IE 361 Module 6. Gauge R&R Studies Part 2: Two-Way ANOVA and Corresponding Estimates for R&R Studies

Transcription

1 IE 361 Module 6 Gauge R&R Studie Part 2: Two-Way ANOVA and Correponding Etimate for R&R Studie Reading: Section 2.2 Statitical Quality Aurance for Engineer (Section 2.4 of Revied SQAME) Prof. Steve Vardeman and Prof. Max Morri Iowa State Univerity Augut 2008 ardeman and Morri (Iowa State Univerity) IE 361 Module 6 Augut / 21

2 The range-baed Gauge R&R etimate of SQAME are fairly imple and erve the purpoe of helping make the analyi goal eay to undertand. But we have no good handle on how reliable thee etimate are. In order to 1) produce Gauge R&R etimate that are typically better than range-baed one, and 2) produce con dence limit, we mut intead ue "ANOVA-baed" etimate. A careful treatment of ANOVA would require it own coure. We ll imply make ue of it main "output" and direct the intereted tudent to book on engineering tatitic (like Vardeman Statitic for Engineering Problem Solving) for more detail. The fact i that an I J m data et of y ijk like that produced in a typical Gauge R&R tudy i often ummarized in a o-called ANOVA table. A generic verion of uch a table i Vardeman and Morri (Iowa State Univerity) IE 361 Module 6 Augut / 21

3 Source SS df MS Part SSA I 1 MSA = SSA/ (I 1) Operator SSB J 1 MSB=SSB/ (J 1) PartOperator SSAB (I 1) (J 1) MSAB=SSAB/ (I 1) (J 1) Error SSE IJ (m 1) MSE = SSE /IJ (m 1) Total SSTot IJm 1 Any decent tatitical package (and even EXCEL) will proce a Gauge R&R data et and produce uch a ummary table. In thi table the "mean quare" are eentially ample variance (quare of ample tandard deviation). (MSA i eentially a ample variance of part average, MSB i eentially a ample variance of operator average, MSE i an average of within cell-ample variance, "MSTot" in t typically calculated, but i a grand ample variance of all obervation,...) The mean quare indicate how much of the overall variability i accounted for by the variou ource. Vardeman and Morri (Iowa State Univerity) IE 361 Module 6 Augut / 21

4 Example 6-1 We ll ue the data et with I = 4, J = 3, m = 2 from the in-cla R&R tudy (ued a a numerical example in Module 5) to illutrate. The JMP data table and ome creen hot for uing the program to get the um of quare follow. Figure: JMP Data Sheet for the In-Cla R&R Study ardeman and Morri (Iowa State Univerity) IE 361 Module 6 Augut / 21

5 Example 6-1 Figure: JMP Dialogue Box for Fit Model Two-Way ANOVA on the Gauge R&R Data Vardeman and Morri (Iowa State Univerity) IE 361 Module 6 Augut / 21

6 Example 6-1 Figure: JMP Two-Way ANOVA Report for the In-Cla Gauge R&R Study Vardeman and Morri (Iowa State Univerity) IE 361 Module 6 Augut / 21

7 Example 6-1 Although we certainly don t recommend uing EXCEL (a preadheet i no ubtitute for a tatitical package and, beide, EXCEL ha terribly unreliable numerical analyi) we found intruction on uing the program two-way ANOVA plug-in at Two creen hot from uing thee intruction on the in-cla two-way data follow. Vardeman and Morri (Iowa State Univerity) IE 361 Module 6 Augut / 21

8 Example 6-1 Figure: EXCEL Two-Way Data Spreadheet for the In-Cla R&R Study ardeman and Morri (Iowa State Univerity) IE 361 Module 6 Augut / 21

9 Figure: EXCEL Two-Way Data Spreadheet for the In-Cla R&R Study For our preent purpoe, we will take mean quare and degree of freedom out of uch an ANOVA table and make Gauge R&R etimate baed on them. Point etimator for the quantitie of mot interet in a Gauge R&R tudy are partially ummarized on the bottom of page 27 in SQAME. Thee are Vardeman and Morri (Iowa State Univerity) IE 361 Module 6 Augut / 21

10 ˆσ repeatability = bσ = p MSE and ˆσ reproducibility = max 0, MSB + mi (I 1) MSAB mi 1 m MSE Althoughq it i not preented in SQAME, an appropriate etimator for σ R&R = σ 2 γ + σ 2 αγ + σ 2 (that i called σ overall in SQAME) i ˆσ R&R = r 1 mi MSB + I 1 mi MSAB + m 1 m MSE Vardeman and Morri (Iowa State Univerity) IE 361 Module 6 Augut / 21

11 It i further poible to ue thee etimate to make an exact con dence interval for σ repeatability = σ and Satterthwaite approximate con dence limit for σ reproducibility and σ R&R. Let ν repeatability = IJ (m 1) Then, con dence limit for σ repeatability are ˆσ repeatability νrepeatability χ 2 ν repeatability, upper and ˆσ repeatability νrepeatability χ 2 ν repeatability, lower For etimating σ reproducibility, let Vardeman and Morri (Iowa State Univerity) IE 361 Module 6 Augut / 21

