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Unformatted text preview: IE 361 Module 5
Gauge R&R Studies Part 1: Motivation, Data, Model and RangeBased Estimates Reading: Section 2.2 Statistical Quality Assurance for Engineers (Section 2.4 of Revised SQAME ) Prof. Steve Vardeman and Prof. Max Morris
Iowa State University Vardeman and Morris (Iowa State University) IE 361 Module 5 1 / 15 Standard R&R Data and Descriptive Statistics (Based on Ranges) for Partitioning Measurement Variation
A very common type of industrial measurement study is one where a single gauge or piece of measuring equipment is used (according to a standard protocol) by multiple operators to measure multiple parts, with the primary end goal of quantifying repeatability and reproducibility measurement variation, and comparing measurement imprecision to the basic engineering requirements that a part must satisfy in order to be functional. Remember from what we have already said in Modules 3 and 4, that "Repeatability" variation is variation characteristic of one operator/analyst remeasuring one specimen "Reproducibility" variation is variation in operator biases, i.e. variation characteristic of many operators measuring a single specimen after accounting for (or somehow mathematically eliminating) repeatability variation
Vardeman and Morris (Iowa State University) IE 361 Module 5 2 / 15 R&R Data and Descriptive Statistics
In a typical (balanced data) industrial Gauge R&R study, each of I items is measured m times by each of J operators. For example, a typical data layout for I = 2 parts, J = 3 operators, and m = 2 repeats per "cell" might be represented as in this ...gure. Figure: Hypothetical Gauge R&R Data
Vardeman and Morris (Iowa State University) IE 361 Module 5 3 / 15 R&R Data and Descriptive Statistics If only one part/measurand were involved, the oneway model and analyses of Module 4 could be used to do inference for what we have been calling and device . (See again Example 43 of Module 4.) But (presumably in order to have some check on how measurement performs across a spectrum of parts) it is common in Gauge R&R studies to use multiple parts. The next ...gure shows some real R&R data collected inclass in IE 361 on I = 4 parts, by J = 3 operators, making m = 2 repeats per cell, and some summary statistics based on ranges (rather than standard deviations). (The data are measurements of the sizes of some Styrofoam peanuts, and rangebased methods are presented here because of their connection to fairly standard industry practice.) Vardeman and Morris (Iowa State University) IE 361 Module 5 4 / 15 R&R Data and Descriptive Statistics Figure: Styrofoam Peanut Size Measurements and Summary Statistics (Inches)
Vardeman and Morris (Iowa State University) IE 361 Module 5 5 / 15 R&R Data and Descriptive Statistics R is a very simple descriptive measure of withincell variability and is related to repeatability variation. Similarly, is a measure of betweenoperator variation and is related to reproducibility variation. In order to be more more precise about these (and to do statistical inference) one must adopt a probability model for the data collected in an R&R study. Vardeman and Morris (Iowa State University) IE 361 Module 5 6 / 15 The "TwoWay Random Eects" Model for Gauge R&R Data
Typical analyses of Gauge R&R studies are based on the socalled "twoway random eects" model. With yijk = the kth measurement made by operator j on specimen i this model is that yijk is made up as a sum of independent contributions, yijk = + i + j + ij + where is an (unknown) constant, an average (over all possible operators and all possible parts/specimens) measurement the i are normal with mean 0 and variance 2 , (random) eects of dierent parts/specimens the j are normal with mean 0 and variance 2 , (random) eects of dierent operators
Vardeman and Morris (Iowa State University) IE 361 Module 5 7 / 15 ijk The TwoWay Random Eects Model the ij are normal with mean 0 and variance 2 , (random) joint eects peculiar to particular part/operator combinations the ijk are normal with mean 0 and variance 2 , (random) errors that are peculiar to a particular attempt to make a measurement (they change measurementtomeasurement, even if the part and operator remain the same) 2 , 2 , 2 , and 2 are called "variance components" and their sizes govern how much variability is seen in the measurements yijk . Vardeman and Morris (Iowa State University) IE 361 Module 5 8 / 15 The TwoWay Random Eects Model
