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This report describes an application of
the Canadian Strategic Highway Research Program's
(CSHRP) Bayesian statistical analysis methodology for
the development of a pavement deterioration model for
asphaltic concrete surfaces containing steel slag
aggregates. The application is the result of an agreement
between CSHRP and the Ministry of Transportation of
Ontario to evaluate and utilize CSHRP Bayesian
methodology using a technical problem of interest to the
Ministry. The asphaltic concrete mixes containing steel
slag have been used in Ontario on major highways since
the late '70s. In 1992, their use was discontinued
because of premature pavement deterioration. The purpose
of the model was to facilitate timely scheduling of
effective rehabilitation treatments for projects
containing steel slag mixes. The Bayesian model combines
information derived from field observations of 79
existing projects with information elicited from experts.
The resulting model predicts the pavement deterioration
in terms of a Distress Index which is a function of age,
asphalt content of the mix, and traffic volume. The
results indicate that CSHRP Bayesian statistical
analysis approach is useful in that (a) it provides an
independent review and endorsement of prediction models
by experts, (b) it can increase the application scope,
reliability and predictive power of the models, and (c)
it facilitates quantification of the influence and
contribution of field data and expert judgment in the
modelling process. TABLE OF CONTENTS 2.1 Project Team Members
4.0 PAVEMENT PERFORMANCE MODELLING
5.0 CONCLUSIONS AND RECOMMENDATIONS TABLES
FIGURES
This report is the result of an
agreement between the Canadian Strategic Highway Research
Program (CSHRP) and the Ministry of Transportation,
Ontario (MTO) to evaluate and utilize CSHRP Bayesian
statistical analysis methodology which was developed
under CSHRP auspices by VEMAX Management Inc. between
1989 and 1995. The agreement provided the MTO the
opportunity to address a technical problem of interest to
the Ministry under the guidance of VEMAX personnel. The CSHRP Bayesian methodology
enables one to express the judgement of experienced
individuals in a quantified format and combine it with
field data. The format used to express expert judgment is
a linear regression model. The same format is used to
represent information based on data and for the
combination of expert judgement and data. The utilization
of expert judgement is particularly useful when available
data is insufficient to develop reliable prediction
models, as is often the case in the area of pavement
performance prediction. The use of the Bayesian statistical
approach for pavement performance prediction is not new
[1,2]. What is new, and where the CSHRP Bayesian
methodology excels, is the availability of a detailed
stepbystep developmental procedure (a template) and
customized, taskoriented computer software. The objective of the joint CSHRP and
MTO Bayesian application project was twofold: 1. To develop a model for predicting
deterioration of asphaltic concrete surfaces containing
steel slag using Bayesian methodology. The purpose of the
model is to facilitate timely scheduling of effective
pavement rehabilitation treatments. 2. To assess the usefulness and
applicability of CSHRP Bayesian software for the MTO,
particularly in the area of pavement performance
modelling. The MTO started experimenting with
using steel slag aggregate in asphaltic concrete surface
courses on freeways and major highways in the late '70s.
At that time, steel slag, a byproduct of a local
steelmaking industry, was viewed as a good alternative to
highquality natural aggregates (such as trap rock)
needed to provide a durable skidresistant pavement
surface. In the mid '80s, surfaces containing
steel slag aggregates began showing signs of premature
deterioration, and some locations required rehabilitation
sooner than initially anticipated. In 1991, the MTO
initiated a comprehensive field performance evaluation
study of pavements with premium surface courses, dense
friction courses (DFC) and open friction courses (OFC),
containing steel slag or highquality natural aggregates
[3]. This study concluded that DFC and OFC containing
100% steel slag aggregates deteriorated faster than the
corresponding courses containing only natural aggregates.
The latter performed according to expectations. A
moratorium on the use of steel slag aggregate in
asphaltic concrete paving mixes was then put into effect.
