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D-1 APPENDIX D Statistical Analyses of the Steady Shear Flow and Phase Angle Methods

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D-2 Results from Regression Analyses This section describes a statistical analysis to determine if either the Steady Shear Flow method or the Phase Angle method provides a better fit to experimental data from the mixture tests and the producers' recommendations for mixing temperature. Given below are the Minitab outputs for regressions between the bucket mixing temperature (BuckMixT) and the Steady Shear Flow Mixing Temperature (SSFMT), and between bucket mixing temperature (BuckMixT) and the Phase Angle Mixing Temperature (PAMT). Regression Analysis: BuckMixT versus SSFMT The regression equation is BuckMixT = 37 + 0.893 SSFMT (1) 12 cases used, 1 cases contain missing values Predictor Coef SE Coef T P Constant 36.9 105.1 0.35 0.733 SSFMT 0.8928 0.3399 2.63 0.025 S = 22.1202 R-Sq = 40.8% R-Sq(adj) = 34.9% Analysis of Variance Source DF SS MS F P Regression 1 3375.9 3375.9 6.90 0.025 Residual Error 10 4893.0 489.3 Total 11 8268.9

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D-3 Regression Analysis: BuckMixT versus PAMT The regression equation is BuckMixT = 58 + 0.797 PAMT (2) 12 cases used, 1 cases contain missing values Predictor Coef SE Coef T P Constant 58.5 149.8 0.39 0.705 PAMT 0.7970 0.4697 1.70 0.121 S = 25.3382 R-Sq = 22.4% R-Sq(adj) = 14.6% Analysis of Variance Source DF SS MS F P Regression 1 1848.7 1848.7 2.88 0.121 Residual Error 10 6420.2 642.0 Total 11 8268.9 In order to ascertain that the two regressors, SSFMT and PAMT, explain the same amount of variation in Buck-Mix Temperature, we first note that Neter's test for equality of two fatigue curves is not applicable because Neter's test requires that the two curves describe mixes from two different mixtures. Clearly, the two models: BuckMixT = 37 + 0.893SSFMT, and BuckMixT = 58 + 0.797PAMT, have the same response variable, namely Buck-Mix-Temperature, and hence are not from two different mixtures. However, it is widely known in statistical literature that the ratio 2 / 1 1 , 2 / 2 2 has the Fisher's F sampling distribution, where 1 = n1 -1 is the df of the chi- square in the numerator and 2 = n2 1 is the df of the chi-square in the denominator, but the two chi-squared random variables must be stochastically independent. Form ANOVA theory, it is well known that (SSError / 2 ) / 2 has a

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D-4 chi-square sampling distribution with error degrees of freedom equaling 2. The 2 represents the process error variance. That is, for model (1), which describes Buck-Mix Temperature versus SSFMT, SS ( Residual Error )SSFMT / 2 has the 2 sampling distribution, and similarly 10 10 for model (2) we have another 2 10 , but unfortunately the two chi-squared are not independent because the responses, Buck-Mix-Temperature, are the same for both models. The following test is at the very best an approximation, because it violates the independence assumption, and does not possess as much statistical power as the case when the two chi-squared random variables are independent. To this end, the ratio SS ( Residual Error ) PAMT / (102 ) SS ( Residual Error )SSFMT / (102 ) has very roughly an F-distribution with 10 degrees of freedom for the numerator and 10 for the denominator. That is to say, SS ( Residual Error ) PAMT / (102 ) F10,10 = SS ( Residual Error )SSFMT / (102 ) MS ( Residual Error ) PAMT MS ( Residual Error )SSFMT F10,10 642.0/489.3 = 1.3121 (P-value 2*0.337883 0.676; the PAMT = SSFMT is 2-sided.) multiplier 2 is needed because the hypothesis H0: 2 2 Because, the F0.30,10,10 =1.406, we may draw a tenuous conclusion that there is not sufficient statistical evidence that SSFMT explains more variability in Buck- Mix-Temp that does PAMT, even if the model R-Sq for SSFMT is 40.8%, while R-Sq for PAMT is only 22.4%. Note that even if we set the type I error rate as high as 60% for the above 2-sided test of equality of two residual variances, we PAMT = SSFMT . Further, the following correlation matrix still cannot reject H0: 2 2

