Validation of Urban Freeway Models (2014)

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217 Overview This appendix presents the results of the recalibration of the L03 data-poor models and the development of new model forms for the 95th-, 90th-, and 80th-percentile predictive equations for the travel time index (TTI). Recalibration For each recalibration, model statistics along with diagnostic figures are presented. The model statistics show the overall model performance measures, such as the mean squared error (MSE) and the F-test. R2 is also presented if applicable, but it may be redefined in different cases. The model statistics also include parameter-associated results, such as the param- eter estimates and the t-test results. The figures include (if applicable) 1. Fit plot in the original scale. This scatter plot shows the measured data samples and the recalibrated model line. 2. Fit plot showing a similar scatter plot, but the x-axis is in logarithm scale. This is the linear recalibrated model form. The 95% prediction confidence limits are also indicated. 3. Observed-by-predicted plot. If the model fits well, the scat- tered points should fall on the two sides of the reference line. 4. Residual-by-predicted plot. This is an important method to analyze the residuals; same method is used in the model validation analysis to evaluate the model performance. 5. Outlier and leverage plot. This is to analyze the outliers and leverage points that may raise questions in model building. 6. Histogram. This is used to display the distribution of the residuals. 7. Normality plot. This is used to show how close the residual distribution is to a normal distribution. Note that these statistics and plots are presented to evaluate the model from different perspectives; however, it is almost always impossible to build a real world model that perfectly satisfies all the criteria. Some of the measures are more important than others. From the recalibration results we can see that the non- constant variance is still a common problem for almost all cases; however, due to the characteristics of the data, it may not be fully resolvable. 95th-Percentile TTI The recalibration of the 95th-percentile TTI model is con- ducted by building a linear model between the response variable, 95th-percentile TTI, and the independent variable, log-transformed mean TTI—ln(meanTTI)—with the restric- tion that the intercept is fixed at 1. Due to the restriction, the R2 is redefined. AllData The recalibrated model for the 95th-percentile TTI with the AllData set is ( )= +95th-percentile TTI 1 3.4201 ln meanTTIAllData p Overall, this model can predict the measured data with good accuracy (Table E.1). The F-test (Table E.2) shows strong evidence that the model can explain most of the variance of the response variable. The t-test (Table E.3) for individual predictors also shows strong evidence for significance of the independent variable. We need to further analyze the goodness-of-fit of the model. The scatter plot (Figure E.1) shows how the regional data identified by different colors are scattered along the model line in the original scale (nontransformed). The observation that most blue dots (representing Minnesota data) are beyond the model line while most of the red dots (representing the California data) are below the model line indicates that regional calibration may be necessary. A p p e n d i x e Model Enhancements

218 Figures E.1 through E.7 evaluate the model performance from different perspectives. The scatter plot (Figure E.1) shows that the model can predict the data trend closely. From the fit plot (Figure E.2) we can see that some of the data samples fall outside the confidence limits, which may be outliers. The pre- dicted versus measured plot (Figure E.3) shows in another way how the model fits the data. In the case of perfect fit, the data samples would fall on the equal value line; however, with noise in the measured data, they are supposed to scatter along the diagonal line. The pattern in the figure shows that the model can predict the data samples with a good satisfactory level. The residual plot (Figure E.4) shows that the residuals are generally randomly scattered, although the increasing residual variance problem still exists, and there is some nonrandom pattern around the origin. The outlier and leverage plot (Figure E.5) shows the identified outliers and leverage points. The histogram and the normality plot (Figures E.6 and E.7) show that even with recalibration, the residuals may still not satisfy the normal distribution assumption perfectly. Table E.1. Root MSE and R-Square, Recalibrated 95th-Percentile TTI Model, AllData Root MSE 0.17230 R-Square 0.9884 Table E.2. Analysis of Variance, Recalibrated 95th-Percentile TTI Model, AllData Source DF Sum of Squares Mean Square F-Value Pr > F Model 1 813.98823 813.98823 27418.0 <0.0001 Error 322 9.55957 0.02969 Uncorrected total 323 823.54780 Table E.3. Parameter Estimates, Recalibrated 95th-Percentile TTI Model, AllData Variable DF Parameter Estimate Standard Error t-Value Pr > t 95% Confidence Limits Intercept 1 1.00000 0 Infty <0.0001 1.00000 1.00000 ln(mean TTI) (AllData) 1 3.42007 0.04034 84.78 <0.0001 3.34071 3.49944 RESTRIC -1 -2.54913 2.60948 -0.98 0.3294a -7.66023 2.56198 a The model restricts the intercept to be 1 (unity). Note: Infty = infinity. Figure E.1. Scatter plot in original scale, recalibrated 95th-percentile TTI model, AllData.

219 Figure E.2. Fit plot in x-axis log scale, recalibrated 95th-percentile TTI model, AllData. Figure E.3. Observed-by-predicted plot, recalibrated 95th-percentile TTI model, AllData.

220 Figure E.4. Residual-by-predicted plot, recalibrated 95th-percentile TTI model, AllData. Figure E.5. Outlier and leverage plot, recalibrated 95th-percentile TTI model, AllData.

221 Figure E.6. Distribution of residuals, recalibrated 95th-percentile TTI model, AllData. Figure E.7. Quantile-quantile (Q-Q) plot of residuals, recalibrated 95th-percentile TTI model, AllData.

222 Table E.6. Parameter Estimates, Recalibrated 95th-Percentile TTI Model, California Variable DF Parameter Estimate Standard Error t-Value Pr > t 95% Confidence Limits Intercept 1 1.00000 0 Infty <0.0001 1.00000 1.00000 ln(mean TTI) (CA) 1 3.18182 0.04671 68.12 <0.0001 3.08966 3.27397 RESTRICT -1 -1.46715 1.83910 -0.80 0.4265a -5.06758 2.13327 a The model restricts the intercept to be 1 (unity). Table E.4. Root MSE and R-Square, Recalibrated 95th-Percentile TTI Model, California Root MSE 0.16425 R-Square 0.9898 Table E.5. Analysis of Variance, Recalibrated 95th-Percentile TTI Model, California Source DF Sum of Squares Mean Square F-Value Pr > F Model 1 484.90978 484.90978 17974.2 <0.0001 Error 186 5.01793 0.02698 Uncorrected total 187 489.92771 Figure E.8. Fit plot in original scale, recalibrated 95th-percentile TTI model, California. California The recalibrated model for the 95th-percentile TTI with the California (CA) set is ( )= +95th-percentile TTI 1 3.1818 ln meanTTICA p The root MSE and R-Square are given in Table E.4. The F-test (Table E.5) yields nearly zero p-values, indicating strong con- fidence in the model validity. Student’s t-tests (Table E.6) also show the significance of the model parameters. The scatter plot, fit plot, and the observed-by-predicted plot all show that the model can predict the data trend well (refer to Figures E.8 through E.14 for interpretation of the observations). How- ever, the residual-by-predicted plot indicates the possibility of inadequate model form as the residuals seem not to be randomly scattered along the reference line. The outlier and leverage plot show that outliers and leverage points may exist, while the histogram and the normality plot show that the residual distribution does not perfectly follow a normal distribution.

223 Figure E.9. Fit plot in x-axis log scale, recalibrated 95th-percentile TTI model, California. Figure E.10. Observed-by-predicted plot, recalibrated 95th-percentile TTI model, California.

224 Figure E.11. Residual-by-predicted plot, recalibrated 95th-percentile TTI model, California. Figure E.12. Outlier and leverage plot, recalibrated 95th-percentile TTI model, California.

225 Figure E.13. Distribution of residuals, recalibrated 95th-percentile TTI model, California. Figure E.14. Q-Q plot of residuals, recalibrated 95th-percentile TTI model, California.

226 Table E.9. Parameter Estimates, Recalibrated 95th-Percentile TTI Model, Minnesota Variable Label DF Parameter Estimate Standard Error t-Value Pr > t 95% Confidence Limits Intercept Intercept 1 1.00000 0 Infty <0.0001 1.00000 1.00000 MN_TTImean_ln ln(mean TTI) (MN) 1 3.97871 0.06673 59.63 <0.0001 3.84587 4.11156 RESTRICT RESTRICT -1 0.27267 1.13445 0.24 0.8118a -1.94456 2.48990 a The model restricts the intercept to be 1 (unity). Table E.7. Root MSE and R-Square, Recalibrated 95th-Percentile TTI Model, Minnesota Root MSE 0.15298 R-Square 0.9928 Table E.8. Analysis of Variance, Recalibrated 95th-Percentile TTI Model, Minnesota Source DF Sum of Squares Mean Square F-Value Pr > F Model 1 252.12485 252.12485 10773.6 <0.0001 Error 78 1.82537 0.02340 Uncorrected total 79 253.95022 Figure E.15. Fit plot in original scale, recalibrated 95th-percentile TTI model, Minnesota. Minnesota The recalibrated model for the 95th-percentile TTI with the Minnesota (MN) set is ( )= +95th-percentile TTI 1 3.9787 ln meanTTIMN p Table E.7 gives the root MSE and R-Square values. The F-test (Table E.8) yields a large F-value and a nearly zero p-value, showing strong confidence that the model can explain most of the variance of the response variable. The Student’s t-test (Table E.9) also shows strong evidence that the model parameter is not zero. From the plots we can see that the model can predict the data trend well: the residual-by-predicted plot shows a generally random pattern although the nonconstant variance problem still exists; some of the samples may be outliers and leverage points; and the residual distribution does not perfectly follow a normal distribution (refer to Figures E.15 through E.21 for interpretation of the observations).

227 Figure E.16. Fit plot in x-axis log scale, recalibrated 95th-percentile TTI model, Minnesota. Figure E.17. Observed-by-predicted plot, recalibrated 95th-percentile TTI model, Minnesota.

228 Figure E.18. Residual-by-predicted plot, recalibrated 95th-percentile TTI model, Minnesota. Figure E.19. Outlier and leverage plot, recalibrated 95th-percentile TTI model, Minnesota.

229 Figure E.20. Distribution of residuals, recalibrated 95th-percentile TTI model, Minnesota. Figure E.21. Q-Q plot of residuals, recalibrated 95th-percentile TTI model, Minnesota.