A subsequent 1992 study [4], confirmed the results of the
previous 1991 study. In 1994, the MTO undertook a third
study which concentrated on the performance of pavements
with surface courses containing 100% steel slag
aggregates [5]. The objective of the study was to develop
a prioritized program for the rehabilitation of all
existing pavements with surface courses containing steel
slag aggregate. The study included (a) inventory of all
relevant paving projects, (b) their detailed pavement
performance evaluation, (c) establishment of a
computerized data base, (d) development of appropriate
pavement performance and prioritization models. When the study described herein was
initiated, it was envisaged that it would build directly
on the results of the 1994 study [5] and utilize its data
base without any major modifications. The tasks envisaged
were as follows: a) to enhance conventional pavement
performance regression models, developed during the third
study, by incorporating into these models expert
judgement using the CSHRP Bayesian statistical
methodology, and b) to quantify the benefits of the
Bayesian methodology by comparing the results achieved by
the conventional classical regression with those achieved
by Bayesian regression. However, a review of the previously
developed data base revealed that several variables
expected to significantly influence the performance of
steel slag mixes were not included in the data base (and
were, therefore, not part of the existing prediction
model). Consequently, it was decided to expand and
improve the data base, and to develop new, more
comprehensive and better conventional pavement
performance models before proceeding with the development
of Bayesian models. 2.1 Project Team Members The "Joint CSHRP/MTO Bayesian
Application Project" was carried out by a project
team consisting of: Isaac Afrani, Project Engineer,
Concrete, Concrete Section; Alison Bradbury, Senior
Pavement Design Engineer, Pavements and Foundations
Section; and Jerry Hajek, Senior Pavement Management
Engineer, Pavements and Foundations Section. Lyle Kajner,
VEMAX Management Inc., Saskatoon, Saskatchewan, provided
guidance on the application of Bayesian methodology and
software. Expert engineering judgement was
provided by: Guy Cautillo Senior Manager, Engineering Materials Office (20 years of experience) Tom Kazmierowski Manager, Pavements and Foundations Section ( 18 years of experience) Kai Tam Manager, Bituminous Section (15 years of experience) Rob Kohlberger Geotechnical Engineer, Central Region (10 years of experience) William Phang Program Manager, Pavement
Management Systems Limited (30 years of experience) All projects with 100% steel slag
aggregate in the surface course, constructed in southern
Ontario, were included in the 1994 study [4]. Altogether,
there were 79 projects with a dense friction course (DFC)
and only 15 projects with an open friction course (OFC).
DFC contains coarse aggregate with a maximum size of 16
mm and unwashed fine aggregate; its design thickness is
40 mm. OFC contains coarse aggregate with a maximum size
of 12 mm and washed fine aggregate; its design thickness
is 25 mm. Because of the significant physical
differences between DFC and OFC mixes, their different
field performance, and a relatively small number of OFC
projects, only the 79 DFC projects were used in this
study to develop the DFC deterioration model. The average
project length was 4.4 km, and the number of lanes of the
individual projects ranged from 4 to 16. 3.1 Pavement Performance Evaluation During the 1994 study, detailed
pavement performance evaluation of all projects was
carried out using procedures described Reference 6.
Briefly, the evaluation consisted of distress and
roughness assessments. These two components were combined
into a Pavement Condition Rating (PCR) [7], which
represents an overall measure of pavement serviceability. Distress evaluation involved the
identification and rating of 15 separate pavement surface
distresses, summarized in Table 1, in terms of their
severity and extent. Distress severity and extent were
both measured on a 5point scale as described in Table
1. Only extant projects were evaluated and included in
the study. There were six projects resurfaced prior to
the 1994 survey which were not included in the study. 3.2 Establishment of Data Base The objective of the data collection
effort was to obtain and store information on all
relevant variables in a computerized data base. The
selection of the variables was based on experience,
literature review, and consultation with experts. The
data collection effort was thorough and involved
extensive search of mix design documentation.
