1. 1 The Analysis and Comparison of Gauge Variance Estimators Peng-Sen Wang and Jeng-Jung Fang Southern Taiwan University of Technology Tainan, Taiwan

2. 2 Content  3 Criterions for comparison  8 estimators for comparison  Background  Objectives  Assumptions  Literatures － Definitions － References － Methods for Estimating Gauge Variance

3. 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 Studies Long Form Before doing GR&R research, 3 parameters must be decided. n: number of parts, p: number of operators, k :number of repetitions

4. 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. 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. Repeatabilty Reproducibility

6. 6 Definition Gauge Repeatability and Reproducibility : (GR&R ）： The overall performance of gauge capability, call it measurement variation. Literature

7. 7 GR&R related reference  AIAG Editing Group (1991), “Measurement Systems Analysis-Reference Manual （ MSA ） ”,1 nd 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. 8 Methods for estimating gauge variance ANOVA  Based 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 ：  X ijk is the k th repeated measurement on the i th part by the j th operator. P i is the i th part effect. O j is the j th operator effect. PO ij is the interaction. R ijk is the error term. Random factors are normally distributed with mean 0 and constant variances. Literature

9. 9 ANOVA  When the interaction exists, the unbiased estimators for gauge capability is: Literature ANOVA of random effects model

10. 10 ANOVA  If, usually define it 0. Assume that no interaction exists. A reduced model is fitted as:  Without interaction existing, the estimators for gauge capability are: Literature

11. 11 Methods for estimating gauge variance Classical GR&R  Montgomery and Runger （ 1993a ） called it “Classical Gauge Repeatability and Reproducibility Study” 。  Estimator for repeatability ： where d 2 is determined by the number of repetitions k.  Estimator for reproducibility ： where, is the overall average of the j th operator and d 2 is determined by the number of operators. Literature

12. 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

13. 13 Repeatability and Reproducibility Estimators Literature

14. 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:

15. 15 Revised Classical GR&R and Long Form Methods Measurement Layout

16. 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.

17. 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:

18. 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 Variance  Variance of Estimated Gauge Variance  Mean Squares Error of Estimated Gauge Variance, (MSE ）。

19. 19 Criterions for comparing GR&R estimators  Mean Ratio  To evaluate accuracy of estimator to its true value (Unbiasedness)  The equation is ：  Decision ： The closer the ratio to 1, the more accurate the estimator is.

20. 20 Criterions for comparing GR&R estimators  Variance of gauge variance estimate  After 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.

21. 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 the estimator. MSE combines accuracy and precision into one index.  Equation ：  Decision ： The smaller the MSE, the more accurate and precise the estimator is.

22. 22 Criterions for comparing GR&R estimators  MSE  Bickel and Doksum （ 1977 ） points out that MSE both 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. 23 Simulation result and comparison of estimators 程式模擬流程圖 n 為 15, 20 和 25 p 為 2, 3 和 4 k 為 2 和 3

24. 24 Eight gauge variance estimators for comparison

25. 25 Simulation result and comparison of estimators Data from the case study of Montgomery (1993a ）

26. 26 Simulation result and comparison of estimators Data from the case study of Montgomery (1993a ）

27. 27 For the case with interaction Mean 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.

28. 28 For the case with interaction Variance of estimated gauge variances under various npk values comparison of estimators 不同參數組合數之總變異的變異數比較圖 MLF N1, MLF L,and MLF N2 methods have the smallest variances. ANOVA and LF are the second. MCRR N, MCRR L, 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.

29. 29 For the case with interaction MSE 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均方誤差之比較圖 MLF N2, MLFL, and MLF N1 methods have the smallest MSE values while ANOVA and LF methods are the second. MCRR N, MCRR L, 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.

30. 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.

31. 31 For the case without interaction Mean 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 the most closest to 1 and is the best one. LF, MLF N1, MLF L, and MLF N2 methods are close to one another, and there is only little difference among them and ANOVA method. CRR, MCRR N, and MCRR L methods are the worst. LF, ANOVA, MLF N1, MLF L, and MLF N2 won’t change as the npk increases while MCRR L, MCRR N, and CRR get closer to true value.

32. 32 For the case without interaction Variance of estimated gauge variances under various npk values comparison of estimators 不同參數組合數之總變異的變異數比較圖 ANOVA, MLF N1, MLF L, MLF N2, and LF methods are close to one another. CRR, MCRR N, MCRR L 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.

33. 33 For the case without interaction MSE 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均方誤差之比較圖 ANOVA, MLF N1, MLF L, MLF N2, and LF methods are the same good. CRR, MCRR N, and MCRR L 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.

34. 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.

35. 35 Conclusion MLF N1 and MLF N2 are good estimators both in the cases of with interaction and without interaction. MLF N2 method is a little better than MLF N1. Under the case with interaction, MLF N1, MLF N2, and MCRR N methods are better than Classical R&R and Long Form methods. MLF N2 estimator is the same good as ANOVA method. Suggest using MLF N2 method, both its accuracy and precision are the same good as ANOVA method no matter there is interaction or not.

36. 36 Conclusion 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. 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. 37 Thanks for your attention