8.5
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8.5 Confidence Intervals for Cp and Cpk bp and C bpk are point estimates of Cp and Cpk and therefore are inherently • Remember that C variable. To determine how reliable a point estimate is, we can also generate an interval estimate. • If the quality characteristic is normally distributed, then a 100(1 − α)% confidence interval for Cp is s s χ21−α/2,n−1 χ2α/2,n−1 b b Cp ≤ Cp ≤ Cp n−1 n−1 where χ21−α/2,n−1 and χ2α/2,n−1 are the lower and upper α/2 critical values from a χ2 distribution with n − 1 degrees of freedom. • An approximate 100(1 − α)% confidence interval for Cpk is " # " # s s 1 1 1 1 bpk 1 − zα/2 bpk 1 + zα/2 C + ≤ Cpk ≤ C + 2 b b2 2(n − 1) 2(n − 1) 9nCpk 9nC pk • Confidence intervals for Cpl and Cpu are found in the SAS Help. • An alternative approach is to use bootstrapping techniques to generate confidence intervals. 8.6 Process Capability Using a Control Chart • Although histograms, probability plots, and process capability ratios summarize the performance of a process, they do not necessarily display the potential capability of the process because they do not address the issue of statistical control (that is, how the variability of the quality characteristic would be reduced if systematic patterns or assignable causes in the process were reduced or eliminated). • Montgomery recommends that the control chart should be the primary technique of a process capability analysis because we can study processes without regard to specifications. • Control charts also allow the study of both short-term and long-term process capability. • In particular, it is helpful if the data in a process capability study are collected over several time periods (e.g., shifts, day, etc.). The control chart can be used as a monitoring device which can show the effects of changes in process in both the short-term and long-term. • When a process capability analysis indicates an out-of-control process, it is unsafe to estimate process capability. The process must be stable to produce a reliable estimate of process capability. • When the process is determined to be out-of-control early in a process capability analysis, the first objective is to find and eliminate the assignable causes so the process is returned to an in-control state. 140 8.7 Process Capability Using Designed Experiments • A designed experiment can be – A systematic approach to varying the controllable input process variables so the effects of changing these variables on the output can be estimated. – Useful for screening process variables. That is, determine which set of process variables most influence the output. – Used to determine operating conditions to optimize process performance. • Designed experiments can be used to determine process capability by isolating and estimating the sources of variability (which are characterized by variance components). • Estimation of variance components is a commonly-used technique in gage capability studies. 8.7.1 Gage Capability Studies • Gages are the measuring devices used to take measurements in a process. They are, therefore, a critical link between turning data into information for decision-making. • An improperly functioning gage may yield process measurements that bear little relationship to what the process is actually doing. • A measurement system consists of gages and the procedure used to take a measurement (including operator handling, set-up techniques, data entry, gage calibration, etc.). • A gage study can help identify areas of improvement in the measurement process and to ensure that the process signal is not obscured by noise produced by the measurement gage. • A gage study or a gage repeatability and reproducibility (GRR) study can be designed to 1. Estimate the total variability or error inherent in the measurement process. 