Nevertheless, not all variables which might influence
performance of mixes containing steel slag aggregate were
available or could be obtained. Missing variables
include, for example, the actual (as build) asphalt
cement content of the mix, the manufacturing process of
steel slag, and the chemical composition of steel slag
(such as the presence of free lime). The results of the field evaluation of
the 15 pavement distresses were also stored in the data
base, which included the following variables: 1. Variables describing physical features of the projects
2. Pavement performance variables
3. Traffic variables
4. Variables describing properties of the DFC mix
5. Environmental variables
4.0
PAVEMENT PERFORMANCE MODELLING 4.1 Selection of Dependent Variable What to Predict? The results of previous studies showed
that asphaltic concrete mixes containing steel slag
aggregate perform in a unique way. The two predominant
deterioration modes identified during the field surveys
were: a) Cracking, particularly map cracking
(defined as interconnecting cracks forming a series of
large polygons which resemble a map [6]), and b) Ravelling (characterized by
progressive loss of pavement materials, both coarse and
fine aggregates, which, in its most severe state, may
result in potholes of various sizes [6]). In general steel slag mixes do not fail
because of rutting or flushing (steel slag is harsh and
absorbs asphalt cement), or subgrade distortion, or
loadassociated cracking (steel slag mixes were used on
major highways with structurallyadequate pavements). An
example of typical performance of a 1 0year old
pavement surface containing steel slag is shown in Figure
1. The first signs of defects are usually
in the form of greyish secretions forming veinlike
marking on the pavement surface. These grey secretions,
attributed to the presence of free lime in the steel slag
aggregate, appear to be a precursor to development of map
cracking, and disappear as the cracking becomes fully
developed. Ravelling is associated not only with
cracking, making the cracks progressively wider, but it
also occurs independently. The selection of the dependent variable
took into account the following considerations. 1. The need for the variable to
describe not only a pavement condition or serviceability,
but also to assess the need for rehabilitation. For this
reason, the dependent variable was related to the two
predominant deterioration modes, ravelling and cracking,
in a way which: a) Placed more importance on the
severity of a distress rather than on its extent. For
example, pavements with slight ravelling occurring throughout
do not require a speedy rehabilitation treatment as
do pavements with very severe ravelling occurring intermittently.
Yet, a simple addition index for combining the effect of
severity and extent, based on numerical values given in
Table 1, yields the same index value of 5 in both cases
(e.g., 1+4 versus 4+1). b) Highlighted the importance of
ravening compared to cracking. Cracking, per se, does not
result in rough pavement or an urgent need for
rehabilitation, provided that it is not stepped or
ravelled. Also, in the past, ravelling has led to an
accelerated development of potholes during freezethaw
conditions. 2. The need for the variable to be
understandable and meaningful to experts in view of its
anticipated use during the solicitation of knowledge from
experts. Several alternative dependent variables were constructed and evaluated. The Distress Index (DI), defined by Eq. 1, was selected as the variable which best meets the requirements described above. DI depends only on the two predominant deterioration modes, ravelling and cracking.
where DI = Distress Index which can range from 0 to 180. The maximum contribution from ravelling is 60 and the maximum contribution from cracking is 120. Sr = Severity of ravelling measured on the 0 to 4 scale (Table 1) Er = Extent of ravellling measured on the 0 to 4 scale (Table 1) i = Cracking distress type identified in Table I Sci = Severity of cracking distress type i on the 0 to 4 scale (Table 1) ECi = Extent of cracking type i on the
0 to 4 scale (Table 1) 4.2 Selection of Independent
Variables All available independent variables
(variables describing mix properties, traffic, and
environment) were considered for inclusion in the
prediction model, with an understanding that the number
of independent variables should not exceed four, based on
previous experience [ 1]. For example, the inclusion of
four independent variables means that experts, during the
knowledge acquisition phase, would be required to
consider simultaneously the influence of all four
variables on the Distress Index. Other considerations for
the selection of independent variables included their
predictive power as judged by conventional regression
analysis, and need to exclude highly mutually correlated
variables. The following independent variables were
selected.
The freezing index, frost penetration,
and percentage of aggregates passing 4.75 mm sieve were
not used because they lacked predictive power, perhaps
because their values had a limited range and variation.
