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Federal Highway Administration Research and Technology
Coordinating, Developing, and Delivering Highway Transportation Innovations
REPORT 
This report is an archived publication and may contain dated technical, contact, and link information 

Publication Number: FHWAHRT12030 Date: August 2012 
Publication Number:
FHWAHRT12030
Date: August 2012 
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Parameter 
Degrees of Freedom (DF) 
Estimate 
Standard Error 
tValue 
P_{r} > t 
VIF 
Intercept 
1 
9,907.383 
2,732.919 
3.63 
0.0023 
0 
w/c 
1 
4,893.05 
2,532.455 
1.93 
0.0712 
3.01113 
Cementitious content 
1 
3.30331 
1.56188 
2.11 
0.0505 
3.76626 
Coarse_Aggregate_ Mix_Design 
1 
1.67238 
0.61169 
2.73 
0.0147 
1.38486 
Fine_Aggregate_Mix_Design 
1 
1.51914 
0.78059 
1.95 
0.0694 
1.79848 
Note: Italicized text indicates that the parameter and statistic do not satisfy the criteria adopted for model development.
The model statistics for table 17 are as follows:
Parameter 
DF 
Estimate 
Standard Error 
tValue 
P_{r} > t 
VIF 
Intercept 
1 
10,789 
2,181.11 
4.95 
<0 .0001 
0 
w/c 
1 
2,050.86 
2,200.846 
0.93 
0.3607 
2.78251 
Cementitious content 
1 
3.57161 
1.36819 
2.61 
0.0153 
3.23079 
Coarse_Aggregate_ Mix_Design 
1 
2.34227 
0.51775 
4.52 
0.0001 
1.25735 
Fine_Aggregate_Mix_Design 
1 
2.35301 
0.64777 
3.63 
0.0013 
1.39035 
Note: Italicized text indicates that the parameter and statistic do not satisfy the criteria adopted for model development.
The model statistics for table 18 are as follows:
Parameter 
DF 
Estimate 
Standard Error 
tValue 
P_{r} > t 
VIF 
Intercept 
1 
9,381.832 
1,569.631 
5.98 
< 0.0001 
0 
Cementitious 
1 
4.57228 
0.84557 
5.41 
< 0.0001 
1.24054 
Coarse_Aggregate_Mix_Design 
1 
2.50707 
0.48533 
5.17 
< 0.0001 
1.11065 
Fine_Aggregate_Mix_ Design 
1 
2.23659 
0.63393 
3.53 
0.0016 
1.33863 
Note: Italicized text indicates that the parameter and statistic do not satisfy the criteria adopted for model development.
The model statistics for table 19 are as follows:
Parameter 
DF 
Estimate 
Standard Error 
tValue 
P_{r} > t 
VIF 
Intercept 
1 
4,897.511 
1,105.332 
4.43 
0.0002 
0 
Cementitious content 
1 
5.80657 
0.92386 
6.29 
< 0.0001 
1.02819 
Coarse_Aggregate_Mix_Design 
1 
2.0405 
0.56042 
3.64 
0.0012 
1.02819 
Note: Italicized text indicates that the parameter and statistic do not satisfy the criteria adopted for model development.
The model statistics for table 20 are as follows:
In establishing and optimizing a model, each variable selected has to be significant (p < 0.05) and not show an interaction effect with other variables (VIF > 5). However, the opposite is not true. It is not necessary that a variable with a pvalue less than 0.05 and VIF less than 5 be included in a model if it is not meaningful from an engineering standpoint or if it does not show promise based on a sensitivity analysis.
While this evaluation process can be performed in a systematic manner, it cannot be performed in a fully automated manner. Each parameter in each model needs to be assessed manually. Table 17 and table 18 show the regression statistics for the fourvariable model shown to produce the best correlation (R^{2}) in table 16. Note that the number of data points in the model is different in the two tables (N = 21 and N = 29 in table 17 and table 18, respectively). Table 17 shows the subset of data that was used in the C_{p} analysis, wherein 21 observations have data in all fields evaluated; however, 29 observations have data for the parameters selected for the model. R^{2} in table 17 matches that shown against the fourparameter model in table 16. However, the regressed coefficients and R^{2} in table 18 correspond to the variables selected for this model, and the contents of table 18 are the proper statistics to report for the model.
The results in table 18 indicate the following:
Removal of the w/c ratio parameter in the threevariable model results in regression statistics shown in table 19. Note that the coarse and fine aggregate contents show trends that counter engineering knowledge even though the parameters are significant to the model. The best two variable model, shown in table 20, also shares the same concern. Thus, the iterative process needs to evaluate several parameters and balance both statistical and engineering needs. Often, a trial and error method has to supplement the pure statistical approach. The model selection is not based solely on the best R^{2} value, either.
The final model selected for the estimation of 28day compressive strength is shown in table 21 and includes the w/c ratio and CMC as the regressors. All 42 observations have been included. The R^{2} value is 54.4 percent. Although it is compromised relative to the models discussed above, it provides a more meaningful model with a superior predictive ability. RMSE for the model is 871 psi. Table 22 provides details of the range of data used to develop the model. Figure 133 and figure 134 show the predicted versus measured values and the residuals plot for the model, respectively.