALYSIS OF TRANSIT BUS DRIVER DISTRACTIONS

The Hampton University Transportation Center Bus Driver Distraction Survey collected

information on the source, extent, and duration of distraction. The transit bus drivers rated how

distracting they found the listed activities and the approximate duration they experience the

distracting activities in a typical 8-hour shift. The ratings and durations for each activity was

averaged and ranked from highest to lowest. The five highest distracting activities listed in

Table 1 were graded relative to the highest rating (2.48). The range of rating for Risk zone I was

set at above 90% of the highest rating [ > 2.2]. Similarly, the range for Risk Zone II was set at

between 70% and 90% of the highest rating [1.8 – 2.2], and the range for Risk Zone III was for

50% to 70% of the highest rating [1.2 – 1.8]. The range for Risk Zone IV was set below 50% of

the highest rating [< 1.2]. The five highest distracting durations listed in Table 2 were graded

relative to the highest duration (2.66 hrs). The range of duration for Risk Zone I was set at above

90% of the highest duration [ > 2.4 Hrs]. Similarly, Risk Zone II was set at between 70% and

90% of the highest duration [1.9 – 2.4 Hrs], and Risk Zone III was for 50% to 70% of the highest

duration [1.3 – 1.9 Hrs]. The range for Risk Zone IV was set below 50% of the highest duration

[< 1.3 Hrs]

The U. S. DOT (2010) has categorized distractions as Visual, Manual, and Cognitive —

the severity of distractions increases as it involves more number of categories. The bus drivers

were asked to categorize each distracting activity according to their perception. The total

responses from the bus drivers were ranked from highest to lowest. The five highest number of

driver responses for each category listed in Table 3 were graded relative to the highest visual (19

driver responses), cognitive (33 driver responses), and manual (11 driver responses). The range

of rating for Risk Zone I was set at above 90% of the highest visual [ > 17 driver responses],

highest cognitive [> 30 driver responses], and highest manual [> 10 driver responses]. Similarly,

the range for Risk Zone II was set at 70% of the highest driver responses, and the range for Risk

Zone III was set at 50% of the highest driver responses. The range for Risk Zone IV was set

below 50% of the highest driver responses.

The estimated ranges for average rating, average duration, and perceived effects of

distraction were combined to develop a risk range system (Table 4) for classifying all the

distracting activities into respective risk zones.. The risk range system considered the rating,

duration, and perceptions of each distracting activity resulting in classification of seven high risk

5

activities into Risk Zones I, II, and III, and remaining 11 activities classified in Zone IV (Figure

1).

Table 1: The Top Five Distracting Activities (Rating)

Rank Activity Average Distraction

Rating

Distraction Category

1 Passengers using a mobile phone 2.48 Passenger

2 Passengers not following etiquette (eating, drinking,

smoking, noisy)

2.35 Passenger

3 Passengers trying to talk to driver 2.23 Passenger

4 Fatigue/Sickness 2.1 Personal

5 Passengers 2.08 Passenger

Table 2: The Top Five Distracting Activities (Duration).

Rank Activity Avg Distraction Duration/Shift (Hrs) Distraction Category

1 Passengers using a mobile phone 2.66 Passenger

2 Other Road Users 2.24 External

3 Passengers 2.23 Passenger

4 Passengers trying to talk to driver 1.96 Passenger

5 Passengers not following etiquette

(eating, drinking, smoking, noisy)

1.84 Passenger

Table 3: Top Five Distraction Activities (Driver Perception)

Activity Rank Visual

Effects of

Distraction

Rank Manual

Effects of

Distraction

Rank Cognitive

Effects of

Distraction

Distraction

Category

Passengers using a mobile phone - 1 Passenger

Reading (eg Route Sheet) 1 - - Operational

Ticket Machine 2 3 - Technology

Climate Control 3 4 - Technology

Passengers 4 - 3 Passenger

Disabled Passengers 5 5 5 Passenger

Fatigue/Sickness - 1 - Personal

Pedestrians - 2 - Infrastructure

Passengers not following etiquette

(eating, drinking, smoking, noisy)

- - 5 Passenger

Passengers trying to talk to driver - 2 Passenger

General Broadcast - - 4 Operational

Table 4: Risk Range System.

Average Rating Average Duration Number of Driver’s Perception (out of total 48 drivers) Risk Zone

> 2.2 > 2.4 hrs visual > 17; cognitive > 30; manual > 10 I

1.8 - 2.2 1.9 - 2.4 hrs visual 13-17; cognitive 23-30; manual 8-10 II

1.2 - 1.8 1.3 - 1.9 hrs visual 10-13; cognitive 17-23; manual 6-8 III

< 1.2 < 1.3 hrs visual < 10; cognitive < 17 manual < 6 IV

6

Figure 1: Classification of Distracting Activities into Risk Zones

R = Distracting Rating; D = Distracting Duration; V = Visual Perception; C = Cognitive Perception; M = Manual/Physical

Perception. Bolding indicates the scores meet or exceed range of the assigned risk zone.

Modeling High Risk Activities

Multinomial logistic regression (MLR) was applied as an extension of binary logistic regression

to model distracting activities in Risk Zone I, II, and III, where, each response variable level is

compared to a reference level providing three binary logistic regression models. The general

MLR model proposed by Moutinho and Hutcheson (2011) is expressed as:

(1) +

Where j is the identified distraction level, and j’ is the reference distraction level.