Example 51 The reader should conduct a "Thought Experiment" generating a Gauge R&R data set, and ...ll in formulas for the 12 measurements in the table below. (For example, y111 = + 1 + 1 + 11 + 111 .) Operator 2 y121 = y122 = y221 = y222 = 1 y111 = 1 y112 = Part y211 = 2 y212 = 3 y131 = y132 = y231 = y232 = Vardeman and Morris (Iowa State University) IE 361 Module 5 9 / 15 The TwoWay Random Eects Model
In this (twoway random eects) model measures withincell/repeatability variation q reproducibility = 2 + 2 is the standard deviation that would be experienced by many operators measuring the same specimen once each, in the absence of repeatability variation q q 2 + 2 = 2 + 2 + is the standard R&R = 2 reproducibility deviation that would be experienced by many operators measuring the same specimen once each (this is called overall in SQAME ) To make connections to what we have done earlier, consider what these twoway model parameters mean if we restrict attention to part #1. The twoway random eects model says that measurements on part #1 can be thought of as y1jk = + 1 + j + 1j + 1jk
Vardeman and Morris (Iowa State University) IE 361 Module 5 10 / 15 The TwoWay Random Eects Model
What then varies operatortooperator is j + 1j This quantity thus plays the role of what we before called j (operator bias for operator j ... for part #1) and 2 j +1j = 2 + 2 plays the role of what we before called 2 (the reproducibility variance). The fact that j + 1j is speci...c to part #1 (for example changes to j + 2j if part #2 is considered instead) has the interesting interpretation that the terms ij play the role of "device" nonlinearities! That is, in the twoway random eects model where multiple parts are considered, a large variance component 2 is indicative of substantially nonconstant bias of the various operators in their use of the gauge ... a most unpleasant circumstance indeed.
Vardeman and Morris (Iowa State University) IE 361 Module 5 11 / 15 The TwoWay Random Eects Model
The most common analyses (both those based on ranges and those based on ANOVA) (e.g. following the AIAG manual) are wrong, in that they purport to produce estimates of reproducibility and R&R but fail to do so. SQAME presents correct rangebased and ANOVAbased methods. We will use primarily the generally more eective ANOVAbased estimates and con...dence intervals that can be based on them (these limits are not found in SQAME ). If you end up doing a gauge R&R study for your project client, you will almost certainly be asked to use company standard formulas or a company spreadsheet that implements the (WRONG) AIAG formulas. If you do this you should compare those results to ones obtained using correct formulas from these modules! Failure to do so will be frowned upon when projects are graded. Vardeman and Morris (Iowa State University) IE 361 Module 5 12 / 15 Simple RangeBased Point Estimates of Repeatability and Reproducibility Standard Deviations
For motivation sake (and because of the connection to standard formulas), ...rst briey consider rangebased estimates. Possible estimates are: b = d2 Rm ) for R the average withincell range and d2 (m ) a "control ( chart constant" based on "sample size" m s ^ reproducibility = max 0, d 2 (J ) 2 1 m part ranges of cell means and d2 (m ) a "control chart constant" based on "sample size" J (The second of these is NOT the AIAG estimate of reproducibility standard deviation.) b ( )2 for the average of Vardeman and Morris (Iowa State University) IE 361 Module 5 13 / 15 RangeBased Point Estimates
Example 52 Below are some simple calculations based on measurements of a geometric dimension of a machined part in a study with I = 3, J = 3, and m = 2.
Operator 1 Part 1 Part 2 Part 3 Operator 2 Operator 3 y 11 = .34730 R11 = 0 y 21 = .34710 R21 = 0 y 31 = .34720 R31 = 0 y 12 = .34660 R12 = .0002 y 22 = .34645 R22 = .0001 y 32 = .34655 R32 = .0003 y 13 = .34715 R13 = .0001 y 23 = .34710 R23 = 0 y 33 = .34710 R33 = 0 1 = .00070 2 = .00065 3 = .00065 So R = .0007/9 = .000078 and = .00067 and b = R .000078 = = .000069 in d2 ( m ) 1.128
IE 361 Module 5 14 / 15 Vardeman and Morris (Iowa State University) RangeBased Point Estimates
Example 52 and v 0 u u u = tmax @0, d2 ( J )
2 ^ reproducibility !2 = s .00067 1.693 1 (.000069)2 = .000391 in 2 1 1 b ( )2 A m A natural way to estimate R&R is as q ^ R&R = (.000069)2 + (.00039)2 = .000396 in and the calculations here suggest that the bulk of measurement imprecision is traceable to dierences between operators.
IE 361 Module 5 15 / 15 Vardeman and Morris (Iowa State University) ...
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