Measurement Error Models (MEM) regression method to Harmonize Friction Values from Different Skid Testing Devices.
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Transport Research Arena 2014, Paris
Measurement Error Models (MEM) regression method to Harmonize
Friction Values from Different Skid Testing Devices.
Azzurra Evangelistib*, Samer W. Katichaa, Edgar de León Izeppia, Gerardo W. Flintscha,
Mauro D'Apuzzob, Vittorio Nicolosic.
a
Center for Sustainable Transportation Infrastructure, VTTI, Blacksburg, Virginia, USA
b
University of Cassino and Southern Lazio, Cassino, Italy
c
University of Rome "Tor Vergata", Rome, Italy
Abstract
Skid measurement errors are unavoidable for each kind of skid testing device. The Simple Linear Regression
(SLR), used worldwide to harmonize friction measuring devices, does not consider that measurement errors
affect both devices. For this reason its use provides biased and not unique estimate of the relationship between
devices.
The Measurement Error Models (MEM) regression method is proposed as a better method to harmonize any two
skid testing devices. SLR and MEM regressions between repeated measurements (from the same device) and
between measurements obtained from two different skid testing devices are performed. A comparison of the
results is shown and MEM regression appears to be a more appropriate tool to harmonize friction measuring
devices instead of SLR.
Keywords: Skid measurement, Friction, Harmonization, Simple Linear Regression, Measurement Error Models.
Résumé
Les erreurs de mesure de dérapage sont inévitables pour chaque type de dispositif de contrôle de dérapage. La
régression linéaire simple (SLR), utilisée dans le monde entier pour harmoniser les dispositifs de mesure de
friction, ne considère pas que les erreurs de mesure affectent les deux appareils. Pour cette raison, son utilisation
fourni une estimation biaisée et non unique de la relation entre les dispositifs.
Le modèle d'erreur de mesure (MEM) de la méthode de régression est présenté comme la meilleure méthode
pour harmoniser n’importe quelle paire de dispositifs de contrôle de dérapage. Les régressions SLR et MEM
entre les mesures répétées (à partir du même dispositif) et les mesures obtenues à partir de deux dispositifs de
contrôle de dérapage différent sont réalisées. La comparaison des résultats entre les méthodes a montré que la
régression MEM semble être l'outil le plus approprié pour harmoniser les dispositifs de mesure de la friction.
Mots-clé: Mesure de dérapage, friction, harmonisation, régression linéaire simple, modèle de mesure d'erreur.
*
Azzurra Evangelisti, Ph.D. Student. Tel.: (+39) 0776 2993893; fax: (+39) 0776 2993939.
E-mail address: azzurra.evangelisti@unicas.it, ae941985@vt.edu.Katicha et al. / Transport Research Arena 2014, Paris 2 1. Introduction To improve and maintain high safety level on the roads, a good skid resistance is required. Skid resistance is estimated by means of the Skid Number (SN) or Friction Number (FN) and it is used for the specifications of new pavements, for network maintenance management programs and for accident investigations. The SN or FN is directly affected by the texture of the pavement and the estimate of the macrotexture, using a prediction model (D’Apuzzo et al., 2012), can be an indirect estimation of the pavement’s friction. More than twenty different types of measuring devices, are currently used to measure skid resistance. Most often, each of these devices measures skid values that are differently affected by several factors: basic measuring principles (longitudinal friction tester, transverse friction tester, slider tester), tires (smooth, ribbed), thickness of the water film, test speed, etc. Because of the different measuring principles used in different devices, SN or FN values obtained by two different devices can be significantly different. This suggests that a device dependent error which, in this paper, is called Model Error affects all friction measurements. In addition to Model Error, each device in itself is not perfectly repeatable which gives rise to what, in this paper, is called Measurement Error. Consequently the skid values are different from one device to another and, for this reason harmonization or comparison of different devices measurements is a difficult but crucial task. Previous efforts to harmonize skid resistance measurements include the Permanent International Association of Road Congresses (PIARC) International Experiment, where the International Friction Index (IFI) was proposed (PIARC Report 01.04, 1995) and the Harmonisation of European Routine and Research Measurement