Selecting Ramp Design Speeds, Volume 2: Conduct of Research Report (2021)

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68 Section 4. Development of Ramp Speed Prediction Models Based upon a review of the speed data collected as part of this project (see Section 3) and speed data collected along ramps as part of previous research (Torbic et al., 2012), a database was assembled to develop ramp speed prediction models that could be incorporated into a spreadsheet-based RSPM tool to aid in ramp design. This section describes the database and statistical analysis used to develop the ramp speed prediction models. Separate statistical analyses were conducted to develop ramp speed prediction models for curves and tangents. 4.1 Ramp Speed Prediction Models for Curves The objective of the statistical analysis was to develop ramp speed prediction models for curves using a database that included speed data from ramps located in Kansas, Missouri, and Texas. Several models were fit to predict vehicle speeds at the midpoint (MC) and end (PT) of curves on entrance ramps, exit ramps, and connector ramps. Only the best fitting models, based on both statistical criterion (BIC) and engineering judgment, are provided below. These are the models incorporated into the spreadsheet-based RSPM tool. Due to limited data on freeway-to-freeway ramps at system interchanges, no models were found that provided statistically reliable results that appeared valid from an engineering perspective. Therefore, only the models developed to predict vehicle speeds at the midpoint (MC) and end (PT) of curves on entrance and exit ramps at service interchanges are provided below. The number of curves and observations included in the final database used to develop models to predict vehicle speeds at the midpoint (MC) and end (PT) of curves on entrance and exit ramps is summarized in Table 17. Table 17. Number of curves and observations in database to model speeds at midpoint and end of curves Entrance Ramps Exit Ramps Number of Curves Number of Observations Number of Curves Number of Observations Database Used to Model Vehicle Speeds at Midpoint of Curve (VMC) 11 1615 17 2086 Database Used to Model Vehicle Speeds at End of Curve (VPT) 11 1614 5 474 The dependent variables in the analyses were vehicle speed at the midpoint of the curve (MC) and vehicle speed at the end of the curve (PT). The candidate predictor variables included: • Vehicle speed at the beginning of the curve (VPC). • Curve radius (R). • Curve length (LC). • Ramp length (Lr). • Gradient (level, upgrade, downgrade). • Posted speed limit of the freeway (VSL Frwy). Table 18 provides summary statistics of the dependent and predictor variables in the ramp speed database for curves.

69 Table 18. Descriptive statistics of ramp speed database for curves Variable Descriptive Entrance Ramp Exit Ramp De pe nd en t V ar ia bl es Vehicle speed at midpoint of curve (VMC) (mph) Maximum 86.1 69.5 Mean 46.3 33.8 Standard deviation 7.3 10.1 Median 45.9 32.0 Minimum 17.3 12.0 Vehicle speed at end of curve (VPT) (mph) Maximum 87.0 69.3 Mean 47.0 43.5 Standard deviation 8.2 11.6 Median 46.7 46.1 Minimum 11.7 10.0 Pr ed ict or V ar ia bl es Vehicle speed at beginning of curve (VPC) (mph) Maximum 85.1 69.6 Mean 44.5 40.6 Standard deviation 7.5 8.9 Median 44.8 40.8 Minimum 15.5 15.0 Curve radius (R) (ft) Maximum 12101 1848 Mean 2376 475 Standard deviation 2957 475 Median 1531 264 Minimum 158 106 Curve length (LC) (mi) Maximum 0.12 0.20 Mean 0.04 0.11 Standard deviation 0.03 0.05 Median 0.04 0.09 Minimum 0.01 0.01 Ramp length (Lr) (mi) Maximum 0.70 0.37 Mean 0.26 0.21 Standard deviation 0.17 0.07 Median 0.21 0.19 Minimum 0.13 0.14 Gradient (count/proportion) Level 1306 (0.17) 873 (0.30) Upgrade 122 (0.07) 1165 (0.40) Downgrade 300 (0.76) 855 (0.30) Posted speed limit of the freeway (VSL Frwy) (mph) Maximum 75 80 Mean 61 62 Standard deviation 4.9 5.5 Median 60 60 Minimum 55 50

