r&r talk at q&p june 2002-burdick

7/27/2019 R&R Talk at Q&P June 2002-Burdick

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Recent Extensions in GaugeCapability Studies

2002 Q&P Research Conference

June 6, 2002

Richard K. Burdick

Arizona State University


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The Big Picture Statistical techniques have played an

important role in maintaining competitive

position for the U. S. manufacturing industry. In order to maintain a competitive position,

companies must manufacture products withalmost perfect consistency and repeatability.

This requires an ability to measure thevariability of highly complex manufacturingprocesses.


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3

Cell Phone TestA cell phone manufacturer tests

whether each phone is functioning

properly just prior to shipment.

The test system consists of a fixturewhich secures the phone and a rack of

measurement equipment which testsover 40 functions.


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4

Kick Panel AssemblyZ2002-83208-1

Opt 908 Rack Kit(Rails Supplied)

Opt AX4 Rack KitE3663A Rail Kit

Fixture Drawer Opens Out

Two Hand Tie Down Buttons CloseFixture And Start Alignment

E4079A Keyboard-Mouse Tray(Retractable)

E1301A "B" Size

83206A TDMACellular Adapter

8920BRF

Test Set

System Configuration Layout

83236AUP-Converter

E3909A Vectra Rack Kit

Opt 908 Rack KitE3663A Rail Kit40102A Filler Panel

D2806B 15" Monitorw / E4475A RackMount Kit

6626A PS

437B 5062-4080 Rack KitE3663A Rail Kit

Barcode

Reader

46298N Work Surface

PNEUMATICSHUTOFF

EMERGENCYFINAL ALIGNMENT Emergency stop button

Opt AXK Rack KitE3663A Rail Kit


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Cell Phone Test Questions arose on an installed base of 20

test systems as to whether the measurement

process had excessive noise. In particular, the phone manufacturer felt the

false failure rate was too high.

A gauge capability study (measurementsystem analysis) was conducted to examinethis issue.


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Gauge Capability Studies The major objective of a gauge study is to

determine if a measurement procedure isadequate for monitoring a process.

If the measurement error is small relative tothe total process variation, then themeasurement procedure is deemed

adequate. Montgomery and Runger (1994), Burdick and

Larsen (1997), Vardeman and VanValkenburg(1999)


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Traditional Two-Factor Design 24 phones were randomly selected

6 test systems were randomly selected

2 repeated measurements were takenon each phone and test system

42 phone parameters were measured

on each of the 288 design points


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ijkijjiijk E)PO(OPY

i=1,.,p; j=1,.,o; k=1,.r

Pi, Oj, (PO)ij, Eijkare iid normal with means of zero and

is a constant

variances2

P ,

2

O ,

2

PO ,

2

E

Traditional Two-Factor DesignA typical model in a gauge study is the

random two-factor model


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Cell Phone Test Phones are parts

Test systems are operators

Parts and operators are crossed sinceeach phone and test systemcombination is replicated two times.


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Expected Mean SquaresSV df EMS

Parts p-1 2 2 2P E PO Pr orq

Operators o-1 2 2 2O E PO Or prq

P x O (p-1)(o-1)2 2

PO E POrq

Error po(r-1) 2E Eq


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Measures of AdequacyMeasure Symbol I n terms of EM S

Repeatability 2E

Eq

Measurement

Error

2 2 2

O PO E g [ ( 1) ( 1) ] /( )O PO E p p r prqqq--

S/N Ratio 2

/Pwg

2 ( ) /( )P P PO

orqq-

P/T Ratio /( )USL LSLhg-


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Criteria S/N Ratio

Criterion: Lower bound of a 90%

confidence interval for w >5

P/T Ratio

Criterion: Upper bound of a 90%

confidence for h < .05


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Confidence intervals for measures ofadequacy are needed to apply the

criteria. Measures of adequacy are functions of

variance components


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Confidence Intervals for

Variance Components Modified Large-Sample (MLS) methods

Graybill and Wang (1980)

Burdick and Graybill (1992)

Interval for w based on Gui, Graybill,Burdick, Ting (1995)

Interval for h based on Graybill and Wang(1980)


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Advantages of MLS Closed form intervals

Require only F-values

Easy to compute in spreadsheet

Excel program that computes MLS intervalsin two-factor random, mixed, and three-

factor random available from Burdick [email protected]

C t ti f R&R P t


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Computation of R&R ParametersUser Inputs are in yellow:

Desired Confidence Level (%) 90

2-sided tolerance width 20

Two Factor Random Effects Model

Factors Number of Levels Mean Squares (MS)