230 Table E.12. Parameter Estimates, Recalibrated 90th-Percentile TTI Model, AllData Variable DF Parameter Estimate Standard Error t-Value Pr > t 95% Confidence Limits Intercept 1 1.00000 0 Infty <0.0001 1.00000 1.00000 ln(mean TTI) (AllData) 1 2.81886 0.02780 101.39 <0.0001 2.76416 2.87355 RESTRICT -1 -6.40766 1.79838 -3.56 0.0003a -9.93010 -2.88523 a The model restricts the intercept to be 1 (unity). Table E.10. Analysis of Variance, Recalibrated 90th-Percentile TTI Model, AllData Source DF Sum of Squares Mean Square F-Value Pr > F Model 1 688.14766 688.14766 48802.7 <0.0001 Error 322 4.54040 0.01410 Uncorrected total 323 692.68806 Table E.11. Root MSE and R-Square, Recalibrated 90th-Percentile TTI Model, AllData Root MSE 0.11875 R-Square 0.9934 90th-percentile TTI The recalibration of the 90th-percentile TTI model is con- ducted by building a linear model between the response variable, 90th-percentile TTI, and the independent variable, log-transformed mean TTI—ln(mean TTI)—with the restric- tion that the intercept is fixed at 1. Due to the restriction, the R2 is redefined. AllData The recalibrated model for the 90th-percentile TTI with the AllData set is ( )= +90th-percentile TTI 1 2.8189 ln meanTTIAllData p The statistical test results (Tables E.10 through E.12) show that both the model and the parameters are significant. We need to further analyze the goodness-of-fit of the model. The scatter plot (Figure E.22) shows that the model can predict the trend of the data samples, but there may be two question- able patterns. First, the blue points representing the MN data samples are mostly beyond the model line, while the red points representing the CA data are mostly below the model line, indicating regional disparity. Another pattern (Figure E.23) is that there are more points beyond the upper 95% predic- tion confidence limit than below the lower 95% prediction confidence limit, which may indicate that the model form may not be adequate; perhaps the 90th-percentile TTI increases faster than the log-line. Figure E.24 shows the predicted versus observed plot. The residual-by-predicted plot (Figure E.25) shows a generally random pattern with the nonconstant vari- ance problem. Other plots show the existence of outlier and leverage points (Figure E.26) and that the residual distribu- tion does not perfectly follow a normal distribution because skewness exists (Figures E.27 and E.28). California The recalibrated model for the 90th-percentile TTI with the CA set is ( )= +90th-percentile TTI 1 2.6631 ln meanTTICA p The statistical tests results (Tables E.13 through E.15) show that both the model and the parameters are significant. The scatter plots (Figures E.29 through E.31) show that the model predicts the data samples well. The residual plots (Figures E.32 and E.35) show that the nonconstant variance problem still exists, and the residuals seem to present a nonrandom pattern. Figure E.33 shows the outlier and leverage plot. The histogram and the nor- mality plot (Figures E.34 and E.35) show that the residual dis- tribution does not perfectly follow a normal distribution. Minnesota The recalibrated model for the 90th-percentile TTI with the MN set is ( )= +90th-percentile TTI 1 3.2008 ln meanTTIMN p The statistical test results (Tables E.16 through E.18) show that both the model and the parameters are significant. The scatter plots and the residual-by-predicted plot (Figures E.36 through E.40) show that the model can predict the data trend well, and the residuals present a random pattern along the zero refer- ence line, although the nonconstant variance problem still exists. The histogram and the normality plot (Figures E.41 and E.42) show that the residual may not perfectly follow a normal distribution. (text continues on page 242)

231 Figure E.22. Scatter plot in original scale, recalibrated 90th-percentile TTI model, AllData. Figure E.23. Fit plot in x-axis log scale, recalibrated 90th-percentile TTI model, AllData.

232 Figure E.25. Residual-by-predicted plot, recalibrated 90th-percentile TTI model, AllData. Figure E.24. Observed-by-predicted plot, recalibrated 90th-percentile TTI model, AllData.

233 Figure E.26. Outlier and leverage plot, recalibrated 90th-percentile TTI model, AllData. Figure E.27. Distribution of residuals, recalibrated 90th-percentile TTI model, AllData.

234 Table E.15. Parameter Estimates, Recalibrated 90th-Percentile TTI Model, California Variable Label DF Parameter Estimate Standard Error t-Value Pr > t 95% Confidence Limits Intercept Intercept 1 1.00000 0 Infty <0.0001 1.00000 1.00000 CA_TTImean_ln ln(mean TTI) (CA) 1 2.66311 0.03273 81.37 <0.0001 2.59854 2.72767 RESTRICT -1 -3.44219 1.28862 -2.67 0.0072a -5.96493 -0.91945 a The model restricts the intercept to be 1 (unity). Table E.13. Analysis of Variance, Recalibrated 90th-Percentile TTI Model, California Source DF Sum of Squares Mean Square F-Value Pr > F Model 1 414.83485 414.83485 31320.3 <0.0001 Error 186 2.46355 0.01324 Uncorrected total 187 417.29840 Table E.14. Root MSE and R-Square, Recalibrated 90th-Percentile TTI Model, California Root MSE 0.11509 R-Square 0.9941 Figure E.28. Q-Q plot of residuals, recalibrated 90th-percentile TTI model, AllData.

235 Figure E.29. Fit plot in original scale, recalibrated 90th-percentile TTI model, California. Figure E.30. Fit plot in x-axis log scale, recalibrated 90th-percentile TTI model, California.

236 Figure E.32. Residual-by-predicted plot, recalibrated 90th-percentile TTI model, California. Figure E.31. Observed-by-predicted plot, recalibrated 90th-percentile TTI model, California.

237 Figure E.33. Outlier and leverage plot, recalibrated 90th-percentile TTI model, California. Figure E.34. Distribution of residuals, recalibrated 90th-percentile TTI model, California.

238 Table E.18. Parameter Estimates, Recalibrated 90th-Percentile TTI Model, Minnesota Variable Label DF Parameter Estimate Standard Error t-Value Pr > t 95% Confidence Limits Intercept Intercept 1 1.00000 0 Infty <0.0001 1.00000 1.00000 MN_TTImean_ln ln(mean TTI) (MN) 1 3.20080 0.04570 70.03 <0.0001 3.10981 3.29179 RESTRICT -1 -1.43443 0.77701 -1.85 0.0645a -2.95306 0.08421 a The model restricts the intercept to be 1 (unity). Table E.16. Analysis of Variance, Recalibrated 90th-Percentile TTI Model, Minnesota Source DF Sum of Squares Mean Square F-Value Pr > F Model 1 201.88201 201.88201 18388.9 <0.0001 Error 78 0.85632 0.01098 Uncorrected total 79 202.73833 Table E.17. Root MSE and R-Square, Recalibrated 90th-Percentile TTI Model, Minnesota Root MSE 0.10478 R-Square 0.9958 Figure E.35. Q-Q plot of residuals, recalibrated 90th-percentile TTI model, California.

239 Figure E.36. Fit plot in original scale, recalibrated 90th-percentile TTI model, Minnesota. Figure E.37. Fit plot in x-axis log scale, recalibrated 90th-percentile TTI model, Minnesota.

240 Figure E.39. Residual-by-predicted plot, recalibrated 90th-percentile TTI model, Minnesota. Figure E.38. Observed-by-predicted plot, recalibrated 90th-percentile TTI model, Minnesota.

241 Figure E.41. Distribution of residuals, recalibrated 90th-percentile TTI model, Minnesota. Figure E.40. Outlier and leverage plot, recalibrated 90th-percentile TTI model, Minnesota.

242 Table E.19. Analysis of Variance, Recalibrated 80th-Percentile TTI Model, AllData Source DF Sum of Squares Mean Square F-Value Pr > F Model 1 573.17609 573.17609 122817 <0.0001 Error 322 1.50274 0.00467 Uncorrected total 323 574.67883 Table E.20. Root MSE and R-Square, Recalibrated 80th-Percentile TTI Model, AllData Root MSE 0.06831 R-Square 0.9974 Figure E.42. Q-Q plot of residuals, recalibrated 90th-percentile TTI model, Minnesota. 80th-Percentile TTI The recalibration of the 80th-percentile TTI model is con- ducted by building a linear model between the response variable, 80th-percentile TTI, and the independent variable, log-transformed mean TTI—ln(mean TTI)—with the restric- tion that the intercept is fixed at 1. Due to the restriction, the R2 is redefined. AllData The recalibrated model for the 80th-percentile TTI with the AllData set is ( )= +80th-percentile TTI 1 2.1598 ln meanTTIAllData p The statistical test results (Tables E.19 through E.21) show that both the model and the parameters are significant, which indicates that overall the model is valid. We need to further analyze the goodness-of-fit of the model. The scatter plot (Figure E.43), fit plot (Figure E.44), and observed-by- predicted plot (Figure E.45) show that although the model can generally predict the data trend, the model form may not be adequate. The residual-by-predicted plot (Figure E.46) also indicates the same problem because a nonrandom pattern exists. The outlier and leverage plot is shown in Figure E.47. The histogram and the normality plot (Figures E.48 and E.49) show that the residuals may not closely follow a normal distribution.

243 Table E.21. Parameter Estimates, Recalibrated 80th-Percentile TTI Model, AllData Variable Label DF Parameter Estimate Standard Error t-Value Pr > t 95% Confidence Limits Intercept Intercept 1 1.00000 0 Infty <0.0001 1.00000 1.00000 AllData_TTImean_ln ln(mean TTI) (AllData) 1 2.15981 0.01599 135.04 <0.0001 2.12835 2.19128 RESTRICT -1 -6.72653 1.03461 -6.50 <0.0001a -8.75299 -4.70007 a The model restricts the intercept to be 1 (unity). Figure E.43. Scatter plot in original scale, recalibrated 80th-percentile TTI model, AllData. Figure E.44. Fit plot in x-axis log scale, recalibrated 80th-percentile TTI model, AllData.

244 Figure E.46. Residual-by-predicted plot, recalibrated 80th-percentile TTI model, AllData. Figure E.45. Observed-by-predicted plot, recalibrated 80th-percentile TTI model, AllData.

245 Figure E.47. Outlier and leverage plot, recalibrated 80th-percentile TTI model, AllData. Figure E.48. Distribution of residuals, recalibrated 80th-percentile TTI model, AllData.

246 Table E.24. Parameter Estimates, Recalibrated 80th-Percentile TTI Model, California Variable Label DF Parameter Estimate Standard Error t-Value Pr > t 95% Confidence Limits Intercept Intercept 1 1.00000 0 Infty <0.0001 1.00000 1.00000 CA_TTImean_ln ln(mean TTI) (CA) 1 2.08443 0.01835 113.59 <0.0001 2.04823 2.12063 RESTRICT -1 -3.73107 0.72250 -5.16 <0.0001a -5.14551 -2.31663 a The model restricts the intercept to be 1 (unity). Table E.22. Analysis of Variance, Recalibrated 80th-Percentile TTI Model, California Source DF Sum of Squares Mean Square F-Value Pr > F Model 1 348.33950 348.33950 83662.1 <0.0001 Error 186 0.77444 0.00416 Uncorrected total 187 349.11394 Table E.23. Root MSE and R-Square, Recalibrated 80th-Percentile TTI Model, California Root MSE 0.06453 R-Square 0.9978 Figure E.49. Q-Q plot of residuals, recalibrated 80th-percentile TTI model, AllData. California The recalibrated model for the 80th-percentile TTI with the CA set is ( )= +80th-percentile TTI 1 2.0844 ln meanTTICA p The statistical test results (Tables E.22 through E.24) show that both the model and the parameters are significant. From the scatter plots (Figures E.50 through E.52) and the residual plot (Figure E.53) we can see that the model form may not be adequate because the 80th-percentile TTI seems to increase faster than the model line when the mean TTI is large. Fig- ure E.54 shows the outlier and leverage plot. The histogram and the normality plot (Figures E.55 and E.56) show that the residual distribution does not closely follow a normal distribution.

