Upload
letitia-sutton
View
238
Download
0
Tags:
Embed Size (px)
Citation preview
1
The Analysis and Comparison of Gauge Variance Estimators
Peng-Sen Wang and Jeng-Jung Fang
Southern Taiwan University of Technology
Tainan, Taiwan
2
Content3 Criterions for
comparison8 estimators for comparison
BackgroundObjectivesAssumptionsLiteratures
- Definitions- References- Methods for Estimating Gauge Variance
3
Background and Objectives
The precision of measurement system will affect the quality of statistical analysis.3 methods for estimating GR&R varaince : ANOVA Classical GR&R StudiesLong Form
Before doing GR&R research, 3 parameters must be decided. n: number of parts, p: number of operators, k :number of repetitions
4
Assumptions
Parts can be measured repeatedly.Quality characteristic is quantitative.Single quality characteristic.Quality characteristic is normally distributed.Independent measurements among parts.Other factors are controllable.
5
Definition
Repeatability : The variability of gauge itself. Same operator measures same part.
Literature
Reproducibility : The variability due to different operators using the same gauge. Different operators measures same part.
重複性
Repeatabilty
量測人員B
再現性
量測人員A
量測人員C
Reproducibility
rept
6
Definition
Gauge Repeatability and Reproducibility : (GR&R ): The overall performance of gauge capability, call it measurement variation.
Literature
7
GR&R related reference
AIAG Editing Group (1991), “Measurement Systems Analysis-Reference Manual ( MSA )” ,1nd ed., Automotive Industries Action Group.
Barraentine, L. B. (1991), “Concepts for R&R Studies”, ASQC Quality Press , Milwaukee, Wisconsin.
Montgomery, D. C. and Runger, G. C. (1993a), “Gauge Capability Analysis and Designed Experiments. Part I : Basic Methods”, Quality Engineering, Vol.6, No.1, pp.115-135.
Montgomery, D. C. and Runger, G. C. (1993b), “Gauge Capability Analysis and Designed Experiments. Part II : Experimental Design Models and Variance Component Estimation”, Quality Engineering, Vol.6, No.2, pp.289-305.
Literature
8
Methods for estimating gauge variance ANOVABased on the ANOVA model of Montgomery and Runger
( 1993b.).Two-factor random effects model : One factor is part
( P ) with n levels, the other is operator (O) with p level. With k repeated measurements for each combination, the linear model is :
Xijkis the kth repeated measurement on the ith part by the jth operator. Pi is the ith part effect. Oj is the jth operator effect. POij is the interaction. Rijk is the error
term. Random factors are normally distributed with mean 0 and constant variances.
Literature
k k
pj
ni
RPOOPX ijkijjiijk
, 2, 1,
, 2, 1,
, 2, 1,
9
ANOVA
When the interaction exists, the unbiased estimators for gauge capability is:
Literature
變異來源Source of
Variability
平方和
Sum of
Squares
自由度
Degrees of
Freedom
均方
Mean
Squares
期望均方
Expected
Mean Squares
產品
Parts SSp N-1 MSp
222PPORP pkkMSE
量測人員Operators
SSo P-1 MSo 222
OPORO nkkMSE
產品×量測人員
Parts×Operators SSpo (n-1)(p-1) MSpo 22
PORPO kMSE
誤差項
Error SSR np(k-1) MSR 2
RRMSE
總和
Total SST npk-1
ANOVA of random effects model
nkMSknMSnMS
nknMSMSnMS
MS
RPOOilityreproducibityrepeatabilgauge
RPOOPOOilityreproducib
RRityrepeatabil
11ˆˆˆ
1ˆˆˆ
ˆˆ
222
222
22
10
ANOVA If , usually define it 0. Assume that no interaction e
xists. A reduced model is fitted as:
Without interaction existing, the estimators for gauge capability are:
Literature
0ˆ 2 po
κ k
p j
n i
ROPX ijkjiijk
,2,1
,,2,1
,,2,1
nkMSnkMS
nkMSMS
MS
ROilityreproducibityrepeatabilgauge
ROOilityreproducib
RRityrepeatabil
1ˆˆˆ
ˆˆ
ˆˆ
222
22
22
11
Methods for estimating gauge variance Classical GR&R Montgomery and Runger ( 1993a ) called it “Classical Gauge R
epeatability and Reproducibility Study” 。
Estimator for repeatability :
where d2 is determined by the number of repetitions k.