2. Provide information on the magnitude and component sources of measurement error. 3. Estimate the closeness of the measurements to a target or true value. 4. Determine the adequacy of the gage for the intended purpose. 5. Check gage stability over time. • Two major properties of a measurement process are the location and variability of the measurements. The variability of the observed measurements is the composite of two effects: 1. The variability in the product being measured. 2. The variability from the measurement process. • In other words, in any problem involving gages, a portion of the observed variability will be due to variability in the product itself while the rest is due to measurement or gage variability: 2 σtotal = 2 2 where σtotal is the total variance observed, σproduct is the variance component associated with 2 the product, and σgage is the variance component associated with measurement error. 141 • The variability from the measurement process can come from a variety of sources such as operator technique, ambient temperature and humidity, etc. • A gage study can be used to – Reduce the variability of the measurement process so the process can be adequately measured. – Estimate the variability of the measurement process which can then be used to separate the real variability of the process from the measured variability. Determining Gage Capability – The Simple Case 1. Take a sample of N items. 2. Take n measurements on each of the N items using the same gage. All other factors are held constant (e.g. operators). • Montgomery provides an analysis based on estimating σgage using the sample ranges (R/d2 ). • An alternative approach is to estimate variance components using an ANOVA. • Once we have an estimate σ bgage , it is usually compared to the tolerance band (U SL − LSL) in the precision-to-tolerance ratio P/T : P = T • Values of P/T ≤ 0.1 are often taken to imply adequate gage capability. This is based on the generally used rule that a measurement device should be calibrated in units one-tenth as large as the required accuracy in the final measurement. • Note that is only a rule-of-thumb and can vary with respect to the process of interest. SAS Code for Gage Capability – Simple Case *********************************************; *** GAGE CAPABILITY STUDY -- SIMPLE CASE ***; *********************************************; DATA rrs; DO part = 1 TO 20; DO msrmnt = 1 TO 2; INPUT diameter @@; OUTPUT; END; END; LINES; 21 20 24 23 20 21 27 27 19 18 23 21 24 23 25 23 21 20 18 19 23 25 24 24 20 20 19 21 25 26 19 19 ; PROC GLM DATA=rrs PLOTS=(ALL); CLASS part ; MODEL diameter = part / SS3; TITLE ’GAGE CAPABILITY STUDY -- Simple Case’; ************************************; *** VARIANCE COMPONENTS ANALYSIS ***; ************************************; PROC VARCOMP DATA=rrs METHOD=REML; CLASS part ; MODEL diameter = part ; RUN; 142 22 21 29 30 19 17 26 26 GAGE CAPABILITY STUDY -- Simple Case The GLM Procedure Dependent Variable: diameter Source Sum of Squares Mean Square F Value Pr > F DF Model 19 377.4000000 Error 20 Corrected Total 39 392.4000000 19.8631579 15.0000000 26.48 <.0001 0.7500000 R-Square Coeff Var Root MSE diameter Mean 0.961774 3.883522 0.866025 22.30000 GAGE CAPABILITY STUDY -- Simple Case Source DF Type III SS Mean Square F Value Pr > F The GLM Procedure part 19 377.4000000 Dependent Variable: diameter 