247 Figure E.50. Fit plot in original scale, recalibrated 80th-percentile TTI model, California. Figure E.51. Fit plot in x-axis log scale, recalibrated 80th-percentile TTI model, California.

248 Figure E.53. Residual-by-predicted plot, recalibrated 80th-percentile TTI model, California. Figure E.52. Observed-by-predicted plot, recalibrated 80th-percentile TTI model, California.

249 Figure E.54. Outlier and leverage plot, recalibrated 80th-percentile TTI model, California. Figure E.55. Distribution of residuals, recalibrated 80th-percentile TTI model, California.

250 Table E.27. Parameter Estimates, Recalibrated 80th-Percentile TTI Model, Minnesota Variable Label DF Parameter Estimate Standard Error t-Value Pr > t 95% Confidence Limits Intercept Intercept 1 1.00000 0 Infty <0.0001 1.00000 1.00000 MN_TTImean_ln ln(mean TTI) (MN) 1 2.35394 0.03104 75.84 <0.0001 2.29214 2.41573 RESTRICT -1 -1.86088 0.52771 -3.53 0.0003a -2.89227 -0.82950 a The model restricts the intercept to be 1 (unity). Table E.25. Analysis of Variance, Recalibrated 80th-Percentile TTI Model, Minnesota Source DF Sum of Squares Mean Square F-Value Pr > F Model 1 157.28090 157.28090 31059.9 <0.0001 Error 78 0.39498 0.00506 Uncorrected total 79 157.67587 Table E.26. Root MSE and R-Square, Recalibrated 80th-Percentile TTI Model, Minnesota Root MSE 0.07116 R-Square 0.9975 Figure E.56. Q-Q plot of residuals, recalibrated 80th-percentile TTI model, California. Minnesota The recalibrated model for the 80th-percentile TTI with the MN set is ( )= +80th-percentile TTI 1 2.3539 ln meanTTIMN p The statistical test results (Tables E.25 through E.27) show that both the model and the parameters are significant. The scatter plots (Figures E.57 through E.59) show that the model can predict the trend in data samples. However, the residual plot (Figure E.60) shows that the data samples are not evenly distributed along the zero reference line. Fig- ure E.61 shows the outlier and leverage diagnostics. The his- togram and the normality plot (Figures E.62 and E.63) show that the residual distribution does not closely follow a nor- mal distribution.

251 Figure E.57. Fit plot in original scale, recalibrated 80th-percentile TTI model, Minnesota. Figure E.58. Fit plot in x-axis log scale, recalibrated 80th-percentile TTI model, Minnesota.

252 Figure E.60. Residual-by-predicted plot, recalibrated 80th-percentile TTI model, Minnesota. Figure E.59. Observed-by-predicted plot, recalibrated 80th-percentile TTI model, Minnesota.

253 Figure E.61. Outlier and leverage plot, recalibrated 80th-percentile TTI model, Minnesota. Figure E.62. Distribution of residuals, recalibrated 80th-percentile TTI model, Minnesota.

254 Table E.29. Parameter Estimates, Recalibrated Standard Deviation TTI Model, AllData Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a 0.7775 0.0158 0.7464 0.8086 b 0.6810 0.0242 0.6334 0.7287 Table E.28. Analysis of Variance, Recalibrated Standard Deviation TTI Model, AllData Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 2 25.3386 12.6693 1896.51 <0.0001 Error 321 2.1444 0.00668 Uncorrected total 323 27.4829 Figure E.63. Q-Q plot of residuals, recalibrated 80th-percentile TTI model, Minnesota. Standard Deviation of TTI The recalibration of the standard deviation of the TTI model is conducted by building a nonlinear model between the response variable, the standard deviation of TTI, and the independent variable, mean TTI. The model form is the same as the L03 data-poor model form, described as ( )= −StdDevTTI meanTTI 1a bp where a and b are the model parameters to be calibrated. Note that a similar model can be built by adopting a linear model form using log-transformed dependent and independent vari- ables; however, it is found that the nonlinear model and the transformed linear model are quite different, and the nonlinear model line is closer to the measured data trend. AllData The recalibrated model for the standard deviation of TTI with the AllData set is ( )= −StdDevTTI 0.7775 meanTTI 1AllData 0.6810p The F-test results (Table E.28) show that the model is signifi- cant. The parameter estimation results are shown in Table E.29. The scatter plot (Figure E.64), the fit plot (Figure E.65), and the observed-by-predicted plot (Figure E.66) all show that the model can generally predict the trend of the measured data. From the scatter plot we can also see that most of the blue points representing the MN data samples are beyond the model line, while a greater portion of the red points representing the CA samples are below the model line, indicating regional differ- ence. The residual-by-predicted plot (Figure E.67) indicates the problem of nonconstant variance; however, this problem may not be totally fixed due to the characteristics of the data. The histogram and the normality plot (Figures E.68 and E.69) show that the residual distribution is generally close to a normal dis- tribution but with long tails.

255 Figure E.64. Scatter plot in original scale, recalibrated standard deviation TTI model, AllData. Figure E.65. Fit plot in original scale, recalibrated standard deviation TTI model, AllData.

256 Figure E.67. Residual-by-predicted plot, recalibrated standard deviation TTI model, AllData. Figure E.66. Observed-by-predicted plot, recalibrated standard deviation TTI model, AllData.

257 Figure E.68. Distribution of raw residuals, recalibrated standard deviation TTI model, AllData. Figure E.69. Q-Q plot of residuals, recalibrated standard deviation TTI model, AllData.

258 California The recalibrated model for the standard deviation of TTI with the CA set is ( )= −StdDevTTI 0.6886 meanTTI 1CA 0.6444p The F-test (Table E.30) results show that the model is signifi- cant. Table E.31 shows the parameter estimates. The fit plot (Figure E.70) and the observed-by-predicted plot (Figure E.71) show that the model generally fits the data. The residual- by-predicted plot (Figure E.72) shows that the nonconstant variance problem still exists. The histogram and the normal- ity plot (Figures E.73 and E.74) show that the residual distri- bution closely follows a normal distribution but with a long right tail. Minnesota The recalibrated model for the standard deviation of TTI with the MN set is ( )= −StdDevTTI 0.9611 meanTTI 1MN 0.6961p The F-test results (Table E.32) show that the model is signifi- cant. Table E.33 shows the parameter estimation. The fit plot Figure E.70. Fit plot in original scale, recalibrated standard deviation TTI model, California. Table E.31. Parameter Estimates, Recalibrated Standard Deviation TTI Model, California Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a 0.6886 0.0184 0.6523 0.7250 b 0.6444 0.0310 0.5833 0.7055 Table E.30. Analysis of Variance, Recalibrated Standard Deviation TTI Model, California Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 2 13.9468 6.9734 1107.15 <0.0001 Error 185 1.1652 0.00630 Uncorrected total 187 15.1121 (Figure E.75) and the observed-by-predicted plot (Figure E.76) show that the model generally fits the data. The residual-by- predicted plot (Figure E.77) presents a random pattern, but the nonconstant variance problem still exists. The histogram and the normality plot (Figures E.78 and E.79) show that the residual distribution closely follows a normal distribution but with a long left tail. (text continues on page 264)

259 Figure E.71. Observed-by-predicted plot, recalibrated standard deviation TTI model, California. Figure E.72. Residual-by-predicted plot, recalibrated standard deviation TTI model, California.

260 Figure E.74. Q-Q plot of residuals, recalibrated standard deviation TTI model, California. Figure E.73. Distribution of raw residuals, recalibrated standard deviation TTI model, California.

261 Table E.33. Parameter Estimates, Recalibrated Standard Deviation TTI Model, Minnesota Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a 0.9611 0.0217 0.9179 1.0043 b 0.6961 0.0306 0.6353 0.7570 Table E.32. Analysis of Variance, Recalibrated Standard Deviation TTI Model, Minnesota Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 2 10.8257 5.4129 1485.34 <0.0001 Error 77 0.2806 0.00364 Uncorrected total 79 11.1063 Figure E.75. Fit plot in original scale, recalibrated standard deviation TTI model, Minnesota.

262 Figure E.77. Residual-by-predicted plot, recalibrated standard deviation TTI model, Minnesota. Figure E.76. Observed-by-predicted plot, recalibrated standard deviation TTI model, Minnesota.

263 Figure E.79. Q-Q plot of residuals, recalibrated standard deviation TTI model, Minnesota. Figure E.78. Distribution of raw residuals, recalibrated standard deviation TTI model, Minnesota.

264 Figure E.80. Scatter plot in original scale, recalibrated PctTripsOnTime50mph model, AllData. Table E.35. Parameter Estimates, Recalibrated PctTripsOnTime50mph Model, AllData Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a -2.0293 0.0514 -2.1305 -1.9281 Table E.34. Analysis of Variance, Recalibrated PctTripsOnTime50mph Model, AllData Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 1 247.3 247.3 40121.7 <0.0001 Error 322 1.9848 0.00616 Uncorrected total 323 249.3 PctTripsOnTime50mph The recalibration of the PctTripsOnTime50mph model is conducted by building a nonlinear model between the response variable, PctTripsOnTime50mph, and the indepen- dent variable, mean TTI. The model form is the same as the L03 data-poor model form, described as = [ ]−PctTripsOnTime50mph meanTTI 1ea where a is the model parameter to be calibrated. AllData The recalibrated model for the PctTripsOnTime50mph with the AllData set is = [ ]− −PctOnTimeTrip50mphAllData 2.0293 meanTTI 1e The F-test results (Table E.34) show that the model is signifi- cant. The parameter estimates are shown in Table E.35. The scatter plot (Figure E.80) shows that the regional pattern may still exist; that is, when the mean TTI is larger than 1.25, the blue points representing MN data tend to scatter beyond the model line while the red points representing CA data tend to scatter below the blue line. Note that around the origin the measured samples are scattered out forming a cone shape, but almost all of them are beyond the model line. This results in the nonrandom pattern on the right side of the residual- by-predicted value plot (Figure E.83). The nonconstant variance problem still exists in the residual-by-predicted plot. These unsatisfactory patterns may mainly be due to the char- acteristics of the data. The fit plot (Figure E.81) and observed- by-predicted plot (Figure E.82) are also given. The histogram and the normality plot (Figures E.84 and E.85) show that the residual distribution is close to a normal distribution when the residual is positive but has an unusual long tail on the negative side.

265 Figure E.81. Fit plot in original scale, recalibrated PctTripsOnTime50mph model, AllData. Figure E.82. Observed-by-predicted plot, recalibrated PctTripsOnTime50mph model, AllData.

266 Figure E.83. Residual-by-predicted plot, recalibrated PctTripsOnTime50mph model, AllData. Figure E.84. Distribution of raw residuals, recalibrated PctTripsOnTime50mph model, AllData.