Equipment for Skid Resistance of Roads and Runways (HERMES) Project, where the European Friction Index (EFI) was presented (Descornet, 2004). In both studies, the objective was to find valid conversions from different skid testing devices’ measured values, taken under generic test conditions, to equivalent values identified on a Common Scale, which represents the friction under defined reference test conditions. It was observed that the Common Scale-converted measurements were not consistent and unacceptably poor; for this reason, the Limits of Agreement (LOA) Method, adapted from the biomedical field, was recently proposed and applied to analyse the degree of agreement between friction measurement devices (de León Izeppi et al., 2012-3; de León Izeppi et al., 2012-4) and deflection testing devices (Katicha et al., 2013). The LOA is based on the concepts of repeatability within the same device and reproducibility between different devices and it compares the friction measurements of two devices quantifying the distance between pairs of their measurements. The LOA method evaluates the agreement is adequate when the relationship between two sets of measurements follows the line of equality; if this requirement is not satisfied, the method results in LOA that are too large. In this case, the Measurement Error Models (MEM) regression method, can be used to evaluate the relationship between the two devices. Once this relationship is obtained, the LOA method can then be applied to evaluate device agreement. 2. Objective The objective of this paper is to present the MEM method to harmonize skid measurements obtained by two different types of skid measuring devices. It will be shown that the MEM method provides an asymptotically unbiased relationship between pairs of skid testing devices. In the same time, it will be shown that the Simple Linear Regression (SLR), which does not consider that Measurement Error and Model Error affect each device, results in a biased estimate of the relationship between the two devices. A comparison of MEM and SLR applications shows that the first gives accurate estimate of the true regression parameters in the relationship between two devices. The MEM use is recommended to transform skid measurements of one device into the other. Then the LOA method can be recommended, to estimate the agreement between the two devices. 3. MEM: background and modeling Classical harmonization models, as IFI or EFI, use SLR to evaluate the relationship between different skid measuring devices. The SLR approach assumes that errors belong to the dependent variable chosen for the regression, but in reality, errors exist for both skid measuring devices. As illustrated in Figure 1a, SLR
Author name / Transport Research Arena 2014, Paris
minimizes the squared vertical distance from the points to the regression line, but this is suitable when only one
of the two variables is random (in this case affected by error). The SLR produces estimate of the regression’s
parameters biased and not unique, in fact the relationship depends on which device measurements are selected as
the dependent variable.
Fig. 1. (a) The SLR approach. (b) The MEM approach.
The MEM approach, instead, takes into account that errors are present in both devices. The Orthogonal
regression, represented in Figure 1b, which is a particular case of MEM, minimizes the orthogonal distance from
the measured data points to the regression line. This method allows identifying the exact relationship between
different friction measuring devices when they are both affect by the same error standard deviation; in addition,
in orthogonal regression, the variables can be interchanged without compromise the fitted line's relation.
The preliminary observation is that the relation between two measurements’ sets should be unique and
independent from the choice of response or explanatory variables. It means that, if X1 and X2 are two sets of
repeated measurements from different devices, obtained on n pavements sections, and the relationship between
these devices is X2=A1X1+B1 or X1=A2X2+B2, then to satisfy the condition of uniqueness, it should be A2=1/A1
and B2=-B1/A1.
Because of Measurement Error, if the SLR is used, the estimate of the model parameters will be: a2≠1/a1 and
b2≠ - b1/a1, therefore SLR coefficients a1, a2, b1 and b2 are not satisfactory estimators of A1, A2, B1 and B2.
The MEM regression results are more consistent in that a2=1/a1 and b2= - b1/a1, and asymptotically unbiased. In
fact, the bias is negligible for small or medium sample sizes, and for n it produces unbiased results:
E{a1}=A1, E{a2}=A2, E{b1}=B1 and E{b2}=B2.