19.8631579 26.48 <.0001 2 0.5 1 1 0.0 -0.5 RStudent 2 RStudent Residual Fit Diagnostics for diameter 1.0 0 -1 -1.0 -1 -2 -2 17.5 20.0 22.5 25.0 27.5 30.0 17.5 20.0 22.5 25.0 27.5 30.0 0.5 0.6 0.7 0.8 0.9 1.0 Predicted Value Predicted Value Leverage 30.0 0.125 27.5 0.0 -0.5 0.100 Cook's D 0.5 diameter Residual 1.0 25.0 22.5 20.0 -1.0 -1 0 1 22.5 27.5 Predicted Value Quantile Fit–Mean 25 0.050 0.000 17.5 2 0.075 0.025 17.5 -2 0 10 20 Residual 5.0 15 Observations 40 Parameters 20 Error DF 20 MSE 0.75 R-Square 0.9618 Adj R-Square 0.9255 2.5 10 0.0 5 -2.5 -5.0 0 -1.8 -1 -0.2 0.6 Residual 1.4 0.0 0.4 0.8 0.0 0.4 0.8 Proportion Less 143 30 Observation 7.5 20 Percent 0 40 The GLM Procedure Dependent Variable: diameter Distribution of diameter 30.0 F 26.48 Prob > F <.0001 27.5 diameter 25.0 22.5 20.0 17.5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 GAGE CAPABILITY STUDY -- Simple Case part 19 Variance Components Estimation Procedure Class Level Information Class Levels Values part 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Number of Observations Read 40 Number of Observations Used 40 Dependent Variable: diameter REML Iterations Iteration Objective Var(part) Var(Error) 0 51.0348249726 9.5565789474 0.7500000000 1 51.0348249726 9.5565789474 0.7500000000 Convergence criteria met. REML Estimates Variance Component Estimate Var(part) 9.55658 Var(Error) 0.75000 Asymptotic Covariance Matrix of Estimates 144 Var(part) Var(Error) 20 8.7.2 Gage Repeatability and Reproducibility (GRR) Studies • The concepts of precision, repeatability, and reproducibility relate to the variability of the observations. • The concepts of accuracy, bias, and calibration relate to the location of the observations. • The precision of a gage is the total measurement error and is given by σgage , the standard deviation of the measurements. Precision can be separated into two components: repeatability and reproducibility. – Reproducibility is the variability that results from using the measurement system (same gages) to make measurements under different operating conditions of normal use (e.g., different operators, different times or shifts, etc.). ∗ Reproducibility, given by σreproducibility , is the long-term variability that captures the changes in operating conditions. – Repeatability is the variability in the measurement system. ∗ Repeatability, given by σrepeatability , is the short-term variability that occurs under identical operating conditions. – Repeatability and reproducibility can be estimated using a designed experiment where the potential sources of variation are changed systematically. 2 • Thus, σmeasurement error = • The typical design: 1. Take a sample of N items. 2. Take a sample of k operating conditions. 3. For each of the k × N operating/item conditions, take n measurements on each item using the same gage. 4. All other factors are held constant. • Montgomery provides an analysis based on estimating σgage using the sample ranges (R/d2 ). An alternative approach is to estimate variance components using an ANOVA. • Once we have an estimate of σgage , it is common to compare it to the tolerance band (U SL − LSL) in the precision-to-tolerance ratio P/T : P = T • Example: A gage study was performed to study the variability due to different operators (reproducibility), part-to-part variability plus measurement variability (repeatability). There were 3 operators, 20 parts, and 2 measurements taken on each part by each operator using the same gage. The data comes from Exercise 8.28 on page 394 of the Montgomery text. Four analyses will be performed on the data: 1. 2. 