Risk Levels

Non-Etiquette

Passenger

[R=2.35,D=1.84,

V=9,C=27,M=5]

Passenger Talk to

Driver

[R=2.23,D=1.96,

V=4,C=33,M=5]

Risk Zone II High Risk

Risk Zone III Moderate Risk Risk Zone I Very High Risk

Ticket Machine

[R=1.52,D=1.48,

V=17,C=19,M=7]

Passengers

[R=2.08,D=2.23,

V=11,C=29,M=5]

Climate Control

[R=1.38,D=0.90,

V=11,C=17,M=7

]

,

Fatigue/Sick

[R=2.3, D=1.32,

V=4, C=21,M=11]

Risk Zone IV Low Risk

Other Distracting

Activities

Passengers Using

Mobile Phones

[R=2.48, D=2.66,

V=5, C=33, M=5]

V=11,C=29,M=5]

Ris

k L

evel

s

7

Logit model 2 comparing Slightly Distracted with Not Distracted is stated as:

(2) +

Logit model 3 comparing Distracted with Not Distracted is stated as:

(3) +

Logit model 4 comparing Very Distracted with Not Distracted is stated as:

(4) +

The logit models 2, 3, and 4 provide three estimates for the impact each independent variable has

on the dependent variable, allowing the impact of dependent variable Xk to be computed for each

logit model and for the whole model.

The intercept β0 is the value of Y when all the independent variables are equal to zero.

β1, β2, β3,… βk are the regression coefficients of X1, X2, X3 ….. Xk. Each of the regression

coefficients describes the size of the contribution of risk factor Xi relative to a reference

category. A positive regression coefficient means that the explanatory variable increases the

probability of the outcome, while a negative regression coefficient means that the variable

decreases the probability of that outcome (Bhadra 2005 and Washington et al. 2011); a large

regression coefficient means that the risk factor strongly influences the probability of that

outcome, while a near-zero regression coefficient means that that risk factor has little influence

on the probability of that outcome (Petrucci 2009). The multinomial dependent variable Yi

(logit) which measures the total contribution of the five factors (independent variables) is

expressed as:

(5) +

Where,

LOCAT: Location of driver, a categorical variable, 1 = Northside, 2 = Southside.

SEX: Gender of driver, a categorical variable, 1 = Male, 2 = Female.

AGE: Reported age of driver in years, a continuous variable.

EXP: Number of years of experience driving a bus, a continuous variable.

DRIVING/WK: Weekly driving hours, a continuous variable.

The coefficients computed by the MLR models are relative to the reference category and are

utilized to predict the probability of the extent that a driver finds an activity distracting versus the

reference category from the following binary logistic function (Field 2009):

(6)

8

Where, f (Yi) is the probability of a driver getting Slightly Distracted, Distracted, or Very

Distracted.

Each level is referenced versus Not Distracted. The event Y is very unlikely to occur if f

(Yi) is close to 0 and very likely to occur if it is close to 1. The output is split into three tables

since the dependent categorical variables are compared in pairs. Due to space restriction, the

statistical test ratios and parameter estimates for Passenger Using Mobile Phone (Risk Zone I)

has been presented in Table 5. The statistical package (SPSS 17.0 2008) includes direct entry of

all variables in the MLR model. It is not necessary to create dummy variables for categorical

variables LOCAT and SEX since the software will do this automatically when these variables are

inputted as “factors” in the software package. The analysis was conducted for each of the seven

distracting activity in the Risk Zone I, II, and III (Figure 1) to estimate the utility function Yij

(logit) of the MLR model that best fits the data for each distracting activity. Out of seven MLR

models developed for the Risk Zone activities, six were found to be highly significant and

exhibiting a good fit (Table 6). The utility function Yij (logit) for distracting levels included

independent variable coefficients that were not-significant (N/S) at the 0.05 level.

Table 5: MLR Model Outputs for Passenger Using Mobile Phone.

Model Chi-Square (χ2) =

71.56 (24)****

Pearson Stat (NS)

Deviance Stat(NS)

R2 = 0.833 (Cox &

Snell); 0.897

(Nagelkerke);

0.678(McFadden)

AIC initial/final values: 119.76/87.93

BIC initial/final values: 150.16/133.53

Independent Predictor

Variables and Interactions

Coeff β (SE)

Wald

Statistic

Odds Ratio

Exp (B)