70 Entrance Ramps Several models were fit to predict vehicle speeds at the midpoint (MC) and end (PT) of curves on entrance ramps. Only the best fitting models are provided here based on both statistical criterion (BIC) and engineering judgment. In the final models, vehicle speed at the beginning of the curve (VPC), curve radius (R), curve radius squared (R2), and the posted speed limit of the freeway (VSL Frwy) were treated as fixed effects, while the Curve ID was treated as a random effect. The final model to predict vehicle speeds at the midpoint of curves on entrance ramps is provided as Equation 33. It is based on data from 1615 vehicles observed on 11 curves. Table 19 provides a summary of the statistical results of the final model to predict vehicle speeds at the midpoint of curves on entrance ramps, and Figure 15 provides a plot of the actual to predicted values. VMC = -8.7255 + 1.0125VPC + 4.7053R – 2.0183R2 + 0.1316VSL Frwy (33) where: VMC = Predicted vehicle speed at the MC, mph. VPC = Vehicle speed at the beginning of a curve (i.e., initial point of horizontal curve – PC), mph. R = Curve radius, mi. VSL Hwy = Regulatory speed limit on the highway (or freeway) mainline, mph. Table 19. Statistical parameters of final model to estimate vehicle speed at midpoint of curve on an entrance ramp Summary of Fit RSquare 0.983638 RSquare Adj 0.983598 Root Mean Square Error 0.932094 Mean of Response 46.29867 Observations (or Sum Wgts) 1615 Parameter Estimates Term Estimate Std Error DFDen t Ratio Prob>|t| Intercept -8.725519 2.755146 7.15 -3.17 0.0153 VPC (mph) 1.012546 0.004218 1609 240.05 <.0001 R (mi) 4.705275 0.967322 7.084 4.86 0.0018 R2 (mi) -2.018305 0.388952 7.018 -5.19 0.0013 VSL Frwy (mph) 0.131626 0.046524 7.104 2.83 0.0250 REML Variance Component Estimates Random Effect Var Ratio Var Component Std Error 95% Lower 95% Upper Pct of Total Curve_ID 0.226312 0.1966198 0.1087033 -0.016435 0.4096744 18.455 Residual 0.8687993 0.0306873 0.811656 0.9322278 81.545 Total 1.0654191 0.1128967 0.874650 1.3265609 100.000 -2 LogLikelihood = 4405.6999379 Note: Total is the sum of the positive variance components. Total including negative estimates = 1.0654191 Fixed Effect Tests Source Nparm DF DFDen F Ratio Prob > F VPC (mph) 1 1 1609 57624.11 <.0001* R (mi) 1 1 7.084 23.6607 0.0018* R2 (mi) 1 1 7.018 26.9266 0.0013* VSL Frwy (mph) 1 1 7.104 8.0044 0.0250*

71 Figure 15. Plot of actual versus predicted values of final model to estimate vehicle speed at midpoint of curve on an entrance ramp The final model to predict vehicle speeds at the end of curves on entrance ramps is provided as Equation 34. It is based on data from 1614 vehicles observed on 11 curves. Table 20 provides a summary of the statistical results of the final model to predict vehicle speeds at the end of curves on entrance ramps, and Figure 16 provides a plot of the actual to predicted values. VPT = -12.1179 + 1.0127VPC + 8.3558R – 3.5183R2 + 0.1882VSL Frwy (34) where: VPT = Predicted vehicle speed at the end of a curve (PT), mph.

