The Analysis and Comparison of Gauge Variance Estimators

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

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

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

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.

Literature Definition • Repeatability ：The variability of gauge itself. Same operator measures same part. Repeatabilty • Reproducibility：The variability due to different operators using the same gauge. Different operators measures same part. Reproducibility

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

Literature 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 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： • Xijkis the kth repeated measurement on the ith part by the jthoperator. Pi is the ith part effect. Oj is the jth operator effect. POijis the interaction. Rijk is the error term. Random factorsarenormally distributed with mean 0 and constant variances.

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

Literature 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 Methods for estimating gauge variance ClassicalGR&R • Montgomery and Runger （1993a）called it “Classical Gauge Repeatability 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 the jth operator and d2 is determined by the number of operators.

Literature 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： whereis in appendix B（g=1，m=number of operators）

Literature Repeatability and Reproducibility Estimators

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:

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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.

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:

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）。

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.

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.

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.

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.

n 為 15 , 20 和 25 p 為 2 , 3 和 4 k 為 2 和 3 Simulation result and comparison of estimators 程式模擬流程圖

Eight gauge variance estimators for comparison

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

comparison of estimators For the case with interaction • Mean ratios of estimated gauge variances under various npk values • 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. 不同參數組合數之量測總變異的平均真值比之比較圖

comparison of estimators For the case with interaction • Variance of estimated gauge variances under various npk values • 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. 不同參數組合數之總變異的變異數比較圖

comparison of estimators For the case with interaction • MSE of estimated gauge variances under various npk values • 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. 不同參數組合數之量測總變異的均方誤差之比較圖

comparison of estimators 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（20,2,3）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. （20,3,2）量測總變異的均方誤差比較圖

comparison of estimators For the case without interaction • Mean ratios of estimated gauge variances under various npk values • 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. 不同參數組合數之量測總變異的平均真值比之比較圖

For the case without interaction comparison of estimators • Variance of estimated gauge variances under various npk values • 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. 不同參數組合數之總變異的變異數比較圖

comparison of estimators For the case without interaction • MSE of estimated gauge variances under various npk values • 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. 不同參數組合數之量測總變異的均方誤差之比較圖

comparison of estimators For the case without 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（20,2,3）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. （20,3,2）量測總變異的均方誤差比較圖

Conclusion • MLFN1 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.

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.

Thanks for your attention

The Analysis and Comparison of Gauge Variance Estimators