95% CI includes

1

Slightly distracting vs. Not distracting

Intercept -105.49 (47.85)** 4.86

LOCAT = 1 -9.48 (3.05)*** 9.64 < 1 No

LOCAT = 2 0.00

SEX =1 82.41 (21.78)**** 14.31 > 1 No

SEX = 2 0.00

AGE 1.65 (0.82)** 4.01 > 1 No

EXP 2.57 (1.23)** 4.34 > 1 No

DRIVING/WK 1.89 (1.01)* 3.49 > 1 Yes

AGE*DRIVING/WK -0.03 (0.02)* 2.90 < 1 Yes

SEX=1*DRIVING/WK -1.70 (0.45)**** 14.45 < 1 No

AGE*EXP -0.04 (0.02)* 3.46 < 1 Yes

Distracting vs. Not distracting

Intercept +156.58

(51.15)***

9.37

LOCAT = 1 -5.82 (2.02)*** 8.29 < 1 No

LOCAT = 2 0.00

SEX =1 20.06 (9.68)** 4.29 > 1 No

SEX = 2 0.00

AGE -2.72 (0.88)*** 9.51 < 1 No

EXP 3.69 (1.38)*** 7.09 > 1 No

DRIVING/WK -3.79 (1.23)*** 9.54 < 1 No

AGE*DRIVING/WK 0.070 (0.02)*** 9.80 > 1 No

SEX=1*DRIVING/WK 0.36 (0.20)* 3.16 < 1 Yes

AGE*EXP -0.06 (0.02)** 6.55 < 1 Yes

Very distracting vs. Not distracting N/S

*p < 0.10; **p < 0.05; ***p < 0.01; ****p < 0.001. N/S = Not Significant.

9

MLR Model Results. The magnitude and direction of each constant term and significant

independent variable coefficient of the MLR utility functions in Table 6 along with odds ratios

were used to identify the impact of each change on the categorical dependent variable (UCLA

2007 and Washington et al. 2011). A higher or positive amount of a coefficient of the

independent variable would result in a higher dependent variable outcome and vice versa.

Where the confidence interval does not include 1.0 and odds ratio is greater than 1 indicates a

greater likelihood for a driver to get distracted versus not getting distracted by the risk zone

activity for a unit increase in the independent variable. Similarly, where the confidence interval

does not include 1.0 and odds ratio is less than 1 indicates a lower likelihood for a driver to get

distracted versus not getting distracted by the risk zone activity for a unit increase in the

independent variable.

CONSTANT TERM: The logit Y12 for Passenger Using Mobile Phone has a negative constant

term and Y13 has a positive constant term. Hence, keeping everything else the same, the driver

is less likely to get slightly distracted than distracted. The logit Y23 for Passenger has a negative

constant term. Hence, keeping everything else the same, the driver is less likely to get distracted

than slightly distracted followed by very distracted. The logit Y43 for activity Passenger Not

Following Etiquette has a positive constant term. Hence, keeping everything else the same, the

driver is more likely to get distracted than slightly distracted followed by very distracted. Some

of the logit Yij have missing constant terms resulting in rescaling of the remaining constants

(Washington et al. 2011).

Table 6: MLR Utility Functions for Risk Zone Activities.

Activity Slightly Distracting (2) Distracting (3) Very Distracting (4)

Passenger Using

Mobile Phone (1)

(Y12) = -105.49 – 9.48LOCAT +

82.41SEX+1.65AGE

+2.57EXP+1.89DRIVING/WK

(Y13) = 156.58 – 5.82LOCAT + 20.06SEX –

2.72AGE +3.67EXP – 3.79DRIVING/WK

N/S**

Passengers (2) (Y22) = – 2.20LOCAT +

16.05SEX+0.13DRIVING/WK

(Y23) = -224.35 + 235.99SEX

+0.20EXP+4.53DRIVING/WK

(Y24) =

0.47DRIVING/WK

Fatigue/Sick (3) N/S N/S (Y34) = 137.74SEX

Passenger Not

Following

Etiquette (4)

(Y42) = – 4.47LOCAT + 53.49SEX (Y43) = 323.22 – 6.52LOCAT -6.26AGE -

7.99DRIVING/WK

(Y44) = 152.61SEX

Ticket Machine

(5)

(Y52) = 1050.21 – 11.51LOCAT - 68.67SEX-

20.30AGE +23.86EXP-26.51DRIVING/WK

N/S N/S

Climate

Control(6)

(Y62) = -1.46LOCAT - 0.05DRIVING/WK N/S N/S

Passenger Trying

to Talk to Driver

(7)

Model N/S Model N/S Model N/S

Note: SPSS 17.0 sets the reference level Not Distracted = 0 with Slightly Distracting (2), Distracting (3), and Very

Distracting (4) set = 1; Peninsula = 1 and Southside = 0; Male = 1 and Female = 0.

(Yij) is the estimated utility function that measures the total contribution of each significant factor where, column i =

1 to 6, and row j = 2 to 4. N/S = MLR Final Model or individual independent variables were not significant.

10

LOCATION: The driving location had a significant impact on the level at which a driver would

find the risk zone activity distracting. The multinomial independent variable LOCAT compares

Northside driving routes to Southside driving routes for the dependent variable. Since, the

coefficient of the variable LOCAT is consistently negative and odds ratios are < 1, one could

conclude that Southside drivers are more likely to get Slightly Distracted followed by Distracted

than Northside drivers by the passenger and technology related risk zone activities. It appears

that the higher distraction could possibly lead to increased safety hazard thereby confirming

occurrence of higher accident rate in the Southside (62 accident/million miles) compared to the

Northside (54 accident/million miles).

SEX: The literature reports mixed results for the impact of gender on driver distraction. The

multinomial independent variable SEX compares male drivers to female drivers for the

dependent variable. Since, the coefficient of the variable SEX is consistently positive and odds

ratios are > 1, one could conclude that male drivers are more likely than female drivers to get

Slightly Distracted followed by Distracted and Very Distracted by any of the passenger related

activities. Other studies have shown that male truck drivers are more likely to be involved in

rear-ended crashes (possibly due to distraction) compared to female drivers (Yan, et. al. 2009).

