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CSHRP/NEW BRUNSWICK BAYESIAN APPLICATION 4.1 ITERATION 1 4.1.1 Definition of Model Individual models of rutting are to be
developed for each of the three rehabilitation methods
used by NBDOT to place asphalt concrete overlays on
asphalt concrete pavements. The models represent rutting
performance for a: (1) thin overlay; (2) thick overlay
with milling; (3) thick overlay with padding. 4.1.2 Dependent Variable The dependent variable (Yi ) is the
depth of rutting measured in millimeters as per
SHRPP333 spec (1.2 M straight edge) 4.1.3 Independent Variables The independent variables selected for
the three models were based on variables which were
included in the CLTPP rutting model regression
equation. All of these variables were for the top lift of
a virgin AC overlay. The variables selected are: % Air
Voids; % Retained on 4.75 mm sieve; % Crushed particles;
Age in years; Traffic Log10 annual lane; and Thickness of
overlay in millimeters. 4.1.4 Model Type and Functional
Form The model type is the empirical model
type with a curvilinear functional form (Log Traffic): 4.1.5 Model Inputs 4.1.5.1 Actual Data "Data" The database was compiled by first
gathering tender documents and specifications for paving
contracts on arterial highways from the Design Branch.
Once a list of contracts was compiled, the list was
sorted to identify the three types of rehabilitation for
which the models were to be developed. The paving
summaries for each of these contracts were then obtained
from the Construction Branch. These summaries provided
the exact stationing of the contract, the rate of asphalt
application, the percent air voids in the mix and the
percent passing the 4.75mm sieve in the mix. Rutting data was then gathered from the
Planning Branch. The rut data is collected by the mobile
data recorder (MDR) and recorded at 50m intervals for
each control section for each arterial highway in the
Province. Therefore, knowing the exact contract
stationing an average rut over the contract limits was
obtained. Traffic data was obtained from the
Maintenance and Traffic Branch. Intersection counts
obtained from specific traffic studies were collected as
close to the paving contract stationing as possible.
These intersection counts gave the actual truck traffic
AADTT and also classified the trucks according to the
NBDOT Classification which is referenced to the FHWA
Classification. This gave an accurate measurement of the
truck distribution for every paving contract. This
information was entered into the NBDOT ESAL Forecaster
program to obtain the ESAL value. The ESAL value obtained
from the ESAL Forecaster program is based on actual truck
distributions and truck factors defined to be sensitive
to four specific types of hauling use the highway would
receive in different sections of the Province. Once all the data was collected into a
spreadsheet (see Appendix B) for each model, only those
contracts with complete sets of independent variables and
corresponding dependent variable observations were
included. Any outlying data was removed after evaluating. 4.1.5.2 Encoding Expert Judgement Data "Prior" Nine experts from the NBDOT were
encoded. These experts were gathered together for their
input on appropriate limits for each independent
variable. After meeting with the experts, an encoding
package similar to the CLTPP encoding package was
prepared for each of the three different models. Each of
the nine experts was encoded on a 48 cell matrix for each
model. After the experts were encoded and their data
analyzed, there was a group review of the encoded
information to see if results actually reflected the
experts opinions. Each expert's ability to predict
rutting was compared (See Appendix C). The purpose of
this exercise was to select one representative expert out
of the nine (9) for expedient performance of this
project. A classical regression was run on each expert's
judgement database for each rehabilitation strategy to
develop predictive models. Then using actual data from a
corresponding database a rut value was predicted and
compared to the corresponding actual rut data. From this
type of analysis Fred MacFarlane was chosen initially as
the best representative for the "prior" models.