Estimator for reproducibility :
where , is the overall average of t
he jth operator and d2 is determined by the number of operators.
2
ˆd
Rityrepeatabil
Literature
2
ˆd
RX
ilityreproducib
jj
jjX
XXR minmax jX
12
Methods for estimating gauge variance Long Form Method Introduced in the MSA manual of QS 9000 system without interaction being considered.The repeatability and reproducibility estimators are:
where is in appendix B(g=1,m=number of operators)
Literature
nk
dR
d
RX
ilityreproducib
2
2
2
*2
ˆ
2d
2d
Rityrepeatabil
13
Repeatability and Reproducibility Estimators
Literature
標準差
方法 ityrepeatabil ilityreproducib
變異數分析法
RMS nknMSMSnMS RPOO 1 (交互作用)
nkMSMS RO (交互作用不顯著)
傳統 GR&R 2d
R
2d
RX
長表格 2d
R
nk
dR
d
RX
2
2
2
*2
14
Revised Classical GR&R and Long Form Methods
Classical GR&R and Long Form methods can’t be used under the cases with interaction between operators and parts.
Adjust the estimator of reproducibility as:
n
R
R
n
iX
X
ij
ij
1
其中,
pj
ni
k
X
X
k
kijk
ij ,,1
,,1,1
niXXR ijj
ijjX ij
,,1,minmax
15
Revised Classical GR&R and Long Form Methods
Measurement Layout
量測人員 1 量測人員 2 … 量測人員 p 人員
重
產 複
品 量測值 平均 全距 量測值 平均 全距 量測值 平均 全距
x111 x112 x121 x122 … x1p1 x1p2 1
… x11k 11X 11R
… x12k 12X 12R
… x1pk pX 1 pR1
x211 x212 x221 x222 x2p1 x2p2 2
… x21k 21X 21R
… x22k 22X 22R
… x2pk pX 2 pR2
… … … … … … … … … … … … … …
xn11 xn12 xn21 xn22 xnp1 xnp2 n
… xn1k 1nX 1nR
… xn2k 2nX 2nR
… xnpk npX npR
1X 1R 2X 2R … pX pR
16
Revised Classical GR&R and Long Form Methods
Lin(2005) revised Classical GR&R and Long Form methods as:
Montgomery and Runger (1993a) mentioned . Thus in the research, the estimators for GR&R are revised as the following to make them unbiased.
2
'ˆd
R ijX
ilityreproducib
nk
dR
d
RijX
ilityreproducib
2
2
2
*2
"ˆ
21222' )ˆ(E nRPOOilityreproducib
n
dR
d
R ijXilityreproducib
2
2
2
2
'ˆ
n
dR
d
RijX
ilityreproducib
2
2
2
*2
"ˆ
17
Revised Classical GR&R and Long Form Methods
Burdick and Larsen ( 1997 ) found the number of operators have major effect on the confidence interval of repeatability and reproducibility. Jiang ( 2002 ) proposed more operators under the same npk vlaue. Based on the two researches, the reproducibility estimator of Long Form method is revised as:
npk
dR
d
RijX
ilityreproducib
2
2
2
*2
'"ˆ
18
Criterions for comparing GR&R estimators
Assume repeatability and reproducibility are known, simulate N runs to calculate the average values of repeatability, reproducibility, and total gauge variance.The criterions were used in the research:Mean Ratio of Estimated Gauge VarianceVariance of Estimated Gauge VarianceMean Squares Error of Estimated Gauge Variance,
(MSE )。
19
Criterions for comparing GR&R estimatorsMean Ratio To evaluate accuracy of estimator to its true value (Unbiased
ness)The equation is :
Decision : The closer the ratio to 1, the more accurate the estimator is.