Female drivers are more likely than male drivers to get Slightly Distracted with the Ticket

Machine.

AGE: Age is an important factor related to accidents with younger drivers more prone to

distracted driving and accidents. The average age of the regional transit bus driver is 49 years

with Southside drivers slightly older (51 years) compared to Northside drivers (47 years).

Earlier studies conducted on passenger vehicle concluded that age of a driver had a significant

impact on distraction, with younger and older drivers more prone to distraction. This study

reveals positive and negative impact of age on distraction. The coefficient of the variable AGE

for technology-related activities such as Ticket Machine is a negative value and odds ratio is < 1

indicating that as age of the driver increases, drivers are less likely to get Slightly Distracted by

the ticket machine. The coefficient of the variable AGE for passenger-related distractions are

negative (except for slightly distracted) and odds ratio < 1 indicating that as age of the driver

increased, drivers are less likely to get slightly distracted followed by distracted due to

passenger-related activities.

EXPERIENCE: The transit bus drivers had an average of 12 years experience driving buses.

There was a significance difference in driving experience in both locations with drivers on the

Southside (15 years) having more driving experience as compared to the Northside drivers (8

years). It is believed that experienced drivers would get less distracted by the distracting

activities. The coefficient of EXP variables are consistently positive values and the odds ratios

are > 1, indicating that added experience increases the likelihood of getting Slightly Distracted

followed by Distracted due to passenger and ticket machine related activities. This appears

contrary to popular belief, where experience made a driver better at handling distraction and

possibly due to the more experienced Southside drivers also has a high accident rate.

DRIVING HOURS/WEEK: The bus drivers reported that they drive a bus for an average of 43

hours per week and that they typically drive the buses mostly during the day (65%) peak and

non-peak times and during the night (35%). The driving hours per week has been found to be

related to driver error although it was not reported as driver distraction. This study finds positive

and negative impact of driving load on distraction. The coefficients of the variable

DRIVING/WEEK for technology-related distraction such as Ticket Machine and Climate

Control are negative values and odds ratios < 1 indicating that as drivers with higher driving load

11

are less likely to get slightly distracted by technology related activities. The coefficient of the

variable DRIVING/WEEK for passenger-related activities is negative and positive values, hence

no conclusions could be drawn.

MODEL ASSESSMENT AND VALIDATION OF RESULTS

The likelihood ratio test using model fitting information for Passenger Using Mobile Phone is

presented in Table 5. It shows that the difference in the -2Log Likelihood between the intercept

only (without any independent variables) and the final model (with all the independent variables)

provides the chi-square (χ2) = 71.56 (24) signifying a good improvement in the model fit. It

follows that the independent variables contribute significantly to the outcome of the distraction

level. The values of the AIC (119.76/87.93) and the BIC (150.16/133.53) gets smaller during the

stepwise process indicating a good fit for the final model. Table 5 shows the model’s Goodness

of Fit as indicated by the multiple statistics. The p values for Pearson and Deviance (both test

the same results) chi-square (χ2) = 1.00 (p = 1) proving no significance. Hence, the predicted

values of the model are not significantly different from the observed values at all outcome levels

– the model fits the data well. The measures of Pseudo R2 (0,833, 0.897, and 0.678) are

reasonably similar and high values resulting in a good fit. The impact of each coefficient is

summarized in Table 7.

Table 7: Impact of Independent Variable Coefficients on Risk Zone Activities.

High Risk

Activity

Location Sex Age Driving Exp Driving Hrs/Wk

Passenger

Using

Mobile

Southside Drivers are

more likely to get

Slightly Distracted

followed by Distracted

Male Drivers are

more likely to get

Slightly Distracted

followed by

Distracted.

Older Drivers

are more likely

to get Slightly

Distracted but

less likely to

get Distracted.

Drivers with more

driving experience

are more likely to

get Slightly

Distracted followed

by Distracted

Drivers doing more

driving hours are

more likely to get

Slightly Distracted

but less likely to get

Distracted

Passenger Southside Drivers are

more likely to get

Slightly Distracted.

Male Drivers are

more likely to get

Distracted followed

by Slightly

Distracted.

N/S Drivers having more

driving experience

are more likely to

get Distracted

Drivers doing more

driving hours are

more likely to get

Distracted followed

by Very Distracted,

or Slightly Distracted.

Fatigue/Sick N/S Male Drivers are

more likely to get

Very Distracted.

N/S N/S N/S

Passenger

Not

Following

Etiquette

Southside Drivers are

more likely to get

Slightly Distracted

followed by Distracted

Male Drivers are

more likely to get

Very Distracted

followed by Slightly

Distracted

Older Drivers

are less likely

to get

Distracted.

N/S Drivers doing more

driving hours are less

likely to get

Distracted

Ticket

Machine

Southside Drivers are

more likely to get

Slightly Distracted.

Female Drivers are

more likely to get

Slightly Distracted

Older Drivers

are less likely

to get Slightly

Distracted.

Drivers having more

driving experience

are more likely to

get Slightly

Distracted.

Drivers doing more

driving hours are less

likely to get Slightly

Distracted

Climate

Control

Southside Drivers are

more likely to get

Slightly Distracted.