3. 4. A A A A gauge gauge gauge gauge R&R R&R R&R R&R study study study study with with with with an an an an additive model. additive model and part 15 removed. interaction model. interaction model and part 15 removed. 145 SAS Code for Repeatability and Reproducibility Study DM ’LOG; CLEAR; OUT; CLEAR;’; * ODS PRINTER PDF file=’C:\COURSES\ST528\SAS\GAGE2.PDF’; OPTIONS NODATE NONUMBER; *************************************************************************; *** REPEATABILITY AND REPRODUCIBILITY STUDY (Montgomery Table 8E.9) ***; *************************************************************************; DATA rrs; DO part = 1 TO 20; DO operator = 1 TO 3; DO msrmnt = 1 TO 2; INPUT diameter @@; OUTPUT; END; END; END; LINES; 21 20 20 20 19 21 24 23 24 24 23 24 20 21 19 21 20 22 27 27 28 26 27 28 19 18 19 18 18 21 23 21 24 21 23 22 22 21 22 24 22 20 19 17 18 20 19 18 24 23 25 23 24 24 25 23 26 25 24 25 21 20 20 20 21 20 18 19 17 19 18 19 23 25 25 25 25 25 24 24 23 25 24 25 29 30 30 28 21 20 26 26 25 26 25 27 20 20 19 20 20 20 19 21 19 19 21 23 25 26 25 24 25 25 19 19 18 17 19 17 ; ***************************************************************; *** (I) ASSUMING NO INTERACTION BETWEEN parts AND operators ***; ***************************************************************; PROC GLM DATA=rrs PLOTS=(ALL); CLASS part operator ; MODEL diameter = part operator / SS3; RANDOM part operator / TEST; TITLE ’REPEATABILITY AND REPRODUCIBILITY STUDY’; TITLE2 ’(model without a PART*OPERATOR interaction)’; PROC VARCOMP DATA=rrs METHOD=REML; CLASS part operator ; MODEL diameter = part operator; ****************************************************************; *** (II) ASSUMING AN INTERACTION BETWEEN parts AND operators ***; ****************************************************************; PROC GLM DATA=rrs PLOTS=(ALL); CLASS part operator ; MODEL diameter = part|operator / SS3; RANDOM part|operator / TEST; TITLE2 ’(model with a PART*OPERATOR interaction)’; PROC VARCOMP DATA=rrs METHOD=REML; CLASS part operator ; MODEL diameter = part|operator; RUN; Model (III) and Model (IV) correspond to Model (I) and Model (II), respectively, but with Part = 15 removed. 146 The F-test results in the ANOVA table assume all effects are fixed. Without the part*operator interaction in the model, this ANOVA table is okay because the denominators of the F-tests use the MSE which would be the correct error to use if there really is no part*operator interaction effects. REPEATABILITY AND REPRODUCIBILITY STUDY (model without a PART*OPERATOR interaction) The GLM Procedure Dependent Variable: diameter Sum of Squares Mean Square F Value Pr > F Source DF Model 21 958.708333 45.652778 Error 98 188.216667 1.920578 23.77 <.0001 Corrected Total 119 1146.925000 R-Square Coeff Var Root MSE diameter Mean 0.835895 Source 6.235542 1.385849 22.22500 DF Type III SS Mean Square F Value Pr > F REPEATABILITY AND REPRODUCIBILITY STUDY part 19 957.7583333 50.4083333 26.25 <.0001 (model without a PART*OPERATOR interaction) operator 2 0.9500000 0.4750000 0.25 0.7814 The GLM Procedure Source Type III Expected Mean Square REPEATABILITY AND REPRODUCIBILITY STUDY part Var(Error) + 6 Var(part) (model without a PART*OPERATOR interaction) operator Var(Error) + 40 Var(operator) The GLM Procedure Tests of Hypotheses for Random Model Analysis of Variance Dependent Variable: diameter Source part operator Error: MS(Error) DF Type III SS Mean Square F Value Pr > F 19 957.758333 50.408333 26.25 <.0001 2 0.950000 0.475000 0.25 0.7814 98 188.216667 1.920578 147 REPEATABILITY AND REPRODUCIBILITY STUDY (model without a PART*OPERATOR