1 the analysis and comparison of gauge variance estimators peng-sen wang and jeng-jung fang southern...

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


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

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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


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 ）。

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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


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


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


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


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


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


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


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


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.


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.


（ 15,4,2 ）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


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


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.


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


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.


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.





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　

1 the analysis and comparison of gauge variance estimators peng-sen wang and jeng-jung fang southern...

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