Parts (DUT) 9 74.095

Operators 6 9.021

Interaction 1.802

Reps 2 0.191

Outputs:

Factor DF MS Estimate Lower CL Upper CL Percent

Parts 8 74.095 6.024417 3.032825 17.92288 81.17Operators 5 9.021 0.401056 0.120174 2.084657 28.70

Interaction 40 1.802 0.8055 0.548039 1.26415 57.64

Repeatability 54 0.191 0.191 0.142946 0.270594 13.67

Meas. Error 1.397556 1.040718 3.132987 18.83

S/N Ratio 2.076218 1.259792 3.618473

P/T Ratio 0.059109 0.051008 0.088501

Criteria for an "adequate" test process:

S/N Lower CL > 5P/T Upper CL < .05


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Other Methods Satterthwaite/Welch/Cochran

Generalized inference/Surrogate

variables Zhou and Mathew (1994)

Hamada and Weerahandi (2000)

Chiang (2001) Bayesian methods

Bootstrap methods


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Extensions Comparisons across time/location

Fixed vs. random operators

More complex designs Three-factor crossed

Nested

Other models Attribute (pass-fail) data Truncated data

Destructive testing


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Comparisons Across

Time/Location Compare total variation after attempt to

improve the measurement process

Comparison of same measurementprocess used at two different locations


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( ) ( )( ) ( )lijk l i j l li ij l ijkl Y M P O MP PO E

i=1,.,p; j=1,.,o; k=1,.r; l=1,2

Pi, Oj(l), (MP)li, (PO)ij(l), Eijklare iid normal with means of zero and

variances2

P ,

2

Ol ,

2

MP , 2

El

Comparisons Across

Time/Location The model is now

andMlare fixed withM1+M2=0

2,

POl


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Comparisons Across

Time/Location To compare the different processes, we

can compute a confidence interval for

the ratio

2 2 2

1 1 12 2 2

2 2 2

O PO E

O PO E


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Comparisons Across

Time/Location Confidence intervals can be constructed

using Cochran/Satterthwaite interval or

MLS interval proposed by Ting, Burdick,and Graybill (1991).

Details in Burdick, Allen, and Larsen

(2002).


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Fixed vs. Random OperatorsAlthough it is customary to assume all

effects are random, such an assumption

is not always warranted.Although parts are typically random, in

many cases operators are more

properly considered as fixed effects.


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Fixed vs. Random OperatorsA simple modification to the operator

degree of freedom allows one to use

the same formulas used for the case ofrandom operators.

This modification is based on a chi-squared approximation of a non-centralchi-square random variable.

Dolezal, Burdick, and Birch (1998)


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More Complex Designs Three-factor crossed designs

Nested designs


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Three-Factor Design Consider the fabrication of magnetic tape for

computerized data storage.

In this study, o automated test stations(operators) are used to evaluate the qualityof p tape heads (parts).

In order to measure characteristics of the

heads, t tapes are used in each test station. In this design, all p heads are measured with

each of the test station/tape combinations.


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i=1,.,p; j=1,.,o; k=1,.t; l=1,,r

Pi, Oj, Tk, (PO)ij, (PT)ik, (OT)jk, (POt)ijk, Eijklare iid normal

with means of zero and variances

Three-Factor Design The ANOVA model is

fixed

( ) ( ) ( ) ( )ijkl i j k ij ik jk ijk ijkl Y P O T PO PT OT POT E

2

P ,2

O ,2

T ,2

PO ,2

PT ,2

OT ,2

POT and2

E


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Three-Factor DesignAn appropriate measure of adequacy

(parts to total measurement error) is

2

2 2 2 2 2 2 2

P

O T PO PT OT POT E


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Three-Factor DesignA confidence interval based on a

combination of Satterthwaite and MLS

methods appears to maintain statedconfidence level in most situations.

A generalized confidence interval also

performs well. Details in Adamec and Burdick (2003)


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Nested Designs John (1994, p. 12) provides an example

where bbatches of wafers are manufactured,wwafers are sampled from each batch, eachwafer is placed on a machine for poccasions,and rrepeated measurements are collectedon each placement.

In this experiment, wafers are nested withinbatches, the placements are nested withinwafers, and observations are nested withinplacements.


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( ) ( )ijkl i j i k ij ijkl Y B W P E

i=1,.,b; j=1,.,w; k=1,.p; l=1,,r

Bi, Wj(i), Pk(ij), Eijklare iid normal with means of zero and

variances2

B ,

2

W ,

2

P ,

2

E

Nested Designs The ANOVA model is

fixed


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Nested DesignsAn appropriate measure of adequacy

(process to total measurement error) is

2 2

2 2

B W

P E


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Nested Designs Confidence intervals can be constructed using

either the Satterthwaite/Cochran method orthe MLS method proposed by Gui, Graybill,Burdick, and Ting (1995).