267 California The recalibrated model for the PctTripsOnTime50mph with the CA set is = [ ]− −PctTripsOnTime50mphCA 2.2663 meanTTI 1e The F-test results (Table E.36) show that the model is sig- nificant. Parameter estimates are shown in Table E.37. The fitted-curve-to-data plot (fit plot) (Figure E.86) shows a very similar pattern to that of the AllData set. The most obvious unusual pattern is that when the mean TTI is close to 1, the measured response values are almost all larger than the model predicted values (Figure E.87). The nonconstant variance problem also exists. The residual-by-predicted plot is shown in Figure E.88. The histogram and the normality plot (Figures E.89 and E.90) show that the residual distribution is close to Figure E.85. Q-Q plot of residuals, recalibrated PctTripsOnTime50mph model, AllData. Table E.37. Parameter Estimates, Recalibrated PctTripsOnTime50mph Model, California Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a -2.2663 0.0787 -2.4215 -2.1111 Table E.36. Analysis of Variance, Recalibrated PctTripsOnTime50mph Model, California Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 1 135.0 135.0 17643.1 <0.0001 Error 186 1.4229 0.00765 Uncorrected total 187 136.4 a normal distribution when it is positive but has a long nega- tive tail. Minnesota The recalibrated model for the PctTripsOnTime50mph with the MN set is = [ ]− −PctTripsOnTime50mphMN 1.6422 meanTTI 1e The F-test results (Table E.38) show that the model is signifi- cant. Parameter estimation is shown in Table E.39. The fit plot (Figure E.91) shows that the data samples are in general randomly scattered out along the model line, indicating that the regionally recalibrated model performs better for the MN model than the model built on the AllData set. The observed- by-predicted plot is shown in Figure E.92. The residual plot (Figure E.93) shows that when the mean TTI is close to 1, it is still unbalanced. The histogram and the normality plot (Fig- ures E.94 and E.95) show that the residual distribution is not perfectly following a normal distribution. (text continues on page 273)

268 Figure E.86. Fit plot in original scale, recalibrated PctTripsOnTime50mph model, California. Figure E.87. Observed-by-predicted plot, recalibrated PctTripsOnTime50mph model, California.

269 Figure E.88. Residual-by-predicted plot, recalibrated PctTripsOnTime50mph model, California. Figure E.89. Distribution of raw residuals, recalibrated PctTripsOnTime50mph model, California.

270 Figure E.90. Q-Q plot of residuals, recalibrated PctTripsOnTime50mph model, California. Table E.39. Parameter Estimates, Recalibrated PctTripsOnTime50mph Model, Minnesota Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a -1.6422 0.0486 -1.7390 -1.5454 Table E.38. Analysis of Variance, Recalibrated PctTripsOnTime50mph Model, Minnesota Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 1 59.0003 59.0003 26281.6 <0.0001 Error 78 0.1751 0.00224 Uncorrected total 79 59.1754

271 Figure E.91. Fit plot in original scale, recalibrated PctTripsOnTime50mph model, Minnesota. Figure E.92. Observed-by-predicted plot, recalibrated PctTripsOnTime50mph model, Minnesota.

272 Figure E.93. Residual-by-predicted plot, recalibrated PctTripsOnTime50mph model, Minnesota. Figure E.94. Distribution of raw residuals, recalibrated PctTripsOnTime50mph model, Minnesota.

273 PctTripsOnTime45mph The recalibration of the PctTripsOnTime45mph model is con- ducted by building a nonlinear model between the response variable, PctTripsOnTime45mph, and the independent vari- able, mean TTI. The model form is the same as the L03 data- poor model form, described as = [ ]−PctTripsOnTime45mph meanTTI 1ea where a is the model parameter to be calibrated. AllData The recalibrated model for the PctTripsOnTime45mph with the AllData set is = [ ]− −PctTripsOnTime45mphAllData 1.4874 meanTTI 1e The model form for the PctTripsOnTime45mph is the same as that for the PctTripsOnTime50mph, and the patterns shown in the recalibrated models are also very similar. As for this recalibrated PctTripsOnTime45mph model, the F-test results (Table E.40) show that overall the model is sig- nificant. The parameter estimation is shown in Table E.41. The scatter plot (Figure E.96), fit plot (Figure E.97), and the measured-by-predicted plot (Figure E.98) show that the re- calibrated model line can generally predict the trend in the measured data. The regional difference can also be noticed from the scatter plot where the blue points representing MN data tend to scatter beyond the model line while the red points representing CA data tend to scatter below the blue line when the mean TTI is larger than 1.25. The nonconstant variance problem can be identified in the residual-by-predicted plot (Figure E.99). The histogram and the normality plot (Fig- ures E.100 and E.101) show that the residual distribution has a longer negative tail compared to a normal distribution. Figure E.95. Q-Q plot of residuals, recalibrated PctTripsOnTime50mph model, Minnesota. Table E.41. Parameter Estimates, Recalibrated PctTripsOnTime45mph Model, AllData Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a -1.4874 0.0290 -1.5446 -1.4303 Table E.40. Analysis of Variance, Recalibrated PctTripsOnTime45mph Model, AllData Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 1 257.4 257.4 70942.6 <0.0001 Error 322 1.1685 0.00363 Uncorrected total 323 258.6

274 Figure E.96. Scatter plot in original scale, recalibrated PctTripsOnTime45mph model, AllData. Figure E.97. Fit plot in original scale, recalibrated PctTripsOnTime45mph model, AllData.

275 Figure E.98. Observed-by-predicted plot, recalibrated PctTripsOnTime45mph model, AllData. Figure E.99. Residual-by-predicted plot, recalibrated PctTripsOnTime45mph model, AllData.

276 Figure E.100. Distribution of raw residuals, recalibrated PctTripsOnTime45mph model, AllData. Figure E.101. Q-Q plot of residuals, recalibrated PctTripsOnTime45mph model, AllData.

277 California The recalibrated model for the PctTripsOnTime45mph with the CA set is = [ ]− −PctTripsOnTime45mphCA 1.6125 meanTTI 1e The recalibrated model for the CA data set presents similar results to that for the AllData set. The F-test results (Table E.42) show that the model is significant. The parameter estimation is shown in Table E.43. From the fit plot (Figure E.102) we can see that when the mean TTI is close to 1, the measured response values are almost all larger than the model predicted values (Figure E.103). The nonconstant variance problem also exists. The residual-by-predicted plot is shown in Figure E.104. The histogram and the normality plot (Figures E.105 and E.106) show that the residual distribution is close to a normal distribution when it is positive but has a long negative tail. Minnesota The recalibrated model for the PctTripsOnTime45mph with the MN set is = [ ]− −PctTripsOnTime45mphMN 1.2568 meanTTI 1e Similar to the PctTripsOnTime50mph case, the regionally recalibrated model works better for the MN data set than the model built on the AllData set because the measured samples Table E.43. Parameter Estimates, Recalibrated PctTripsOnTime45mph Model, California Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a -1.6125 0.0423 -1.6959 -1.5291 Table E.42. Analysis of Variance, Recalibrated PctTripsOnTime45mph Model, California Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 1 141.7 141.7 31390.1 <0.0001 Error 186 0.8396 0.00451 Uncorrected total 187 142.5 Figure E.102. Fit plot in original scale, recalibrated PctTripsOnTime45mph model, California. are in general randomly scattered out on both sides of the model line, although when the mean TTI is close to 1, the model tends to underestimate the response variable. The F-test results (Table E.44) show that the model is significant. The param- eter estimation is shown in Table E.45. The fit plot is shown in Figure E.107. The observed-by-predicted plot is shown in Figure E.108. The residual-by-predicted plot is shown in Fig- ure E.109. The histogram and the normality plot (Figure E.110 and Figure E.111) show that the residual distribution is not perfectly following a normal distribution. (text continues on page 283)

278 Figure E.103. Observed-by-predicted plot, recalibrated PctTripsOnTime45mph model, California. Figure E.104. Residual-by-predicted plot, recalibrated PctTripsOnTime45mph model, California.

279 Figure E.105. Distribution of raw residuals, recalibrated PctTripsOnTime45mph model, California. Figure E.106. Q-Q plot of residuals, recalibrated PctTripsOnTime45mph model, California.

280 Table E.45. Parameter Estimates, Recalibrated PctTripsOnTime45mph Model, Minnesota Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a -1.2568 0.0281 -1.3128 -1.2008 Table E.44. Analysis of Variance, Recalibrated PctTripsOnTime45mph Model, Minnesota Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 1 61.7399 61.7399 50278.5 <0.0001 Error 78 0.0958 0.00123 Uncorrected total 79 61.8357 Figure E.107. Fit plot in original scale, recalibrated PctTripsOnTime45mph model, Minnesota.

281 Figure E.108. Observed-by-predicted plot, recalibrated PctTripsOnTime45mph model, Minnesota. Figure E.109. Residual-by-predicted plot, recalibrated PctTripsOnTime45mph model, Minnesota.

282 Figure E.110. Distribution of raw residuals, recalibrated PctTripsOnTime45mph model, Minnesota. Figure E.111. Q-Q plot of residuals, recalibrated PctTripsOnTime45mph model, Minnesota.

283 PctTripsOnTime30mph The recalibration of the PctTripsOnTime30mph model is conducted by building a nonlinear model between the response variable, PctTripsOnTime30mph, and the independent vari- able, mean TTI. The model form is the same as the L03 data- poor model form, described as [ ]( )= + − + − PctTripsOnTime30mph 1 exp meanTTI a b a c dp where a, b, c, d are the four model parameters to be calibrated. AllData The recalibrated model for the PctTripsOnTime30mph with the AllData set is [ ]( )= + + − PctTripsOnTime30mph 0.3401 0.6803 1 exp 4.5026 meanTTI 1.7890 AllData p The F-test results (Table E.46) show that overall the model is significant. The parameter estimation is shown in Table E.47. The scatter plots (Figures E.112 through E.114) show that the model line can generally predict the trend in the data, indicating that the model is satisfactory in general. The residual plot (Figure E.115) presents a general random scatter, except for the slight nonconstant variance pattern on the right side of the figure. The histogram and the nor- mality plot (Figures E.116 and E.117) show that the residual distribution does not closely follow a normal distribution, which may largely be due to the disproportional number of residuals close to zero. Table E.46. Analysis of Variance, Recalibrated PctTripsOnTime30mph Model, AllData Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 3 3.8334 1.2778 2396.81 <0.0001 Error 319 0.1701 0.000533 Corrected total 322 4.0035 Table E.47. Parameter Estimates, Recalibrated PctTripsOnTime30mph Model, AllData Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a 0.3401 0.0312 0.2787 0.4016 b 0.6803 0.0343 0.6129 0.7478 c 4.5026 0.2817 3.9485 5.0568 d 1.7890 0.0221 1.8324 1.7455 Figure E.112. Scatter plot in original scale, recalibrated PctTripsOnTime30mph model, AllData.

