Go/No-Go InspectionSome Simple Probability ModelingSome Simple R&R Point Estimates for 0/1 ContextsApplication of Inference for the Difference in Two Binomial "p's"IE 361 Module 8Repeatability and Reproducibility for "0/1" (or Go/No-Go) ContextsReading: Section 2.7 of Revised SQAMEProf. Steve Vardeman and Prof. Max MorrisIow a State UniversityVarde man and Morris (Iowa State University) IE 361 Module 8 1 / 16Go/No-Go InspectionIdeally, observation of a process results in quantitative measurements.But there are some contexts in which all that is determined is whether ornot an item or process condition is of one of two types, that we will for thepresent call "conforming" and "non-conforming." It is, for example,common to check the conformance of machined metal parts to someengineering requirements via the use of a "go/no-go gauge." (A part isconforming if a critical dimension …ts into the larger of two check …xturesand does not …t into the smaller of the two.) And it is common to taskhuman beings with making visual inspections of manufactured items andproducing a "OK/Not-OK" call on each.Engineers are sometimes then called upon to apply the qualitative"repeatability" and "reproducibility" concepts of metrology to suchGo/No-Go or "0/1" contexts. One wants to separate some measure ofoverall inconsistency in 0/1 "calls" on items into pieces that can beVarde man and Morris (Iowa State University) IE 361 Module 8 2 / 16Go/No-Go Inspectionmentally charged to inherent inconsistency in the equipment or method,and the remainder that can be charged to di¤erences between howoperators use it. Exactly how to do this is, in fact, presently notwell-established. The best available statistical methodology for this kindof problem is more complicated than can be presented here (involvingso-called "generalized linear models"). What we can present is a rationalway of making point estimates of what might be termed repeatability andreproducibility components of varia tion in 0/1 calls. (The estimatespresented here are based on reasoning similar to that employed in SQAMEto …nd correct range-based estimates in usual measurement R&Rcontexts.) We will then remind you of elementary methods of estimatingdi¤erences in population proportions and point to their relevance in thepresent situation.Varde man and Morris (Iowa State University) IE 361 Module 8 3 / 16Some Simple Probability ModelingTo begin, think of coding a "non-conforming" call as "1" and a"conforming" call as "0," and having J operators each make m calls on a…xed part. Suppose that J operators have individual probabilitiesp1, p2, . . . , pJof calling the part non-conforming on any single viewing,and that across m viewingsXj= the number of non-conforming calls among the m made by operator jis Binomial(m, pj). We’ll assume that the pjare random draws fromsome population with mean π and variance v .The quantitypj(1 pj)is a kind of "per call variance" associated with the declarations of operatorj, and might serve as a kind of repeatability variance for that operator.Varde man and Morris (Iowa State University) IE 361 Module 8 4 / 16Some Simple Probability Modeling(Given the value of pj, elementary probability says that the varian ce of Xjis mpj(1  pj).) The biggest problem here is that unlike what is true inthe usual case of Gauge R&R for measu rements, this variance is notconstant across operators. But its expected value, namelyE(pj(1  pj))= π  Ep2j= π v + π2= π(1  π)vcan be used as a sensible measure of variability inconforming/non-conforming classi…cations chargeable to repeatabilitysources. The variance v serves as a measure of reproducibility variance.Varde man and Morris (Iowa State University) IE 361 Module 8 5 / 16Some Simple Probability ModelingThis ultimately points toπ(1  π)as the "total R&R variance" in this context. That is, we make de…nitionsfor 0/1 contextsσ2R&R= π(1  π)σ2repeatability= E(pj(1  pj))= π(1  π)vandσ2reproducibility= vVarde man and Morris (Iowa State University) IE 361 Module 8 6 / 16Some Simple R&R Point Estimates for 0/1 ContextsStill thinking of a single …xed part, we’ll letˆpj=the number of "non-conforming" calls made by operator jm=Xjmand de…ne the (sample) average of these¯ˆp =1JJ∑j =1ˆpjIt is possible to argue thatE¯ˆp = πand a plausible estimate of σ2R&Ris thenˆσ2R&R=¯ˆp(1 ¯ˆp)Varde man and Morris (Iowa State University) IE 361 Module 8 7 / 16Some Simple R&R Point Estimates for 0/1 ContextsThen, since ˆpj(1  ˆpj)is a plausible estimate of the "per call variance"associated with the declarations of operator j, pj(1  pj), an estimate ofσ2repeatabilityisˆσ2repeatability= ˆp(1  ˆp)(the sample average of the ˆpj(1  ˆpj))Finally, a simple estimate of σ2reproducibility= v isˆσ2reproducibility=ˆσ2R&Rˆσ2repeatability=¯ˆp(1 ¯ˆp) ˆp(1  ˆp)Varde man and Morris (Iowa State University) IE 361 Module 8 8 / 16Some Simple R&R Point Estimates for 0/1 ContextsWhat to do based on multiple parts (say I of them) is not completelyobvious. For our purposes in IE 361, we will simply average estimatesmade one part at a time across multiple parts, presuming that parts inhand are sensibly thought of as a random sample of parts to be checked,and that this averaging is a sensible way to combine information acrossparts.In order for any of this to have a chance of working, m is going to have tobe fairly large. The usual Gauge R&R "m = 2 or 3" just isn’t going toproduce informative results in the present context. And in order for this towork in practice (so that an operator isn’t just repeatedly looking at thesame few parts over and over and remembering how he’s called them in thepast) this seems like it’s going to require a large value of I as well as m.Varde man and Morris (Iowa State University) IE 361 Module 8 9 / 16Some Simple R&R Point Estimates for 0/1 ContextsExample 8-1Suppose that I = 5 parts are inspected by J = 3 operators, m = 10 timesapiece, and that the table below lists sample fractions of "non-conforming"calls made by the operators and estimated per call variances.Operator 1 Operator 2 Operator 3 Averageˆp ˆp(1  ˆp)ˆp ˆp(1  ˆp)ˆp ˆp(1  ˆp)ˆp(1  ˆp)Part 1 .2 .16 .4 .24 .2 .16 .187Part 2 .6 .24 .6 .24 .7 .21 .230Part 3 1.0 0 .8 .16 .7 .21 .123Part 4 .1 .09 .1 .09 .1 .09 .090Part 5 .1 .09 .3 .21 .3 .21 .170Average .160The …nal column of this table gives