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Page Predicting the Compressive JOINT
CSHRP/NEWFOUNDLAND BAYESIAN APPLICATION
Model 2 Model Definition The Model Definition from Model 1
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: Although super plasticizer dosage was
identified as statistically insignificant in Model 1, it
is included as it is thought to have a more vital
function in silica fume concrete. Model Type and Functional Form The model is empirical in nature. The
functional form of the new model is: Model Inputs Appendix 5 and Excel File RUN2INPT.XLS contain
the new model's worksheets NEWDATA2, OLDDATA2 and PRIOR2.
NEWDATA2 contains the same files as worksheet NEWDATA in
Model 1. It consists of mix design and compression
testing data from two bridge projects: Main Brook Bridge
and Holyrood Pond Bridge. The worksheet contains 50
observations. OLDDATA2 contains data collected from
laboratory concrete mix designs performed on silica fume
concrete only. It includes additional observations made
since the development of Model 1 and now contains 26
observations. The classical regression analysis
option in XLBAYES was run on OLDDATA2 to calculate
GPrior data. Estimates of the model coefficients,
degrees of freedom and residual variances were
calculated. The output is displayed in worksheet PRIOR2
of Excel File RUN2INPUT.XLS. Analysis Bayesian regression was run in HAYES
using the GPrior option. The sample data is contained
in worksheet NEWDATA2 while PRIOR2 contains the
"old" data and regression coefficients. The
GPrior was weighted by setting the GFactor to 1. 5.
Setting the GFactor greater than 1 intensifies the
influence of the Prior (old database). More emphasis is
placed on the old data versus the new data because the
former is measured under more controlled conditions. The Posterior model for predicting Compressive Strength was found to be:
Evaluation Table 2 and the following sections compare the Prior, Data and Posterior coefficients and their statistics. Analysis is found in Appendix 6 and is stored in the Excel file XLRUN2A.XLS.
Water Cement Ratio The coefficients for Prior, Data and
Posterior have the correct signs. The magnitude of the
Posterior coefficient lies between that of the Data and
the Prior. This Posterior's coefficient is statistically
significant, t1 < t 025 and therefore contributes information to the
prediction of compressive strength. Meanwhile, the
coefficient for the Prior is not significant. The normal probability distribution
plot shows that the Posterior lies between the Data and
Prior. Although the Data refutes the Prior, confidence
has increased, making the certainty of the Posterior
increase. This is a favourable case. Air Content The coefficient has a rational sign. As
t2 is less than t.025 = 1.996, the coefficient is
statistically significant: the coefficient b2 differs
from zero. As the plot shows, the Data is definitive. The Posterior reflects the Data. The Posterior and Data means are almost equal (2.23 vs. 2.57). The only effect of the Prior has been to decrease the uncertainty of the Posterior compared with the Data. Note in Evaluation Table 2 that the tstatistic for the Prior is low.
Fine to Total Aggregate Ratio The signs for the Prior and Posterior
coefficients are incorrect again. Furthermore, the
tstatistic suggests that the Posterior coefficient is
not statistically significant. The Prior data now
contains several more fine to total aggregate ratios. Its
coefficient is significant. Evaluation Table 2 and the probability
plot indicate that the Posterior is most influenced by
the Prior. There is a possibility that either the Data or
Prior information is incorrect. The amount of fine and
coarse aggregate proportions used in the construction of
the girders ("new" data), may not be as
specified. The laboratory data ("old" data) may
be affected by another variable that is not measured. The probability distribution plots indicate that the Posterior falls in the interval between the Prior and Data. The Data pulls the Posterior away and increases the uncertainty.
Super Plasticizer The coefficient for the super
plasticizer dosage variable has a rational sign and is
statistically significant. Note that the value of the
coefficient is greater than the values for the Prior and
Data regressions which are 0.33 and 1. 11, respectively. The PDF plot shows the Posterior coefficient's position lying outside both the Prior and Data. This may have happened because as the regression tried to move one variable to fit the model, it moved this variable out of range. It may suggest a poor model. Both the Prior and Data coefficients were statistically insignificant and may equal zero.
Constant The magnitude of the coefficient appears reasonable. Even the Prior has a more rational magnitude than in Model 1. The ttest suggests that the coefficient is significant.
Model Statistics and Comparison of
Models Table 2 shows that the coefficients of
determination for the Prior and Data are 0.57 and 0.52,
respectively. Referring to Model 1, it can be seen that R2
for the Prior regression has decreased from 0.86 to 0.57.
Model 1 can account for more variability in compressive
strength predictions because it has more data points than
Model 2 (98 vs. 26). The "new" data has not
changed so the value of R2 remains the same. The standard error for the Prior
regression is lower in Model 2 than in Model 1 (2.72 vs.
3.60) because Model 2 deals only with high compressive
strength silica fume concrete. Model l also dealt with
normal portland cement and therefore had a greater range
in compressive strengths. Standard error for the Posterior model
has increased slightly from 5.12 to 5.64 between the
models. Base case, high and low predictions for Models 1
and 2 are almost identical. Sensitivity Analysis The Prior, Data and Posterior models
were compared. See Appendix 7 and Excel file SENSTVY2.XLS For
the base case, the Prior and Data regressions predict
nearly the same compressive strength (72.44 MPa vs. 72.34
MPa). The Posterior predicts a lower strength of 69.69
MPa. Most of the variables contribute
equally to the models except the super plasticizer
dosage. Increasing the dosage does not change the
predicted strength much. There is conflict again about the effect of the fine to total aggregate ratio. The Data regression predicts a decrease in compressive strength with an increase in the ratio whereas the other two regression curves still predict otherwise.
Recommendations for Model 2 Models 1 and 2 were presented at the
MidCourse Workshop for Joint CSHRP/Agency Bayesian
Applications, held in Ottawa on May 78, 1995. Along
with these models, a third model was presented that would
predict the compressive strength based on 7day
strengths. After the workshop, it was decided to drop
this model as it had a high correlation with most of the
variables. Model 2 is a more desirable model than
Model 1 though the standard error is slightly larger.
More of Model 2's coefficients are statistically
significant. The coefficient for silica fume content in
Model 1 was definitely too low. However, the coefficient for fine to
total aggregate ratio for Model 2 still does not have the
correct sign. Literature suggests that the coefficient
should have a negative value. It was recommended that an expert
judgement prior should be used in the modeling. By asking
the experts for their predictions of compressive
strength, their results can be encoded to form a
"true" GPrior and another analysis could be
performed. If the experts agree with convention about the fine to total aggregate ratio, both the Prior and Posterior coefficients may finally be negative.
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