THE ANALYSIS OF GAUGE REPEATABILITY AND
REPRODUCIBILITY WITH INTERACTION BETWEEN OPERATORS
AND PARTS
JENG-JUNG FANG
Department of Management and Information Technology
Southern Taiwan University of Technology
710 Yung-Kung City, Tainan County, Taiwan
PENG-SEN WANG
Department of Management and Information Technology
Southern Taiwan University of Technology
710 Yung-Kung City, Tainan County, Taiwan
The purpose of the research is to improve the analysis of the measurement system nowadays. The
current measurement system usually utilizes the method of the analysis of variance (ANOVA) to
analyze the measurement system while there is interaction between the parts and operators. There
are other methods can be used to analyze the measurement systems such as Classical GR&R and
Long Form methods. Although the two methods are easy in calculation, they are not suitable to
analyze the measurement system while there is interaction between parts and operators. The research
modified the two methods to make them suitable to the cases no matter there is interaction between
parts and operators or not.
In the mean time, the research also compared the gauge variance estimators of five different
estimation methods by simulating a measurement system based on the criterions of the biasness,
variance, and mean squares error (MSE) of the gauge variance estimate. The research verified that
the Modified Long Form method developed in the research gives smaller MSE and variance of
gauge variance estimator both in the cases of with interaction and without interaction even though it
gives biased estimator of the gauge variance.
Keywords: Gauge Analysis, Repeatability, Reproducibility, Interaction
Introduction
Due to the rapid development of the high-tech industry and the requirement of the high
quality from the customers, the need of accuracy and precision on the measurement
system is highly demanding. A gauge must be sufficiently capable to measure product
accurately enough and precisely enough so that the analyst can make the correct decision.
In the QS 9000 quality assurance system, the analysis of repeatability and reproducibility
of the measurement variation is to apply the verified measurement system to the
measurement process. Most of the industries apply the QS 9000 criterions to judge the
acceptance of their measurement systems.
1
2
Tsai (1988) defined the gauge reproducibility as the variability due to different operators
using the same gauge and the gauge repeatability as the variability reflecting the basic
precision of the gauge itself. The gauge variability is composed of the gauge
reproducibility and the gauge repeatability. The variance components of the total
observed variability includes both the part variability and gauge variability. The random
effects model used for analyze a measurement system is presented as
i  1, 2, ..., n;

 j  1, 2, ..., p;
k  1, 2, ..., k

Yijk    Pi  O j  POij  Rijk
(1)
The Yijk is the kth repeated measurement on the ith part by the jth operator.  is the
process average. Pi is the ith part effect. Oj is the jth operator effect. POij is the interaction
between the ith part and the jth operator. Rijk is the error term. The source of variation can
be divided into four components which are parts, operators, interaction between parts and
operators, and the random error term. Pi 、 Oj 、 POij, and, Rijk are random factors
normally distributed with mean 0 and constant variances equal to  P ,
2
2
 O2 ,  PO
, and
 R2
respectively. The effects of parts and operators, the interaction between parts and
operators, and the error terms are assumed to be additive. The estimated gauge variance is
the sum of variances of repeatability and reproducibility. That is
2
2
ˆ2
ˆ gauge
 ˆ repeatabil
ity   reproducibility
where
and
2
ˆ repeatabil
ity
=
ˆ R2
2
ˆ2
ˆ reproducib
ility =  O
2
+ ˆ PO
(2)
(3)
(4)
Methods to Analyze Gauge Variation
There are three analysis methods widely used to estimate the components of variance in
the gauge system. They are ANOVA, Classical GR&R, and Long Form methods. Tsai
(1988) developed the ANOVA method to compute the variance components.
Montgomery and Runger (1993a, 1993b) emphasized the repeatability and reproducibility
study is equivalent to an experiment designed to estimate certain components of variance.
Negative estimate of variance component
2
 PO
is
possible. One method to solve the
problem is to set it to zero and recalculate the gauge variance estimate. A reduced model
is then fitted and the estimates of repeatability and reproducibility are then adjusted.
Barrentine (1991) introduced the control chart tabular methods to estimate the variance
components of repeatability and reproducibility. Montgomery and Runger （1993a）
called the methods “Classical Gauge Repeatability and Reproducibility Study” (Classical
3
GR&R). The tabular method tabulates the collected data for k equal to 2 as the form of
Table 1. The Classical GR&R method estimates repeatability as