All projects were located in southern Ontario, which has
quite uniform wetfreeze environment. Several
combinations of car and truck volumes were evaluated in
order to develop a traffic variable which would best
capture different contributions of cars and trucks to the
pavement surface damage. Since none of these
traffic variables appeared to noticeably improve the
model, the simplest traffic variable, AADT per lane, was
used. The 79 projects used to develop the
model are described in Table 2 in terms of highway
number, location, length, distress index, age, % asphalt
cement, and AADT per lane. 4.3 Prediction Model Based on Data
Alone Due to the complexity of the steel slag mix deterioration process and the availability of pavement evaluation data, the modelling approach was empirical. The selected model is linear in the coefficients as required for CSHRP Bayesian analysis. Several prediction models were constructed and evaluated and the best model, represented by Equation 2, was selected for the subsequent Bayesian analysis. The model selection considered the choice, transformation and interaction of independent variables to be included in the model, statistical properties of the model, and practical implications:
where:
The model accounts for about 84% of the
total variance in the DI, and its standard error of
estimate is 15.4 DI units. Statistical parameters of the
model are summarized in Appendix A. As expected, DI
increases with increasing age, and decreases as the
asphalt cement content increases. The statistically
significant influence of the asphalt cement content on DI
indicates that steel slag mixes with higher asphalt
content performed better. On the other hand, the minus
sign for TRAFFIC indicates that with the increasing
traffic volumes the DI decreases, i.e., pavement
performance improves. However, while the contributions of
AGE and AC are highly statistically significant (p <
0.00 I for a null hypothesis), the contribution of
TRAFFIC is not statistically significant (p = 0.44 for a
null hypothesis). One of the reasons for the unexpected
influence of traffic may be a lack of variation in
traffic volumes. All 79 projects evaluated are on
freeways and major highways with similar, high traffic
volumes per lane. Another reason may be that the
deterioration of surface courses containing steel slag is
simply caused primarily by environmental exposure rather
than traffic exposure. At any rate, the traffic
volume variable was left in the prediction equation in
order to study its significance as perceived by experts. 4.4 Prediction Models Based on
Expert Judgement (Prior Models) The key feature of the CSHRP Bayesian
statistical modelling is its ability to enhance or
significantly improve the databased prediction models
by incorporating expert judgement. In general, the
process of interviewing experts and encoding their
knowledge is also beneficial for the following reasons: I . It provides information for an
independent expert review of databased models. Such a
review may also increase acceptance of the models by
their potential users. This type of contribution of
Bayesian modelling is important when the available data
base is already quite extensive and comprehensive, as it
was in this case. There were 79 observations for the 3
independent variables, and data was available for all
potential basic independent variables. 2. It increases the inference space of
the databased model which is limited by the range of
available data. In this case, the data base did not
include projects which failed and were resurfaced before
1994. Experts know about these projects and may even have
an increased awareness of past failures. By including
their judgement in the predictive model, the model
applicability should increase. 4.4.1 Encoding of Expert Judgement The encoding of expert judgement was
guided by the CSHRP Bayesian implementation template
[8]. The main challenge was to explain to the experts
what was meant by different values of the DI so that the
experts could link deterioration of steel slag mixes in
terms of DI to the contributory (independent) variables
of AGE, AC, and TRAFFIC. Because DI is defined as a sum of two
distresses, ravelling and cracking (Eq. 1), two separate
scales, one for ravelling and one for cracking, were
constructed. Both scales ranged from 0 to 10, where 0
means no visible distress is present and 10 represents
the distress stage which unmistakeably requires a
rehabilitation treatment. The scales were constructed
using a series of pavement photographs showing typical
steel slag mixes in progressively advance stages of
ravelling and cracking deterioration. The photographs were a part of the
encoding package which was distributed to the experts
during the initial interview. Other items in the package
included separate coding sheets for ravelling and
cracking, histograms showing the distribution of the
model variables, and brief coding instructions. An example of a completed coding matrix
for ravelling is given in Table 3. The encoding matrix
for cracking was similar to that for ravelling. AGE and
AC variables were coded at three levels. For example, the
AGE was coded at 3, 6, and 12 years. In general, the
middle level was close to the median of the coded
variable, while the two outside levels were roughly plus
or minus one standard deviation from the mean value of
the variable. Because of a relatively small variation in
traffic volumes, the TRAFFIC variable was coded at only
two levels representing roughly the I 0th and 90th
percentile of its cumulative distribution. The histograms of the model variables
were included in the package to show experts the overall
range and distribution of data and to help them define
the scope of the model. Histograms for all model
variables are given in Figure 2. The encoding package is
fully described in Appendix B. Five experts with 10 to 30 years of
relevant experience, identified in Section 2.1, were
invited to complete the encoding matrices. All experts
were assembled in one room where they received
information about the project and encoding instructions,