3.1. MEM modeling for pavement friction applications
As previously mentioned, the Measurement Error and the Model Error constitute the Error Structure of the
MEM method. The first is due to the device repeatability, which can be defined as the difference between two
measurements on the same pavement, in a short space of time, performed with the same device, with the same
tire and the same operator, with a specified probability, usually of 95% (Vos et al.,2009). The Model Error is
defined as the error between the true friction, which is not known, and the friction measured by the devices,
excluding the repeatability error (i.e. perfect repeatability conditions). This hypothesis is more general than the
SLR assumptions.
Before presenting the MEM method, a description of the variables used is presented:
X, is the “true” pavement friction (unknown); Y, is the estimated friction by device 1; Z, is the estimated friction
by device 2. It is assumed that Y and Z are related to X through a linear relationship as follows:
Y aY X bY Y (1)
Z a Z X bZ Z (2)
With: aY, aZ slopes of the relationship between Y- X and Z -X, respectively;
bY, bZ intercepts of the relationship between Y- X and Z –X, respectively;
εY, εZ the Model Errors of device Y and Z respectively, with εY ~ N(0,σY) and εZ ~ N(0,σZ).
Combining Equation 1 and Equation 2, the relation between the two devices can be written as:Katicha et al. / Transport Research Arena 2014, Paris 4
aY a a
Y Z bY Y bz Y Y z AZ B (3)
aZ aZ aZ
With: E ~ N(0,πYZ) and
YZ Y2 aY2 Z2 a Z2 . (4)
The Y and Z values can be affected by the Measurement Error (repeatability), so the friction measured by device
Y and device Z is respectively ψ and ς, that are related to Y and Z as:
Y rY (5)
Z rZ (6)
With: rY ~ N(0,τY) and rZ ~ N(0,τZ).and are Measurement Errors of device Y and Z, respectively.
Using the MEM regression, from these noisy measurements ψ and ς, A and B can be estimated as (Fuller,1987
and Carroll et al.,2006):
s2 s2 s
2
s2
2
4s
2
Aˆ (7)
2s
Bˆ Y Aˆ Z (8)
With: s2ψ=variance(ψ); s2ς=variance(ς); sςψ=covariance(ς,ψ); Y Y
2 2
A and overhead 2
Z
2
Z
bar denotes average values.
The value is requested to estimate A and B. If Y1 and Y2 are measurements obtained with the same device or
they are ideal measurements from two different devices (devices are theoretically exactly the same, with same
operator), the relationship between them should be E{Y1}=E{Y2}. In this case, to calculate Â, 1 can be used
in Equation 7.
Instead, if Y1 and Y2 are measurements from two different devices, it is possible to estimate the repeatability
errors τY and τZ from replicate measurements of each devices (de León Izeppi et al.,2012-2; de León Izeppi et
al.,2012-3). The value of πYZ can be estimated from the residuals of the MEM regression as (Fuller,1987 and
Carroll et al.,2006):
YZ
2
max(sv2 ˆZ2 Aˆ ˆY2 ,0) (9)
Y Y Aˆ Z
n
sv2 n 2 Z
1 2
with i i (10)
i 1
Is it always possible to express the relationship, between the errors σY and σZ of the two devices, as: σY =α σz=σ
and substituting into πYZ expression, it is obtained:
YZ
(11)
1 2 Aˆ 2
If there is no additional information about the devices being compared, the value remains unknown and a
reasonable assumption on its value needs to be made. The knowledge of  implies the knowledge of πYZ , and
vice-versa; for this purpose an iterative procedure to calculate  and πYZ is proposed:
1. Assume
ˆ 2
ˆ Y2 ˆ
and calculate an initial estimate A
s2 ˆ s2 s
2
ˆ s2 4ˆs
2
2
ˆZ 2s
sv2 ˆZ2 Aˆ ˆY2
Calculate ˆ max ,0
2
2.
1 ˆ2
A
ˆY ˆ
2 2
3. Calculate ˆ 2
ˆ Aˆ ˆ 2
ZAuthor name / Transport Research Arena 2014, Paris
4. Calculate Aˆ
s2 ˆ s2 s
2
2
ˆ s2 4ˆs
2
2s
Repeat steps 2, 3, and 4 until convergence (usually 1 to 2 iterations).
