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Page Predicting the Compressive JOINT
CSHRP/NEWFOUNDLAND BAYESIAN APPLICATION
Model 3 Model Definition Definition of the model remains the
same. A model is required to predict the compressive
strength of highperformance silica fume concrete. Dependent Variable The dependent variable is the
compressive strength of silica fume concrete at 28 days. Independent Variables The selected independent variables are: When asked to encode the effect of
super plasticizer dosage on compressive strength, two of
the experts asked that the variable should be dropped.
Mr. Keith Foster and Mr. Colin Crane stated that the
variable is highly correlated to the water cement ratio.
To achieve a low water cement ratio, super plasticizer is
added to the mix. Review of the posterior correlation
matrix for Model 2 showed a correlation value of 0.495
between the coefficients for water cement ratio and super
plasticizer dosage. A correlation calculation in Excel of
all the data for Model 2 showed a correlation of 0.468
between the variables. The variable for super plasticizer
dosage should be suspect. Model Type and Functional Form The form of the model was examined and revised. Residual analysis was carried out on both the "new" and "old" data to decide if transforms were required. A plot of residual errors Ypredicted Yactual) was drawn for each variable to see if there was a pattern.
Examination of the residuals for air
content did suggest such a pattern. Figure 14 is a plot
of the residuals for the "old" data for air
content. For small values of x, the residuals were below
the 0 (mean) line. The residuals corresponding to the
middle values of x were above the 0 line and the
residuals for the largest value of x were again below the
0 line. Such a pattern usually indicates that curvature
should be added to the model. A residual plot for the
"new" data exhibited the same non random
pattern. Examination of the residuals for water
cement ratio or fine to total aggregate ratio did not
suggest a pattern. Several transforms were carried out until the following "curvilinear" model was selected:
Model Inputs Worksheet NEWDATA3 of Excel File RUN3INPT.XLS contains
the same sample files as worksheets NEWDATA and NEWDATA2.
It consists of mix designs and compressive strength
testing data from the Main Brook Bridge and Holyrood Pond
Bridge projects. The values for the variable AIRCONT have
been squared and the transformation has been saved as the
new variable AIRCONT2. An expert judgement Prior was set up.
An encoding form was developed for the three experts to
encode their estimations of compressive strengths.
Estimates were based on high and low values for the three
independent variables: water cement ratio, air content,
and fine to total aggregate ratio. See Appendix 8 for a
copy of the form. To help the experts in their
estimations, plots were made of each variable versus
compressive strength. Data was obtained from the 50
observations of "new" data and 26 observations
of "old" data. Although this additional
information could bias the experts' responses, two of the
experts had limited experience with silica fume concrete
and requested the assistance. Classical regression was carried out on
each estimate. The results are found in the Excel File EXPERTS.XLS and in
Appendix 9. A "True GPrior" was used in
the analysis. The means of the regression coefficients
and the residual variances, Se^2 were determined from the
classical regressions of each expert's predictions. The
Degrees of Freedom were assumed to equal the number of
observations from the "old" data minus the
number of independent variables minus one for the
constant. (26 3 1 = 22) These inputs are found in
Appendix 10 and stored in RUN3INPT.XLS
as worksheets KFPRIOR, CCPRIOR
AND DCFPRIOR Variance/covariance data was obtained
from the independent xdata of worksheet OLDDATA3.