284 Figure E.113. Fit plot in original scale, recalibrated PctTripsOnTime30mph model, AllData. Figure E.114. Observed-by-predicted plot, recalibrated PctTripsOnTime30mph model, AllData.

285 Figure E.115. Residual-by-predicted plot, recalibrated PctTripsOnTime30mph model, AllData. Figure E.116. Distribution of raw residuals, recalibrated PctTripsOnTime30mph model, AllData.

286 California The recalibrated model for the PctTripsOnTime30mph with the CA set is [ ]( )= + + − PctTripsOnTime30mph 0.3263 0.6827 1 exp 5.4915 meanTTI 1.7916 CA p The F-test (Table E.48) has a small p-value, indicating that overall the model is valid. The parameter estimates are shown in Table E.49. The fit plot (Figure E.118) and the measured- by-predicted plot (Figure E.119) confirm that the model can predict the trend in the data samples. The residual-by-predicted plot (Figure E.120) presents a random pattern in general. The histogram and the normality plot (Figures E.121 and E.122) show that the residual distribution does not closely follow a normal distribution. Minnesota The recalibrated model for the PctTripsOnTime30mph with the MN set is [ ]( )= + + − PctTripsOnTime30mph 0.5795 0.4341 1 exp 6.1112 meanTTI 1.5715 MN p Figure E.117. Q-Q plot of residuals, recalibrated PctTripsOnTime30mph model, AllData. Table E.48. Analysis of Variance, Recalibrated PctTripsOnTime30mph Model, California Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 3 2.6242 0.8747 1744.94 <0.0001 Error 183 0.0917 0.000501 Corrected total 186 2.7159 Table E.49. Parameter Estimates, Recalibrated PctTripsOnTime30mph Model, California Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a 0.3263 0.0248 0.2774 0.3752 b 0.6827 0.0263 0.6307 0.7346 c 5.4915 0.2995 4.9005 6.0824 d 1.7916 0.0157 1.8226 1.7605 The F-test (Table E.50) has a small p-value, indicating that overall the model is valid. The parameter estimates are shown in Table E.51. The fit plot (Figure E.123) and the measured- by-predicted plot (Figure E.124) confirm that the model can predict the trend in the data samples, resulting in the random pattern in the residual-by-predicted plot (Figure E.125) although the nonconstant variance problem may still slightly exist (Figures E.126 and E.127). (text continues on page 292)

287 Figure E.118. Fit plot in original scale, recalibrated PctTripsOnTime30mph model, California. Figure E.119. Observed-by-predicted plot, recalibrated PctTripsOnTime30mph model, California.

288 Figure E.120. Residual-by-predicted plot, recalibrated PctTripsOnTime30mph model, California. Figure E.121. Distribution of raw residuals, recalibrated PctTripsOnTime30mph model, California.

289 Figure E.122. Q-Q plot of residuals, recalibrated PctTripsOnTime30mph model, California. Table E.50. Analysis of Variance, Recalibrated PctTripsOnTime30mph Model, Minnesota Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 3 1.0652 0.3551 1168.55 <0.0001 Error 75 0.0228 0.000304 Corrected total 78 1.0880 Table E.51. Parameter Estimates, Recalibrated PctTripsOnTime30mph Model, Minnesota Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a 0.5795 0.0191 0.5414 0.6175 b 0.4341 0.0247 0.3850 0.4833 c 6.1112 0.7944 4.5287 7.6937 d 1.5715 0.0157 1.6028 1.5402

290 Figure E.124. Observed-by-predicted plot, recalibrated PctTripsOnTime30mph model, Minnesota. Figure E.123. Fit plot in original scale, recalibrated PctTripsOnTime30mph model, Minnesota.

291 Figure E.125. Residual-by-predicted plot, recalibrated PctTripsOnTime30mph model, Minnesota. Figure E.126. Distribution of raw residuals, recalibrated PctTripsOnTime30mph model, Minnesota.

292 new Models This chapter presents the results of the new model development for the 95th-, 90th-, and 80th-percentile TTI SHRP 2 L03 mod- els. The new model development was performed on three of the L33 data sets: California, Minnesota, and AllData (representing roadway sections in both CA and MN). Three enhanced models were developed: (1) a single-parameter power form model; (2) a two-parameter power form model; and (3) a two-parameter polynomial model. For each of the measures, this appendix contains a summary table of the mean square error (MSE) for the recalibrated model and the three new models. Note that all modeling results in this chapter are built on a reduced data set that excludes two outlier samples. As such, the MSE values for the recalibration models shown in this chapter may be different from those presented in the previous chapter. The new models show the most significant improvements for the 80th-percentile TTI model and the least improvement for the 95th-percentile TTI model. Based on the results of the data-poor recalibration and enhancement, the L33 project team recommends that the SHRP 2 program recommend the new models for adoption. This recommendation is based on the following reasons: 1. The residual-by-predicted value plot shows improvement in the shape and in the balance of scatter around the origin, indicating that the new models better satisfy the assump- tions of regression. Note that there is still a problem with the nonconstant variance that, due to data characteristics, may not be possible to fully solve. 2. The new models allow for a consistent model form between different percentile TTI measures. 3. Since the variance of travel times tends to increase with the mean travel time, reliability model curves should show an increasing pattern at an increasing rate. The new models satisfy this characteristic in a way that the original L03 data- poor models (which show a decreasing rate of increase) do not. 95th-Percentile TTI Summary The MSE summary table (Table E.52) shows that the recali- brated 95th-percentile TTI model has a smaller MSE value than the new models. However, due to the reasons stated above and the fact that the MSE values are similar between the recalibrated and new models, we still propose that the SHRP 2 program adopt the new models. When evaluating the MSE tables, it is important to keep in mind that there is no benchmark to decide how much smaller Figure E.127. Q-Q plot of residuals, recalibrated PctTripsOnTime30mph model, Minnesota.

293 Table E.52. MSE Summary Table (Comparable) Model Name Formula AllData CA MN Recalibration y = 1 + a p ln(x) 0.0277 0.0255 0.0234 1-parameter power y = xb 0.0300 0.0273 0.0408 2-parameter power y = a p xb 0.0286 0.0264 0.0345 2-parameter polynomial y = a p x + b p x2 0.0289 0.0268 0.0352 of an MSE makes the model better. In this case, due to the fact that the validation data sets have a relatively small range in the mean TTI, each of the four types of models has predictive power. However, the research team believes the new models are more consistent with professional knowledge and real world experience. The following three sections present the new models built for the 95th-percentile TTI with the AllData (Tables E.54, E.56, and E.58), CA (Tables E.60, E.62, and E.64), and MN (Tables E.66, E.68, and E.70) data sets. The three models present very similar results in terms of fit plot (Figures E.128, E.129, E.134, E.135, E.140, E.141, E.146, E.151, E.156, E.161, E.166, and E.171) and residual-by-predicted plot (Figures E.131, E.137, E.143, E.148, E.153, E.158, E.163, E.168, and E.173). Regional differences can be identified in the plots. All the models passed the F-test (Tables E.53, E.55, E.57, E.59, E.61, E.63, E.65, E.67, and E.69), indicating overall validity. The fit plots and the observed-by-predicted plots (Figures E.130, E.136, E.142, E.147, E.152, E.157, E.162, E.167, and E.172) show that the model can predict the trend in the data set. The residual-by-predicted plots show an improvement in the nonrandom pattern around the origin, although the nonconstant variance problem still exists. The histograms (Figures E.132, E.138, E.144, E.149, E.154, E.159, E.164, E.169, and E.174) and the normality plots (Figures E.133, E.139, E.145, E.150, E.155, E.160, E.165, E.170 and E.175) indicate that the normality assumption may still be violated. Due to the validation data characteristics, the nonconstant variance problem and the violation of normality problem may be hard to get rid of. The MN modeling results raise a questionable pattern because the residual-by-predicted plots for all three new model forms show an unbalanced distribution of residual points and a concave shape. In fact, the original data-poor model for the 95th-percentile TTI model fits better. However, such results may be due to insufficient data points. Although the MN data set has 79 samples, most of them are around the origin area (low mean TTI and high reliability). To make the model form consistent, the team still proposes adopting the three new model forms. Table E.53. Analysis of Variance, New Model: Power Form with a Single Parameter, 95th-Percentile TTI, AllData Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 1 793.5 793.5 26492.1 <0.0001 Error 320 9.5845 0.0300 Uncorrected total 321 803.1 Table E.54. Parameter Estimates, New Model: Power Form with a Single Parameter, 95th-Percentile TTI, AllData Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits b 1.9566 0.0146 1.9279 1.9853 AllData poWeR foRM Model With a Single paRaMeteR Model: =95th-percentile TTI meanTTIAllData 1.9566 poWeR foRM Model With tWo paRaMeteRS Model: =95th-percentile TTI 1.0406 meanTTIAllData 1.8821p polynoMial foRM Model With tWo paRaMeteRS Model: =95th-percentile TTI 1.1494 meanTTI + 0.8902 meanTTI AllData 2 p p (text continues on page 304)

294 Figure E.129. Fit plot, new model: power form with a single parameter, 95th-percentile TTI, AllData. Figure E.128. Fit plot by region, new model: power form with a single parameter, 95th-percentile TTI, AllData.

295 Figure E.130. Observed-by-predicted plot, new model: power form with a single parameter, 95th-percentile TTI, AllData. Figure E.131. Residual-by-predicted plot, new model: power form with a single parameter, 95th-percentile TTI, AllData.

296 Figure E.133. Q-Q plot of residuals, new model: power form with a single parameter, 95th-percentile TTI, AllData. Figure E.132. Distribution of residuals, new model: power form with a single parameter, 95th-percentile TTI, AllData.

297 Table E.55. Analysis of Variance, New Model: Power Form with Two Parameters, 95th-Percentile TTI, AllData Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 2 793.9 397.0 13865.6 <0.0001 Error 319 9.1328 0.0286 Uncorrected total 321 803.1 Table E.56. Parameter Estimates, New Model: Power Form with Two Parameters, 95th-Percentile TTI, AllData Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a 1.0406 0.0103 1.0203 1.0609 b 1.8821 0.0236 1.8358 1.9285 Figure E.134. Fit plot by region, new model: power form with two parameters, 95th-percentile TTI, AllData.

298 Figure E.135. Fit plot, new model: power form with two parameters, 95th-percentile TTI, AllData. Figure E.136. Observed-by-predicted plot, new model: power form with two parameters, 95th-percentile TTI, AllData.

299 Figure E.137. Residual-by-predicted plot, new model: power form with two parameters, 95th-percentile TTI, AllData. Figure E.138. Distribution of residuals, new model: power form with two parameters, 95th-percentile TTI, AllData.