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D-5 from Minitab clearly shows that the two regressor variables SSFMT and PAMT are highly correlated (r = 0.810 with a P-value = 0.001.) Correlations: SSFMT, PAMT, BuckMixT SSFMT PAMT PAMT 0.810 0.001 BuckMixT 0.639 0.473 0.025 0.121 Cell Contents: Pearson correlation P-Value In conclusion, it should be noted that perhaps the two regressors, SSFMT and PAMT, explain different amount of variation in Buck-Mix-Temp from a practical standpoint, but due to the small sample sizes (12 to 13) the practical differences cannot be detected from statistical standpoint. We now perform the same analyses for Pugmill Mixing Temperature (PugMixT). The Minitab regression outputs are given below.

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D-6 Regression Analysis: PugMixT versus SSFMT The regression equation is PugMixT = 28 + 0.943 SSFMT (3) 12 cases used, 1 cases contain missing values Predictor Coef SE Coef T P Constant 28.4 128.0 0.22 0.829 SSFMT 0.9426 0.4096 2.30 0.044 S = 27.3356 R-Sq = 34.6% R-Sq(adj) = 28.1% Analysis of Variance Source DF SS MS F P Regression 1 3958.6 3958.6 5.30 0.044 Residual Error 10 7472.3 747.2 Total 11 11430.9 Unusual Observations Obs SSFMT PugMixT Fit SE Fit Residual St Resid 3 289 250.00 300.81 12.26 -50.81 -2.08R R denotes an observation with a large standardized residual.

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D-7 Regression Analysis: PugMixT versus PAMT The regression equation is PugMixT = - 86 + 1.27 PAMT (4) 12 cases used, 1 cases contain missing values Predictor Coef SE Coef T P Constant -86.2 152.2 -0.57 0.584 PAMT 1.2745 0.4743 2.69 0.023 S = 25.7628 R-Sq = 41.9% R-Sq(adj) = 36.1% Analysis of Variance Source DF SS MS F P Regression 1 4793.7 4793.7 7.22 0.023 Residual Error 10 6637.2 663.7 Total 11 11430.9 Unusual Observations Obs PAMT PugMixT Fit SE Fit Residual St Resid 3 305 250.00 302.55 10.48 -52.55 -2.23R R denotes an observation with a large standardized residual. F10,10 663.7/747.2= 0.888; F0.60,10,10 = 0.848, showing that the F0- statistic = 0.888 is not significant at the 80% level. The P-value 2*0.4275 = 0.855, and 0.4275 = Pr(F10,10 0.888). Correlations: SSFMT, PAMT, PugMixT SSFMT PAMT PAMT 0.810 0.001 PugMixT 0.588 0.648 0.044 0.023 Cell Contents: Pearson correlation P-Value It should be noted that because of positive correlations amongst SSFMT, PAMT, and PugMixT, the MSRESIDUALS from models (3) and (4), in all likelihood, would

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D-8 tend to be closer to each other than the required case of independence if the two variables SSFMT and PAMT were independent. Hence, the F-test at best is very conservative (larger P-value ) than if an exact test could be performed. The Minitab output for Equivalent Workability Temperature (EWT) is given below. Regression Analysis: EWT versus SSFmid The regression equation is EWT = - 27 + 1.13 SSFmid Predictor Coef SE Coef T P Constant -27.3 152.4 -0.18 0.861 SSFmid 1.1276 0.5119 2.20 0.050 S = 34.0257 R-Sq = 30.6% R-Sq(adj) = 24.3% Analysis of Variance Source DF SS MS F P Regression 1 5617 5617 4.85 0.050 Residual Error 11 12735 1158 Total 12 18352

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D-9 Regression Analysis: EWT versus PAMid The regression equation is EWT = - 241 + 1.78 PAMid Predictor Coef SE Coef T P Constant -241.3 186.0 -1.30 0.221 PAMid 1.7849 0.6041 2.95 0.013 S = 30.4978 R-Sq = 44.3% R-Sq(adj) = 39.2% Analysis of Variance Source DF SS MS F P Regression 1 8121.0 8121.0 8.73 0.013 Residual Error 11 10231.3 930.1 Total 12 18352.3 Correlations: SSFmid, PAMid, EWT SSFmid PAMid PAMid 0.830 0.000 EWT 0.553 0.665 0.050 0.013 Cell Contents: Pearson correlation P-Value F11,11 930.1/1157.73 = 0.803; F0.65,11,11 = 0.788, showing that the F0- statistic = 0.803 is not significant at the 70% level. The P-value 2*0.361 = 0.723, and 0.361= Pr(F11,11 0.803). The Minitab output for Compaction Temperature is given below.