N
N
i gauge
gauge
12
2ˆ
模擬次數真值
量測總變異估計變量
20
Criterions for comparing GR&R estimators
Variance of gauge variance estimateAfter simulating N runs, N gauge variance estimates are
obtained and its variance is computed. It is used to evaluate the precision of the gauge variance estimator. The equation is:
Decision : The smaller the variance, the more precise the estimator is, and the narrower its confidence is.
1
ˆˆ
1
2
1
2
22
N
NN
N
i
N
igauge
gauge
21
Criterions for comparing GR&R estimators Mean Square Errors ( MSE )
MSE is composed of two parts:
shows the precision while bias measures the accuracy of th
e estimator. MSE combines accuracy and precision into one index.
Equation :
Decision : The smaller the MSE, the more accurate and precise the estimator is.
2
222
BiasˆVar
ˆˆˆˆMSE
EEEEE
Var
N
N
igauagegauge
2
1
222 ˆ
模擬次數真值量測總變異估計變異
22
Criterions for comparing GR&R estimators
MSEBickel and Doksum ( 1977 ) points out that MSE bo
th considers accuracy and precision. The estimator with minimum MSE indicates that it is a best estimator.
The research used MSE as a major criterion for comparing estimators while considering mean ratio and variance of estimated gauge variance as supplementary rules.
23
Simulation result and comparison of estimators
程式模擬流程圖
模擬開始
設定產品數 n、量測人員數 p、量測重複次數 k
設定 、 及 2O 2
PO 2R
產生模擬量測值
估算重複性變異再現性變異量測總變異
量測再現性變異和總變異的平均真值比量測再現性變異和總變異的變異數量測再現性變異和總變異的均方誤差
模擬10000次
模擬結束
n 為 15 , 20 和 25p 為 2 , 3 和 4k 為 2 和 3
5.025.0
5.0
21
2
2
2
和 為
1和 為
和 為
R
po
O
24
Eight gauge variance estimators for comparison
標準差
方法 ityrepeatabil ilityreproducib
變 異 數 分 析 法(ANOVA)
RMS
nknMSMSnMS RPOO 1 (交互作用)
nkMSMS RO (交互作用不顯著)
傳統 GR&R
(CRR) 2d
R
2d
RX
長表格(LF) 2d
R
nk
dR
d
RX
2
2
2
*2
林郁智(2005)Modified Classical
GR&R(MCRRL) 2d
R
2d
R ijX
標準差
方法 ityrepeatabil ilityreproducib
林郁智(2005)Modified Long
Form(MLFL) 2d
R
nk
dR
d
RijX
2
2
2
*2
Modified Classical
GR&R(MCRRN) 2d
R
n
dR
d
R ijX2
2
2
2
Modified Long
Form(MLFN1) 2d
R
n
dR
d
RijX
2
2
2
*2
Modified Long
Form(MLFN2) 2d
R
npk
dR
d
RijX
2
2
2
*2
25
Simulation result and comparison of estimators
Data from the case study of Montgomery (1993a )
GR&R 估算方法的之比較(交互作用不顯著)
ANOVA CRR LF MCRRL MLFL MCRRN MLFN1 MLFN2 2ˆ ityrepeatabil 0.88316 1.03883 1.03883 1.03883 1.03883 1.03883 1.03883 1.03883
2ˆ ilityreproducib 0.01063 0.03687 0.00298 0.33182 0.23461 0.27988 0.20864 0.25192 2ˆ gauge 0.89379 1.07570 1.04182 1.37065 1.27344 1.31871 1.24747 1.29076
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Repeatability Reproducibility Gauge
變異數估計值
ANOVA
CRR
LF
MCRRL
MLFL
MCRRN
MLFN1
MLFN2
26
Simulation result and comparison of estimators
Data from the case study of Montgomery (1993a )
GR&R 估算方法的之比較(交互作用顯著)
ANOVA CRR LF MCRRL MLFL MCRRN MLFN1 MLFN2 2ˆ ityrepeatabil 0.81111 0.85673 0.85673 0.85673 0.85673 0.85673 0.85673 0.85673
2ˆ ilityreproducib 1.95556 0.32617 0.22759 1.90040 1.46385 1.81473 1.40673 1.48289 2ˆ gauge 2.76667 1.18290 1.08432 2.75713 2.32058 2.67146 2.26346 2.33962
0
0.5
1
1.5
2
2.5
3
Repeatability Reproducibility Gauge
變異數估計值
ANOVA
CRR
LF
MCRRL
MLFL
MCRRN
MLFN1
MLFN2
27
For the case with interactionMean ratios of estimated gauge variances under various npk values
comparison of estimators
不同參數組合數之量測總變異的平均真值比之比較圖
ANOVA estimator is most closest to 1 and is the best one. LF estimator is the worst one.