300 Figure E.139. Q-Q plot of residuals, new model: power form with two parameters, 95th-percentile TTI, AllData. Table E.57. Analysis of Variance, New Model: Polynomial Form with Two Parameters, 95th-Percentile TTI, AllData Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 2 793.8 396.9 13716.8 <0.0001 Error 319 9.2308 0.0289 Uncorrected total 321 803.1 Table E.58. Parameter Estimates, New Model: Polynomial Form with Two Parameters, 95th-Percentile TTI, AllData Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a 0.1494 0.0321 0.0862 0.2125 b 0.8902 0.0241 0.8427 0.9378

301 Figure E.140. Fit plot by region, new model: polynomial form with two parameters, 95th-percentile TTI, AllData. Figure E.141. Fit plot, new model: polynomial form with two parameters, 95th-percentile TTI, AllData.

302 Figure E.142. Observed-by-predicted plot, new model: polynomial form with two parameters, 95th-percentile TTI, AllData. Figure E.143. Residual-by-predicted plot, new model: polynomial form with two parameters, 95th-percentile TTI, AllData.

303 Figure E.144. Distribution of residuals, new model: polynomial form with two parameters, 95th-percentile TTI, AllData. Figure E.145. Q-Q plot of residuals, new model: polynomial form with two parameters, 95th-percentile TTI, AllData.

304 California poWeR foRM Model With a Single paRaMeteR Model: =95th-percentile TTI meanTTICA 1.8922 poWeR foRM Model With tWo paRaMeteRS Model: =95th-percentile TTI 1.0363 meanTTICA 1.8232p polynoMial foRM Model With tWo paRaMeteRS Model: = + 95th-percentile TTI 0.2245 meanTTI 0.8119 meanTTI CA 2 p p Minnesota poWeR foRM Model With a Single paRaMeteR Model: =95th-percentile TTI meanTTIMN 2.0516 poWeR foRM Model With tWo paRaMeteRS Model: =95th-percentile TTI 1.0871 meanTTIMN 1.9081p Table E.59. Analysis of Variance, New Model: Power Form with a Single Parameter, 95th-Percentile TTI, California Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 1 464.4 464.4 16991.0 <0.0001 Error 184 5.0293 0.0273 Uncorrected total 185 469.4 Table E.60. Parameter Estimates, New Model: Power Form with a Single Parameter, 95th-Percentile TTI, California Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits b 1.8922 0.0184 1.8559 1.9285 Figure E.146. Fit plot, new model: power form with a single parameter, 95th-percentile TTI, California. polynoMial foRM Model With tWo paRaMeteRS Model: = + 95th-percentile TTI 0.1122 meanTTI 0.9701 meanTTI MN 2 p p (text continues on page 322)

305 Figure E.147. Observed-by-predicted plot, new model: power form with a single parameter, 95th-percentile TTI, California. Figure E.148. Residual-by-predicted plot, new model: power form with a single parameter, 95th-percentile TTI, California.

306 Figure E.149. Distribution of residuals, new model: power form with a single parameter, 95th-percentile TTI, California. Figure E.150. Q-Q plot of residuals, new model: power form with a single parameter, 95th-percentile TTI, California.

307 Table E.61. Analysis of Variance, New Model: Power Form with Two Parameters, 95th-Percentile TTI, California Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 2 464.6 232.3 8797.71 <0.0001 Error 183 4.8322 0.0264 Uncorrected total 185 469.4 Table E.62. Parameter Estimates, New Model: Power Form with Two Parameters, 95th-Percentile TTI, California Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a 1.0363 0.0134 1.0099 1.0627 b 1.8232 0.0311 1.7619 1.8845 Figure E.151. Fit plot, new model: power form with two parameters, 95th-percentile TTI, California.

308 Figure E.152. Observed-by-predicted plot, new model: power form with two parameters, 95th-percentile TTI, California. Figure E.153. Residual-by-predicted plot, new model: power form with two parameters, 95th-percentile TTI, California.

309 Figure E.154. Distribution of residuals, new model: power form with two parameters, 95th-percentile TTI, California. Figure E.155. Q-Q plot of residuals, new model: power form with two parameters, 95th-percentile TTI, California.

310 Table E.63. Analysis of Variance, New Model: Polynomial Form with Two Parameters, 95th-Percentile TTI, California Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 2 464.5 232.3 8670.53 <0.0001 Error 183 4.9023 0.0268 Uncorrected total 185 469.4 Table E.64. Parameter Estimates, New Model: Polynomial Form with Two Parameters, 95th-Percentile TTI, California Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a 0.2245 0.0414 0.1428 0.3062 b 0.8119 0.0307 0.7514 0.8725 Figure E.156. Fit plot, new model: polynomial form with two parameters, 95th-percentile TTI, California.

311 Figure E.157. Observed-by-predicted plot, new model: polynomial form with two parameters, 95th-percentile TTI, California. Figure E.158. Residual-by-predicted plot, new model: polynomial form with two parameters, 95th-percentile TTI, California.

312 Figure E.159. Distribution of residuals, new model: polynomial form with two parameters, 95th-percentile TTI, California. Figure E.160. Q-Q plot of residuals, new model: polynomial form with two parameters, 95th-percentile TTI, California.

313 Table E.65. Analysis of Variance, New Model: Power Form with a Single Parameter, 95th-Percentile TTI, Minnesota Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 1 250.8 250.8 6151.90 <0.0001 Error 78 3.1795 0.0408 Uncorrected total 79 254.0 Table E.66. Parameter Estimates, New Model: Power Form with a Single Parameter, 95th-Percentile TTI, Minnesota Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits b 2.0516 0.0268 1.9983 2.1048 Figure E.161. Fit plot, new model: power form with a single parameter, 95th-percentile TTI, Minnesota.

314 Figure E.162. Observed-by-predicted plot, new model: power form with a single parameter, 95th-percentile TTI, Minnesota. Figure E.163. Residual-by-predicted plot, new model: power form with a single parameter, 95th-percentile TTI, Minnesota.

315 Figure E.164. Distribution of residuals, new model: power form with a single parameter, 95th-percentile TTI, Minnesota. Figure E.165. Q-Q plot of residuals, new model: power form with a single parameter, 95th-percentile TTI, Minnesota.

316 Table E.67. Analysis of Variance, New Model: Power Form with Two Parameters, 95th-Percentile TTI, Minnesota Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 2 251.3 125.6 3638.20 <0.0001 Error 77 2.6592 0.0345 Uncorrected total 79 254.0 Table E.68. Parameter Estimates, New Model: Power Form with Two Parameters, 95th-Percentile TTI, Minnesota Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a 1.0871 0.0228 1.0416 1.1325 b 1.9081 0.0443 1.8200 1.9963 Figure E.166. Fit plot, new model: power form with two parameters, 95th-percentile TTI, Minnesota.

317 Figure E.167. Observed-by-predicted plot, new model: power form with two parameters, 95th-percentile TTI, Minnesota. Figure E.168. Residual-by-predicted plot, new model: power form with two parameters, 95th-percentile TTI, Minnesota.

318 Figure E.169. Distribution of residuals, new model: power form with two parameters, 95th-percentile TTI, Minnesota. Figure E.170. Q-Q plot of residuals, new model: power form with two parameters, 95th-percentile TTI, Minnesota.

319 Table E.69. Analysis of Variance, New Model: Polynomial Form with Two Parameters, 95th-Percentile TTI, Minnesota Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 2 251.2 125.6 3572.65 <0.0001 Error 77 2.7075 0.0352 Uncorrected total 79 254.0 Table E.70. Parameter Estimates, New Model: Polynomial Form with Two Parameters, 95th-Percentile TTI, Minnesota Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a 0.1122 0.0652 -0.0176 0.2419 b 0.9701 0.0466 0.8772 1.0630 Figure E.171. Fit plot, new model: polynomial form with two parameters, 95th-percentile TTI, Minnesota.

320 Figure E.172. Observed-by-predicted plot, new model: polynomial form with two parameters, 95th-percentile TTI, Minnesota. Figure E.173. Residual-by-predicted plot, new model: polynomial form with two parameters, 95th-percentile TTI, Minnesota.

321 Figure E.174. Distribution of residuals, new model: polynomial form with two parameters, 95th-percentile TTI, Minnesota. Figure E.175. Q-Q plot of residuals, new model: polynomial form with two parameters, 95th-percentile TTI, Minnesota.

322 90th-Percentile TTI Summary For the 90th-percentile TTI models, the MSE table (Table E.71) shows that the MSE values for the AllData and the CA data sets are smaller with the new models. For the MN data set, the recalibration model has the smallest MSE. The new models (Tables E.73, E.75, E.77, E.79, E.81, E.83, E.85, E.87, and E.89) all passed the F-test (Tables E.72, E.74, E.76, E.78, E.80, E.82, E.84, E.86, and E.88), indicating over- all validity. The fit plots (Figures E.176, E.177, E.182, E.183, E.188, E.189, E.194, E.199, E.204, E.209, E.214, and E.219) show that the new models can predict the data trend well and also show the regional differences. The residual-by-predicted plots (Figures E.179, E.185, E.191, E.196, E.201, E.206, E.211, E.216, and E.221) show improved residual pattern compared with the recalibrated models in that the residual samples are more ran- domly balanced on the two sides of the zero reference line and around the origin. The nonconstant variance problem still exists. The histograms (Figures E.180, E.186, E.192, E.197, E.202, E.207, E.212, E.217, and E.222) and the normality plots (Figures E.181, E.187, E.193, E.198, E.203, E.208, E.213, E.218, and E.223) show that the residuals may not perfectly follow a normal distribution. The observed-by-predicted plots are shown in Figures E.178, E.184, E.190, E.195, E.200, E.205, E.210, E.215, and E.220. AllData poWeR foRM Model With a Single paRaMeteR Model: =90th-percentile TTI meanTTIAllData 1.7324 poWeR foRM Model With tWo paRaMeteRS Model: =90th-percentile TTI 1.0099 meanTTIAllData 1.7137p polynoMial foRM Model With tWo paRaMeteRS Model: = + 90th-percentile TTI 0.3528 meanTTI 0.6591 meanTTI AllData 2 p p California poWeR foRM Model With a Single paRaMeteR Model: =90th-percentile TTI meanTTICA 1.6826 poWeR foRM Model With tWo paRaMeteRS Model: 90th-percentile TTI 1.0090 meanTTICA 1.6651 = p polynoMial foRM Model With tWo paRaMeteRS Model: =90th-percentile TTI 0.4060 meanTTI + 0.6055 meanTTI CA 2 p p Table E.71. MSE Summary Table (Comparable) Model Name Formula AllData CA MN Recalibration y = 1 + a p ln(x) 0.0137 0.0131 0.0110 1-parameter power y = xb 0.0122 0.0118 0.0151 2-parameter power y = a p xb 0.0121 0.0118 0.0144 2-parameter polynomial y = a p x + b p x2 0.0125 0.0121 0.0153 Table E.72. Analysis of Variance, New Model: Power Form with a Single Parameter, 90th-Percentile TTI, AllData Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 1 670.6 670.6 55066.8 <0.0001 Error 320 3.8971 0.0122 Uncorrected total 321 674.5 Table E.73. Parameter Estimates, New Model: Power Form with a Single Parameter, 90th-Percentile TTI, AllData Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits b 1.7324 0.0106 1.7116 1.7532 (text continues on page 342)

323 Figure E.176. Fit plot by region, new model: power form with a single parameter, 90th-percentile TTI, AllData. Figure E.177. Fit plot, new model: power form with a single parameter, 90th-percentile TTI, AllData.

