Qual Quant (2012) 46:55–69 DOI 10.1007/s11135-010-9326-8 RESEARCH PAPER
Confidence intervals for two-dimensional data with circular tolerances in a gauge R&R study Fu-Kwun Wang · H. L. Chern
Published online: 31 March 2010 © Springer Science+Business Media B.V. 2010
Abstract In this paper, we discuss our application of the Bootstrap method to construct the confidence interval of the diameter for two-dimensional data with circular tolerances in a gauge repeatability and reproducibility study. Factors simulated to validate performance include: the variance component, and sample size. The simulation results show that the Bootstrap method can cover the stated nominal coefficient in most scenarios. There exists a positive correlation between width of confidence intervals and variance components; the width of confidence intervals for diameters is increased when the variance components (σˆ x2 , σˆ y2 or σˆ x2y ) are increased. The coverage proportion is not significantly affected by variance-components. Also, the width of confidence interval for the diameter and coverage proportion is not significantly affected by sample size. One real example based on a nested design is used to demonstrate the application of the proposed method. Keywords Bootstrap · Confidence interval · Two-dimensional data · Gauge repeatability and reproducibility
1 Introduction The purpose of a measurement system analysis (MSA) is to separate the variation among devices being measured from the error in the measurement system. Here, the measurement system error could be the combination of gauge bias, repeatability, reproducibility, stability, and linearity (Automotive Industry Action Group 2002). Most gauge repeatability and reproducibility (GR&R) studies are focused on a single quality characteristic. However, multiple quality characteristics in a GR&R study can occur. The development of new technologies, as well as upgrades in industry, can be linked with the success of the metrology. In order to improve quality, it is necessary to obtain accurate measurements; thus, it is important for
F.-K. Wang (B) · H. L. Chern Department of Industrial Management, National Taiwan University of Science and Technology, 43 Keelung Rd., Sec.4, Taipei 106, Taiwan e-mail:
[email protected]
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quality engineers to ensure accurate measurements. Burdick et al. (2003) provided a beneficial review of possible methods that can be used to conduct and analyze measurement system capability studies; these methods are based on the analysis of variance approach. Larsen (2003) presented a study that extended the univariate single-instance case to common manufacturing test scenarios; in this study, multiple parameters are tested on each device with a sequence of tests, and may include retest, test, and repair steps. Voelkel (2003) proposed a new circle-diameter measure to estimate the gauge variability for two-dimensional data with circular tolerance. Engineers are often interested in constructing a confidence interval of the interest parameter to provide a range for the measurement system. Wang and Li (2003) showed that the Bootstrap method can be used to obtain the confidence intervals of gauge variability when the control chart is used to find the point estimates. A practical example is provided to show the application of this control chart with the Bootstrapping method, and comparisons are made; comparisons are also made with three experimental design procedures in terms of point estimates and confidence intervals for repeatability, reproducibility and total gauge variability. Burdick et al. (2005a,b) provided the generalized inference method for constructing the misclassification rates of confidence intervals in a GR&R study. The simulation results suggest a minimum of six operators and 20–50 parts for random two-factor designs. In this paper, we use the Bootstrap method to investigate the accuracy of the confidence interval for the circle-diameter with circular tolerances. Section 2 describes the diameter for two-dimensional data with circular tolerance. Section 3 presents the Bootstrap method. Section 4 contains the simulation study and results. Section 5 provides a pragmatic example based on a nested design in order to demonstrate the application of the proposed method. Finally, we provide our conclusion and suggest future research directions.