The estimators of LF and ANOVA won’t changed with the increase of npk values. Other estimators will be closer to the true value as the npk values increase.
0.600
0.800
1.000
1.200
1.400
1.600
60 80 90 100 120 135 150 160 180 200 225 240 300
npk參數組合
平均真值比
ANOVA
CRR
LF
MCRRL
MLFL
MCRRN
MLFN1
MLFN2
28
For the case with interactionVariance of estimated gauge variances under various npk v
alues
comparison of estimators
不同參數組合數之總變異的變異數比較圖
MLFN1, MLFL,and MLFN2 methods have the smallest variances. ANOVA and LF are the second. MCRRN, MCRRL, and CRR are the worst. All the variances decreases as the npk values increase.
When the npk value equals 160, all the variance estimators decrease rapidly and then become steady thereafter.
0.000
2.000
4.000
6.000
8.000
10.000
12.000
14.000
60 80 90 100 120 135 150 160 180 200 225 240 300
npk參數組合
變異數
ANOVA
CRR
LF
MCRRL
MLFL
MCRRN
MLFN1
MLFN2
29
For the case with interactionMSE of estimated gauge variances under various npk value
s
comparison of estimators
不同參數組合數之量測總變異的均方誤差之比較圖
MLFN2, MLFL, and MLFN1methods have the smallest MSE values while ANOVA and LF methods are the second. MCRRN, MCRRL, and CRR are the worst ones. All the MSE values decrease with the increase of npk vlaues.
When the npk value equals 160, all the variance estimators decrease rapidly and then become steady thereafter.
0.000
2.000
4.000
6.000
8.000
10.000
12.000
14.000
16.000
60 80 90 100 120 135 150 160 180 200 225 240 300
npk參數組合
均方誤差
ANOVA
CRR
LF
MCRRL
MLFL
MCRRN
MLFN1
MLFN2
30
For the case with interaction The MSE values of estimated gauge variances while npk equals 120.
comparison of estimators
( 15,4,2 )量測總變異的均方誤差比較圖
( 20,2,3 )量測總變異的均方誤差比較圖
( 20,3,2 )量測總變異的均方誤差比較圖
Given npk value being fixed, increasing the number of operators is suggested first. The second choice is to increase the number of parts. Increasing the number of repetitions is not recommended.
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
1.75 2 2.25 2.5 2.75 3 3.25 3.5
量測總變異
均方誤差
ANOVA
CRR
LF
MCRRL
MLFL
MCRRN
MLFN1
MLFN2
0
1
2
3
4
5
6
7
8
1.75 2 2.25 2.5 2.75 3 3.25 3.5
量測總變異
均方誤差
ANOVA
CRR
LF
MCRRL
MLFL
MCRRN
MLFN1
MLFN2
0
5
10
15
20
25
1.75 2 2.25 2.5 2.75 3 3.25 3.5
量測總變異
均方誤差
ANOVA
CRR
LF
MCRRL
MLFL
MCRRN
MLFN1
MLFN2
31
For the case without interaction Mean ratios of estimated gauge variances under various n
pk values
comparison of estimators
不同參數組合數之量測總變異的平均真值比之比較圖
ANOVA estimator is the most closest to 1 and is the best one. LF, MLFN1, MLFL, and MLFN2 methods are close to one another, and there is only little difference among them and ANOVA method. CRR, MCRRN, and MCRRL methods are the worst.