N/S N/S N/S Drivers doing more

driving hours are less

likely to get Slightly

Distracted

12

The graphical methods were found suitable to assess the best fit of the binary logistic regression

model (Landwehr et al. 1984), since model inadequacies are generally reflected in the pattern of

residuals generated by the statistical package (SPSS 17.0 2008). The model’s output data was

analyzed by plotting the Pearson residuals versus the Predicted Probability. The residuals are

scattered randomly above and below the line without any pattern over the entire range of the

predicted values (Figure 2).

Figure 2: Scatter plot of Pearson residuals

The MLR models developed for the risk zone activities provide estimates of the current levels of

distraction at the transit agency. Are the results generated from the MLR models for each risk

zone activity listed in Table 7 valid for a large random population of transit bus drivers?

Simulation of the models using probabilistic distributions generates driver distraction events that

would occur in practice over a range of random factors. Monte Carlo simulation using discrete

distribution that incorporate random variability into the model was applied to validate a model’s

output results. The MLR model for each risk zone activity was repeatedly simulated by a

different random set of values (inputs) drawn from the cumulative probability distribution of the

independent variable coefficients resulting in a set of possible distraction outcomes (outputs).

Each MLR logit in Table 6 were validated for 1,000 drivers by Monte Carlo simulation. The

simulated outputs presented graphically were compared to the MLR results in Table 7.

The Simulation Spreadsheet Model

The MLR model for each risk zone activity was defined along with their probability functions.

The probability function f(Yi) represented by Equation 6, for binary logistic regression models is

applicable to MLR models by breaking down the dependent variable into a series of pair wise

comparisons between two categories (Moutinho and Hutcheson 2011 and Field 2009). Each of

the MLR utility function Yij (Table 6) for the risk zone activities was simulated for 1,000

replications (bus drivers) with randomly selected location, age, sex, driving experience, and

driving hours/week. The cumulative probability distributions for each independent variable used

by the simulation model to generate the pseudorandom numbers were developed from the survey

data.

13

Following the approach of Smith et al. (2005), the five independent variable were

simulated for 1,000 replications one at a time, keeping the remaining variables constant (Albright

and Winston 2005). For each set of 1,000 replications, the simulation spreadsheet model

generated average probability values for Yij. The impact of independent variable coefficients on

high risk activities (Table 7) were validated by comparison with the simulated outputs.

MLR Model Validation

For each risk zone activity, the simulation spreadsheet model generated average probability

values for 1,000 drivers getting Slightly Distracted, Distracted, and Very Distracted by the

impact of the factors. They are presented as follows:

LOCATION. The MLR results for the five risk zone activities in Table 8 indicates that the

Southside drivers have a higher chance of getting Slightly Distracted and Distracted versus Not

Distracted. The simulation output in Figure 3 validates these results for all the passenger-related

activities and Climate Control. In the case of Ticket Machine, the probability of getting Slightly

Distracted is the same for Southside and Northside drivers.

SEX. The MLR results for the five risk zone activities in Table 9 indicates that male drivers

have a higher chance of getting Slightly Distracted, Distracted, and Very Distracted versus Not

Distracted. The simulation output in Figure 4 validates these results for all the passenger-related

activities and Fatigue/Sickness. In the case of Ticket Machine, the probability of getting Slightly

Distracted is the same for male and female drivers.

AGE. The MLR results for the three risk zone activities in Table 10 indicates that older drivers

have a higher chance of getting Slightly Distracted but less chance of getting Distracted by

Passenger Using Mobile Phone, Passenger Not Following Etiquette, and Ticket Machine. The

simulation output in Figure 5 validates these results for all the passenger-related activities. In the

case of Ticket Machine, the probability of getting Slightly Distracted is the same for older and

younger drivers.

DRIVING EXP. The MLR results for the three risk zone activities in Table 11 indicates that

more experienced drivers have a higher chance of getting Slightly Distracted and Distracted by

Passenger-related activities and Ticket Machine. The simulation output in Figure 6 validates

these results for Passenger Using Mobile Phone. In the case of Passenger and Ticket Machine,

the probability of getting Slightly Distracted and Distracted is the same for more experienced and

less experienced drivers.

DRIVING/WEEK. The MLR results for the five risk zone activities in Table 12 indicates that

drivers with more driving hours/week have a higher chance of getting Slightly Distracted and

Distracted by Passenger Using Mobile Phone but less Distracted by Passenger Not Following

Etiquette, Ticket Machine, and Climate Control. The simulation output in Figure 7 validates

these results for Passenger Using Mobile Phone and Passenger. In the case of Passenger Not

Following Etiquette and Ticket Machine, the probability of getting Slightly Distracted and

Distracted is the same for drivers with > 40 hours/week or < 40 hours/week.

14

Table 8: MLR Results for Location Risk Zone

Activity

MLR Results – Location Simulation Results (Fig. 3)

Passenger

Using Mobile

Phone

Southside Drivers are more

likely to get Slightly

Distracted followed by

Distracted

Probability of Slightly

Distracted and Distracted is

higher for Southside Drivers

Passenger Southside Drivers are more

likely to get Slightly

Distracted.

Probability of Slightly

Distracted is higher for

Southside Drivers

Passenger Not

Following

Etiquette

Southside Drivers are more

likely to get Slightly

Distracted followed by

Distracted

Probability of Slightly

Distracted and Distracted is

higher for Southside Drivers

Ticket Machine Southside Drivers are more

likely to get Slightly

Distracted.