4. Data collection
As part of the annual event "Surface Properties RODEO", organized by the Virginia Transportation Research
Council (VTRC) and the Virginia Tech Transportation Institute (VTTI), a friction measurement campaign was
undertaken on May 30, 2008, at the Virginia Smart Road. The friction testing included three locked-wheel skid
trailers units and one Grip Tester (fixed slip with 16% slip ratio) unit. One of the locked-wheel trailer used only
the smooth test tire (ASTM E-524), another tested only the ribbed test tire (ASTM E-501) while the third tested
both smooth and ribbed. All the Locked-wheel tests were performed in accordance with ASTM E-274, whereas
the Grip Tester was conducted according to ASTM E 2340. Five repetitions were scheduled at 20, 40, and 50
mph, both downhill and uphill (only the locked-wheel tester which used both tires, didn’t test the ribbed tire at
20 mph). Table 1 summarizes the tests’ schedules and Table 2 describes the 12 tests sections, present on Virginia
Smart Road and used for the test.
Table 1. Test matrix.
Device
Test Configuration
locked-wheel 1 locked-wheel 2 locked-wheel 3 Grip Tester
Tire Ribbed and Smooth Smooth Ribbed Smooth
20, 40 and 50 (smooth)
Speed [mph] 20, 40 and 50 20, 40 and 50 20, 40 and 50
40 and 50 (ribbed)
Uphill and
Direction Uphill and downhill Uphill and downhill Uphill and downhill
downhill
Number of runs 5 5 5 5
Table 2. Section texture and material properties of the pavement surfaces tested.
Section No. Mix Type Asphalt Binder NMS
1 SMA 19.0 PG 70-22 19
2 SM-12.5D PG 70-22 12.5
3 SM-9.5D PG 70-22 9.5
4 SM-9.5E PG 76-22 9.5
5 SM-9.5A PG 64-22 9.5
6 SM-9.5A(h) PG 64-22 9.5
7 SM-9.5D PG 70-22 9.5
8 OGFC PG 76-22 12.5
9 SMA-12.5D PG 70-22 12.5
10 Epoxy O/L Proprietary -
11 Epoxy O/L VDOT EP – 5 -
12 CRCP Tined -
Note: NMS = nominal maximum aggregate size, O/L = overlay
5. Results and Analysis
5.1. Repeatability evaluation
As mentioned in section 3, when developing a relationship between repeated measurements, it is expected that
this relationship is the same irrespective of which replicate is chosen as the dependent variable and which
replicate is chosen as the independent variable.Katicha et al. / Transport Research Arena 2014, Paris 6
Table 3 shows the results of SLR as well as MEM regression performed with a set of 5 replicate measurements
performed by a locked wheel tester with a smooth tire and testing speed of 50 mph. With 5 replicate
measurements, 10 combinations of pairs of runs can be obtained and used to develop a linear relationship. The
first interesting observation is that SLR does not give the same relationship if the two runs are switched; this
shows inconsistencies in the developed relationship. On the other hand, MEM regression gives the same result.
TABLE 3 Results of the linear regression and orthogonal regression for runs obtained from the same device.
Linear Regression Orthogonal Regression
Yi = f(Yj) ij ij
Y2 = f(Y1) 1.7286 2.0015 1.9388 1.9388
Y3 = f(Y1) 1.1235 1.1928 1.1627 1.1627
Y4 = f(Y1) 0.5785 1.0221 0.9324 0.9324
Y5 = f(Y1) 0.8605 1.8984 1.7998 1.7998
Y3 = f(Y2) 0.4266 0.6695 0.6020 0.6020
Y4 = f(Y2) 0.7579 0.5959 0.4540 0.4540
Y5 = f(Y2) 1.5285 0.9614 0.9118 0.9118
Y4 = f(Y3) 0.8724 0.8659 0.7986 0.7986
Y5 = f(Y3) 1.3562 1.5966 1.5180 1.5180
This example shows how, when SLR is used, different relationships are obtained between two sets of
measurements, depending on which one is considered as dependent variable and which one as independent
variable.
