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65 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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66 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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67 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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68 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)