ANOVA for an R&R Data Set and Corresponding Estimates and Confidence IntervalsIE 361 Module 6Gauge R&R Studies Part 2: Two-Way ANOVA and CorrespondingEstimates for R&R StudiesReading: Section 2.2 Statistical Quality Assurance for Engineers(Section 2.4 of Revised SQAME )Prof. Steve Vardeman and Prof. Max MorrisIow a State UniversityAugust 2008Vardeman and Morris (I o wa State U niversity) IE 361 Module 6 A ugust 2008 1 / 21ANOVA and R&R AnalysisThe range-based Gauge R&R estimates of SQAME are fairly simple andserve the purpose of helping make the analysis goals easy to understand.But we have no good handle on how reliable these estimates are. In orderto 1) produce Gauge R&R estimates that are typically better thanrange-based ones, and 2) produce con…dence limits, we must instead use"ANOVA-based" estimates.A careful treatment of ANOVA would require its own course. We’ll simplymake use of its main "output" and direct the interested student to bookson engineering statistics (like Vardeman’s Statistics for EngineeringProblem Solving ) for more details. The fact is that an I J m data setof yijk’s like that produced in a typical Gauge R&R study is oftensummarized in a so-called ANOVA table. A generic version of such atable isVardeman and Morris (I o wa State University) IE 361 Module 6 August 2008 2 / 21ANOVA and R&R AnalysisSource SS df MSPart SSA I 1 MSA = SSA/(I 1)Operator SSB J  1 MSB=SSB/(J  1)PartOperator SSAB(I 1) (J  1)MSAB=SSAB/(I 1) (J  1)Error SSE IJ(m 1)MSE = SSE /IJ(m 1)Total SSTot IJm 1Any decent statistical package (and even EXCEL) will process a GaugeR&R data set and produce such a summary table. In this table the "meansquares" are essentially sample variances (squares of sample standarddeviations). (MSA is essentially a sample variance of part averages, MSBis essentially a sample variance of operator averages, MSE is an average ofwithin cell-sample variances, "MSTot" isn’t typically calculated, but is agrand sample variance of all observations, ...) T he mea n squares indicatehow much of the overall variability is accounted for by the various sources.Vardeman and Morris (I o wa State University) IE 361 Module 6 August 2008 3 / 21ANOVA and R&R AnalysisExample 6-1We’ll use the data set with I = 4, J = 3, m = 2 from the in-class R&Rstudy (used as a numerical example in Module 5) to illustrate. The JMPdata table and some screen shots for using the program to get the sums ofsquares follow.Figure: JMP Data Sheet for the In-Class R&R StudyVardeman and Morris (I o wa State University) IE 361 Module 6 August 2008 4 / 21ANOVA and R&R AnalysisExample 6-1Figure: JMP Dialogue Box for Fit Model Two-Way ANOVA on the Gauge R&RDataVardeman and Morris (I o wa State University) IE 361 Module 6 August 2008 5 / 21ANOVA and R&R AnalysisExample 6-1Figure: JMP Two-Way ANOVA Report for the In-Class G auge R&R StudyVardeman and Morris (I o wa State University) IE 361 Module 6 August 2008 6 / 21ANOVA and R&R AnalysisExample 6-1Although we certainly don’t recommend using EXCEL (a spreadsheet isno substitute for a statistical package and, besides, EXCEL has terriblyunreliable numerical analysis) we found instructions on using the program’stwo-way ANOVA plug-in athttp://www.cvgs.k12.va.us/digstats/main/Guides/g_2anovx.html. Twoscreen shots from using these instructions on the in-class two-way datafollow.Vardeman and Morris (I o wa State University) IE 361 Module 6 August 2008 7 / 21ANOVA and R&R AnalysisExample 6-1Figure: EXCEL Two-Way Data Spreadsheet for the In-Class R&R StudyVardeman and Morris (I o wa State University) IE 361 Module 6 August 2008 8 / 21ANOVA and R&R AnalysisFigure: EXCEL Two-Way Data Spreadsheet for the In-Class R&R StudyFor our present purposes, we will take mean squares and degrees offreedom out of such an ANOVA table and make Gauge R&R estimatesbased on them. Point estimators for the quantities of most interest in aGauge R&R study are partially summarized on the bottom of page 27 inSQAME. These areVardeman and Morris (I o wa State University) IE 361 Module 6 August 2008 9 / 21ANOVA and R&R Analysisˆσrepeatability=bσ =pMSEandˆσreproducibility=smax0,MSBmI+(I 1)mIMSAB 1mMSEAlthough it is not presented in SQAME, an appropriate estima tor forσR&R=qσ2γ+ σ2αγ+ σ2(that is called σoverallin SQAME ) isˆσR&R=r1mIMSB +I 1mIMSAB +m 1mMSEVardeman and Morris (I o wa State University) IE 361 Module 6 A ugust 2008 10 / 21ANOVA and R&R AnalysisIt is further possible to use these estimates to make an exact con…denceinterval for σrepeatability= σ and Satterthwaite approximate con…dencelimits for σreproducibili tyand σR&R. Letνrepeatability= IJ(m 1)Then, con…dence limits for σrepeatabilityareˆσrepeatabilitysνrepeatabilityχ2νrepeatability, upperandˆσrepeatabilitysνrepeatabilityχ2νrepeatability, lowerFor estimating σreproducibility, letVardeman and Morris (I o wa State University) IE 361 Module 6 A ugust 2008 11 / 21ANOVA and R&R Analysisˆνreproducibili ty=ˆσ4reproducibili tyMSBmI2J  1+(I 1)MSABmI2(I 1) (J  1)+MSEm2IJ(m 1)=ˆσ4reproducibili ty1m2MSB2I2(J  1)+(I 1)MSAB2I2(J  1)+MSE2IJ(m 1)Then approximate con…dence limits for σreproducibili tyareˆσreproducibilitysˆνreproducibilityχ2ˆνreproducibility,uppe randˆσreproducibilitysˆνreproducibili tyχ2ˆνreproducibility,l o werFor estimating σR&R, letVardeman and Morris (I o wa State University) IE 361 Module 6 A ugust 2008 12 / 21ANOVA and R&R AnalysisˆνR&R=ˆσ4R&RMSBmI2J  1+(I 1)MSABmI2(I 1) (J  1)+(m1)MSEm2IJ(m 1)=ˆσ4R&R1m2MSB2I2(J  1)+(I 1)MSAB2I2(J  1)+(m 1)MSE2IJthen approximate con…dence limits for σR&RareˆσR&RsˆνR&Rχ2ˆνR&R,uppe randˆσR&RsˆνR&Rχ2ˆνR&R,l o werVardeman and Morris (I o wa State University) IE 361 Module 6 A ugust 2008 13 / 21ANOVA and R&R AnalysisThese formulas are tedious (but hardly impossible) to use with a pocketcalculator. Or a ve ry small program could be written to evaluate theestimates of standard deviations and approximate degrees of freedom.Vardeman uses a simple MathCAD worksheet to do the computing. Thefollowing …gure illustrates the use of that worksheet beginning fromSSB, SSAB, SSE, I , J, and m for the in-class R&R study.Vardeman and Morris (I o wa State University) IE 361 Module 6 A ugust 2008 14 / 21ANOVA and R&R AnalysisExample 6-1Figure: MathCAD