Probability of Slightly

Distracted Equal for both

locations

Climate Control Southside Drivers are more

likely to get Slightly

Distracted.

Probability of Slightly

Distracted is higher for

Southside Drivers

Figure 3: Simulation Results for Location.

Table 9: MLR Results for Sex Risk Zone Activity MLR Results – Sex Simulation Results (Fig.

4)

Passenger Using

Mobile Phone

Male Drivers are more

likely to get Slightly

Distracted followed by

Distracted.

Probability of Slightly

Distracted and Distracted

is higher for Male Drivers

Passenger Male Drivers are more

likely to get Distracted

followed by Slightly

Distracted.

Probability of Slightly

Distracted and Distracted

is higher for Male Drivers

Fatigue/Sick Male Drivers are more

likely to get Very

Distracted.

Probability of Very

Distracted is higher for

Male Drivers

Passenger Not

Following Etiquette

Male Drivers are more

likely to get Very

Distracted followed by

Slightly Distracted

Probability of Slightly

Distracted and Very

Distracted is higher for

Male Drivers

Ticket Machine Female Drivers are more

likely to get Slightly

Distracted

Probability of Slightly

Distracted Equal for both

locations

Figure 4: Simulation Results for Sex

15

Table 10: MLR Results for Age Risk Zone Activity MLR Results – Age Simulation Results

(Fig. 5)

Passenger Using

Mobile Phone

Older Drivers are

more likely to get

Slightly Distracted

but less likely to get

Distracted.

Probability of Slightly

Distracted increases

with age and

Probability of

Distracted reduces

with age

Passenger Not

Following Etiquette

Older Drivers are less

likely to get

Distracted.

Probability of

Distracted reduces for

older drivers

Ticket Machine Older Drivers are less

likely to get Slightly

Distracted.

Probability of Slightly

Distracted Equal for

all ages

Figure 5: Simulation Results for Age.

Table 11: MLR Results for Driving Experience Risk Zone

Activity

MLR Results – Driving

Exp

Simulation Results (Fig. 6)

Passenger Using

Mobile Phone

Drivers with more driving

experience are more likely

to get Slightly Distracted

followed by Distracted

Probability of Slightly

Distracted and Distracted

increases with Driving

Experience

Passenger Drivers having more

driving experience are

more likely to get

Distracted

Probability of Distracted

Equal for all driving

experience

Ticket Machine Drivers having more

driving experience are

more likely to get Slightly

Distracted.

Probability of Slightly

Distracted Equal for all

driving experience

Figure 6: Simulation Results for Driving Experience

16

Table 12: MLR Results for Driving Hours/Week Risk Zone Activity MLR Results – Driving Hrs/Week Simulation Results (Fig. 7)

Passenger Using Mobile

Phone

Drivers doing more driving hours are more likely

to get Slightly Distracted but less likely to get

Distracted

Probability of Slightly Distracted is

higher for >40 hours/week but lower for

Distracted

Passenger Drivers doing more driving hours are more likely

to get Distracted followed by Very Distracted, or

Slightly Distracted.

Probability of Distracted and Slightly

Distracted is higher for >40 hours/week

but probability for Very Distracted is

equal for >40 and ≤40

Passenger Not

Following Etiquett

Drivers doing more driving hours are less likely to

get Distracted

Probability of Distracted is equal for >40

and ≤40 hour/week

Ticket Machine Drivers doing more driving hours are ess likely

to g t Slightly Distracted

Probability of Slightly Distracted is equal

for >40 and ≤40 hour/week

Climate Control Drivers doing more driving hours are less likely to

get lightly Dist acted

Probability of Slightly Distracted is lower

for >40 hours/week

Figure 7: Simulation Results for Driving

Hours/Week

Figure 8: Sample Size

Simulation results matches the predicted results for certain distraction causing factors like

“Passenger Using Mobile Phone” and “Climate Control” for all independent variables namely

sex, age, location, experience and driving per week. Two other distraction factors--“Passenger

Not Following Etiquette” and “Passenger” matches simulation and predicted results for all but

one independent variable. However, distraction factor “Ticket Machine” does not show any

convergence between predicted and simulated results for any independent variable. It is possible

that survey respondents have confounded between two distraction factors: “Ticket Machine” and

“Passenger”. In practice most of the “Ticket Machine’ distraction can be attributed to

“Passenger”. Therefore, those two factors can easily be confounded. Thus, some degree of

disagreement between simulation and statistical prediction is possible. A larger sample and

better explanation of survey to the respondents should improve result convergence between

statistical model and simulation.

In order to determine an appropriate sample size, simulation replications were conducted

for 1,000 drivers and correlation coefficient for the independent variables was computed for

17

various sample sizes. The current sample size of 48 results in a high correlation coefficient of

0.54 for variables Age and Experience. The correlation coefficient reduces as the sample size is

increased (Figure 8). A larger sample size would result in low standard error; and will be above

the preferred a case-to-variable ratio of 20:1 recommended by researchers (Petrucci 2009).

