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probability of responses within each LOS based on a combi- What the increasing value of the intercept guarantees in
nation of explanatory variables. This characteristic also this case is that for each (integer) value of RatingNum the se-
allowed the research team to make use of the nearly 1,650 quence of cumulative probabilities for a certain value l of
observations contained in the modeling database rather stops_per_mile are in the right order, meaning that,
than just 35 mean estimates of LOS (1 mean LOS for each
P(Y 1 | stops _ by _ mile = l) P(Y 2 | stops _ by _ mile = l)
video clip shown), and to predict the discrete outcome (i.e.
P(Y 3 | stops _ by _ mile = l) P(Y 4 | stops _ by _ mile = l)
1,2, . . ., 6) as generated from the video lab surveys.
P(Y 5 | stops _ by _ mile = l) 1
(Eq. 15)
Cumulative Logistic Regression
A positive slope is evident in Exhibit 70 as the increment in
For the auto LOS survey, the overall ratings (RatingNum) the cumulative probability for a particular RatingNum score
have a hierarchical ordering that varies from 1 (worst rating, when stops_per_mile value increases. The difference between
or LOS F ) to 6 (best rating, or LOS A). The discrete nature of successive curves for RatingNum scores determines the prob-
RatingNum rules out the use of ordinary Linear Regression, ability P(Y = j | stops_per_mile) for an individual RatingNum
because it requires the response to be a continuous variable. score given a fixed value of stops_per_mile. For instance, the
Cumulative logistic regression addresses the issue of model- value of P(Y 1) = P(Y = 1) is higher when stops_per_mile =
ing discrete variables with hierarchical ordering. 18 than when stops_per_mile = 1 and the value of P(Y = 5) =
Consider the following cumulative probability P(Y j | x) P(Y 5) - P(Y 4) is higher when stops_per_mile = 1 than
and define the logistic model for this probability as for stops_per_mile = 18, so it appears that, with higher prob-
ability, high ratings in LOS are given to trips with fewer stops
P(Y j | x )
Ln = (x ) (Eq. 12) per mile. Also shown in the exhibit are the marginal proba-
1 - P(Y j | x )
bilities of the various levels of service (A to F) when the num-
In general, P(Y = j | x) = 1 - P(Y j - 1 | x), so estimated ber of stops per mile is fixed at 2.
probabilities for all scores can be obtained. The vector rep-
resents the vector of coefficients for both LOS ranges (there
are 6 - 1 = 5 such intercept coefficients designated as s) as Best Candidate Auto LOS Models
well as the coefficients of the independent variables consid-
Preliminary modeling analysis has resulted in two fairly
ered in the model (designated as s). Equation 12 can be
strong models, one which uses number of stops per mile and
rewritten as
the other which uses average space mean speed as the primary
exp( x ) explanatory variable. Both models perform well. Both mod-
P(Y j | x ) = (3) (Eq. 13)
1 + exp( x ) els are presented with "trees" or "no trees" options. The rec-
ommended models are shown in Exhibits 72 through 74.
Each cumulative probability has its own intercept j; the
The Pearson Correlation measures the ability of each
values of j are increasing in j since P(Y j | x) increases in j
model to reproduce the observed video clip ratings of level of
for fixed x. The model assumes the same effects tree_presnce,
service. A higher value indicates a better model fit. The high-
stops_per_milet and Pres__Of_Ex__LT_Lane for each j.
est possible value is 100%.
In order to have an appropriate interpretation of the inter-
The Akaike Information Criterion (AIC) represents how
cept values, consider Model 3 for two scores j and k with j < k
close fitted values are to actual values taking into account the
and assume values = 0 for the two dummies tree_presence
number of parameters included in each model (Agresti, 2002
and Pres__Of_Ex__LT_lane. After some algebraic manipula-
[97]). Lower values indicate a superior model of the data.
tions we have
Model 4 (stops per mile, presence of an exclusive left-turn
P(Y k | stops_per_mile)= P(Y j | stops_per_mile lane and presence of trees) reported the lowest AIC measure
+ (k - j) / ). (Eq. 14) and the highest Pearson correlation.
Cumulative probability for j is the same as the cumulative
probability for k but evaluated at a stops_per_mile value dis- Performance of Candidates
placed by an amount dependent on the positive difference be-
tween intercepts at score j and k, and the parameter . A preliminary analysis of the ability of Models 4, 5, and 6
Exhibit 70 illustrates this model with increasing values of to predict the distribution of ratings of LOS as reported by
each j and positive value of for stops_per_mile. The model participants was also undertaken. (The performance of
coefficients, estimated using the Maximum Likelihood estima- Model 7 is presented in a later section describing refinement
tion methods, and their significance are shown in Exhibit 71. options for the recommended Auto LOS Model.)