OLDDATA3 contains 26 observations collected from
laboratory mix designs of silica fume concrete. The
values for AIRCONT have been transformed and the variable
renamed as AIRCONT2. Worksheet OLDDATA3 is also found in
Excel file RUN3INPT.XLS. Analysis Bayesian analysis was run in XLBAYES
using the GPrior option. The sample data is contained
in worksheet NEWDATA3. GPrior data is stored in
worksheet OLDDATA3. The regression coefficients, degrees
of freedom and residual variances are saved in the
following worksheets: KFPRIOR Regression analysis of
expert Keith Foster CCPRIOR Regression analysis of
expert Colin Crane DCFPRIOR Regression analysis of
expert Dennis Coffin A GFactor of 1.5 was applied to
intensify the influence of the GPrior data in the
analysis. Greater emphasis was placed on the
"old" data because it was measured under more
controlled conditions. The analyses for the three experts
are found in XLKEITH.XLS, XLCOLIN.XLS and XLDENNIS.XLS
and in Appendices 11 to 13. Table 3 compares the coefficients for
the Prior, Posterior and Data regressions. Coefficients
with the wrong sign are boldfaced and underlined. For the
Prior analysis, Keith Foster and Colin Crane assigned a
positive value to the fine to total aggregate ratio
whereas Dennis Coffin and the Data suggested otherwise. When asked for an explanation, Keith
and Colin mentioned that the examination of laboratory
mix designs suggested an increase in compressive
strengths. Dennis has had little experience with
different ratios and decided to go with common practice. After the XLBAYES program was run, only the sign of Keith Foster's coefficient remains positive. In Colin Crane's case, the Data was more definitive and had changed the sign of his coefficient. Notice the very high coefficient for the Data regression (144.86). Table 3: Comparison of Coefficients From Experts
Table 4 compares tstatistics for each coefficient. For the Prior regression, the fine to total aggregate ratio was found statistically insignificant for each expert. The three experts had put more emphasis on the other two variables in predicting strengths. After running XLBAYES, only the Posterior coefficient for Keith Foster was insignificant. Table 4: Comparison
of T-Statistics From Experts Sensitivity Analysis Two plots were generated comparing predictions ofthe experts. See Appendix 14 and Excel file SNSTVTY3.XLS for the results. The first plot shows the predictions using the Prior models, with the Data model as a baseline. Figure 15:
Sensitivity of Prior Predictions Across Experts It can be seen that the compressive
strength is most affected by the water cement ratio while
there is little change among the experts for the fine to
total aggregate ratio. Experts Colin Crane and Dennis
Coffin predict the same compressive strengths for the
water cement ratio and air contents values, while Keith
Foster predicts lower strengths. Only the model proposed by Dennis
Coffin and the Data show a decrease in compressive
strength with an increase in the fine to total aggregate
ratio, yet his coefficient of31.25 is much lower than
the classical regression's coefficient of144.86. Another plot was generated to compare predictions across each variable for each expert's posterior. Figure 16:
Sensitivity of Posterior Predictions Across Experts Colin Crane and Dennis Dennis agree
with the Data and each other on the magnitudes of
predicted compressive strengths and on the sensitivity of
the variables. Keith Foster predicts lower strengths for
each variable and an increase in strength for an increase
in fine aggregate ratio. Dennis Coffin's model is selected as
the best model for the prediction of compressive
strength. Coefficients for the Prior and Posterior have
the correct sign and all coefficients except the fine to
total aggregate ratio for the Prior are statistically
significant. The following pages examine the coefficients
for Dennis Coffin's' model. Water Cement Ratio: Expert Dennis
Coffin The coefficients for Prior, Data and Posterior coefficients have the correct signs. Furthermore, all three coefficients are statistically significant. According to the plot, the Posterior reflects the Data. The only effect of the Prior has been to decrease the uncertainty.
Air Content2 The coefficients for Prior, Data, and Posterior are almost the same and have the correct signs. All three coefficients are statistically significant. The expert judgement confirms the Data, but the Data is more definitive and pulls the Posterior slightly away from the Prior.
Fine to Total Aggregate Ratio:
Expert Dennis Coffin Finally, all three coefficients have
negative signs though the Prior coefficient may equal 0
(t = 1.34). The Posterior falls between the Prior and
the Data and the uncertainty has decreased. This decrease
in uncertainty may be driven by another variable not
measured. Constant (intercept):For Expert
Dennis Coffin The coefficients have a rational sign but their values are high. The Posterior interval lies inside the Prior and Data intervals and has a higher mean. All coefficients are statistically significant.
Recommendations for Model 3 The model based on Dennis Coffin's
expert judgement is the best model to date. The
coefficients have the correct sign and most are
statistically significant. The standard error for the
Posterior is lower than that for Model 2 (5.60 vs. 5.64).
The predictive capacity of the model is
still only +/ 11 MPa for a 95% confidence interval.
Additional variables, more elaborate transforms or more
data are required if it is hoped that the model would
have a predictive capacity of +/ 5 MPa for similiar
concrete mixes having a Silica Fume content of 8 %. It is strongly recommended that further research should be conducted at the Central Laboratory to examine the relationship between the fine to total aggregate ratio and compressive strength. After additional data has been collected, the experts should be approached again and asked to encode their predictions based on the new information..
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