72 Table 20. Statistical parameters of final model to estimate vehicle speed at end of curve on an entrance ramp Summary of Fit RSquare 0.943958 RSquare Adj 0.943818 Root Mean Square Error 1.835069 Mean of Response 47.25775 Observations (or Sum Wgts) 1614 Parameter Estimates Term Estimate Std Error DFDen t Ratio Prob>|t| Intercept -12.11791 3.976288 7.247 -3.05 0.0179 VPC (mph) 1.01265 0.008338 1596 121.44 <.0001 R (mi) 8.35582 1.393358 7.127 6.00 0.0005 R2 (mi) -3.51828 0.559098 7.002 -6.29 0.0004 VSL Frwy (mph) 0.18820 0.067053 7.152 2.81 0.0257 REML Variance Component Estimates Random Effect Var Ratio Var Component Std Error 95% Lower 95% Upper Pct of Total Curve_ID 0.1168546 0.3935055 0.2247703 -0.047036 0.8340472 10.463 Residual 3.3674784 0.1189807 3.1459274 3.6134073 89.537 Total 3.7609839 0.2539168 3.3090162 4.3128333 100.000 -2 LogLikelihood = 6578.4878322 Note: Total is the sum of the positive variance components. Total including negative estimates = 3.7609839 Fixed Effect Tests Source Nparm DF DFDen F Ratio Prob > F VPC (mph) 1 1 1596 14748.62 <.0001 R (mi) 1 1 7.127 35.9627 0.0005 R2 (mi) 1 1 7.002 39.5991 0.0004 VSL Frwy (mph) 1 1 7.152 7.8781 0.0257

73 Figure 16. Plot of actual versus predicted values of final model to estimate vehicle speed at end of curve on an entrance ramp Exit Ramps Several models were fit to predict vehicle speeds at the midpoint (MC) and end (PT) of curves on exit ramps. Only the best fitting models are provided here based on both statistical criterion (BIC) and engineering judgment. In the final models, vehicle speed at the beginning of the curve (VPC), curve radius (R), curve radius squared (R2), curve length (LC), gradient, and the posted speed limit of the freeway (VSL Hwy) were treated as fixed effects, while the Curve ID was treated as a random effect. The final model to predict vehicle speeds at the midpoint of curves on exit ramps is provided as Equation 35. It is based on data from 2086 vehicles observed on 17 curves. Table 21 provides a summary of the statistical results of the final model to predict vehicle speeds at the midpoint of curves on exit ramps, and Figure 17 provides a plot of the actual to predicted values. VMC = -13.4726 + 0.5951VPC + 208.5633R – 521.3073R2 + 0.2361VSL Hwy + 2.1981IDN + 0.7507 ILV – 2.9488 IUP (35)

74 where: IDN = Indicator variable for downward slope (= 1 if the overall ramp slopes downward, 0 otherwise). ILV = Indicator variable for level slope [= 1 if the overall ramp is level (i.e., ± 2 percent), 0 otherwise]. IUP = Indicator variable for upward slope (= 1 if the overall ramp slopes upward, 0 otherwise). Table 21. Statistical parameters of final model to estimate vehicle speed at midpoint of curve on an exit ramp Summary of Fit RSquare 0.931439 RSquare Adj 0.931241 Root Mean Square Error 2.645201 Mean of Response 33.75379 Observations (or Sum Wgts) 2086 Parameter Estimates Term Estimate Std Error DFDen t Ratio Prob>|t| Intercept -13.47259 8.7789 11.08 -1.53 0.1529 VPC (mph) 0.59513 0.0126 2075 47.13 <.0001 R (mi) 208.56333 44.2822 11.08 4.71 0.0006 R2 (mi) -521.30730 134.9935 11.07 -3.86 0.0026 VSL Hwy (mph) 0.23606 0.1343 11.06 1.76 0.1064 IDN 2.19809 1.4102 11.15 1.56 0.1470 ILV 0.75067 1.6935 11.15 0.44 0.6661 IUP -2.94880 REML Variance Component Estimates Random Effect Var Ratio Var Component Std Error 95% Lower 95% Upper Pct of Total Curve_ID 1.1879258 8.31202 3.5684894 1.31791 15.306132 54.295 Residual 6.99709 0.2175978 6.58945 7.443952 45.705 Total 15.30911 3.5750283 10.15883 25.690158 100.000 -2 LogLikelihood = 10040.753648 Note: Total is the sum of the positive variance components. Total including negative estimates = 15.30911 Fixed Effect Tests Source Nparm DF DFDen F Ratio Prob > F VPC (mph) 1 1 2075 2221.6480 <.0001 R (mi) 1 1 11.08 22.1828 0.0006 R2 (mi) 1 1 11.07 14.9129 0.0026 VSL Hwy (mph) 1 1 11.06 3.0886 0.1064 Gradient 2 2 11.10 5.0777 0.0272 Effect Details (Gradient) Least Squares Means Table Level Least Sq Mean Std Error IDN 36.578765 1.4599695 ILV 35.131343 2.1113467 IUP 31.431912 1.0579163 LSMeans Differences Tukey HSD α = 0.050 Level Least Sq Mean IDN A 36.578765 ILV A B 35.131343 IUP B 31.431912 Levels not connected by same letter are significantly different.