12 ˆν reproducibility = = MSB 2 mi J 1 + ˆσ 4 reproducibility (I 1)MSAB 2 mi 2 MSE (I 1) (J 1) + m IJ (m 1) ˆσ 4 reproducibility 1 MSB 2 (I 1) MSAB2 m 2 I 2 + (J 1) I 2 + MSE 2 (J 1) IJ (m 1) Then approximate con dence limit for σ reproducibility are ˆσ reproducibility For etimating σ R&R, let ˆνreproducibility χ 2ˆν reproducibility,upper and ˆσ reproducibility ˆνreproducibility χ 2ˆν reproducibility,lower Vardeman and Morri (Iowa State Univerity) IE 361 Module 6 Augut / 21

13 ˆν R&R = = MSB 2 mi J 1 + (I ˆσ 4 R&R 1)MSAB 2 mi (I 1) (J 1) + ˆσ 4 R&R (m 1)MSE m IJ (m 1) 1 MSB 2 (I 1) MSAB2 m 2 I 2 + (J 1) I 2 + (J 1) 2 (m 1) MSE 2 IJ then approximate con dence limit for σ R&R are ˆνR&R ˆνR&R ˆσ R&R χ 2ˆν and ˆσ R&R R&R,upper χ 2ˆν R&R,lower Vardeman and Morri (Iowa State Univerity) IE 361 Module 6 Augut / 21

14 Thee formula are tediou (but hardly impoible) to ue with a pocket calculator. Or a very mall program could be written to evaluate the etimate of tandard deviation and approximate degree of freedom. Vardeman ue a imple MathCAD workheet to do the computing. The following gure illutrate the ue of that workheet beginning from SSB, SSAB, SSE, I, J, and m for the in-cla R&R tudy. Vardeman and Morri (Iowa State Univerity) IE 361 Module 6 Augut / 21

15 Example 6-1 Figure: MathCAD Workheet for Example 6-1 ardeman and Morri (Iowa State Univerity) IE 361 Module 6 Augut / 21

16 Example 6-1 The reult in panel 6,9, and 15 how that 95% con dence limit for σ repeatability are νrepeatability νrepeatability ˆσ repeatability χ 2 and ˆσ repeatability ν repeatability, upper χ 2 ν repeatability, lower i.e. i.e r 4 3 (2 1) and in and.0089 in r 4 3 (2 1) Similarly, approximate 95% con dence limit for σ reproducibility are ˆνreproducibility ˆνreproducibility ˆσ reproducibility χ 2ˆν and ˆσ reproducibility reproducibility,upper χ 2ˆν reproducibility,lower Vardeman and Morri (Iowa State Univerity) IE 361 Module 6 Augut / 21

17 Example 6-1 i.e. i.e. r r 4 and in and.0259 in And nally, approximate 95% con dence limit for σ R&R are ˆνR&R ˆνR&R ˆσ R&R χ 2ˆν and ˆσ R&R R&R,upper χ 2ˆν R&R,lower i.e. i.e. r r and in and.0224 in Vardeman and Morri (Iowa State Univerity) IE 361 Module 6 Augut / 21

18 Example 6-1 Thee interval how that none of thee tandard deviation are terribly well-determined (degree of freedom are mall and interval are wide). If better information i needed, more data would have to be collected. But there i at leat ome indication that σ repeatability and σ reproducibility are roughly of the ame order of magnitude. The caliper ued to make the meaurement wa a fairly crude one, and there were detectable di erence in the way the tudent operator ued that caliper. Vardeman and Morri (Iowa State Univerity) IE 361 Module 6 Augut / 21

19 Quite often indutrial Gauge R&R tudie are done to invetigate the adequacy of a meauring device (and the operator that ue it) to check conformance of item produced to engineering peci cation (value that delineate limit of what i required of the item for it to be functional). Suppoe that ome feature of a product need to have a value, x, that i at leat L and no more than U in order for it to be functional. (L i the lower peci cation for x and U i the upper peci cation.) In thi context, it i common to want to compare one perception of the ize of σ R&R to "how tight L and U are." (For example, trying to compare x that can be een only through a large amount of meaurement noie to very tight peci cation i a hopele tak.) A way of quantifying thi kind of comparion i thi. If one think of meaurement error a normally ditributed, in the abence of (average acro operator) meaurement bia (µ δ = 0), a meaurement y made by a "randomly elected" operator in ome ene repreent x to within Vardeman and Morri (Iowa State Univerity) IE 361 Module 6 Augut / 21

20 3σ R&R and o 6σ R&R might be taken a a kind of meaurement uncertainty. The di erence U L repreent the allowable variation in x. So the ratio GCR = 6σ R&R U L i ometime called a Gauge Capability Ratio or a Preciion to Tolerance Ratio and ued a an index of the adequacy of a meaurement ytem to verify the functionality of product. Of coure, thi can only be etimated uing the output of a Gauge R&R tudy, o an etimated verion of thi i [GCR = 6 ˆσ R&R U L Notice too that having computed con dence limit for σ R&R, one need only multiply thee by 6 U L in order to produce con dence limit for GCR. Vardeman and Morri (Iowa State Univerity) IE 361 Module 6 Augut / 21

21 A common rule of thumb i that one need to be fairly ure that GCR <.1 (and preferably that GCR <.01) before a gauge can be conidered adequate for the purpoe of checking conformance of x to peci cation L and U. Vardeman and Morri (Iowa State Univerity) IE 361 Module 6 Augut / 21