LF, ANOVA, MLFN1, MLFL, and MLFN2 won’t change as the npk increases while MCRRL, MCRRN, and CRR get closer to true value.
0.600
0.800
1.000
1.200
1.400
1.600
60 80 90 100 120 135 150160 180 200 225 240 300
npk參數組合
平均真值
比
ANOVA
CRR
LF
MCRRL
MLFL
MCRRN
MLFN1
MLFN2
32
For the case without interactionVariance of estimated gauge variances under various npk v
alues
comparison of estimators
不同參數組合數之總變異的變異數比較圖
ANOVA, MLFN1, MLFL, MLFN2, and LF methods are close to one another. CRR, MCRRN, MCRRLare the worst.
All the variances decreases as the npk values increase.
When the npk value equals 160, all the variance estimators decrease rapidly and then become steady thereafter.
0.000
2.000
4.000
6.000
8.000
10.000
12.000
14.000
60 80 90 100 120135 150160180 200225 240300
npk參數組合
變異數
ANOVA
CRR
LF
MCRRL
MLFL
MCRRN
MLFN1
MLFN2
33
For the case without interactionMSE of estimated gauge variances under various npk value
s
comparison of estimators
不同參數組合數之量測總變異的均方誤差之比較圖
ANOVA, MLFN1, MLFL, MLFN2, and LF methods are the same good. CRR, MCRRN, and MCRRL are the worst.
All the variances decreases as the npk values increase.
When the npk value equals 160, all the variance estimators decrease rapidly and then become steady thereafter.
0.0002.0004.0006.0008.00010.00012.00014.00016.000
60 80 90 100 120135 150160180 200225 240300
npk參數組合
均方誤差
ANOVA
CRR
LF
MCRRL
MLFL
MCRRN
MLFN1
MLFN2
34
For the case without interaction The MSE values of estimated gauge variances while npk equals 120.
comparison of estimators
( 15,4,2 )量測總變異的均方誤差比較圖
( 20,2,3 )量測總變異的均方誤差比較圖
( 20,3,2 )量測總變異的均方誤差比較圖
Given npk value being fixed, increasing the number of operators is suggested first. The second choice is increasing the number of parts. Increasing the number of repetitions is not recommended.
0
1
2
3
4
5
1.25 1.5 2.25 2.5
量測總變異
均方誤差
ANOVA
CRR
LF
MCRRL
MLFL
MCRRN
MLFN1
MLFN2
0
5
10
15
20
25
1.25 1.5 2.25 2.5
量測總變異
均方誤差
ANOVA
CRR
LF
MCRRL
MLFL
MCRRN
MLFN1
MLFN2
0
2
4
6
8
1.25 1.5 2.25 2.5
量測總變異
均方誤差
ANOVA
CRR
LF
MCRRL
MLFL
MCRRN
MLFN1
MLFN2
35
ConclusionMLFN1 and MLFN2 are good estimators both in the
cases of with interaction and without interaction. MLFN2 method is a little better than MLFN1.
Under the case with interaction, MLFN1, MLFN2, and
MCRRN methods are better than Classical R&R and
Long Form methods. MLFN2 estimator is the same good as
ANOVA method.
Suggest using MLFN2 method, both its accuracy and
precision are the same good as ANOVA method no matter there is interaction or not.
36
ConclusionGiven npk value being fixed, increasing the num
ber of operators is suggested first. The second choice is increasing the number of parts. Increasing the number of repetitions is not recommended.
At least three operators is suggested so that the variance and MSE of estimated gauge variance will be small enough.An npk value of 160 is suggested so that the variance
and MSE of estimated gauge variance decrease rapidly and then become steady thereafter.
37
Thanks for your attention