5.2. Relationship between two same measuring principles devices
In this example, skid measurements, obtained with the two skid testers equipped with smooth tires, are used. Six
test configuration each with five replicates on the 12 pavement sections resulted in 360 measurements. The 5
replicates are averaged to obtain a single set of 72 measurements for each device. Fig.2 shows the averaged
measurements along with the un-averaged measurements.
90
All data 360 measurements
80 Averaged 5 replicates 72 measurements
70
Device Friction
60
2
50
40
30
20
10 20 30 40 50 60 70 80 90
Device Friction
1
Fig.2: Comparison of smooth tire test results for all test configurations
Performing a SLR, a slope of 0.8927 with a 95% confidence interval of [0.8253; 0.9601] and an intercept of
10.3757 with a 95% confidence interval of [6.9259; 13.8255] were obtained. With the MEM regression a slope
of 0.9341 with a 95% bootstrap confidence interval of [0.8702; 0.9965] and an intercept of 8.3281 with a 95%Author name / Transport Research Arena 2014, Paris
bootstrap confidence interval of [5.0931; 11.52], were obtained. In both cases the hypothesis that the slope is
equal to 1 and that the intercept is equal to 0 are rejected (although for the MEM, a confidence interval of 96%
for the slope will cover the case the slope equals to 1). This suggests that the two devices are not equivalent and
there is a difference between readings obtained from either device. Although the coefficients of the SLR and the
MEM regression might not be considered that different, the SLR estimate is biased. This can practically be easily
shown by performing the regression on pairs of runs from each device and then averaging all the parameters
(instead of averaging the runs and doing the regression). In this case, the average slope obtained from the simple
linear regression is 0.8590 while the average slope obtained from the MEM regression is 0.9346. The bias of the
linear regression is evident (Bias = 0.8927-0.8590 = 0.337) while the bias of the MEM regression is practically
zero (Bias = 0.9341-0.9346 = -0.005).
5.3. Relationship between two different measuring principles devices
In the MEM model, the error between device (σy or σz in Equation 4) is essential to evaluate the model
parameters. In the case of two same-principle-measuring Testers, this standard deviation between device error
can be assumed to be divided equally between the two devices. This between device error could be interpreted as
follows: the error between the two devices is greater than what can be explained from each device repeatability
error. The extra error could be due to the fact that both testers are not perfect and even if the devices are perfectly
repeatable this error can be explained as the error of the devices with respect to an “ideal” perfectly designed
tester (with the same principle of friction measuring).
On the other hand, when comparing two devices with different measuring principles, each device is affected by:
the repeatability error (Measurement Error), the device error compared to a perfectly designed device, and the
error that is due to the different measuring principles implemented in each device (Model Error).
In this paragraph, the relationship between the Grip Tester and the Locked-Wheel Tester, using SLR as well as
the MEM regression, is evaluated. One of the major differences in measuring principle is that the locked-wheel
Tester completely locks the test wheel while the Grip Tester has a fixed slip (there are also other important
differences between the devices such as tire size).
a) 360 measurements data b) 72 measurements data (averaged data)
Fig. 3. Comparison between the Grip Tester and the Locked-Wheel Tester.
The skid measurements obtained by the Grip Tester and the Locked-Wheel Tester, were used. For each device
360 measurements resulted: three speeds, two directions and for each, five replicates on the 12 pavement
sections. The 5 replicates are averaged to obtain a single set of 72 measurements for each device and the model
is fitted to the measurements. Figure 3 shows the data used for the analysis.
