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Federal Highway Administration Research and Technology
Coordinating, Developing, and Delivering Highway Transportation Innovations
REPORT |
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Publication Number: FHWA-HRT-12-030 Date: August 2012 |
Publication Number: FHWA-HRT-12-030 Date: August 2012 |
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Parameter |
Degrees of Freedom (DF) |
Estimate |
Standard Error |
t-Value |
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 |
t-Value |
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 |
t-Value |
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 |
t-Value |
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 p-value 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 four-variable 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 four-parameter 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 three-variable 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 28-day 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.