75 Figure 17. Plot of actual versus predicted values of final model to estimate vehicle speed at midpoint of curve on an exit ramp The final model to predict vehicle speeds at the end of curves on exit ramps is provided as Equation 36. It is based on data from 474 vehicles observed on five curves. Table 22 provides a summary of the statistical results of the final model to predict vehicle speeds at the end of curves on exit ramps, and Figure 18 provides a plot of the actual to predicted values. VPT = -0.6272 + 0.8637VPC + 108.0929R – 265.9747R2 – 190.89.8941LC (36) where: Lc = Curve segment length, mi.

76 Table 22. Statistical parameters of final model to estimate vehicle speed at end of curve on an exit ramp Summary of Fit RSquare 0.95242 RSquare Adj 0.95201 Root Mean Square Error 2.55227 Mean of Response 43.54403 Observations (or Sum Wgts) 474 Parameter Estimates Term Estimate Std Error DFDen t Ratio Prob>|t| Intercept -0.6272 2.50145 1.396 -0.25 0.8339 VPC (mph) 0.8637 0.02292 441.2 37.69 <.0001 R (mi) 108.0929 23.70501 0.999 4.56 0.1376 R2 (mi) -265.9747 62.03266 0.971 -4.29 0.1514 LC (mi) -190.8941 14.25417 1.007 -13.39 0.0467 REML Variance Component Estimates Random Effect Var Ratio Var Component Std Error 95% Lower 95% Upper Pct of Total Curve_ID 0.0571489 0.3722732 0.6345339 -0.87139 1.615937 5.406 Residual 6.5140974 0.4258399 5.75387 7.436460 94.594 Total 6.8863706 0.7613931 5.60761 8.661051 100.000 -2 LogLikelihood = 2223.6163292 Note: Total is the sum of the positive variance components. Total including negative estimates = 6.8863706 Fixed Effect Tests Source Nparm DF DFDen F Ratio Prob > F VPC (mph) 1 1 441.2 1420.360 <.0001 R (mi) 1 1 0.999 20.793 0.1376 R2 (mi) 1 1 0.971 18.384 0.1514 LC (mi) 1 1 1.007 179.350 0.0467

77 Figure 18. Plot of actual versus predicted values of final model to estimate vehicle speed at end of curve on an exit ramp 4.2 Ramp Speed Prediction Models for Tangents The objective of this statistical analysis was to develop ramp speed prediction models for tangents using an assembled database that included speed data from ramps located in Kansas, Missouri, and Texas. Speed data consisted of observations at the beginning of each tangent, at a point one-third of the tangent length, at a point two-thirds of the tangent length, and at the end of the tangent. Several models were fit, and it was decided to focus on predicting vehicle speeds at the end of the tangent for tangents with subsequent curves and for tangents without subsequent curves on entrance ramps, exit ramps, and connector ramps. Only the best fitting models, based on both statistical criterion (BIC) and engineering judgment, are provided below. These are the models incorporated into the spreadsheet-based RSPM tool. Due to limited data for tangents on freeway-to-freeway ramps, no models were found that provided statistically reliable results that appeared valid from an engineering perspective. Therefore, only the models developed to predict vehicle speeds at the end of the tangent, for tangents with and without subsequent curves on entrance and exit ramps at service interchanges are provided below. The number of tangents and observations included in the final database used to develop models to predict vehicle speeds at the end of the tangent for tangents with subsequent curves and without subsequent curves on entrance and exit ramps is summarized in Table 23.