His models predicted more accurately than the other
experts and he was more consistent in defending his
selections made during the encoding. 4.1.6 Analysis of Data "Posterior" During the initial analysis, results
were confusing and it was discovered that for each of the
models, the thickness variable was acting like a
constant. This resulted from the experts being encoded on
a specific thickness to designate thick ( 125mm thick
overlay) and thin (35mm thick overlay) overlays . As a
result, a diffuse prior on thickness was used on each
"prior" model. This involved removing thickness
from the expert judgement and running a classical
regression. The resultant vector of means and the
variance covariance matrix then had to be modified to
reflect a perfectly diffuse prior estimate with respect
to thickness. The degrees of freedom and the residual
variance did not need to be modified. Because it was
desirable to maintain thickness in the final model the
thickness variable had to be reentered back into the
analysis after the classical regression had been
performed on the expert judgement data. As can be seen
from the results in Appendix "D" the thickness
variable was reentered in the vector of means table as
a value of zero . It was also reentered in the variance covariance table as a
covariance value of zero and a variance value of 10. The
Thickness variable remained in the actual database for
each model. Also included in each model, at this stage,
was zero distress data (data at age 0). 4.1.7 Model Runs The data base for the thin overlay was
the largest. Therefore, it was decided to concentrate on
analyzing this model initially and proceed through its
development (see Appendix D for Bayesian Analysis). This
allowed working through any unforeseen potential problems
before proceeding to the thick overlay with padding and
the thick overlay with milling models in the second
iteration. 4.1.7.1 First Model Run After running the analysis on the
original model the results were tabulated in the
evaluation form on the following page. The
"posterior" results indicate that the model has
a high intercept Bo =10.13 and the standard error Se
=3.577 is a little high ,however all the variables except
thickness are statistically significant if Tvalue >
1.96 (see section 4.1.9). For this model it is
recommended that Bo be in the 12 range and Se =
approximately 2.0. 4.1.7.2 Second Run Because the poor Tvalue for the
thickness variables indicate that the variables are
statistically insignificant a second run was made with
the thickness variable removed from the expert judgment
database and the actual database to see how the resultant
model predicted. The results, see Appendix "D",
indicate that removing the thickness variable did not
cause any drastic changes to the prediction power of the
model. 4.1.7.3 Third Run A third run was performed with the
variable %retained removed and thickness put back in the
model. This was done to evaluate the effect of removing a
definitive variable. As can be seen in Appendix "D" the
model was significantly affected indicating that this
model was sensitive to this operation. 4.1.7.4 Fourth Run The final run of this first iteration
was to return to the first results, with all variables in
place. The Joint CSHRP/ Agency Bayesian Application
Midcourse workshop held in Ottawa on May 79, 1995
provided insight needed on the first iteration results to
proceed to the next iteration. 4.1.8 Sensitivity Analysis The same sensitivity analysis was
carried out for all three models, however for brevity,
only the thin model analysis will be explained in detail. 4.1.8.1 Building Predictive Cases To build predictive data in a
spreadsheet 80% confidence limits were found for each
variable in the model databases. Once the 80% confidence
limits were found, a prediction table was constructed for
each model to determine the sensitivity of the models
with respect to predictions to rutting. 4.1.8.1.1 Thin Model The table and sensitivity line plot for the thin model can be seen on the following pages. Using the upper and lower confidence limits and the average value for each independent variable the table was constructed by varying one variable at a time, while holding the other five variables constant. Each row of the data was then multiplied by the model regression coefficients for the Prior, Posterior and Data shown in the columns to the right of the prediction table. These multiplications of the rows of data by the columns of the coefficients gave the lower, average and upper predictions for each independent variable for the Prior, Posterior and the Data. Table: Sensitivity Analysis for Thin Overlay Figure: Prediction
Graph for Thin Overlay This type of sensitivity analysis is
useful to determine which variables have the greatest
contribution (i.e. Dominant variables) and to identify
where your prior, posterior and classical models agree
and disagree. This graph shows that for the thin model
the variable age seems to be the dominant factor while
the variable thickness appears to have no affect on the
model. The prior, posterior and data models all agree on
the slope of the variables age, % crushed and thickness.
The trends indicated are; as the age of the overlay
increases, rut depth increases, and as the asphalt mix
property % crushed increases, the rut depth decreases and
that the variable thickness has no effect on the model.
The models disagree on the slope of the variables for %
air voids and % retained and on the variable traffic. The
"data" shows that as the asphalt mix properties
% air voids and % retained increase the rut depth will
increase, while the "prior" shows that as these
variables increase the rut depth will decrease. The
"prior" and "data" do not agree on
the effect the traffic variable has on the model. The
"data" shows traffic as having no effect on the
model while the experts believe that as the traffic on
the overlay increases rut depth will increase. The
"data" acts like it is in disagreement at this
time because it is early in its development . Pavements
generally change so slowly that nothing substantive can
be learned for many years. The expert judgement,
"prior" data base helps build on the
"data" database to give posterior results
showing what is known to be true, i.e. as the asphalt mix
properties % air voids and % retained increase the rut
depth will decrease. A tornado plot, shown on the following page, was also constructed for the models to illustrate the sensitivity of rutting to variations in the variables. As in the above prediction table and graph, a table was constructed using upper and lower 80% confidence limits and the average value of each independent variable. The upper confidence limit was the maximum likely value while the lower confidence limit was the minimum likely value. Each of these independent variables was set at their average value with each parameter then varied from their minimum likely to their maximum likely values. The tornado plot can be used in conjunction with the line plot but should not be used alone since it can not show the sign or magnitude of the change in the predictive estimates. Figure: Tornado Plot for Thin Overlay The height of the bars illustrates the
sensitivity of the dependent variable rutting to the
variations in each of the independent variables. This
graph confirms the findings of the prediction graph by
showing that the independent variable age is the most
dominant while the variable thickness has no influence on
the prediction of the model. 4.1.8.1.2 Thick
Overlay With Milling Model The 80% Confidence Interval Prediction
Table and graph as well as a Tornado Plot were also
developed for the Thick Overlay with Milling Model. The
results for this model are on the following three pages. The prediction graph for the Thick
Overlay with Milling model shows the same trends as the
Thin model did previously. The graph shows that the
dominant factor is the variable age, while the variable
thickness seemed to have no effect on the prediction of
the model. The variables % air voids and % retained also
disagree with the "prior" in this model. Figure: Prediction Graph for Milling The Tornado plot on page 19 shows that
the variable age is also the dominant factor in the
Milling Model while the variable thickness has no
influence in the prediction of the model. 4.1.8.1.3 Thick Overlay with Padding Model The prediction graph for the Thick
Overlay with Padding Model shows that the dominant factor
is the variable age while the variable thickness seems to
have no effect on the prediction of the model. From the
plot prediction graph it can be seen that the
"prior", "posterior" and
"data" agree on the slopes of the variables age
% retained, % air voids and thickness. They show that as
the mix properties % air voids and % retained increase,
the rut depth will decrease and as the age of the overlay
increases the rut depth will increase . The variables %
crushed and traffic are in disagreement in this model. Figure: Prediction Graph for Padding Figure: Tornado Plot for Padding The Tornado plot again shows that the
variable age is the dominant factor in the Thick with
Padding model, while the variable thickness has no
influence in the prediction of the model. 4.1.8.2 Sensitivity of Input
Assumptions A series of analysis was performed to
determine the sensitivity of the "posterior"
model to various input parameters for the thin model.
