ISU IE 361 - Motivation, Data, Model and Range-Based Estimates

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Standard R&R Data and Descriptive Statistics (Based on Ranges) for Partitioning Measurement VariationThe "Two-Way Random Effects" Model for Gauge R&R DataSimple Range-Based Point Estimates of Repeatability and Reproducibility Standard DeviationsIE 361 Module 5Gauge R&R Studies Part 1: Motivation, Data, Model and Range-BasedEstimatesReading: Section 2.2 Statistical Quality Assurance for Engineers(Section 2.4 of Revised SQAME )Prof. Steve Vardeman and Prof. Max MorrisIow a State UniversityVardeman and Morris (I owa State University) IE 361 Module 5 1 / 15Standard R&R Data and Descriptive Statistics (Based onRanges) for Partitioning Measurement VariationA very common type of industrial measurement study is one where a singlegauge or piece of measuring equipment is used (according to a standardprotocol) by multiple operators to measure multiple parts, with the primaryend goal of quantifying repeatability and reproducibility measurementvariation, and comparing measurement imprecision to the basicengineering 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 oneoperator/analyst remeasuring one specimen"Reproducibility" variation is variation in operator biases, i.e.variation characteristic of many operators measuring a singlespecimen after accounting for (or somehow mathematicallyeliminating) repeatability variationVardeman and Morris (I owa State University) IE 361 Module 5 2 / 15R&R Data and Descriptive StatisticsIn a typical (balanced data) industrial Gauge R&R study, each of I items ismeasured m times by each of J operators. For example, a typical datalayout 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 DataVardeman and Morris (I owa State University) IE 361 Module 5 3 / 15R&R Data and Descriptive StatisticsIf only one part/measurand were involved, the one-way model andanalyses of Module 4 could be used to do inference for what we have beencalling σδand σd evice. (See again Example 4-3 of Module 4.) But(presumably in order to have some check on how measurement performsacross a spectrum of parts) it is common in Gauge R&R studies to usemultiple parts. The next …gure shows some real R&R data collectedin-class in IE 361 on I = 4 parts, by J = 3 operators, making m = 2repeats per cell, and some summary statistics based on ranges (rather thanstandard deviations). (The data are measurements of the sizes of someStyrofoam peanuts, and range-based methods are presented here becauseof their connection to fairly standard industry practice.)Vardeman and Morris (I owa State University) IE 361 Module 5 4 / 15R&R Data and Descriptive StatisticsFigure: Styrofoam Peanut Size Measurements and Summary Statistics (Inches)Vardeman and Morris (I owa State University) IE 361 Module 5 5 / 15R&R Data and Descriptive StatisticsR is a very simple descriptive measure of within-cel l variability and isrelated to repeatability variation. Similarly, ∆ is a measure ofbetween-operator variation and is related to reproducibility variation. Inorder 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 (I owa State University) IE 361 Module 5 6 / 15The "Two-Way Random E¤ects" Model for Gauge R&RDataTypical analyses of Gauge R&R studies are based on the so-called"two-way random e¤ects" model. Withyijk= the kth measurement made by operator j on specimen ithis model is that yijkis made up as a sum of independent contributions,yijk= µ + αi+ βj+ αβij+ eijkwhereµ is an (unknown) constant, an average (over all possible operatorsand all possible parts/specimens) measurementthe αiare n ormal with mean 0 and variance σ2α, (random) e¤ects ofdi¤erent parts/specimensthe βjare normal with mean 0 and variance σ2β, (random) e¤ects ofdi¤erent operatorsVardeman and Morris (I owa State University) IE 361 Module 5 7 / 15The Two-Way Random E¤ects Modelthe αβijare normal with mean 0 and variance σ2αβ, (random) jointe¤ects peculiar to partic ular part/operator combinationsthe eijkare normal with mean 0 and variance σ2, (random) errors thatare peculiar to a particular attempt to make a measurement (theychange measurement-to-measurement, even if the part and operatorremain the same)σ2α, σ2β, σ2αβ, and σ2are called "variance components" and their sizesgovern how much variability is seen in the measurements yijk.Vardeman and Morris (I owa State University) IE 361 Module 5 8 / 15The Two-Way Random E¤ects ModelExample 5-1The reader should conduct a "Thought Experiment" generating a GaugeR&R data set, and …ll in formulas for the 12 measurements in the tablebelow. (For example, y111= µ + α1+ β1+ αβ11+ e111.)Operator1 2 3y111= y121= y131=1y112= y122= y132=Party211= y221= y231=2y212= y222= y232=Vardeman and Morris (I owa State University) IE 361 Module 5 9 / 15The Two-Way Random E¤ects ModelIn this (two-way random e¤ects) modelσ measures within-cell/repeatability variationσreproducibility=qσ2β+ σ2αβis the standard deviation that would beexperienced by many operators measuring the same specimen onceeach, in the absence of repeatability variationσR&R=qσ2reproducibility+ σ2=qσ2β+ σ2αβ+ σ2is the standarddeviation that would be experienced by many operators measuring thesame specimen once each (this is called σoverallin SQAME )To make connections to what we have done earlier, consider what thesetwo-way model parameters mean if we restrict attention to part #1. Thetwo-way random e¤ects model says tha t measurements on part #1 can bethought of asy1jk= µ + α1+ βj+ αβ1j+ e1jkVardeman and Morris (I owa State University) IE 361 Module 5 10 / 15The Two-Way Random E¤ects ModelWhat then varies operator-to-operator isβj+ αβ1jThis quantity thus plays the role of what we before called δj(operator biasfor operator j ... for part #1) andσ2βj+αβ1j= σ2β+ σ2αβplays the role of what we before called σ2δ(the reproducibility variance).The fact that βj+ αβ1jis speci…c to part #1 (for example changes toβj+ αβ2jif part #2 is considered instead) has the interestinginterpretat ion that the terms αβijplay the role of "device" nonlinearities!That is, in the two-way random e¤ects model where multiple parts areconsidered, a large variance component σ2αβis


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ISU IE 361 - Motivation, Data, Model and Range-Based Estimates

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