Another alternative is to use generalizedconfidence intervals.

The Satterthwaite/Cochran method will likelyprovide the shortest intervals, but in somecases will likely provide confidencecoefficients less than the stated level.


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Other Models Attribute (pass-fail) data

Boyles (2001)

Truncated data Lai and Chew (2000)

Destructive testing

Mitchell, Hegemann, and Liu (1997)

Phillips, Jeffries, Schneider, and Frankoski (1997)

Bergeret, Maubert, Sourd, and Puel (2001)


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References

ADAMEC, E. and BURDICK, R. K. (2003). Confidence Intervals for aRatio of Variance Components in a Gauge Study with Three RandomFactors. To appear in Quality Engineering, March, 2003.

BERGERET, F.; MAUBERT, S.; SOURD, P.; and PUEL, F. (2001).Improving and Applying Destructive Gauge Capability. QualityEngineering14(1), pp. 59-66.

BOYLES, R. A. (2001). Gauge Capability for Pass-Fail Inspection.Technometrics 43, pp. 223-229.

BURDICK, R. K.; ALLEN A. E.; and LARSEN, G. A. (2002). Comparing

Variability of Two Measurement Processes Using R&R Studies. Journalof Quality Technology, 34, pp. 97-105.

BURDICK, R. K. and GRAYBILL, F. A. (1992). Confidence Intervals onVariance Components. Marcel Dekker, Inc., New York, New York.

BURDICK, R. K. and LARSEN, G. A. (1997). Confidence Intervals onMeasures of Variability in R&R Studies. Journal of QualityTechnology

29, pp. 261-273.


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References

CHIANG, A. K. L. (2001). A Simple General Method forConstructing Confidence Intervals for Functions of VarianceComponents. Technometrics 43, pp.356-367.

DOLEZAL, K. K.; BURDICK, R. K.; and BIRCH, N. J. (1998).Analysis of a Two-Factor R&R Study with Fixed Operators.Journal of Quality Technology 30, pp. 163-170.

GRAYBILL, F. A. and WANG, C. M. (1980). Confidence Intervalson Nonnegative Linear Combination of Variances. Journal ofthe American Statistical Association 75, pp. 869-873.

GUI, R; GRAYBILL, F. A.; BURDICK, R. K. and TING, N. (1995).Confidence Intervals on Ratios of Linear Combinations of Non-Disjoint Sets of Expected Mean Squares. Journal of StatisticalPlanning and Inference 48, pp. 215-227.


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References

HAMADA, M. and WEERAHANDI (2000). Measurement SystemAssessment via Generalized Inference. Journal of Quality Technology32, pp. 241-253.

JOHN, P. (1994).Alternative Models for Gauge Studies. SEMATECH

report 93081755A-TR. LAI Y. W. and CHEW E. P. (2000). Gauge Capability Assessment for

High-Yield Manufacturing Processes with Truncated Distribution.Quality Engineering 13(2), pp. 203-210.

MITCHELL, T.; HEGEMANN, V.; and LIU, K.C. (1997). GRRMethodology for Destructive Testing and Quantitative Assessment of

Gauge Capability for One-Side Specifications in Statistical Case Studiesfor Industrial Process Improvement; Czitrom, V. and Spagon, P. D.,Eds; SIAM, Philadelphia, pp. 47-59.

MONTGOMERY, D. C. and RUNGER, G. C. (1994). Gauge Capabilityand Designed Experiments. Part II: Experimental Design Models andVariance Component Estimation. Quality Engineering 6, pp. 289-305.


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References

PHILLIPS, A. R.; JEFFRIES, R.; SCHNEIDER, J.; and FRANKOSKI, S. P.(1997). Using Repeatability and Reproducibility Studies to Evaluate aDestructive Test Method. Quality Engineering 10(2), pp. 283-290.

TING, N.; BURDICK, R. K.; and GRAYBILL, F. A. (1991). ConfidenceIntervals on Ratios of Positive Linear Combinations of VarianceComponents. Statistics and Probability Letters 11, pp. 523-528.

VARDEMAN, S. B. and VANVALKENBURG, E. S. (1999). Two-wayRandom-effects Analyses and Gauge R&R Studies. Technometrics 41,pp. 202-211.

ZHOU, L. and MATHEW, T. (1994). Some Tests for VarianceComponents Using Generalized p-values. Technometrics 36, pp. 394-402.

r&r talk at q&p june 2002-burdick

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