 repeatability 
R
d2
(5)
The value d2 depends on the number of repeated measurements taken on the same part by
the same operator.
Table 1. Tabular Form for the Classical Gauge Repeatability and Reproducibility Study
Operator 1
…
Operator p
Measurements
…
Measurements
Part
Number
1
2
x
R
…
1
2
x
R
1
x111
x112
X 11
R11
…
x1p1
x1p2
X 1 p
R1p
2
x211
x212
X 21
R21
…
x2p1
x2p2
X 2 p
R2p
.
.
.
.
.
…
.
.
.
.
n
xn11
xn12
X 21
Rn1
xnp1
xnp2
X np
Rnp
X 1
R1
X  p
R p
Average
…
The estimator of reproducibility is obtained from the difference
R  max X  j   min X  j  , j = 1, 2, …, p
X
(6)
j
j
The estimator of reproducibility is
ˆ reproducibility 
R
X
(7)
d2
MSA (1991) reference manual introduced another tabular approach similar to control
chart tabular method which is called Long Form method. The estimators of
repeatability and reproducibility are computed as
ˆ repeatability
ˆ reproducibilty
=
= R
R
 X
 d2

(8)
d2
R 
 d 

2
 

nk

2
2
(9)
4
The repeatability and reproducibility of the three analysis methods are summarized
in the Table 2.
Table 2. Repeatability and Reproducibility Estimators of Classical GR&R, ANOVA,
and Long Form Methods
ˆ repeatability
Classical GR&R
ANOVA
Long Form
ˆ reproducibility
R
R d2
X
d2
MS O  n  1MS PO  nMS R  nk
MS R
 RX

 d2

R d2




2
R 
 d 
2


nk
2
Other related gauge analysis methods can be found in the articles such as Dolezal
(1998), and Burdick and Larsen (2002).
Table 3 and Table 4 are the results of gauge variability analysis for measurements
without and with interaction between operators and parts respectively. Comparing the
results of the two tables, one can observe that Classical GR&R and Long Form methods
cannot precisely estimate the gauge variance while there is interaction between parts and
operators.
Table 3. Repeatability, Reproducibility, and Gauge Variance Estimate
for Data without Interaction Between Parts and Operators
Classical GR&R
Long Form
ANOVA
ˆ repeatability
ˆ reproducibility
ˆ gauge
3.783
3.783
2.576
1.418
1.285
3.983
4.039
3.995
4.743
Table 4. Repeatability, Reproducibility, and Gauge Variance Estimate
for Data with Interaction Between Parts and Operators
Classical GR&R
Long Form
ANOVA
ˆ repeatability
ˆ reproducibility
ˆ gauge
2.261
2.261
3.506
1.462
1.418
3.863
2.692
2.668
5.216
5
In order to make both Classical GR&R and Long Form Methods suitable to the cases of
with interaction between parts and operators, referring to the tabular form in Table 1, we
first calculate
R X ij  max X ij  min X ij , i = 1, 2, …, n
 n

R X ij     R X ij   n
 i 1

and
By replacing
R X into R X
ij 
(10)
j
j
(11)
in the reproducibility variance estimator of Classical
GR&R and Long Form methods, the two methods can be applied to estimate the
gauge variance for the cases there are interactions between parts and operators. The
modified reproducibility estimator of the two analysis methods are summarized in
the Table 5.
Table 5. Modified Reproducibility Estimators of
Classical GR&R and Long Form Methods
ˆ reproducibility
Modified
Classical GR&R
Modified
Long Form
R X ij  d 2
 R X ij 

 d2

R 
  d2 

 