interaction) The GLM Procedure Dependent Variable: diameter Fit Diagnostics for diameter 4 2 0 -2 2 0 -2 -6 -4 -6 18 20 22 24 -6 26 18 20 Predicted Value 22 24 26 0.20 Predicted Value 0.25 2 27.5 0.20 -2 25.0 22.5 -4 20.0 -6 17.5 -2 -1 0 1 Cook's D 30.0 0 0.15 0.10 22.5 27.5 0 40 Predicted Value Fit–Mean 80 Observation Residual 5.0 40 2.5 30 Observations 120 Parameters 22 Error DF 98 MSE 1.9206 R-Square 0.8359 Adj R-Square 0.8007 0.0 20 -2.5 10 -5.0 REPEATABILITY AND REPRODUCIBILITY STUDY a PART*OPERATOR -3.6 (model -1.2 1.2 without 3.6 0.0 0.4 0.8 0.0 0.4 interaction) 0.8 0 -6 Residual Proportion Less The GLM Procedure Interaction Plot for diameter 30.0 27.5 diameter 25.0 22.5 20.0 17.5 1 2 3 4 5 6 7 8 0.35 0.00 Quantile 50 0.30 0.05 17.5 2 0.25 Leverage 4 diameter Residual -2 -4 -4 Percent 0 RStudent RStudent Residual 2 9 10 11 12 13 14 15 16 part operator 1 148 2 3 17 18 19 20 120 REPEATABILITY AND REPRODUCIBILITY STUDY (model without a PART*OPERATOR interaction) Variance Components Estimation Procedure Class Level Information Class part Levels Values 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 operator 3 123 Number of Observations Read 120 Number of Observations Used 120 Dependent Variable: diameter REML Iterations Iteration Objective Var(part) Var(operator) Var(Error) 0 138.2306440848 7.9790636709 0 1.8962827988 1 138.2288010822 8.0863763068 0 1.8916554022 2 138.2288010711 8.0861111127 0 1.8916666666 3 138.2288010711 8.0861111127 0 1.8916666666 Convergence criteria met. REML Estimates Variance Component Var(part) Estimate 8.08611 Var(operator) Var(Error) 0 1.89167 Note the very large residual variability associated with part 15. This inflates the M SE Asymptotic which affects our estimate of σ 2 . Covariance Matrix of Estimates Var(part) Var(Error) Also note that σ b2 6= M SE. This will happenVar(operator) using REML estimation which constrains all variance component estimates to7.43181 be ≥ 0. Var(part) 0 -0.01193 Maximum likelihoodVar(operator) (ML) estimation will estimates for variance com0 allow negative 0 0 ponents, and therefore is not recommended by many statisticians. Var(Error) -0.01193 0 149 0.07157 Another approach would be to use a model containing a Part*Operator interaction. 2 . This would mean estimation of an addition variance component σpart∗operator The F-test results in the ANOVA table assume all effects are fixed which is incorrect when the effects are random. You need to use the results for the random model ANOVA. REPEATABILITY AND REPRODUCIBILITY STUDY (model with a PART*OPERATOR interaction) The GLM Procedure Dependent Variable: diameter Source DF Sum of Squares Mean Square F Value Pr > F Model 59 1087.425000 Error 60 18.430932 59.500000 18.59 <.0001 0.991667 Corrected Total 119 1146.925000 R-Square Coeff Var Root MSE diameter Mean 0.948122 Source 4.480651 0.995825 22.22500 DF Type III SS Mean Square F Value Pr > F part 19AND 957.7583333 50.4083333 50.83 <.0001 REPEATABILITY REPRODUCIBILITY STUDY (model with a PART*OPERATOR interaction) operator 2 0.9500000 0.4750000 0.48 0.6218 part*operator 38 128.7166667 3.3872807 The GLM Procedure 3.42 Source Type III Expected Mean Square part Var(Error) + 2 Var(part*operator) + 6 Var(part) <.0001 REPEATABILITY AND REPRODUCIBILITY STUDY operator Var(Error) + 2 Var(part*operator) + 40 Var(operator) (model with a PART*OPERATOR interaction) part*operator Var(Error) + 2 Var(part*operator) The GLM Procedure Tests of Hypotheses for Random Model Analysis of Variance Dependent Variable: diameter Source part operator Error DF Type III SS Mean Square F Value Pr > F 19 957.758333 50.408333 14.88 <.0001 2 0.950000 0.475000 0.14 0.8696 38 128.716667 3.387281 Error: MS(part*operator) Source DF Type III SS Mean Square F Value Pr > F part*operator 38 128.716667 3.387281 Error: MS(Error) 60 59.500000 0.991667 150 3.42 <.0001 REPEATABILITY AND REPRODUCIBILITY STUDY (model with a PART*OPERATOR interaction) The GLM Procedure Dependent Variable: diameter Fit Diagnostics for diameter 1.5 2 2 1 1 0.0 -0.5 RStudent 0.5 RStudent Residual 1.0 0 0 -1 -1 -2 -2 -1.0 -1.5 17.5 20.0 22.5 25.0 27.5 30.0 17.5 20.0 22.5 25.0 27.5 30.0 Predicted Value Predicted Value 2 25.0 22.5 17.5 -2 -1 0 1 0.04 0.00 17.5 2 22.5 27.5 0 40 Predicted Value Quantile Fit–Mean 20 7.5 15 5.0 120 Residual Observations 120 Parameters 60 Error DF 60 MSE 0.9917 R-Square 0.9481 Adj R-Square 0.8971 0.0 5 80 Observation 2.5 10 -2.5 REPEATABILITY-5.0 AND REPRODUCIBILITY STUDY (model 0.2 1.4 with a PART*OPERATOR 0.0 0.4 0.8 0.0 0.4 interaction) 0.8 0 -2.2 -1 Residual Proportion Less The GLM Procedure Interaction Plot for diameter 30.0 27.5 diameter 25.0 22.5 20.0 17.5 1 2 3 4 5 6 7 1.0 0.02 20.0 -2 0.9 0.06 Cook's D diameter Residual -1 0.7 0.8 0.08 27.5 0 0.6 Leverage 30.0 1 Percent 0.5 8 9 10 11 12 13 14 15 16 part operator 1 151 2 3 17 18 19 20 REPEATABILITY AND REPRODUCIBILITY STUDY (model with a PART*OPERATOR interaction) Variance Components Estimation Procedure Class Level Information Class Levels Values part 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 operator 3 123 Number of Observations Read 120 Number of Observations Used 120 Dependent Variable: diameter REML Iterations Iteration Objective Var(part) Var(operator) Var(part*operator) Var(Error) 0 121.0498928641 7.7236004587 0 1.1804988165 0.9773371747 1 121.0243926123 7.8643555272 0 1.1258700096 0.9912645392 2 121.0243826544 7.8611124257 0 1.1250003526 0.9916665036 3 121.0243826544 7.8611124257 0 1.1250003526 0.9916665036 Convergence criteria met. REML Estimates Variance Component Estimate Var(part) 7.86111 Var(operator) 0 Var(part*operator) 1.12500 Var(Error) 0.99167 Asymptotic Covariance Matrix of Estimates Var(part) Var(operator) Var(part*operator) Var(Error) Var(part) Var(operator) Var(part*operator) Var(Error) 7.44442 0 -0.04379 -4.3453E-8 0 0 0 0 -0.04379 0 0.13955 -0.01639 0 -0.01639 0.03278 -4.3453E-8 152 The data was re-analyzed without Part 15 removed. The following analysis is for the model without the part*operator interaction. 2 Note the drop in the estimates from σ b2 = 1.8917 and σ bpart = 8.086 in Model (I) to 2 2 σ b = 0.8825 and σ bpart = 7.734 in Model (III) when part 15 is removed. Note also the increase in the model r2 from .836 to .912 when part 15 is removed. Before removing part 15, the experimenter should try to find an assignable cause for the unusual response values. Note the large change in the vertical scale with part 15 removed. It decreased from −6 to 3.5 in Model I to −2 to 2.5 . REPEATABILITY AND REPRODUCIBILITY STUDY (model without PART*OPERATOR and part=15 removed) The GLM Procedure Dependent Variable: diameter Source DF Sum of Squares Mean Square F Value Pr > F Model 20 852.9122807 Error 93 42.6456140 82.0789474 48.32 <.0001 0.8825693 Corrected Total 113 934.9912281 R-Square Coeff Var Root MSE diameter Mean 0.912214 4.268533 0.939452 22.00877 REPEATABILITY ANDIIIREPRODUCIBILITY STUDY Source DF Type SS Mean Square F Value Pr > F (model part without PART*OPERATOR and part=15 removed) 18 851.1578947 47.2865497 53.58 <.0001 operator Source 2 0.8771930 The1.7543860 GLM Procedure 0.99 0.3740 Type III Expected Mean Square REPEATABILITY AND REPRODUCIBILITY STUDY part Var(Error) + 6 Var(part) (model without