324 Figure E.178. Observed-by-predicted plot, new model: power form with a single parameter, 90th-percentile TTI, AllData. Figure E.179. Residual-by-predicted plot, new model: power form with a single parameter, 90th-percentile TTI, AllData.

325 Figure E.180. Distribution of residuals, new model: power form with a single parameter, 90th-percentile TTI, AllData. Figure E.181. Q-Q plot of residuals, new model: power form with a single parameter, 90th-percentile TTI, AllData.

326 Table E.74. Analysis of Variance, New Model: Power Form with Two Parameters, 90th-Percentile TTI, AllData Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 2 670.6 335.3 27631.9 <0.0001 Error 319 3.8712 0.0121 Uncorrected total 321 674.5 Table E.75. Parameter Estimates, New Model: Power Form with Two Parameters, 90th-Percentile TTI, AllData Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a 1.0099 0.00677 0.9965 1.0232 b 1.7137 0.0166 1.6810 1.7464 Figure E.182. Fit plot by region, new model: power form with two parameters, 90th-percentile TTI, AllData.

327 Figure E.183. Fit plot, new model: power form with two parameters, 90th-percentile TTI, AllData. Figure E.184. Observed-by-predicted plot, new model: power form with two parameters, 90th-percentile TTI, AllData.

328 Figure E.185. Residual-by-predicted plot, new model: power form with two parameters, 90th-percentile TTI, AllData. Figure E.186. Distribution of residuals, new model: power form with two parameters, 90th-percentile TTI, AllData.

329 Figure E.187. Q-Q plot of residuals, new model: power form with two parameters, 90th-percentile TTI, AllData. Table E.76. Analysis of Variance, New Model: Polynomial Form with Two Parameters, 90th-Percentile TTI, AllData Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 2 670.5 335.3 26887.0 <0.0001 Error 319 3.9778 0.0125 Uncorrected total 321 674.5 Table E.77. Parameter Estimates, New Model: Polynomial Form with Two Parameters, 90th-Percentile TTI, AllData Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a 0.3528 0.0211 0.3114 0.3943 b 0.6591 0.0159 0.6279 0.6903

330 Figure E.188. Fit plot by region, new model: polynomial form with two parameters, 90th-percentile TTI, AllData. Figure E.189. Fit plot, new model: polynomial form with two parameters, 90th-percentile TTI, AllData.

331 Figure E.190. Observed-by-predicted plot, new model: polynomial form with two parameters, 90th-percentile TTI, AllData. Figure E.191. Residual-by-predicted plot, new model: polynomial form with two parameters, 90th-percentile TTI, AllData.

332 Figure E.192. Distribution of residuals, new model: polynomial form with two parameters, 90th-percentile TTI, AllData. Figure E.193. Q-Q plot of residuals, new model: polynomial form with two parameters, 90th-percentile TTI, AllData.

333 Table E.78. Analysis of Variance, New Model: Power Form with a Single Parameter, 90th-Percentile TTI, California Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 1 397.0 397.0 33589.4 <0.0001 Error 184 2.1745 0.0118 Uncorrected total 185 399.1 Table E.79. Parameter Estimates, New Model: Power Form with a Single Parameter, 90th-Percentile TTI, California Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits b 1.6826 0.0136 1.6559 1.7094 Figure E.194. Fit plot by region, new model: power form with a single parameter, 90th-percentile TTI, California.

334 Figure E.195. Observed-by-predicted plot, new model: power form with a single parameter, 90th-percentile TTI, California. Figure E.196. Residual-by-predicted plot, new model: power form with a single parameter, 90th-percentile TTI, California.

335 Figure E.197. Distribution of residuals, new model: power form with a single parameter, 90th-percentile TTI, California. Figure E.198. Q-Q plot of residuals, new model: power form with a single parameter, 90th-percentile TTI, California.

336 Table E.80. Analysis of Variance, New Model: Power Form with Two Parameters, 90th-Percentile TTI, California Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 2 397.0 198.5 16793.9 <0.0001 Error 183 2.1628 0.0118 Uncorrected total 185 399.1 Table E.81. Parameter Estimates, New Model: Power Form with Two Parameters, 90th-Percentile TTI, California Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a 1.0090 0.00905 0.9911 1.0268 b 1.6651 0.0223 1.6212 1.7091 Figure E.199. Fit plot, new model: power form with two parameters, 90th-percentile TTI, California.

337 Figure E.200. Observed-by-predicted plot, new model: power form with two parameters, 90th-percentile TTI, California. Figure E.201. Residual-by-predicted plot, new model: power form with two parameters, 90th-percentile TTI, California.

338 Figure E.202. Distribution of residuals, new model: power form with two parameters, 90th-percentile TTI, California. Figure E.203. Q-Q plot of residuals, new model: power form with two parameters, 90th-percentile TTI, California.

339 Table E.82. Analysis of Variance, New Model: Polynomial Form with Two Parameters, 90th-Percentile TTI, California Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 2 396.9 198.5 16354.7 <0.0001 Error 183 2.2206 0.0121 Uncorrected total 185 399.1 Table E.83. Parameter Estimates, New Model: Polynomial Form with Two Parameters, 90th-Percentile TTI, California Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a 0.4060 0.0279 0.3510 0.4610 b 0.6055 0.0207 0.5648 0.6463 Figure E.204. Fit plot, new model: polynomial form with two parameters, 90th-percentile TTI, California.

340 Figure E.205. Observed-by-predicted plot, new model: polynomial form with two parameters, 90th-percentile TTI, California. Figure E.206. Residual-by-predicted plot, new model: polynomial form with two parameters, 90th-percentile TTI, California.

341 Figure E.207. Distribution of residuals, new model: polynomial form with two parameters, 90th-percentile TTI, California. Figure E.208. Q-Q plot of residuals, new model: polynomial form with two parameters, 90th-percentile TTI, California.

342 Table E.85. Parameter Estimates, New Model: Power Form with a Single Parameter, 90th-Percentile TTI, Minnesota Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits b 1.8113 0.0188 1.7739 1.8488 Table E.84. Analysis of Variance, New Model: Power Form with a Single Parameter, 90th-Percentile TTI, Minnesota Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 1 201.6 201.6 13391.4 <0.0001 Error 78 1.1740 0.0151 Uncorrected total 79 202.7 Figure E.209. Fit plot, new model: power form with a single parameter, 90th-percentile TTI, Minnesota. Minnesota poWeR foRM Model With a Single paRaMeteR Model: =90th-percentile TTI meanTTIMN 1.8113 poWeR foRM Model With tWo paRaMeteRS Model: =90th-percentile TTI 1.0320 meanTTIMN 1.7562p polynoMial foRM Model With tWo paRaMeteRS Model: =90th-percentile TTI 0.3102 meanTTI + 0.7225 meanTTI MN 2 p p (text continues on page 351)

343 Figure E.210. Observed-by-predicted plot, new model: power form with a single parameter, 90th-percentile TTI, Minnesota. Figure E.211. Residual-by-predicted plot, new model: power form with a single parameter, 90th-percentile TTI, Minnesota.

344 Figure E.212. Distribution of residuals, new model: power form with a single parameter, 90th-percentile TTI, Minnesota. Figure E.213. Q-Q plot of residuals, new model: power form with a single parameter, 90th-percentile TTI, Minnesota.

345 Table E.86. Analysis of Variance, New Model: Power Form with Two Parameters, 90th-Percentile TTI, Minnesota Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 2 201.6 100.8 7018.73 <0.0001 Error 77 1.1060 0.0144 Uncorrected total 79 202.7 Table E.87. Parameter Estimates, New Model: Power Form with Two Parameters, 90th-Percentile TTI, Minnesota Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a 1.0320 0.0149 1.0024 1.0617 b 1.7562 0.0314 1.6937 1.8188 Figure E.214. Fit plot, new model: power form with two parameters, 90th-percentile TTI, Minnesota.

346 Figure E.215. Observed-by-predicted plot, new model: power form with two parameters, 90th-percentile TTI, Minnesota. Figure E.216. Residual-by-predicted plot, new model: power form with two parameters, 90th-percentile TTI, Minnesota.

347 Figure E.217. Distribution of residuals, new model: power form with two parameters, 90th-percentile TTI, Minnesota. Figure E.218. Q-Q plot of residuals, new model: power form with two parameters, 90th-percentile TTI, Minnesota.

348 Table E.88. Analysis of Variance, New Model: Polynomial Form with Two Parameters, 90th-Percentile TTI, Minnesota Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 2 201.6 100.8 6606.74 <0.0001 Error 77 1.1746 0.0153 Uncorrected total 79 202.7 Table E.89. Parameter Estimates, New Model: Polynomial Form with Two Parameters, 90th-Percentile TTI, Minnesota Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a 0.3102 0.0429 0.2247 0.3956 b 0.7225 0.0307 0.6614 0.7837 Figure E.219. Fit plot, new model: polynomial form with two parameters, 90th-percentile TTI, Minnesota.

349 Figure E.220. Observed-by-predicted plot, new model: polynomial form with two parameters, 90th-percentile TTI, Minnesota. Figure E.221. Residual-by-predicted plot, new model: polynomial form with two parameters, 90th-percentile TTI, Minnesota.

350 Figure E.222. Distribution of residuals, new model: polynomial form with two parameters, 90th-percentile TTI, Minnesota. Figure E.223. Q-Q plot of residuals, new model: polynomial form with two parameters, 90th-percentile TTI, Minnesota.