nk

2
2
Table 6 shows the values of the repeatability, reproducibility, and total gauge
estimate for the five methods of ANOVA, Classical GR&R, Modified Classical
GR&R, Long Form, and Modified Long Form based on the same measurements used
in Table4. It showed that Modified Classical GR&R and Modified Long Form
methods are both much better to estimate the gauge variance than Classical GR&R
and Long Form methods while there is interaction between parts and operators.
Table 7 shows Modified Classical GR&R and Modified Long Form methods also
generate good variance estimates for data without interaction between operators and parts.
In summary, the Modified Classical GR&R and Modified Long Form methods are
much better than the Classical GR&R and Long Form methods in estimating the
gauge variance on either case of with or without interaction between parts and
operators.
6
Table 6. Repeatability, Reproducibility, and Gauge Estimate for Data
with Interaction Between Parts and Operators
ANOVA
Classical GR&R
Modified Classical
GR&R
Long Form
Modified Long Form
ˆ repeatability
ˆ reproducibility
ˆ gauge
3.506
2.261
3.863
1.462
5.216
2.692
2.261
4.592
5.119
2.261
2.261
1.418
4.579
2.668
5.1062
Table 7. Repeatability, Reproducibility, and Gauge Variance Estimates for
Data without Interaction Between Parts and Operators
ANOVA
Classical GR&R
Modified Classical
GR&R
Long Form
Modified Long Form
ˆ repeatability
ˆ reproducibility
ˆ gauge
2.576
3.783
3.983
1.418
4.743
4.039
3.783
1.949
4.255
3.783
3.783
1.285
1.855
3.995
4.213
Comparison
Now we will compare the gauge variance estimators of five different measurement
methods by simulating a measurement system based on the criterion of the biasness,
variance, and mean squares error (MSE) of the gauge variance estimate. We generated the
measurements according to the random effects model of equation (1) and calculated the
gauge variance estimates of the five methods. Then compare the gauge variance estimates
with its true value. The procedure of simulating the measurements is as follows:
1.
Choose the values of n, p, and k.
2.
Set the values of  ,
3.
4.
Simulate one set of measurements according to the values in step (1) and (2).
Compute the gauge variance estimates of the five methods (ANOVA, CRR, LF,
MCRR, and MLF).
Repeat step (3) and (4) for 10000 times.
Calculate the criterion values of each 10000 estimates for comparison.
5.
6.
 P2 ,  O ,  PO , and  R2 .
2
2
Criterions for Comparison
There are three criteria for comparison. They are MSE, variance, and mean ratio of
the gauge variance.
7
The MSE of gauge variance estimator is
n
 (ˆ
MSE =
i 1
2
gauge
2
  gauge
)2
(12)
n
where n is the number of simulation runs. In our example n is equal to 10000. The MSE is
a combined index of accuracy and precision of an estimator. The smaller of the MSE
value, the better the estimator is.
The sample variance of the estimated gauge variance is presented as Equation (13). The
smaller of it, the higher precision of the estimator and the narrower width of its
confidence interval are.
 n 2 
  ˆ gauge 
n
2

(ˆ gauge
) 2  n i 1



n
i 1


2


Var ( ˆ gauge ) =
n 1
2
(13)
The mean ratio of the estimated gauge variance to its true value is showed as Equation
(14). If the ratio is equal to 1, the estimator is then unbiased.
n
Mean ratio =
2
ˆ gauge