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D-10 Regression Analysis: CompT versus SSFCompT The regression equation is CompT = - 96.6 + 1.37 SSFCompT 11 cases used, 2 cases contain missing values Predictor Coef SE Coef T P Constant -96.62 89.06 -1.08 0.306 SSFCompT 1.3745 0.3116 4.41 0.002 S = 18.7292 R-Sq = 68.4% R-Sq(adj) = 64.9% Analysis of Variance Source DF SS MS F P Regression 1 6823.5 6823.5 19.45 0.002 Residual Error 9 3157.0 350.8 Total 10 9980.5 Unusual Observations Obs SSFCompT CompT Fit SE Fit Residual St Resid 1 295 272.00 308.86 6.42 -36.86 -2.10R 11 275 317.00 281.37 6.48 35.63 2.03R R denotes an observation with a large standardized residual. Regression Analysis: CompT versus PACompT The regression equation is CompT = - 320 + 2.07 PACompT 11 cases used, 2 cases contain missing values Predictor Coef SE Coef T P Constant -320.3 119.1 -2.69 0.025 PACompT 2.0724 0.4007 5.17 0.001 S = 16.7084 R-Sq = 74.8% R-Sq(adj) = 72.0% Analysis of Variance Source DF SS MS F P Regression 1 7468.0 7468.0 26.75 0.001 Residual Error 9 2512.5 279.2 Total 10 9980.5

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D-11 Correlations: SSFCompT, PACompT, CompT SSFCompT PACompT PACompT 0.850 0.000 CompT 0.827 0.865 0.002 0.001 Cell Contents: Pearson correlation P-Value F9,9 270.2 /350.8= 0.7702; F0.65,9,9 = 0.7676, showing that the F0-statistic = 0.7702 is not significant at the 70% level. The P-value 2*0.3518 = 0.7036, and 0.3518= Pr(F9,9 0.7702). The Minitab output for the Producers' Recommended Mixing Temperature is given below. Regression Analysis: Prod.Midpoint versus SSFMT The regression equation is Prod.Midpoint = 136 + 0.572 SSFMT Predictor Coef SE Coef T P Constant 136.00 35.07 3.88 0.003 SSFMT 0.5724 0.1128 5.08 0.000 S = 7.79965 R-Sq = 70.1% R-Sq(adj) = 67.4% Analysis of Variance Source DF SS MS F P Regression 1 1567.6 1567.6 25.77 0.000 Residual Error 11 669.2 60.8 Total 12 2236.8

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D-12 Regression Analysis: Prod.Midpoint versus PAMixT1 The regression equation is Prod.Midpoint = 116 + 0.627 PAMixT1 12 cases used, 1 cases contain missing values Predictor Coef SE Coef T P Constant 115.73 53.56 2.16 0.056 PAMixT1 0.6269 0.1681 3.73 0.004 S = 8.81614 R-Sq = 58.2% R-Sq(adj) = 54.0% Analysis of Variance Source DF SS MS F P Regression 1 1081.0 1081.0 13.91 0.004 Residual Error 10 777.2 77.7 Total 11 1858.2 Correlations: SSFMT, PAMixT1, Prod.Midpoint SSFMT PAMixT1 PAMixT1 0.954 0.000 Prod.Midpoint 0.837 0.763 0.000 0.004 Cell Contents: Pearson correlation P-Value F10,11 77.7/60.8 = 1.278; F0.70,9,9 = 1.3846, showing that the F0-statistic = 1.278 is not significant at the 60% level. The P-value 2*0.3454 = 0.6907, and 0.3454= Pr(F10,11 >1.278). In summary, none of the above tests for equality of two residual variances were significant even at the 50% level (the minimum 2-sided P-value =0.676). Because all the tests were very conservative (due to positive correlation), even if we do not multiply the Pr. Level of the 2-sided tests by 2, the smallest P-value would be 0.338, which is not significant at an = 0.25.