PART*OPERATOR and part=15 removed) operator Var(Error) + 38 Var(operator) The GLM Procedure Tests of Hypotheses for Random Model Analysis of Variance Dependent Variable: diameter Source part operator Error: MS(Error) DF Type III SS Mean Square F Value Pr > F 18 851.157895 47.286550 53.58 <.0001 2 1.754386 0.877193 0.99 0.3740 93 82.078947 0.882569 153 REPEATABILITY AND REPRODUCIBILITY STUDY (model without PART*OPERATOR and part=15 removed) The GLM Procedure Dependent Variable: diameter Fit Diagnostics for diameter 0 2 RStudent 2 1 RStudent Residual 2 0 0 -1 -2 -2 18 20 22 24 26 -2 18 20 Predicted Value 22 24 26 0.20 Predicted Value 0.25 0.30 0.35 Leverage 28 0.08 26 0 24 Cook's D 1 diameter Residual 2 22 20 -1 -2 -1 0 1 2 0.00 18 Quantile 20 22 24 26 28 0 40 Predicted Value Fit–Mean 30 Percent 0.04 0.02 18 -2 0.06 80 120 Observation Residual 4 20 Observations 114 Parameters 21 Error DF 93 MSE 0.8826 R-Square 0.9122 Adj R-Square 0.8933 2 0 10 -2 REPEATABILITY-4AND REPRODUCIBILITY STUDY PART*OPERATOR removed) 0.0 0.4 0.8 0.0 and 0.4 part=15 0.8 0 (model without -2.4 -1.2 0 1.2 2.4 Residual Proportion Less The GLM Procedure Interaction Plot for diameter 28 26 diameter 24 22 20 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 16 part operator 1 154 2 3 17 18 19 20 REPEATABILITY AND REPRODUCIBILITY STUDY (model without PART*OPERATOR and part=15 removed) Variance Components Estimation Procedure Class Level Information Class Levels Values part 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 16 17 18 19 20 operator 3 123 Number of Observations Read 114 Number of Observations Used 114 Dependent Variable: diameter REML Iterations Iteration Objective Var(part) Var(operator) Var(Error) 0 57.5326780918 7.7331628690 0 0.8824741700 1 57.5326779673 7.7340156120 0 0.8824561400 2 57.5326779673 7.7340156120 0 0.8824561400 Convergence criteria met. REML Estimates Variance Component Var(part) Estimate 7.73402 Var(operator) Var(Error) 0 0.88246 Asymptotic Covariance Matrix of Estimates Var(part) Var(operator) Var(Error) Var(part) Var(operator) Var(Error) 6.90174 0 -0.0027324 0 0 0 -0.0027324 0 0.01639 155 The following analysis if for the data with Part 15, but the model now contains a Part*Operator interaction like Model (II). This would again require estimation of an 2 . addition variance component σpart∗operator REPEATABILITY AND REPRODUCIBILITY STUDY (model with PART*OPERATOR and part=15 removed) The GLM Procedure Dependent Variable: diameter Source DF Sum of Squares Mean Square F Value Pr > F Model 56 878.4912281 Error 57 56.5000000 15.6873434 15.83 <.0001 0.9912281 Corrected Total 113 934.9912281 R-Square Coeff Var Root MSE diameter Mean 0.939572 Source 4.523671 0.995604 22.00877 DF Type III SS Mean Square F Value Pr > F part 18 851.1578947 47.2865497 47.71 <.0001 REPEATABILITY AND REPRODUCIBILITY STUDY operator 2 1.7543860 0.8771930 0.88 0.4183 (model with PART*OPERATOR and part=15 removed) part*operator 36 25.5789474 0.7105263 0.72 The GLM Procedure Source Type III Expected Mean Square part Var(Error) + 2 Var(part*operator) + 6 Var(part) 0.8557 REPEATABILITY AND REPRODUCIBILITY STUDY operator Var(Error) + 2 Var(part*operator) + 38 Var(operator) (model with PART*OPERATOR and part=15 removed) part*operator Var(Error) + 2 Var(part*operator) The GLM Procedure Tests of Hypotheses for Random Model Analysis of Variance Dependent Variable: diameter Source part operator Error DF Type III SS Mean Square F Value Pr > F 18 851.157895 47.286550 66.55 <.0001 2 1.754386 0.877193 1.23 0.3030 36 25.578947 0.710526 Error: MS(part*operator) Source DF Type III SS Mean Square F