2 Measurements of the circle-diameter When a GR&R study is based on multiple quality characteristics (say m) with a multivariate normal distribution that employ a random three-factor nested design usually have parts, operators, and days as factors. Typically, several operators (say I operators) are chosen at random on different days (i.e. day is nested; say J days) to conduct measurements on the randomly selected parts (say K parts) from a manufacturing process. Each part is measured L times (say L trials) by each operator. Thus, the model is given by ⎫ ⎧ i = 1, 2, . . . , I ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ j = 1, 2, . . . , J Y = μ + τi + β j(i) + γk + (τ γ )ik + (βγ ) jk(i) + ε(i jk)l (1) k = 1, 2, . . . , K ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ l = 1, 2, . . . , L T T where the vectors are all of order m × 1, Y = (Y
1 , Y2 , . . . , Ym ) , μ = (μ1 , μ2 , . . . , μm ) is β , γ , τ γ a constant vector, τ ∼ N ∼ N 0, ∼ N 0, ∼ N 0, ), (0, i
τ j(i) β k γ ik τγ , βγik(i) ∼ N 0, βγ and εi jk(l) ∼ N (0, ε ) which all the random vectors are statistically independent. In any process involving the measurement of manufactured products, some of the observed variability occurs as a result of the variability of the product itself; others will be a result of either measurement error or gauge variability. When we translate the above components of variance into GR&R notation, we have
r epeatabilit y = E V = ε
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(2)
Confidence intervals for two-dimensional data
57
where r epeatabilit y is the repeatability of the observed process (i.e. equipment variation; EV). repr oducibilit y = AV = τ + τ γ + β + βγ
(3)
where repr oducibilit y is the reproducibility of the observed process(i.e. appraiser variation; AV). par t = γ
(4)
where par t is the component of variance due to the part. Furthermore, the variability R&R is the sum of two variance components (e.g. E V and AV ), say R&R = E V + AV
(5)
Also, the total variability of the observed process is given by total = R&R + par t
(6)
The components of variance can be estimated using the standard multivariate analysis of variance (MANOVA) method of moments. Thus, the estimators of the random effects model are given as follows: ˆ err or = ˆ EV = ˆ day× par t(operator ) = ˆ day(operator ) = ˆ operator × par t = ˆ par t = ˆ operator =
SSCPerr or IJK (L − 1)
(7)
SSCPday× par t(operator ) SSCPerr or − LI (J − 1) (K − 1) LIJK (L − 1) SSCPday(operator ) SSCPday× par t(operator) − KLI (J − 1) KLI (J − 1) (K − 1) SSCPoperator × par t SSCPday× par t(operator) − JL (I − 1) (K − 1) JLI (J − 1) (K − 1) SSCP par t SSCPoperator × par t − IJL (K − 1) IJL (I − 1) (K − 1) SSCPoperator SSCPoperator × par t SSCPday(operator ) − − JKL (I − 1) JKL (I − 1) (K − 1) JKLI (J − 1) SSCPday× par t(operator ) − JKLI (J − 1) (K − 1)
(8) (9) (10) (11)
(12)
where I
SSCPoperator = JKL ( y¯i... − y¯.... ) ( y¯i... − y¯.... ) , i=1
J I SSCPday(operator ) = KL y¯i j.. − y¯i... y¯i j.. − y¯i... , i=1 j=1 K
SSCP par t = IJL ( y¯..k. − y¯.... ) ( y¯..k. − y¯.... ) , k=1 I
K
SSCPoperator × par t = JL ( y¯i.k. − y¯i... − y¯..k. + y¯.... ) ( y¯i.k. − y¯i... − y¯..k. + y¯.... ) , i=1 k=1
J K I SSCPday× par t(operator ) = L y¯i jk. − y¯i j.. − y¯i.k. + y¯i... i=1 j=1 k=1
× y¯i jk. − y¯i j.. − y¯i.k. + y¯i... ,
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J K L I SSCPerr or = yi jkl − y¯i jk. yi jkl − y¯i jk. , i=1 j=1 k=1 l=1
J K L I SSCPtotal = yi jkl − y¯.... yi jkl − y¯.... , i=1 j=1 k=1 l=1
J K L I K L K L 1 1 yi jkl , y¯. j.. = IKL yi jkl , y¯i j.. = KL yi jkl , j=1 k=1 l=1 i=1 k=1 l=1 k=1 l=1 I I J K L J L 1 1 y¯.... = IJKL yi jkl , y¯..k. = IJL yi jkl , i=1 j=1 k=1 l=1 i=1 j=1 l=1 J L L 1 y¯i.k. = JL yi jkl , and y¯i jk. = L1 yi jkl . j=1 l=1 l=1
y¯i... =
1 JKL
Therefore, we can obtain ˆ repr oducibilit y = ˆτ + ˆ τγ + ˆβ + ˆ βγ ˆ EV + ˆ AV ˆ R&R = ˆ total = ˆ R&R + ˆ par t
(13) (14) (15)
Let us consider a 2D data measured in the r(ounce-inches), θ (◦ ) format. Voelkel (2003) provided the formula to convert the variables r and θ . We can use r = x 2 + y 2 and θ = arctan (y/x) to convert from Cartesian to polar coordinates. That is, the raw data is resolved into the x (ounce-inches), y (ounce-inches) format, where x = r × cos θ and y = r × sin θ . X Then, these data can be modeled with a bivariate normal distribution, ∼ N (μ, ) Y 2 2 σx σx y μx where μ = is a constant vector, and = 2 σ 2 . Here, the simple formula of μy σ yx y circle-diameter can be determined by an approach from Sweeney (2007). The procedures of computing the diameter at 99% capture circle are as follows: 2 2 ˆ = σˆ x2 σˆ x2y , (1) Compute two eigenvalues (λ1 ,λ2 ) from the sample covariance matrix σˆ σˆ √ √ yx y 1 1 2 2 2 2 which are λ1 = 2 σˆ x + σˆ y + D and λ2 = 2 σˆ x + σˆ y − D , where 2 2 2 2 2 2 and λ1 ≥ λ2 ≥ 0. D = σˆ x + σˆ y − 4 σˆ x σˆ y − σˆ x2y (2)
The diameter of interest is given by d = 2 (λ1 + λ2 ) Tˆ99
(3)
61707000y 2 − 302960y + 6059400 3316.6y + 37234 Modification for non-negative definite ˆ :
where Tˆ99 =
(1) (2) (3)
(16)
0.3 and y =
λ1 λ1 +λ2
− 0.5.
If σˆ x2 < 0, set σˆ x2 = σˆ x2y = 0. Go to (2). If σˆ y2
