attribute measurement systems analysis using a binary random

AttributeAttribute MeasurementMeasurement SystemsSystemsAnalysis using A Binary Random Analysis using A Binary Random

Effects ModelEffects Model

Vahid PARTOVI NIAChair of Statistics

Swiss Federal Institute of Technology

Collaboration with: G.H. Shahkar and S.M.M. Tabatabaey

Ferdowsi University of Mashhad

Overview

IntroductionManual Method for Attribute Measurement Systems AnalysisA Binary Random Effects ModelComparison on a sample data

History

Why “attribute” measurement systems are importantHow we started the research

What is an Attribute Measurement System ?

Some Examples

Types of Measurement Systems

Measurement Systems

Variable Attribute

Ruler

Calliper

X-Meter Visual Inspection Go-Nogo Gages

Properties of a Good Measurement System

RepeatabilityReproducibilityUnbiasednessNo Trend in ErrorSensitivity to The Real State

Repeatability

101525

101525

251510

Reproducibility

251510

251510

251510

251510

251510

An R&R Measurement System

251510

251510

251510

251510

251510

251510

251510

251510

Bias

81220

10

15

25

Linearity

8 1220

10

15

25

= -2= -3= -5

REAL

2624222018161412108

ERR

OR

-1.5

-2.0

-2.5

-3.0

-3.5

-4.0

-4.5

-5.0

-5.5

REAL

2624222018161412108

Abso

lute

Err

op

5.5

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

251510

251510

251510

251510

25

15

10

An Ideal Measurement System

251510

251510

251510

251510

17

12

8

17128

17128

17128

17128

251510

251510

251510

251510

25

15

10

251510

251510

251510

251510

17

12

8

17128

17128

17128

17128

A Real Measurement System

231711

261412

22148

25139

25

15

10

An Ideal Attribute Measurement System

CORRECTCORRECT

CORRECTCORRECT

CORRECTCORRECT

CORRECTCORRECT

CORRECT

CORRECT


FAILEDCORRECT

FAILEDCORRECT

FAILEDCORRECT

FAILEDCORRECT

FAILED

FAILED

FAILEDFAILED

FAILEDFAILED

FAILEDFAILED

FAILEDFAILED


FAILEDCORRECT

FAILEDCORRECT

FAILEDCORRECT

FAILEDCORRECT

FAILED

CORRECT

A Real Attribute Measurement System

CORRECTCORRECT

FAILEDCORRECT

FAILEDFAILED

CORRECTFAILED

FAILED

CORRECT

Measures of Agreement

Measures of Inter-rater Agreement

CORRECT

FAILED

CORRECT

FAILED

?

?

FALIEDCORRECT

To

tal

FA

LIE

DC

OR

RE

CT

A+B+C+D=N

A2

B+DA1

A+C

B2

C+DDC

B1

A+BBA

Rate

r B

TotalRater A

E

EO

ppp

−−

=1

κ

NCA

NBApE

+×

+=

NApO =

Agreement Based on Chance

FailedCorrect

To

tal

FailedC

orrect

N=100A2

50A1

50

B2

502525

B1

502525

Rate

B

TotalRater A

0=κ

Almost Perfect Agreement

FailedCorrect

To

tal

FailedC

orrect

N=100A2

1A1

99

B2

110

B1

99099

Rate

B

TotalRater A

99.0=κ

Almost Perfect Disagreement

FailedCorrect

To

tal

FailedC

orrect

N=100A2

99A1

1

B2

101

B1

99990

Rate

B

TotalRater A

99.0−=κ

Inadequacy of Kappa in Marginally Unbalanced Cross tabs and An Alternative

Chance Based Measure

FailedCorrect

To

tal

FailedC

orrect

N=100A2

54A1

46

B2

51456

B1

49940

Rate

B

TotalRater A

FailedCorrect

To

tal

FailedC

orrect

N=100A2

15A1

85

B2

1055

B1

901080

Rate

B

TotalRater A

32.0=κ 7.0=κ

Manual Methodology 3rd EditionDesignDesign

3 Appraiser3 RepeatsTotally 50 Correct and Failed Parts

DecisionDecisionBetween Appraiser’s Agreement (3 Cross tabs)Agreement of Appraisers with Real Status of Parts (3 Cross tabs)Effectiveness’ Homogeneity (3 Confidence Intervals)

A Practical Example

Kappa0.863

976Correct

344Failed

CorrectFailedAppr. AAppr.B

Kappa0.776

927Correct

843Failed

CorrectFailedAppr. AAppr. C

Kappa0.788

945Correct942Failed

CorrectFailedAppr. BAppr.C

Kappa0.879

975Correct345Failed

CorrectFailedAppr. AREF

Continue

Kappa0.923

1002Correct

345Failed

CorrectFailedAppr. BREF

Kappa0.774

939Correct

642Failed

CorrectFailedAppr. CREF

0.690.820.74LCL0.800.900.84Score

0.910.980.94UCLAppr.A Appr.B Appr.C

http://www.carwin.co.uk/qs/english/msaamend.htm

Inadequacies of Manual Method

Kappa is not a reliable measure of agreement The error of decision is unknownSeems by adding appraisers it highly inflates type I error of overall decisionSeems to be Inconsistent!