78 Table 23. Number of tangents and observations in database to model speeds at the end of the tangent Entrance Ramps Exit Ramps Number of Tangents Number of Observations Number of Tangents Number of Observations Database used to model vehicle speed at end of Tangent with subsequent curve (Vt,e,c) 5 546 6 840 Database used to model vehicle speed at end of tangent without subsequent curve (Vt,e,nc) 5 753 1 77 The dependent variables in the analyses were vehicle speed at the end of the tangent with a subsequent curve (Vt,e,c) and without a subsequent curve (Vt,e,nc). The candidate predictor variables included: • Vehicle speed at the beginning of the tangent (VTan-Begin). • Curve radius of subsequent curve (Rnext). • Tangent length (LT). • Ramp length (Lr). • Gradient (level, upgrade, downgrade). • Posted speed limit of the freeway (VSL Hwy). Table 24 provides summary statistics of the dependent and predictor variables in the ramp speed database for tangents.

79 Table 24. Descriptive statistics of ramp speed database for tangents Variable Descriptive Entrance Exit D ep en de nt Va ria bl e Vehicle speed at end of tangent (VTan-End) (mph) Maximum 72.4 64.3 Mean 47.4 42.6 Standard deviation 8.0 9.1 Median 47.3 42.9 Minimum 14.4 5.7 Pr ed ic to r V ar ia bl es Vehicle speed at beginning of tangent (VTan-Begin) (mph) Maximum 69.7 68.8 Mean 41.0 53.0 Standard deviation 10.3 5.4 Median 42.9 53.0 Minimum 10.7 21.3 Curve radius of subsequent curve (Rnext) (mi) Maximum 18.9 18.9 Mean 11.1 2.0 Standard deviation 9.1 5.6 Median 18.9 0.10 Minimum 0.13 0.02 Tangent length (LT) (mi) Maximum 0.1 0.18 Mean 0.06 0.07 Standard deviation 0.02 0.03 Median 0.07 0.07 Minimum 0.03 0.02 Ramp length (Lr) (mi) Maximum 0.70 0.19 Mean 0.27 0.16 Standard deviation 0.17 0.02 Median 0.21 0.17 Minimum 0.13 0.14 Gradient (count/proportion) Level 897 (0.69) 727 (0.78) Upgrade 216 (0.17) 200 (0.22) Downgrade 186 (0.14) 4 (0.00) Posted speed limit of the freeway (VSL Hwy) (mph) Maximum 65 80 Mean 60.1 61.6 Standard deviation 3.0 2.6 Median 60 60 Minimum 55 60 Entrance Ramps Several models were fit to predict vehicle speeds at the end of a tangent for tangents with subsequent curves and for tangents without subsequent curves on entrance ramps. Only the best fitting models are provided here based on both statistical criterion (BIC) and engineering judgment. In the final models, vehicle speed at the beginning of the tangent (VTan-Begin), curve radius of subsequent curve (Rnext), tangent length (LT), ramp length (Lr), gradient, and the posted speed limit of the freeway (VSL Hwy) were treated as fixed effects, while the Tangent_ID was treated as a random effect. The final model to predict vehicle speeds at the end of a tangent followed by a subsequent curve on entrance ramps is provided as Equation 37. It is based on data from 546 vehicles observed on five tangents.