and were asked to complete the matrix for ravelling. The experts indicated that
they found the encoding instructions easy to follow and
completed the encoding within 30 minutes. Two days later,
the same experts were asked to complete the cracking
matrix. The twoday delay was employed so that the
responses to the cracking matrix were not unduly
influenced by the previous responses to the ravelling
matrix. 4.4.2 Analysis of Data Encoded by Experts The separate encoding results obtained
for ravelling and cracking were aggregated into Distress
Index (DI) for each expert using the Equation 3. DI = 6R + 12C (3) where: R = Expert's rating for ravelling on the scale from 0 to 10 C = Expert's rating for cracking on the
scale from 0 to 10 The results obtained for each expert
yielded 18 observations (there are 18 cells in each
coding matrix). Since each expert's contribution was
considered to be independent and unique, the 18
observations obtained from each expert were used to
develop five different expertbased models (one for each
expert) using the CSHRP Bayesian statistical analysis
software, XLBayes. All five models have the same format
and use the same variables as the databased model of
Eq. 2. Selected statistical parameters of the five
expertbased models are summarized in Table 4; detailed
results are given in Appendix C. The results given in Table 4 indicate
that all experts were in agreement on the influence of
variables AGE and AC on the deterioration of steel slag
mixes. The regression coefficients for AGE, obtained for
the five expertbased models, were all positive and
their probability to be equal to zero was very low (p
< 0.001). The partial regression coefficients for AC
variable were all negative and their probability to be
equal to zero was again statistically insignificant (p
ranged from 0.001 to 0.02). Experts' opinions regarding
the influence of the TRAFFIC variable on DI differed. One
of the expert based models (Expert 1) had a negative
regression coefficient for TRAFFIC, and for three out of
five models, the probability of the TRAFFIC regression
coefficient to be equal to zero was quite high (p ranged
from 0.2 to 0.5). Figure 3 compares the five
expertbased models with the databased model using
sensitivity analysis. The graphs in Figure 3 were
obtained by changing values of one variable at a time
while holding all other variables constant. According to the top chart in Figure 3,
all models show an increase of DI with age. However, it
appears that all experts anticipated a higher rate of
deterioration with age than that shown for the
databased model. For example, at the age of 3, DI for
the databased model is 19, and the DI for the
expertbased models ranges from 19 to 45. At the age of
12, DI for the databased model is about 69 but the DI
for the expertbased models now ranges from 1 10 up to
150. The higher rate of deterioration anticipated by the
experts may be attributed to their experience with
projects which failed in the past and could not be
included in the data base. Sensitivity analysis of the influence
of asphalt content on DI, given in Figure 3, indicate an
overall agreement among the experts and between the
experts and data. Again, the experts anticipated a higher
rate of deterioration of steel slag mixes than that
indicated by the databased model. The results of sensitivity analysis for
the TRAFFIC variable reflect the differing opinions of
the experts, and uncertainty in the data regarding the
influence of traffic on Distress Index. Only one expert
predicted a large influence of traffic on Distress Index.
The sensitivity lines for the other experts and for the
data are basically horizontal indicating a very small
influence of TRAFFIC on the Distress Index. 4.5 Prediction Models Combining Data
and Expert Judgment (Posterior Models) While the databased model and the
expertbased models described previously could have been
obtained using a common statistical software package, the
prediction models combining data and expert judgement
using Bayesian Statistics could only be obtained by the
CSHRP Bayesian Statistical Analysis Software [8].
According to Bayesian nomenclature, expertbased models
provide prior data and the combination of the field data
and prior data yields posterior models. The results provided by the
expertbased models (prior data) were combined with
field data using the "Nprior" analysis option
available in CSHRP Bayesian Software. This analysis
yielded posterior prediction models which had the same
form and variables as the databased and expertbased
models. Prior information used in posterior analysis
consisted of mean values of the constant and regression
coefficients, variance/covariance matrix, number of
degrees of freedom, and residual variance. More detailed
description of CSHRP Bayesian methodology is given in
Reference 8. The five expertbased models were
considered to be distinct, unique models, yielding five
posterior models, one for each expert. All posterior
models, summarized in Table 5, are quite similar.
(Completed description of posterior models is given in
Appendix D.) The influence of AGE and AC is invariably
highly statistically significant while the influence of
TRAFFIC is not. The standard error of estimate ranged
from 19.4 to 24.5 and is higher than that obtained for
the databased model (15.4) or for the prior,
expertbased models (5.26 to 15.4 according to Table 4).
This indicates that information conveyed by field data
does not always agree with the information from experts. The sensitivity analysis of the databased model and posterior models are shown in Figure 4. The results indicate the strong influence of the data on the posterior models. For example, when AGE equals 12,
The deterioration predicted by the
posterior model is still somewhat larger than that
predicted by databased models only. One representative model, obtained for
Expert 4, was selected from the five posterior models for
future use and was evaluated in detail. The selection
criteria included:
The recommended posterior model is defined by the following equation. A detailed description of the model is given in Appendix D.