CONCLUSION

The results of the MLR models indicate that passenger-related activities which are beyond the

control of the driver are most distracting to the driver. It is therefore a challenge for the transit

agency to develop effective policies for handling passenger behavior, so that they are less likely

to engage in using cell phones, non etiquette behavior, noisy behavior etc. For example,

personal electronic devices could be allowed to be used beyond the middle section of the bus to

avoid distracting the driver. The front section of the bus could be designated as cell phone free.

Drivers must avoid unnecessary communications with passengers and if conversation cannot be

avoided, it must be done with caution while concentrating on the road ahead, or when the bus is

stopped. Training should focus on male drivers who are more likely to get distracted by

passengers.

Educational training program on the proper use of technological devices mounted in the

cab or issued to the driver, and hazards associated with utilizing these devices while driving

should focus on older and female drivers who are likely to get distracted with technological

devices. The design of ticket machine, control panel, and other devices must be user-friendly,

and not require long glances away from the forward roadway.

How could the transit agency use the MLR models developed in this study? They could

be applied to predict distraction for varying driver characteristics and driving patterns. It is

observed that drivers with differing factors are affected differently by distracting activities which

may be possibly corrected through proper training. The transit agency could develop driver

MLR models for each risk zone activity from its existing driver database. These models could be

used for predicting the probability of a new driver getting distracted by the risk zone activities. If

the probability value is high, the new driver could be scheduled for related training. Other

transit agencies could use this study as a framework for distraction analysis of their drivers.

The relatively small number of surveys completed (48 drivers) out of a total of 265

surveys distributed to bus drivers’ results in a low response rate of 18%. This amounts to a

number of cases per independent variable of 9.6 which is below the preferred a case-to-variable

ratio of 20:1. In addition to the five independent variables, other variables such as

environmental, vehicle, roadway designs etc could also have an impact on driver distraction but

have not been included in the MLR model. (Washington et al. 2011; Morfoulaki et al. 2007; Yan

et al. 2009; Gkritza et al. 2006). The classification of risk zone activities is somewhat subjective

and could be replaced by an objective index system similar to the Hazard Index developed by

Smith et al. (2005).

The presence of Multicollinearity that was detected from the Pearson Correlation analysis

makes it difficult to determine the relative importance of each independent variable on the

regression model and the effects on the dependent variable (Washington et al. 2011).

Furthermore, there may be curvilinear relationship between age and driving experience, resulting

in the violation of the linear assumption. There is no strong evidence at this point in the analysis

for adding or deleting any of the interactions. The increase in sample size (Figure 8) may reduce

the standard error, thereby mitigating the threat of multicollinearity. Furthermore, the MLR data

18

was not tested to show it meets the Independence of Irrelevant Alternative (IIA) specifications

which require the ratio of probabilities of selecting any two alternatives to be independent of the

third choice (Small and Hsiao 1985).

This exploratory study is one of only a few to examine the full range of distractions and

associated risk in transit buses. It has limited applications due to localized sample size and

limitations discussed above; hence inferences may not be drawn for the larger population of bus

drivers in the Commonwealth of Virginia or the United States. It should be followed by an

expanded study covering a larger sample of drivers form other agencies. Knowing the activities

that cause a high risk of distraction as well as the related factors may provide additional inputs to

law and policy makers while crafting legislations and regulations statewide or nationwide.

REFERENCES

AAA Foundation for Traffic Safety. “Distraction in Everyday Driving.” University of North

Carolina at Chapel Hill, Highway Safety Research Center, June, 2003. www.aaafoundation.org.

Albright, S. C, and W. L. Winston. “Introduction to Simulation Modeling.” Spreadsheet

Modeling and Applications: Essentials of Practical Management Science. Thomson

Brooks/Cole (2005): 423 – 488.

Bhadra, Dipasis. “Choice of Aircraft Fleets in the U. S. Domestic Scheduled Air Transportation

System: Findings from a Multinomial Logit Analysis.” Journal of the Transportation Research

Forum 44 (3), (2005): 143 – 162.

Blower, D., P. E. Green, and A. Matteson. “Bus Operator Types and Driver Factors in Fatal Bus

Crashes: Results from the Buses Involved in Fatal Accidents Survey.” U. S. DOT Federal Motor

Carrier Safety Administration. University of Michigan Transportation Research Institute, Ann

Arbor Michigan, June, 2008.

Bonate, Peter L. “A Brief Introduction to Monte Carlo Simulation.” Clin Pharmacokinetics 40

(1), (2001): 15-22.

Carlson, William L. “Application of Simulation to Assessment of Component Changes on rear

End Highway Accidents.” Transportation Science 6 (1), (February, 1972): 1 – 14.

Cross, G. W., H. Hanna, T. Garrison, and C. McKee. “Distracted Driving in Mississippi: A

Statewide Survey and Summary of Related Research and Policies.” Final Report by Family and

Children Research Unit, MSU. The Center for Mississippi Health Policy, September, 2010.

Daily Press Newspaper. “In States Across U. S. New Traffic Laws Gain Traction.” Nation &

World, January 02, 2012.

D’Souza, K. A. and S. K. Maheshwari. “Improving Performance of Public Transit Buses by

Minimizing Driver Distraction.” Urban Transport 2012 Conference, Coruňa, Spain,

(forthcoming) May 13-16, 2012.

19

Field, Andy. “Logistic Regression.” Discovering Statistics Using SPSS. SAGE Publications

Ltd., 3rd

Edition, (2009): 264 - 315.