80 Table 25 provides a summary of the statistical results of the final model to predict vehicle speeds at the end of tangents followed by a subsequent curve on entrance ramps, and Figure 19 provides a plot of the actual to predicted values. VTan-End,C = 0.9667VTan-Begin + 143.9664LT – 5.3122I55 – 2.6028I60 – 7.9150I65 (37) where: VTan-End,C = Predicted vehicle speed at the end of a tangent if the tangent is followed by a curve, mph. VTan-Begin = Predicted vehicle speed at the beginning of a tangent, mph. LT = Tangent length, mi. I55 = Indicator variable for 55-mph regulatory speed limit on the highway (or freeway) mainline (= 1 if the regulatory speed limit is 55 mph, 0 otherwise). I60 = Indicator variable for 60-mph regulatory speed limit on the highway (or freeway) mainline (= 1 if the regulatory speed limit is 60 mph, 0 otherwise). I65 = Indicator variable for 65-mph regulatory speed limit on the highway (or freeway) mainline (= 1 if the regulatory speed limit is 65 mph, 0 otherwise).

81 Table 25. Statistical parameters of final model to estimate vehicle speed at end of tangent with subsequent curve on an entrance ramp Summary of Fit RSquare 0.82122 RSquare Adj 0.82023 Root Mean Square Error 3.21828 Mean of Response 43.67047 Observations (or Sum Wgts) 546 Parameter Estimates Term Estimate Std Error DFDen t Ratio Prob>|t| VTan-Begin (mph) 0.96669 0.031027 504.7 31.16 <.0001 LT (mi) 143.96639 32.3598 3.016 4.45 0.0209 I55 -5.31221 3.32688 2.137 -1.60 0.2435 I60 -2.60280 2.30603 2.016 -1.13 0.3754 I65 7.91500 REML Variance Component Estimates Random Effect Var Ratio Var Component Std Error 95% Lower 95% Upper Pct of Total Tangent_ID 1.4832228 15.362187 15.612177 -15.23712 45.961492 59.730 Residual 10.357303 0.6303256 9.2249577 11.71288 40.270 Total 25.71949 15.624279 10.310942 140.05874 100.000 -2 LogLikelihood = 2836.5425462 Note: Total is the sum of the positive variance components. Total including negative estimates = 25.71949 Fixed Effect Tests Source Nparm DF DFDen F Ratio Prob > F VTan-Begin (mph) 1 1 504.7 970.7041 <.0001 LT (mi) 1 1 3.016 19.7930 0.0209 VSL Hwy 2 2 2.15 3.1259 0.2310 Effect Details (VSL Frwy) Least Squares Means Table Level Least Sq Means Std Error I55 38.466183 4.4690497 I60 41.175590 2.3541431 I65 51.693413 3.5073697

82 Figure 19. Plot of actual versus predicted values of final model to estimate vehicle speed at end of tangent followed by subsequent curve on an entrance ramp The final model to predict vehicle speeds at the end of a tangent without a subsequent curve on entrance ramps is provided as Equation 38. It is based on data from 753 vehicles observed on 5 tangents. Table 26 provides a summary of the statistical results of the final model to predict vehicle speeds at the end of tangents without a subsequent curve on entrance ramps, and Figure 20 provides a plot of the actual to predicted values. VTan-End,NC = 1.0118VTan-Begin + 78.3087LT (38) where: VTan-End,NC = Predicted vehicle speed at the end of a tangent if the tangent is not followed by a curve, mph.

83 Table 26. Statistical parameters of final model to estimate vehicle speed at end of tangent without subsequent curve on an entrance ramp Summary of Fit RSquare 0.89381 RSquare Adj 0.89366 Root Mean Square Error 2.36062 Mean of Response 50.12304 Observations (or Sum Wgts) 753 Parameter Estimates Term Estimate Std Error DFDen t Ratio Prob>|t| VTan-Begin (mph) 1.011792 0.01672 330 60.52 <.0001 LT (mi) 78.308715 19.42207 6.994 4.03 0.0050 REML Variance Component Estimates Random Effect Var Ratio Var Component Std Error 95% Lower 95% Upper Pct of Total Tangent_ID 0.8901097 4.9601523 3.7301438 - 2.350795 12.2711 47.093 Residual 5.5725182 0.2883932 5.0478176 6.183946 52.907 Total 10.532671 3.7393175 5.8301529 24.49974 100.000 -2 LogLikelihood = 3451.9758141 Note: Total is the sum of the positive variance components. Total including negative estimates = 10.532671 Figure 20. Plot of actual versus predicted values of final model to estimate vehicle speed at end of tangent without subsequent curve on an entrance ramp