Three different approaches or subjective decisions can be adopted to harmonize the measurements of two
devices:
to choose one of the two devices as the reference. In this case, all the Model Error is assigned to the second
device and only repeatability error (Measurement Error) is assigned to the first device. This choice while very
simple is somewhat not very effective; it is better to have a measuring principle as the reference. For example
a valid approach can be choosing the measuring principle of the locked-wheel tester as the reference. This isKaticha et al. / Transport Research Arena 2014, Paris 8
different than choosing a specific locked-wheel tester as the reference because no single locked-wheel is
perfect (even if there are no repeatability error) and therefore measure perfectly. For example, for the two
locked-wheel testers investigated in the previous paragraph, the error standard deviation of the devices
compared to the measuring principle of locked-wheel testers (perfect device) is 2.29. So, to harmonize the
Grip Tester to the measuring principle of locked-wheel testers, it is necessary to assign 2.29 of the standard
deviation Model Error (σy or σz) to the locked-wheel tester device and the rest of the Model Error to the Grip
Tester. Calling the locked-wheel tester as Z and the Grip Tester as Y, then Equation 4 becomes:
YZ Y2 2.29 2 A 2 (9)
And the Model Error for the Grip Tester can be determined as
Y2 YZ
2
2.292 A2 (10)
to assume that the two measuring principles estimate the true friction with some error and try to harmonize
the devices with respect to that unknown true friction value. In this case it is necessary to determine the values
of σY and σZ, and it is possible to express the relationship between the two standard deviations as follows
σY=α σZ. Substituting into Equation 4, σY and σZ are obtained:
YZ YZ
Y Z (11)
1 2 Aˆ 2 2 Â2
With two devices, the value of α cannot be determined and would have to be assumed. A reasonable choice is
to take α = 1 which means that the error of the two measuring principles with respect to the “true” friction is
the same.
to assume that one of the two measuring principle gives the “true” friction. This becomes the same as
selecting a specific technology as the reference. If for example it is assumed that locked-wheel testers
technology gives the “true” friction then this is the same as calibrating the Grip Tester with respect to locked-
wheel testers technology as mentioned previously.
The three cases of harmonizing the Grip Tester with the Locked-Wheel Tester are shown in Table 3 along with
the SLR regression.
Table 4. Relationship between the Grip Tester and Locked-Wheel Tester.
GT vs. LWT LWT vs. GT Inverse GT vs. LWT
Linear Regression 0.347 56.093 1.002 -20.577 2.882 -161.651
MEM: Friction 0.435 51.251 2.298 -117.764 2.299 -117.818
MEM: LWT principle 0.365 55.108 2.864 -160.262 2.740 -150.981
MEM: LWT 0.352 55.806 2.881 -161.479 2.841 -158.540
It is possible to note that the SLR does not take into account any of the errors (not even the repeatability error)
and gives inconsistent results depending on which device is set as the dependent variable and which device is set
as the independent variable. This can be seen in the last two columns of the table where the inverse relationship
of GT vs. LWT is obtained. For the SLR, this inverse relationship is significantly different than the relationship
of LWT (Locked-Wheel Tester) vs. GT (Grip Tester) which shows how the SLR is inconsistent. For the MEM,
the inverse relationship of GT vs. LWT is essentially the same as that of LWT vs. GT. There are some
differences which are due to numerical accuracy and to the fact that the MEM is asymptotically unbiased and for
finite (relatively small) sample sizes, the MEM has a small bias (negligible).
The row “MEM: Friction” gives the results of the model which considers the Model Error is equally split
between the two devices; this essentially means that it is trying to estimate the “true” friction and that both
devices have the same error standard deviation compared to the true friction.
The row “MEM: LWT principle” gives the results of the model which gives the conversion of the Grip Tester to
locked-wheel friction values. These locked wheel values are not those of the specific device but rather those of
an ideal device. In this case, the Model Error for the locked-wheel tester is set at 2.29, according to the results of
comparing two locked-wheel testers, while the Model Error for the Grip Tester is obtained using Equation 9 and
Equation 10.Author name / Transport Research Arena 2014, Paris
The row “MEM: LWT principle” gives the results of the model which gives the conversion of the Grip Tester to
specific locked-wheel tester being evaluated. In this case, the Model Error for the locked-wheel tester is set at 0
while the model error for the Grip Tester is obtained using Equation 9. In this case, the result of the MEM is very
close to that of linear regression. The difference is that the MEM still considers the repeatability error of the
locked-wheel tester while the SLR does not. Another advantage of the MEM regression is that it gives a
consistent relationship while SLR does not.
