Appendix
List of Figures
Figure A1
The layout of video camera in the field
Figure A2
Intersection added with a dummy grid system for identifying conflicts data
Figure A3
Figure A4
Figure A5
The white line extension at selected sites
Plot of predicted conflicts against the standardized residuals
Three types of median at selected sites
Figure A6
Cumulative residuals of the conflict predictive model with ±2ϭ* bands
Figure A7
Cumulative residuals of the separate conflict predictive models with ±2ϭ* bands
Figure A8
Plot of the predicted conflicts against the observed conflicts
List of Tables
Table A1
Results of the distributional analysis of conflict frequency
Table A2
Results of the linear regression model
Table A3
Regression results of the overall NB model
Table A4
Regression results for the conflict predictive models for different traffic scenarios
Table A5
Comparisons of the prediction performance of conflict predictive models
Methodology
Generalized linear regression (GLM) models have been widely used in previous studies for
modeling crash frequency data. In this study, generalized linear regression models were used to fit
the frequency of opposing left-turn conflicts observed during a particular time interval to
explanatory variables. The specification of a GLM model usually starts with a Poisson regression
model which assumes that the dependent variable is Poisson distributed. Let μi represent the
expected number of opposing left-turn conflicts on an entity for a specific time period, the
probability of observing yi conflicts is given by:
p(Yi  yi )  p( yi ) 
iy e 
i
yi !
, i  1, 2, 3,..., n
(1)
A logarithm link function connects μ to a linear predictor. The link function and the linear
predictor determine the functional forms of the conflict predictive model. If the linear predictor is a
linear function of the explanatory variables, the fitted conflict predictive model takes an additive
functional form. If the linear predictor is a linear function of the logarithm of the explanatory
variables, the fitted model takes a multiplicative functional form. The Poisson regression model
assumes that the mean of the conflict counts equals the variance. This is usually not a good
assumption because the variance of the conflict counts is often greater than the mean. In this
condition, the estimated coefficients of the Poisson regression model are biased. To deal with the
over dispersed data a negative binomial model can be used. The negative binomial model
assumes that the conflict counts are Poisson-gamma distributed. The probability density function
of the Poisson-gamma structure is given by:
p(Yi  yi ) 
( yi   1 ) i yi 1  1
(
) (
) , i  1, 2,3,..., n
y !( 1 ) 1  i 1  i
(2)
where α is the dispersion parameter which determines the variance of the Poisson-gamma
distribution. Two parameters are often used for evaluating the goodness-of-fit of a generalized
linear model, including the scaled deviance (SD) and the Pearson’s χ2 statistic. The scaled
deviance equals twice the difference between the log-likelihood under the maximum model and
the log-likelihood under the reduced model. The scaled deviance can be calculated as:
SD  2(log(( L )  log( Ls ))
(3)
where Ls is the likelihood under the maximum model and Lβ is the likelihood under the reduced
model. The Pearson’s χ2 statistic can be calculated as
 yi  i 

i 
i 1 
n
2  
2
(4)
where yi is the observed number of traffic conflicts, μi is the expected number conflicts, and σi is
the estimation error.
Figure A1
The layout of video camera in the field
Figure A2
Intersection added with a dummy grid system for identifying conflicts data
Figure A3
The white line extension at selected sites
Figure A4
Plot of predicted conflicts against the standardized residuals
Figure A5
Three types of median at selected sites
Figure A6
Cumulative residuals of the conflict predictive model with ±2ϭ* bands
Figure A7
Cumulative residuals of the separate conflict predictive models with ±2ϭ* bands
Figure A8
Plot of the predicted conflicts against the observed conflicts
Table A1 Results of the distributional analysis of conflict frequency
p-value of the goodness-of-fit test
Conflicting volume
Time interval
Normal
Poisson
regime
NB distribution
distribution
distribution
low
<0.001
0.008
0.799
5-min
medium
<0.001
0.014
0.926
high
<0.001
0.210
<0.001
low
<0.001
<0.001
0.481
15-min
medium
0.027
<0.001
0.256
high
0.003
<0.001
0.231
low
<0.001
<0.001
0.116
30-min
medium
0.801
<0.001
0.431
high
0.440
0.029
0.226
Table A2 Results of the linear regression model
Std.