84 Exit Ramps Several models were fit to predict vehicle speeds at the end of a tangent for tangents with subsequent curves and for tangents without subsequent curves on exit ramps. Only the best fitting models are provided here based on both statistical criterion (BIC) and engineering judgment. In the final models, vehicle speed at the beginning of the tangent (VTan-Begin), curve radius of subsequent curve (Rnext), and tangent length (LT) were treated as fixed effects, while the tangent ID was treated as a random effect. The final model to predict vehicle speeds at the end of a tangent followed by a subsequent curve on exit ramps is provided as Equation 39. It is based on data from 840 vehicles observed on six tangents. Table 27 provides a summary of the statistical results of the final model to predict vehicle speeds at the end of tangents followed by a subsequent curve on exit ramps, and Figure 21 provides a plot of the actual to predicted values. VTan-End,C = 0.7980VTan-Begin + 228.6196Rnext – 575.2145Rnext2 – 645.4397LT + 6733.5203LT2 (39) where: Rnext = Radius of subsequent curve, mi. Table 27. Statistical parameters of final model to estimate vehicle speed at end of tangent with subsequent curve on an exit ramp Summary of Fit RSquare 0.885613 RSquare Adj 0.885066 Root Mean Square Error 2.482274 Mean of Response 44.39558 Observations (or Sum Wgts) 840 Parameter Estimates Term Estimate Std Error DFDen t Ratio Prob>|t| VTan-Begin (mph) 0.7978 0.0183 833.8 43.71 <.0001 Rnext (mi) 228.6196 45.8453 4.228 4.99 0.0065 Rnext2 (mi) -575.2145 146.1954 4.002 -3.93 0.0170 LT (mi) -645.4397 164.7479 4.636 -3.92 0.0130 LT2(mi) 6733.5203 2091.9690 4.172 3.22 0.0304 REML Variance Component Estimates Random Effect Var Ratio Var Component Std Error 95% Lower 95% Upper Pct of Total Tangent_ID 0.2516656 1.5506842 1.5990759 -1.583447 4.6848154 20.106 Residual 6.1616863 0.3019219 5.6102467 6.7990578 79.894 Total 7.7123705 1.6268531 5.3048116 12.238543 100.000 -2 LogLikelihood = 3894.0927503 Note: Total is the sum of the positive variance components. Total including negative estimates = 7.7123705

85 Figure 21. Plot of actual versus predicted values of final model to estimate vehicle speed at end of tangent with subsequent curve on an exit ramp The final model to predict vehicle speeds at the end of a tangent without a subsequent curve on exit ramps is provided as Equation 40. It is based on data from 71 vehicles observed on one tangent. Table 28 provides a summary of the statistical results of the final model to predict vehicle speeds at the end of tangents without a subsequent curve on exit ramps, and Figure 22 provides a regression plot of the speed at the beginning of the tangent and speed at the end of the tangent and the fitted regression line. VTan-End,NC = 0.4890VTan-Begin (40)

86 Table 28. Statistical parameters of final model to estimate vehicle speed at end of tangent without subsequent curve on an exit ramp Summary of Fit Root Mean Square Error 3.756292 Mean of Response 26.36756 Observations (or Sum Wgts) 77 Parameter Estimates Term Estimate Std Error t Ratio Prob>|t| VTan-Begin (mph) 0.4890396 0.007894 61.95 <.0001 Analysis of Variance Source DF Sum of Squares Mean Square F Ratio Model 1 54149.423 54149.4 3837.737 Error 76 1072.339 14.1 Prob > F C Total 77 55221.762 <.0001 Tested against reduced model: Y=0 Figure 22. Regression plot of speed at beginning of tangent and speed at end of tangent for final model to estimate vehicle speed at end of tangent without subsequent curve on an exit ramp