i 1
2
gauge
n
(14)
Result of Comparison
The research first compared the five methods under various combinations of npk values
for the case with interaction between parts and operators. We set process average μ to 0,
 P2
to 0,
2
2
 O2 to 1,  PO
to 1, and  R to 0.25. Ten thousand simulation runs are
conducted for various npk values. Sample variance, MSE, and mean ratio of the gauge
variance estimates are calculated for each of the five methods.
The result of mean ratios of the gauge variance estimates for the five different methods
are presented in Table8. It showed that ANOVA method produces unbiased gauge
variance estimator. The MCRR method generates overestimated estimator while MLF,
CRR, and LF methods generate underestimated estimators. They also revealed MCRR
and MLF estimates are much better than those of CRR and LF.
Table 9 showed the result of sample variances of gauge variance estimates. The MLF
estimate has the smallest variance while the LF estimate is the second smallest.
8
Table 8. Mean Ratios of Gauge Variance Estimates for Various npk Values under the
Case with Interaction Between Parts and Operators
npk
30
40
60
80
90
ANOVA
0.997
1.005
0.997
0.994
0.998
CRR
0.808
0.764
0.717
0.692
0.800
MCRR
1.153
1.124
1.096
1.098
1.113
LF
0.656
0.661
0.652
0.601
0.652
MLF
0.927
0.965
0.992
0.945
0.898
Table9. Sample Variances of Gauge Variance Estimates for Various npk Values under
the Case with Interaction Between Parts and Operators
npk
30
40
60
80
90
ANOVA
1.716
2.660
1.302
2.304
1.219
CRR
2.481
6.072
1.985
5.433
1.899
MCRR
2.461
5.033
1.832
4.337
1.710
LF
1.534
2.493
1.227
2.226
1.174
MLF
1.522
2.067
1.132
1.778
1.057
The results of mean square errors (MSE) of the gauge variance estimates are presented in
Table10. It showed that MLF method generates the best estimator while considering the
accuracy and precision together in spite that MLF estimator is biased.
In summary under the interaction between parts and operators, the ANOVA estimator is
unbiased while MLF estimate has the smallest variance. The MLF estimator is the best
while both accuracy and precision are taken into consideration. The MCRR and MLF
methods are much better than CRR and LF methods.
Table10. Mean Square Errors (MSE) of the Gauge Variance Estimates for Various npk
Values under the Case with Interaction Between Parts and Operators
npk
30
40
60
80
90
ANOVA
1.686
1.130
1.389
0.884
1.179
CRR
2.667
1.735
2.442
1.640
2.263
MCRR
2.522
1.525
2.063
1.168
1.711
LF
2.133
1.623
2.076
1.636
2.026
MLF
1.514
1.042
1.259
0.817
1.073
The research also compared the five methods under various combinations of npk values
for the case without interaction between parts and operators. Under the case without
interaction between parts and operators, ANOVA gauge variance estimator is unbiased as
expectation. Both ANOVA and MLF gauge variance estimators about have the same
precision. While taking accuracy and precision into consideration together, ANOVA and
MLF gauge variance estimators are the best and there is almost no difference between
them.
9
Conclusion
In this article, we modified both the Classical GR&R and Long Form methods to make
them both suitable to either case of with or without interaction between parts and
operators. After comparing the five estimation methods (ANOVA, Classical GR&R, Long
Form, Modified Classical GR&R, and Modified Long Form methods) by simulating a
measurement process and calculating the gauge variance estimates for each method, it
showed the Modified Long Form method is a very good choice to estimate gauge
variability no matter there is interaction between parts and operators or not.
Reference
1. AIAG Editing Group, “Measurement Systems Analysis-Reference Manual (MSA)”,
Automotive Industries Action Group, 1991.
2. Barrentine, L. B., Concepts for R&R Studies, ASQC Quality Press, Milwaukee, WI.
3. Burdick, R. K.; Larsen, G. A. and Allen, A. Elizabeth, “Comparing Variability of
Two Measurement Process Using R&R Studies,” Journal of Quality Technology,
Vol.34, No.1, pp.97-105 2002
4. Dolezal, K. K. and Burdick, R. K. and Birch, N. J., “Analysis of a Two-Factor R&R
Study with Fixed Operators,” Journal of Quality Technology, Vol.30, No.2, pp.163170 1998
5. Montgomery, D. C. and Runger, G. C., “Gauge Capability Analysis and Designed
Experiments. Part II: Experimental Design Models and Variance Component
Estimation”, Quality Engineering, Vol.6, No.2, pp.289-305 1993a
6. Montgomery, D. C. and Runger, G. C., “Gauge Capability Analysis and Designed
Experiments. Part I: Basic Methods”, Quality Engineering, Vol.6, No.1, pp.115-135
1993b