This type of sensitivity analysis is useful to identify
critical and noncritical input parameters to the
Bayesian analysis. 4.1.8.2.1 Thin Overlay Model One of the first analysis that was
performed was to determine the sensitivity of the
posterior model to variations to the Degrees of Freedom
(DOF) of the "prior". A table was constructed with a summary of the regression coefficients and the tstatistic values for the "prior" and "data" models. The degrees of freedom on the prior model was then changed and the resultant "posterior " coefficients and tstatistic values were displayed. A graph of the prior degrees of freedom estimate versus the resultant "posterior" standard error was then plotted. The table and graph showing this is on the following page (page 25). The graph plots as a horizontal line indicating that the DOF is not dominating the model something else is. Assumptions with regards to all the
mean coefficients (prior) were also tested for the thin
overlay model (see appendix E) to watch their effect on
the "posterior" coefficients. If the line was
steep it meant that that variables input was important to
the model and should be monitored carefully. All the
variables, with the exception of the variable thickness,
had close to a one to one slope. 4.1.8.2.2 Thick Overlay with
Milling When the sensitivity of the "posterior" standard error to the "prior" degrees of freedom was plotted for the Thick Overlay with Milling model it also plotted as a horizontal line (see page 26), indicating that the error in the milling model was not sensitive to the prior degrees of freedom. Something else is dominating the model. Table: Sensitivity Analysis for Milling 4.1.8.2.3 Thick Overlay with
Padding The Thick Overlay with Padding model, when tested for the sensitivity of the "posterior" standard error to the "prior" degrees of freedom, also plotted as a horizontal line (see page 27). The error in this model is also not sensitive to the prior degrees of freedom, something else is dominating the model. Table: Sensitivity Analysis for Padding 4.1.9. Inference from Analysis of Iteration 1 Evaluation tables on the following
three pages show a summary of the results of the first
iteration for the thin, thick with milling and thick with
padding models. Each of the variable coefficients were
reviewed to determine if their sign was rational, their
magnitude was rational, if it was statistically
significant to the equation, and if the
"posterior" reflected the "prior" or
the "data" . Each variable in the thin, thick with
milling and thick with padding models were tested for
their contribution to the prediction of rutting using a
tstatistic. A twotailed test was carried out assuming
a normal distribution with a 95 % confidence level. The
null hypothesis, Ho, was set equal to zero (Ho=O). The
TValue for each variable had to be greater than 1.96 (
The critical TValue associated with 95 % confidence for
a normal distribution) to reject the null hypothesis.
Rejection of the null hypothesis means that one can
assume with 95 % confidence that the calculated mean
coefficient value associated with that variable is as
calculated. From this analysis, it was found that
for each of the thin, thick with milling and thick with
padding models, all the variables except thickness were
found to be statistically significant. Therefore, instead
of modifying the prior to reflect a perfectly diffuse
prior estimate with respect to thickness (as previously
performed) the thickness variable was removed from each
model. There was also concern that the expert
judgement was over powering the small databases for the
thick overlays and the results from combining the expert
judgement and the actual data would not yield reasonable
results for this exercise. The narrow band of the data
for each of the variables was also a concern. The result
was to combine the two thick databases and develop one
model for thick overlays and one model for thin overlays.
ANALYSIS AND INTERPRETATION OF RESULTS THIN MODEL The following evaluation table shows the results of the first iteration for the thin model using one representative expert. The model has a high intercept Bo =
10.13 and the standard error Se = 3.577 is a little high
but all variables show strong T values except for
thickness variable. THICK OVERLAY WITH MILLING The evaluation table shows the results from iteration one using one representative expert. The milling model looks good with a
reasonable intercept value Bo = 2.6983 and a low standard
error value of Se = 2.7577 . As with the thin model, the
milling model has strong T values except for thickness
variable. THICK OVERLAY WITH PADDING The evaluation table shows the results from iteration one using one representative expert. The padding model has a very low intercept Bo = .0631 and the standard error value Se = 3.772 is a little high. Again as in the thin and milling models, the padding model has strong T values except for thickness variable. |