The standard error of estimate for the
above model (Expert 4) was 19.4 DI units, based on 14
degrees of freedom for the expertbased model and 75
degrees of freedom for the databased model. By
arbitrarily increasing the number of degrees of freedom
for the expertbased model to 75, the standard error of
estimate for the posterior model can be reduced to 15.4
DI units. It is also possible to consider the
results provided by individual experts as a sample and
combine expert knowledge into one model. The coding results provided by the five
raters were aggregated to form one prior model which was
then combined with the databased model. Since the
TRAFFIC variable was shown to be insignificant in the
analysis, it was not used in either of these models.
Results of this modelling effort are summarized in
Appendix E. Briefly, by combining expert judgement and
eliminating traffic, the remaining independent variables
in this posterior model became more statistically
significant and the standard error of the model increased
to 25.3 DI units. The CSHRP Bayesian statistical
analysis software provides a unique feature which enables
the user to obtain probability density functions for the
regression coefficients (for the databased,
expertbased and the combined models) and plot them in
one composite figure for easy comparison. The probability
density functions for the selected posterior model
(Expert 4) are summarized in Figure 5. Because the
regression coefficients are assumed to have a normal
distribution, the functions shown in Figure 5 have the
familiar bellshape. The mean of a probability density
function is the expected regression coefficient value of
the corresponding variable, and the integration of a
probability density function within given limits gives
the probability that a random variable will assume a
value between these limits. For example, referring to the
normal probability density function for the correlation
coefficient of the variable AGE, given in Figure 5 for
the databased model, the mean of the function is 5.6
years which equals the value of the regression
coefficient for AGE in Equation 2. The total area
underneath any of the probability density functions is
equal to 1, the maximum probability value. The shape of the probability density
functions indicates uncertainty associated with the model
estimates. According to Figure 5, field data do not
provide an assuring answer regarding the size of the
regression coefficient for AC, which can range roughly
from 30 to 5. The expert, according to the posterior
function for AC, appears to be more certain regarding the
beneficial effect of higher asphalt content. The expert's
confidence is indicated by a narrower range of the
expected value for the coefficient: 10 to 2. In Figure 5, the probability density
functions for the posterior model falls between those
obtained for the databased model and the expertbased
model. This is an intuitively expected result indicating
that the posterior, combined model is influenced by both
field data and expert judgment. Overall, the influence of
field data predominates, and the posterior distributions
are always closer to data than to experts (prior data).
Also, the larger uncertainty associated with the data,
indicated by a flatter shape of the databased
probability functions, is reflected in the resulting
shape of the posterior probability functions. The plots of the probability density
function for the traffic variable are wide, indicating
the estimates for the coefficient are uncertain. Further,
the distributions straddle the "zero line"
indicating the estimate of the coefficient could be
either negative, positive, or zero. This implies the
coefficients are not statistically significant. 5.0 CONCLUSIONS AND RECOMMENDATIONS 1. CSHRP Bayesian statistical analysis methodology and software provides a workable, stepby step procedure for developing Bayesian prediction models. 2. The use of Bayesian modelling approach provides additional insights into the modelling process. It highlights the awareness of practical issues, compels the analyst to use relevant, easytounderstand variables, and quantifies the contribution and significance of information provided through field observations and by experts. 3. Even when a data base is considered to be sufficient for the development of conventional prediction models, the inclusion of expert judgement using CSHRP Bayesian methodology can provide a structured independent review and endorsement of the models by experts. 4. The use of the Bayesian statistical approach can improve conventional prediction models by increasing their inference space (for example, if field data are no longer available for failed projects), reliability, and predictive power. 5. Bayesian statistical analysis is not a substitute for hasty or careless conventional regression analysis. It should be used in conjunction with appropriate and fundamentally sound conventional regression analysis. 6. CSHRP Bayesian Statistical Analysis Software is userfriendly but lacks an instructional manual. The existing documentation of the CSHRP Bayesian methodology would benefit from a summary report on the use of the methodology. 7. It is recommended to use the
Bayesian approach for future pavement performance
modelling whenever feasible, particularly for calibration
and verification of the Ministry's pavement design
method. The authors acknowledge help provided
by Mr. Frank Marciello from MTO, who was responsible for
the development of the data base and Mr. Lyle Kajner from
VEMAX Management Inc. for his valuable guidance during
all phases of the project. 1. Smith W., Finn, F. Kulkarni R. et al
"Bayesian Methodology for Verifying Recommendations
to Minimize Asphalt Pavement Distress", NCHRP Report
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June 1979. 2. Harper, W.V., and Majidzadeh, K.,
"Utilization of Expert Opinion in the Two Pavement
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