351 80th-Percentile TTI Summary For the 80th-percentile TTI, the new models all have smaller MSE values than the recalibrated model. The MSE for the AllData set is reduced by almost 50% with the new models (Table E.90). The new models (Tables E.92, E.94, E.96, E.98, E.100, E.102, E.104, E.106, and E.108) perform much better than the L03 data-poor models for the 80th-percentile TTI (F-tests shown in Tables E.91, E.93, E.95, E.97, E.99, E.101, E.103, E.105, and E.107). The significant improvement is shown in both the fit plots (Figures E.224, E.229, E.234, E.239, E.244, E.249, E.254, E.259, and E.264) and the observed-by-predicted plots (Figures E.225, E.230, E.235, E.240, E.245, E.250, E.255, E.260, and E.265). The residual-by-predicted plots (Figures E.226, E.231, E.236, E.241, E.246, E.251, E.256, E.261, and E.266) all show generally random patterns, although the nonconstant problem still exists. The histograms (Figures E.227, E.232, E.237, E.242, E.247, E.252, E.257, E.262, and E.267) and the normality plots (Figures E.228, E.233, E.238, E.243, E.248, E.253, E.258, E.263, and E.268) show that the residual distributions may not perfectly follow a normal distribution. Overall, the new models work much better than the L03 data-poor model for the 80th-percentile TTI. AllData poWeR foRM Model With a Single paRaMeteR Model: =80th-percentile TTI meanTTIAllData 1.4448 poWeR foRM Model With tWo paRaMeteRS Model: =80th-percentile TTI 0.9943 meanTTIAllData 1.4559p polynoMial foRM Model With tWo paRaMeteRS Model: =80th-percentile TTI 0.6166 meanTTI + 0.3809 meanTTI AllData 2 p p California poWeR foRM Model With a Single paRaMeteR Model: =80th-percentile TTI meanTTICA 1.4148 poWeR foRM Model With tWo paRaMeteRS Model: =80th-percentile TTI 0.9943 meanTTICA 1.4264p polynoMial foRM Model With tWo paRaMeteRS Model: =80th-percentile TTI 0.6428 meanTTI + 0.3547 meanTTI CA 2 p p Table E.90. MSE Summary Table (Comparable) Model Name Formula AllData CA MN Recalibration y = 1 + a * ln(x) 0.00469 0.00410 0.00506 1-parameter power y = xb 0.00239 0.00178 0.00384 2-parameter power y = a * xb 0.00237 0.00176 0.00389 2-parameter polynomial y = a * x + b * x2 0.00245 0.00179 0.00436 Table E.91. Analysis of Variance, New Model: Power Form with a Single Parameter, 80th-Percentile TTI, AllData Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 1 558.4 558.4 233468 <0.0001 Error 320 0.7654 0.00239 Uncorrected total 321 559.2 Table E.92. Parameter Estimates, New Model: Power Form with a Single Parameter, 80th-Percentile TTI, AllData Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits b 1.4448 0.00549 1.4340 1.4556 (text continues on page 369)

352 Figure E.224. Fit plot, new model: power form with a single parameter, 80th-percentile TTI, AllData. Figure E.225. Observed-by-predicted plot, new model: power form with a single parameter, 80th-percentile TTI, AllData.

353 Figure E.226. Residual-by-predicted plot, new model: power form with a single parameter, 80th-percentile TTI, AllData. Figure E.227. Distribution of residuals, new model: power form with a single parameter, 80th-percentile TTI, AllData.

354 Figure E.228. Q-Q plot of residuals, new model: power form with a single parameter, 80th-percentile TTI, AllData. Table E.93. Analysis of Variance, New Model: Power Form with Two Parameters, 80th-Percentile TTI, AllData Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 2 558.4 279.2 117639 <0.0001 Error 319 0.7571 0.00237 Uncorrected total 321 559.2 Table E.94. Parameter Estimates, New Model: Power Form with Two Parameters, 80th-Percentile TTI, AllData Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a 0.9943 0.00304 0.9884 1.0003 b 1.4559 0.00810 1.4399 1.4718

355 Figure E.229. Fit plot, new model: power form with two parameters, 80th-percentile TTI, AllData. Figure E.230. Observed-by-predicted plot, new model: power form with two parameters, 80th-percentile TTI, AllData.

356 Figure E.231. Residual-by-predicted plot, new model: power form with two parameters, 80th-percentile TTI, AllData. Figure E.232. Distribution of residuals, new model: power form with two parameters, 80th-percentile TTI, AllData.

357 Figure E.233. Q-Q plot of residuals, new model: power form with two parameters, 80th-percentile TTI, AllData. Table E.95. Analysis of Variance, New Model: Polynomial Form with Two Parameters, 80th-Percentile TTI, AllData Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 2 558.4 279.2 113766 <0.0001 Error 319 0.7829 0.00245 Uncorrected total 321 559.2 Table E.96. Parameter Estimates, New Model: Polynomial Form with Two Parameters, 80th-Percentile TTI, AllData Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a 0.6166 0.00935 0.5983 0.6350 b 0.3809 0.00703 0.3671 0.3948

358 Figure E.234. Fit plot, new model: polynomial form with two parameters, 80th-percentile TTI, AllData. Figure E.235. Observed-by-predicted plot, new model: polynomial form with two parameters, 80th-percentile TTI, AllData.

359 Figure E.236. Residual-by-predicted plot, new model: polynomial form with two parameters, 80th-percentile TTI, AllData. Figure E.237. Distribution of residuals, new model: polynomial form with two parameters, 80th-percentile TTI, AllData.

360 Figure E.238. Q-Q plot of residuals, new model: polynomial form with two parameters, 80th-percentile TTI, AllData. Table E.97. Analysis of Variance, New Model: Power Form with a Single Parameter, 80th-Percentile TTI, California Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 1 333.3 333.3 187478 <0.0001 Error 184 0.3271 0.00178 Uncorrected total 185 333.6 Table E.98. Parameter Estimates, New Model: Power Form with a Single Parameter, 80th-Percentile TTI, California Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits b 1.4148 0.00607 1.4029 1.4268

361 Figure E.239. Fit plot, new model: power form with a single parameter, 80th-percentile TTI, California. Figure E.240. Observed-by-predicted plot, new model: power form with a single parameter, 80th-percentile TTI, California.

362 Figure E.241. Residual-by-predicted plot, new model: power form with a single parameter, 80th-percentile TTI, California. Figure E.242. Distribution of residuals, new model: power form with a single parameter, 80th-percentile TTI, California.

363 Figure E.243. Q-Q plot of residuals, new model: power form with a single parameter, 80th-percentile TTI, California. Table E.99. Analysis of Variance, New Model: Power Form with Two Parameters, 80th-Percentile TTI, California Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 2 333.3 166.6 94557.3 <0.0001 Error 183 0.3225 0.00176 Uncorrected total 185 333.6 Table E.100. Parameter Estimates, New Model: Power Form with Two Parameters, 80th-Percentile TTI, California Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a 0.9943 0.00354 0.9873 1.0013 b 1.4264 0.00936 1.4079 1.4448

364 Figure E.244. Fit plot, new model: power form with two parameters, 80th-percentile TTI, California. Figure E.245. Observed-by-predicted plot, new model: power form with two parameters, 80th-percentile TTI, California.

365 Figure E.246. Residual-by-predicted plot, new model: power form with two parameters, 80th-percentile TTI, California. Figure E.247. Distribution of residuals, new model: power form with two parameters, 80th-percentile TTI, California.

366 Figure E.248. Q-Q plot of residuals, new model: power form with two parameters, 80th-percentile TTI, California. Table E.101. Analysis of Variance, New Model: Polynomial Form with Two Parameters, 80th-Percentile TTI, California Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 2 333.3 166.6 93176.0 <0.0001 Error 183 0.3273 0.00179 Uncorrected total 185 333.6 Table E.102. Parameter Estimates, New Model: Polynomial Form with Two Parameters, 80th-Percentile TTI, California Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a 0.6428 0.0107 0.6217 0.6639 b 0.3547 0.00793 0.3391 0.3704

367 Figure E.249. Fit plot, new model: polynomial form with two parameters, 80th-percentile TTI, California. Figure E.250. Observed-by-predicted plot, new model: polynomial form with two parameters, 80th-percentile TTI, California.

368 Figure E.251. Residual-by-predicted plot, new model: polynomial form with two parameters, 80th-percentile TTI, California. Figure E.252. Distribution of residuals, new model: polynomial form with two parameters, 80th-percentile TTI, California.

369 Figure E.253. Q-Q plot of residuals, new model: polynomial form with two parameters, 80th-percentile TTI, California. Minnesota poWeR foRM Model With a Single paRaMeteR Model: =80th-percentile TTI meanTTIMN 1.4955 poWeR foRM Model With tWo paRaMeteRS Model: =80th-percentile TTI 0.9992 meanTTIMN 1.4969p polynoMial foRM Model With tWo paRaMeteRS Model: =80th-percentile TTI 0.5858 meanTTI + 0.4173 meanTTI MN 2 p p Table E.103. Analysis of Variance, New Model: Power Form with a Single Parameter, 80th-Percentile TTI, Minnesota Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 1 157.4 157.4 41002.4 <0.0001 Error 78 0.2994 0.00384 Uncorrected total 79 157.7 Table E.104. Parameter Estimates, New Model: Power Form with a Single Parameter, 80th-Percentile TTI, Minnesota Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits b 1.4955 0.0115 1.4727 1.5183

370 Figure E.254. Fit plot, new model: power form with a single parameter, 80th-percentile TTI, Minnesota. Figure E.255. Observed-by-predicted plot, new model: power form with a single parameter, 80th-percentile TTI, Minnesota.

371 Figure E.256. Residual-by-predicted plot, new model: power form with a single parameter, 80th-percentile TTI, Minnesota. Figure E.257. Distribution of residuals, new model: power form with a single parameter, 80th-percentile TTI, Minnesota.

372 Figure E.258. Q-Q plot of residuals, new model: power form with a single parameter, 80th-percentile TTI, Minnesota. Table E.105. Analysis of Variance, New Model: Power Form with Two Parameters, 80th-Percentile TTI, Minnesota Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 2 157.4 78.6883 20241.1 <0.0001 Error 77 0.2993 0.00389 Uncorrected total 79 157.7 Table E.106. Parameter Estimates, New Model: Power Form with Two Parameters, 80th-Percentile TTI, Minnesota Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a 0.9992 0.00786 0.9835 1.0149 b 1.4969 0.0183 1.4605 1.5333

373 Figure E.259. Fit plot, new model: power form with two parameters, 80th-percentile TTI, Minnesota. Figure E.260. Observed-by-predicted plot, new model: power form with two parameters, 80th-percentile TTI, Minnesota.

374 Figure E.261. Residual-by-predicted plot, new model: power form with two parameters, 80th-percentile TTI, Minnesota. Figure E.262. Distribution of residuals, new model: power form with two parameters, 80th-percentile TTI, Minnesota.

375 Figure E.263. Q-Q plot of residuals, new model: power form with two parameters, 80th-percentile TTI, Minnesota. Table E.107. Analysis of Variance, New Model: Polynomial Form with Two Parameters, 80th-Percentile TTI, Minnesota Source DF Sum of Squares Mean Square F-Value Approx. Pr > F Model 2 157.3 78.6702 18059.0 <0.0001 Error 77 0.3354 0.00436 Uncorrected total 79 157.7 Table E.108. Parameter Estimates, New Model: Polynomial Form with Two Parameters, 80th-Percentile TTI, Minnesota Parameter Estimate Approx. Std Error Approx. 95% Confidence Limits a 0.5858 0.0229 0.5401 0.6315 b 0.4173 0.0164 0.3846 0.4500

376 Figure E.264. Fit plot, new model: polynomial form with two parameters, 80th-percentile TTI, Minnesota. Figure E.265. Observed-by-predicted plot, new model: polynomial form with two parameters, 80th-percentile TTI, Minnesota.

377 Figure E.266. Residual-by-predicted plot, new model: polynomial form with two parameters, 80th-percentile TTI, Minnesota. Figure E.267. Distribution of residuals, new model: polynomial form with two parameters, 80th-percentile TTI, Minnesota.

378 Figure E.268. Q-Q plot of residuals, new model: polynomial form with two parameters, 80th-percentile TTI, Minnesota.