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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.