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Exhibit 70. Example Cumulative Logit Distribution of LOS.
1.00
P(Y=6| x=2)
0.90
0.80 P(Y=5| x=2)
0.70
0.60
P(Y1)
P(Y j) P(Y2)
P(Y=4| x=2)
0.50 P(Y3)
P(Y4)
P(Y5)
0.40
0.30 P(Y=3| x=2)
0.20
P(Y=2| x=2)
0.10
P(Y=1| x=2)
0.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Stops by mile (x)
Because the strength of the cumulative probability models resulted in the largest number of observations in the test
lies in their ability to predict the distribution of LOS ratings dataset, so only these four clips have been tested with Models
for a particular combination of explanatory variables, the 4, 5, and 6. The remaining clips in the test database do not
models were tested to determine their ability to accurately have enough observations to develop a robust distribution of
predict a distribution of responses as compared with those response data. Clips 2, 15, 52, and 56 are discussed in the fol-
collected at the four study sites. One-third of the auto re- lowing analysis.
sponse dataset was reserved to test the fit of various models Exhibits 75 through 78 compare the observed LOS rating
developed in this study and was not used in the model cali- distributions to those predicted by Model 4 (Stops/Mile; left-
bration. A subset of four clips was chosen to be shown in each turn lane presence; tree presence index), Model 5 (Space
of the four study sites in Phase II of the study. These four clips Mean Speed; median presence index; tree presence index)
Exhibit 71. Maximum Likelihood Estimates for Model #4.
Standard Wald
Parameter DF Estimate Error Chi-Square Pr > ChiSq
Intercept 1 1 -2.9189 0.2270 165.4053 <.0001
Intercept 2 1 -1.8273 0.2075 77.5198 <.0001
Intercept 3 1 -0.8529 0.2009 18.0246 <.0001
Intercept 4 1 0.2832 0.2005 1.9951 0.1578
Intercept 5 1 2.0937 0.2091 100.3006 <.0001
stops_per_mile 1 0.2033 0.0184 122.3357 <.0001
Pres__Of_Ex__LT_Lane 1 -0.5218 0.1111 22.0627 <.0001
Tree_Presence 1 -0.3379 0.0612 30.4761 <.0001
Parameters for Cumulative Regression Model Applied to Auto LOS--Stops per Mile
Model Model 4

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Exhibit 72. Recommended Auto LOS Models.
Parameter Model 4 Model 5 Model 6 Model 7
Intercept LOS E, Alpha1= -2.92 -0.73 -3.80 -1.19
Intercept LOS D, Alpha2= -1.83 0.28 -2.70 -0.20
Intercept LOS C, Alpha3 = -0.85 1.21 -1.74 0.71
Intercept LOS B, Alpha4= 0.28 2.32 -0.62 1.80
Intercept LOS A, Alpha5= 2.09 4.16 1.16 3.62
Stops Per Mile, Beta1 = 0.20 0.25
Presence of Left-Turn Lanes, Beta2 = -0.52 -0.34
Mean Speed (mph), Beta1 = -0.063 -0.084
Median Presence (0-3), Beta2 = -0.33 -0.22
Presence of Trees, Beta3 = -0.34 -0.42
Pearson Correlation= 79% 76% 77% N/A
Akaike Information Criterion (AIC) 4944.0 5034.1 5022.8 5076.4
Exhibit 73. Test Clip Characteristics.
Average Median Presence Tree
Space (0-No Ex. Lt.-Turn- Presence
Mean 1-One-way Pair Lane Presence (1-Few
Clip Number of Speed 2-TWCLTL (0-No 2-Some HCM
Number Stops/Mile (mph) 3-Raised 1-Yes) 3-Many) LOS
2 0 34.5 3 1 2 A=6
15 6 7.86 3 1 1 F=1
52 7 7.9 0 0 1 E=2
56 2 23.1 3 1 3 C=4
Exhibit 74. Correlation Coefficients of Auto LOS Models.
Models Compared Pearson Correlation Coefficient
HCM LOS to Mean Observed LOS 0.465
Mean Observed LOS to Mean LOS Model 4 0.787
Mean Observed LOS to Mean LOS Model 5 0.764
Mean Observed LOS to Mean LOS Model 6 0.770
Exhibit 75. Evaluation of Models Against Clip 2 Ratings.
Clip 2 - Comparing LOS Distributions of Test Data and Models 4, 5 and 6
(N=59) - HCM LOS=6
50.00%
45.00%
TEST MODEL 4
40.00%
MODEL 5 MODEL 6
Probability Mass Function
35.00%
30.00%
25.00%
20.00%
15.00%
10.00%
5.00%
0.00%
1 2 3 4 5 6
LOS Ratings (A=6; F=1)

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Exhibit 76. Evaluation of Models Against Clip 15 Ratings.
Clip 15 - Comparing LOS Distributions of Test Data and Models 4, 5 and 6
(N=42) - HCM LOS=1
35.00%
30.00%
TEST MODEL 4
MODEL 5 MODEL 6
Probability Mass Function
25.00%
20.00%
15.00%
10.00%
5.00%
0.00%
1 2 3 4 5 6
LOS Rating (A=6 F=1)
and Model 6 (Stops per mile; left-turn lane presence) for the closer to the test dataset represented by the periwinkle bar.
four video clips. Exhibit 73 lists the conditions depicted in For Clip 15, there is a definite difference between the HCM
each of the four clips and the current HCM-estimated LOS. LOS of F and the distribution of LOS as provided by the study
Overall the models appear to track comparatively well with participants, which is shifted toward the right, meaning higher
each other and with the data, in that there is a general increase/ LOS ratings. In this clip, there are many stops along a short ar-
decrease in the estimation of LOS probability. Model 4 has terial, however, there is only low to moderate traffic congestion
slightly higher predictive power--it tends to track slightly so that the test vehicle is always in the first position of the queue
Exhibit 77. Evaluation of Models Against Clip 52 Ratings.
Clip 52 - Comparing LOS Distributions of Test Data and Models 4, 5 and 6
(N=44) - HCM LOS=2
35.00%
30.00%
TEST MODEL 4
MODEL 5 MODEL 6
Probability Mass Function
25.00%
20.00%
15.00%
10.00%
5.00%
0.00%
1 2 3 4 5 6
LOS Rating (A=6 F=1)