6. Conclusion
Repeated measurements from the same device (Locked wheel Tester), from two same measuring principles
devices (Locked wheel Testers) and from two different measuring principles devices (Locked wheel Testers and
Grip Tester), to show that the estimated regression parameters from the SLR are biased, are used. To identify the
appropriate model parameters, the MEM regression is presented to give accurate estimates of the true model
parameters. The conclusions of the presented research can be summarized as follows:
SLR does not give a unique relationship between repeated measurements from the same device and from two
any devices. The relationship depends on the choice of which variable is chosen as the dependent variable and
which variable is chosen as the independent variable.
MEM as well as model error consideration regression gives an unbiased estimate of the true relationship
between repeated measurements from the same device and from two any devices. The relationship obtained
from MEM regression is independent of the choice of which variable is the dependent variable and which
variable is the independent variable.
The relationship between two different measuring principles devices (in this paper: Locked wheel Testers and
Grip Tester) depends on how the model error is assigned. It seems that the most logical approach is to divide
the model error equally between the two devices. Another possible approach to take is to develop a
relationship to convert Grip Tester measurement to locked wheel friction values. This approach takes into
account that the locked-wheel tester is not perfect and does not give the true locked-wheel friction values even
beyond the consideration of device repeatability. The last approach to develop a relationship to the specific
locked-wheel tester does not seem to have practical justification.
In all cases, the MEM produces consistent relationship that can be justified by the choice of error structure. The
best choice of error structure would have to be selected based on further extensive studies preferably by a wide
range of investigators.
Acknowledgements
The data used in this paper was obtained during the 2008 equipment rodeo comparison conducted at the Virginia
Smart Road. The authors would like to thank the Virginia, Georgia, Mississippi, Connecticut, South Carolina,
and departments of transportation as well as the Federal Highway Administration for their support of the
equipment rodeo comparison.
References
1. Carroll, R.J., Ruppert, D., Stefanski, L.A., and Crainiceau, C.M. (2006), “Measurement Error in Nonlinear
Models: A Modern Perspective”. Chapman & Hall/CRC, Taylor Francis Group.
2. D’Apuzzo M., Evangelisti A. and Nicolosi V., (2012), “Preliminary Findings for a Prediction Model of
Road Surface Macrotexture.”, Procedia: Social & Behavioral Sciences, Vol. 53C, p.1110-1119, ISSN: 1877-
0428, doi: 10.1016/j.sbspro.2012.09.960.
3. de León Izeppi, E., Flintsch, G.W., McGhee, K.K., (2012). “Limits of Agreement Method for Comparison
of Pavement Friction Measurement”, Transportation Research Record 2306, Washington, DC, 2012, p 188-
195.
4. de León Izeppi, Edgar, Flintsch, Gerardo, and Kevin McGhee, (2012). “Effect of Water, Speed, and Grade
on Continuous Friction Measurement Equipment (CFMEs)”, Pavement Performance: Trends, Advances,Katicha et al. / Transport Research Arena 2014, Paris 10
and Challenges, ASTM STP 1555, Editor Bouzid Choubane, ASTM International, West Conshohocken, PA,
2012, pp. 20-39.
5. Descornet, G., (2004). The HERMES Project, Paper presented at the 5th Symposium on Pavement Surface
Characteristics Conference, SURF 2004 (CD-ROM), World Road Association, Paris.
6. Fuller, W.A. (1987), Measurement Error Models, Wiley Series in Probability and Mathematics, John Wiley
and Sons, Inc.
7. International PIARC Experiment to Compare and Harmonize Texture and Skid Resistance Measurements,
PIARC Report 01.04. Permanent International Association of Road Congresses–World Road Association,
Paris, 1995.
8. Katicha, S.W., Flintsch, G.W., Ferne, B., and Bryce, J. (2013) “Limits of agreement (LOA) method for
comparing TSD and FWD measurements”, International Journal of Pavement Engineering (iFirst).
http://dx.doi.org/10.1080/10298436.2013.782403
9. Vos, E. and Groenendijk, J., Report on analysis of previous skid resistance harmonization research projects,
FEHRL, Brussels, Belgium, 2009, http://tyrosafe.fehrl.org.You can also read