Gkritza, K., D. Niemeier, and F. Mannering. “Airport Security Screening and Changing

Passenger Satisfaction: An Exploratory Assessment.” Journal of Air Transport Management 12,

(2006): 213 – 219.

Hickman, J. S., R. J. Hanowski, and J. Bocanegra. “Distraction in Commercial Trucks and

Buses: Assessing Prevalence and Risk in Conjunction with Crashes and Near-Crashes.” U.S.

DOT, Federal Motor Carrier Safety Administration, Office of Analysis, Research and

Technology, Washington, D.C., September, 2010.

Landwehr, J. M., D. Pregibon, and A. C. Shoemaker. “Graphical Methods for Assessing

Logistic Regression Models.” Journal of the American Statistical Association 79(385), (March,

1984): 61 – 71.

Metro Magazine. “Study: Transit Drivers Involved in Collisions Twice as Likely to Use Phone

While Driving.” Industry News. January 18, 2011.

Morfoulaki, M., Y. Tyrinopoulos, and G. Aifadopoulou. “Estimation of Satisfied Customers in

Public Transport Systems: A New Methodological Approach.” Journal of the Transportation

Research Forum 46(1), (Spring, 2007): 63-72.

Moutinho, L., and G. Hutcheson. “Multinomial Logistic Regression.” The SAGE Dictionary of

Quantitative Management Research. SAGE Publications Ltd. (2011): 208 -212.

Owens, J. M., and R. Lehman. “The Effects Of Age And Distraction On Reaction Time In A

Driving Simulator.” 1st International Driving Symposium on Human Factors in Driver

Assessment, Training and Vehicle Design, Aspen, CO, August 14 – 17, (2001): 147-152.

Oxley, J., J. Charlton, B. Fildes, S. Koppel, J. Scully, M. Congiu, and K. Moore. “Crash Risk of

Older Female Drivers.” Report No. 245. Monash University Accident Research Center,

Victoria, Australia, November, 2005.

Petrucci, Carrie J. “A Primer for Social Worker Researchers on How to Conduct a Multinomial

Logistic Regression. Journal of Social Service Research 35(2), (2009): 193 – 205.

Salmon, P. M., K. L. Young, and M. A. Regan. “Distraction ‘On the Buses’: A Novel

Framework of Ergonomics Methods for Identifying Sources and Effects of Bus Driver

Distraction.” Applied Ergonomics 42, (2011): 602-610.

Salmon, P.M., K. L. Young, and M. A. Regan. “Bus Driver Distraction Stage 1: Analysis of

Risk for State Transit Authority New South Wales Bus Drivers..” Final Report. Monash

University Accident Research Center, Victoria, Australia, June, 2006.

20

Salvucci, D. D., A. K. Chavez, and F. J. Lee. “Modeling Effects of Age in Complex Tasks: A

Case Study in Driving.” 26th

Annual Conference of the Cognitive Science Society, 2004.

Small, K. A. and Hsiao, C. “Multinomial Logit Specification Tests.” International Economic

Review 26 (3), (October, 1985): 619-627.

Smith, D. I., J. Chang, D. Cohen, J. Foley, and R. Glassco. “A Simulation Approach for

Evaluating the Relative Safety Impact of Driver Distraction During Secondary Tasks.” 12th

World Congress on ITS, San Francisco, 6 – 10 November, (2005): Paper 4340.

SPSS 17.0. SPSS Inc., Chicago, IL 60606-6412, 2008.

UCLA Annotated SPSS Output. “Multinomial Logistic Regression.” Academic Technology

Services, Statistical Consulting Group. http://www.ats.ucla.edu/stat/sas/notes2/ , 2007.

U. S. Department of Transportation. “An Examination of Driver Distraction as Recorded in

NHSTA Databases.” DOT HS 811 216, September 2009.

U. S. Department of Transportation. “Highway Accident Report: Gray Summit, MO Collision

Involving Two School Buses, a Bobtail, and a Passenger Vehicle, August 5, 2010.” National

Transportation Safety Board, http://www.ntsb.gov/news/events/2011/gray_summit_mo, 2011.

U. S. Department of Transportation. “Statistics and Facts About Distracted Driving.”

www.distraction.gov/stats-and-facts, 2010.

Virginia Department of Motor Vehicle. “Virginia Traffic Crash Facts.” Virginia Crashes

Involving Buses. Virginia Highway Safety Office, (2008): 58.

Washington, S. P., M. G. Karlaftis, and F. L. Mannering. “Logistic Regression, Discrete

Outcome Models and Ordered Probability Models.” Statistical and Econometric Methods for

Transportation Data Analysis. Chapman & Hall/CRC, (2011): 303 - 359.

Yan, X., E. Radwan, and M. Abdel–Aty. “Characteristics of Rear-End Accidents at Signalized

Intersections Using Multiple Logistic Regression Model.” Accident Analysis and Prevention 37,

(2005): 983-995.

Yan, X., E. Radwan, and K. K. Mannila. “Analysis of Truck-Involved Rear-End Crashes Using

Multinomial Logistic Regression.” Advances in Transportation Studies an International Journal

Section A 17, (2009): 39 – 52.

Yang, C. Y. D. “Trends in Transit Bus Accidents and Promising Collision Countermeasures.”

Journal of Public Transportation 10(3), (2007): 119 – 136.