Variables
Coefficient
Error
Constant
-8.234
1.266
Logarithm of opposing through traffic volume
1.447
0.215
Logarithm of left-turn volume
1.573
0.246
Number of lanes
1 if n1=3, n2=6; 0 otherwise
2.286
0.517
1 if n1=3, n2=4; 0 otherwise
1.455
0.405
1 if n1=2, n2=6; 0 otherwise
0.982
0.397
n1=1, n2=2
0
0.000
Presence of white line extension
1 (Yes)
-1.325
0.244
0 (No)
0
0.000
Average turning radius of left-turn traffic
0.075
0.017
Available green time allocated to left-turn phase
-0.030
0.011
R2
0.498
Adjusted R2
0.490
t-statistic
Sig.
-6.506
6.731
6.396
<0.0001
<0.0001
<0.0001
4.421
3.591
2.473
<0.0001
<0.0001
<0.0001
-5.425
<0.0001
4.333
-2.811
<0.0001
0.005
Table A3 Regression results of the overall NB model
Variables
Coefficient
Std. Error
χ2
Pr>χ2
Constant
Logarithm of opposing through traffic volume
Logarithm of left-turn volume
Number of lanes
1 if n1=3, n2=6; 0 otherwise
1 if n1=3, n2=4; 0 otherwise
1 if n1=2, n2=6; 0 otherwise
n1=1, n2=2
Median type
1 for raised median; 0 otherwise
1 for barrier; 0 otherwise
1 for double yellow line; 0 otherwise
Presence of white line extension
1 (Yes)
0 (No)
Average turning radius of left-turn traffic
Available green time allocated to left-turn phase
Log Likelihood
Scaled Deviance
Pearson χ2
SD/DF
Pearsonχ2/DF
Dispersion Parameter
-2.6919
0.4482
0.4681
0.3766
0.0512
0.0611
51.08
76.57
58.77
<0.0001
<0.0001
<0.0001
0.6270
0.4767
0.3884
0
0.1866
0.1440
0.1452
0.0000
11.29
10.96
7.15
0.0008
0.0009
0.0075
-0.4939
-0.2718
0
0.1727
0.1248
0.0000
8.18
4.75
0.0042
0.0294
-0.1852
0
0.0083
-0.0060
-1121.39
617.90
569.74
1.27
1.17
0.05
0.0510
0.0000
0.0033
0.0025
13.18
0.0003
6.44
5.88
0.0111
0.0153
Model
Table A4 Regression results for the conflict prediction models for different traffic scenarios
Scaled
Pearson SD Pearson
Log(T)
Constant
Log(L)
M
2
2
2
2
Deviance
χ2
/DF
χ2/DF
Value
χ
Value
χ
Value
χ
Value
χ
1
-1.478
(0.52a)
8.09
0.377
(0.05)
57.5
0.589
(0.12)
25.9
2
-2.830
(0.72)
15.6
0.527
(0.16)
10.2
0.618
(0.16)
15.0
3
-0.513
(0.46)
1.22
0.187
(0.06)
10.6
0.440
(0.11)
15.9
4
-0.322
(0.09)
-0.173
(0.09)
5.43
5.86
155.45
147.67
1.18
1.11
138.23
122.96
1.19
1.06
169.57
162.21
1.00
0.95
-2.722
1.151
0.263
-0.323
1.94
25.2
0.38
2.71
80.86
74.04
1.28
(1.96)
(0.23)
(0.42)
(0.19)
Note: athe number in the parentheses represents the standard error of each coefficient
1.17
Table A5 Comparisons of the prediction performance of conflict prediction models
Traffic
GLM models for
Linear
Measure
GLM models
scenarios
different traffic scenarios
regression model
Scenario 1
0.62
2.26
2.55
Scenario 2
1.18
1.38
1.34
MAD
Scenario 3
0.85
1.26
1.27
Scenario 4
1.35
1.57
2.00
MSPE
Scenario 1
Scenario 2
Scenario 3
Scenario 4
1.52
2.41
1.98
3.43
7.05
3.08
2.29
4.46
8.90
3.39
2.49
2.61