Value Pr > F part*operator 36 25.578947 0.710526 Error: MS(Error) 57 56.500000 0.991228 156 0.72 0.8557 REPEATABILITY AND REPRODUCIBILITY STUDY (model with PART*OPERATOR and part=15 removed) The GLM Procedure Dependent Variable: diameter Fit Diagnostics for diameter 1.5 2 2 1 1 0.0 -0.5 RStudent 0.5 RStudent Residual 1.0 0 0 -1 -1 -2 -2 -1.0 -1.5 18 20 22 24 26 28 18 20 Predicted Value 2 26 28 0.5 22 -2 -1 0 1 0.04 0.00 2 18 Quantile 20 22 24 26 28 0 40 Predicted Value Fit–Mean 25 80 Observation Residual 6 20 4 15 Observations 114 Parameters 57 Error DF 57 MSE 0.9912 R-Square 0.9396 Adj R-Square 0.8802 2 10 0 5 -2 0 -2.2 -4 REPEATABILITY AND REPRODUCIBILITY STUDY (model part=15 removed) -1 0.2 with 1.4 PART*OPERATOR 0.0 0.4 0.8 and 0.0 0.4 0.8 Residual Proportion Less The GLM Procedure Interaction Plot for diameter 28 26 diameter 24 22 20 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 16 part operator 1.0 0.02 18 -2 0.9 0.06 24 20 -1 0.7 0.8 0.08 Cook's D 0 0.6 Leverage 26 diameter Residual 24 28 1 Percent 22 Predicted Value 1 157 2 3 17 18 19 20 120 REPEATABILITY AND REPRODUCIBILITY STUDY (model with PART*OPERATOR and part=15 removed) Variance Components Estimation Procedure Class Level Information Class part operator Levels Values 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 16 17 18 19 20 3 123 Number of Observations Read 114 Number of Observations Used 114 Dependent Variable: diameter REML Iterations Iteration Objective Var(part) Var(operator) Var(part*operator) Var(Error) 0 57.6566202134 7.0217298733 0.0039673281 0 0.8966161443 1 57.5326807976 7.7380837652 0 0 0.8823701881 2 57.5326779673 7.7340159905 0 0 0.8824561320 3 57.5326779673 7.7340159905 0 0 0.8824561320 Convergence criteria met. REML Estimates Variance Component Estimate Var(part) 7.73402 Var(operator) 0 Var(part*operator) 0 Var(Error) 0.88246 Asymptotic Matrix oftable Estimates Like Model (II), the F-test results Covariance in the ANOVA assume all effects are fixed which is incorrect when theVar(part) effects are random. You need to use the results for the Var(operator) Var(part*operator) Var(Error) random model ANOVA. Var(part) 6.90175 0 0 -0.0027324 Var(operator) 0 0 0 0 Var(part*operator) 0 0 0 0 0 0 0.01639 Var(Error) -0.0027324 158 8.8 Precision, Accuracy, and Stability • A measurement process is accurate if the measurements are close to a target value. • Bias, which is the difference between the mean of the measurements and the target value, expresses the extent to which the measurements deviate from target. • A gage needs to be calibrated so that it is accurate. That is, remove any bias that could be attributed to the gage. Calibration is the procedure used to adjust the gage so that measurements are accurate. • Assessing the accuracy of a gage through calibration often requires the use of a standard for which the true value of the measured characteristic is known. • If a gage is accurate but not precise, multiple measurements can be taken and then averaged to increase precision. • The two gage capability studies presented deal with the precision of the gage and not its accuracy. • A measurement process is stable if it does not exhibit unusual changes over time, such as sudden shifts, drifts, and cycles. • Therefore, for a stable process, the location and variability of the distribution of measurements remains constant and predictable over time. • Stability can be evaluated by a using control charts, in particular, cusum and EWMA charts. • Once a gage has been deemed adequate to provide reliable measurements, its behavior should be monitored using control charts. 159