Simulation Structure 1

Appraiser 3

Appraiser 2

Appraiser 1

Ppppppppp25Failed

Ppppppppp25Correct

321321321nState

p

Em

piric

al P

roba

bilit

y of

Rej

ecte

d M

easu

rem

ent S

yste

ms

0.5 0.6 0.7 0.8 0.9

0.0

0.2

0.4

0.6

0.8

1.0

ManualDeterministic


Appraiser 3

Appraiser 2

Appraiser 1

ppppppppp5Failed

ppppppppp45Correct

321321321nState

p

Em

piric

al P

roba

bilit

y of

Rej

ecte

d M

easu

rem

ent S

yste

ms

0.5 0.6 0.7 0.8 0.9

0.0

0.2

0.4

0.6

0.8

1.0

ManualDeterministic


Appraiser 3

Appraiser 2

Appraiser 1

0.850.85p0.850.85p0.850.85p100Failed

0.850.85p0.850.85p0.850.85p900Correct

321321321nState

P

Em

piric

al P

roba

bilit

y of

Rej

ecte

d M

easu

rem

ent S

yste

ms

0.5 0.6 0.7 0.8 0.9

0.0

0.2

0.4

0.6

0.8

1.0

DeterministicManual

Binary Random Effects Model( )( )ijiijiij

ijijijk

ROgRO

Biny

++= − µµ

µµ1,|

,1~|

ijikijijiijk ROyROy ,|,| ′⊥

ijiiij OyOy || ** ′⊥

**** ii yy ′⊥

( ) ( )[ ]( )[ ]( )∏ ∫∏ ∫

= = +++

++=

O

i

R

ijP

n

iO

n

jRn

ijiji

ijiji dFdFyROyRO

L1 1 .

.

exp1exp

µµ

θ

Approximated Likelihood

( )( ) ( )

( ) ( )( )[ ]( ) ( )( )[ ]( )

∑∑∏∑

=

===

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

+++

++∝

O

P

RRO

n

in

ijRO

ijRO

R

nqn

u

n

jO

nqn

v

yuRvO

yuRvO

uwvw

1

.00

.00

0

11

0

1

22exp1

22explog~

σσµ

σσµθl

Binary Threshold Model and Its Interpretation in the Nested Model

Logistic Standard~,|iid

ijiijk ROz

).,0(~

),,0(~

,,|

2

2

Rij

Oi

ijkijiijiijk

NR

NO

ROROz

σ

σ

ε++=

µ>ijkz 1=ijky

µ≤ijkz 0=ijky

R&R Measure

( ) ( )3

222 πσσεµ ++=+++= ROijkijiijk ROVarzVar

( )3

2π=ijkzVarR&R

Non R&R

ijkijiijiijk ROROz ε++=,|

3

3& 222

2

πσσ

π

++=

RO

RR

Empirical P-Value

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

2

1

θ

θθ L

⎩⎨⎧ =

0θ0θ

1

1

f::

1

0

HH

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=

2ML

1ML

ML

θ

θθ

ˆ

ˆˆ L

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

=

==

2ML2

1

θθ

0θθ

ˆL

D

mD ( )

( )

( ) ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=m2ML

m1ML

mML

θ

θθ

ˆ

ˆˆ L

( )( )B

ValueP 1MLm1ML θθ ˆˆ# f

=−

Parametric Bootstrap Confidence Intervals

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

2

1

θ

θθ L

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=

2ML

1ML

ML

θ

θθ

ˆ

ˆˆ L

D ( )

( )

( ) ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=m2ML

m1ML

mML

θ

θθ

ˆ

ˆˆ L

mD

Fitting sample data

19.0Value-P Empirical =

10 G.H. Quadrature NodesB=10000

90 % CIParameter

µ

Oσ

ORσ

( )37.3,47.2

( )48.0,0

( )1.1,0

82.0=RR

Slice likelihood

X Angle = 240 X Angle = 330

X Angle = 60 X Angle = 150

( )RO σσ ,

0.1 0.2 0.3 0.4 0.5 0.6SIGMA.O

0.2

0.6

1.0

1.4

SIG

MA

.R

-97.7 -97.5

-97.5

-97.3

-97.3

-97.2

-97.2

-97.0

-97.0

-96.8 -96.6

-96.4

One Dimensional Slice

stvec.sigmaO

mar

vec

0.02 0.04 0.06 0.08 0.10

-96.

249

-96.

247

-96.

246

-96.

244

Oσ


stvec.sigmaR

mar

vec

0.2 0.4 0.6 0.8 1.0 1.2 1.4

-98.

5-9

8.0

-97.

5-9

7.0

-96.

5

Rσ


stvec.mu

mar

vec

0 1 2 3 4 5 6

-130

-120

-110

-100

µ

Bootstrapped Distribution

-0.962784-0.351539

0.2597060.870951

1.4821972.093442

2.7046873.315932

3.9271784.538423

5.149

h0.mu.bre

0.0

0.2

0.4

0.6

0.8

1.0

1.2

µ̂


0.0000000.275758

0.5515150.827273

1.1030311.378788

1.6545461.930303

2.2060612.481819

2.757

h0.sigmaO.bre

0

1

2

3

4

5

Oσ̂


0.0000000.281217

0.5624340.843651

1.1248691.406086

1.6873031.968520

2.2497372.530954

2.8121

h0.sigmaRO.bre

0.0

0.5

1.0

1.5

2.0

2.5

Rσ̂

Summary

We discussed good properties of measurement systemsWe applied a model and responded to requirementsWe showed how Bootstrapping methods could be used where we have lack of small sample theory in GLMMs.

Thanks to

Meeting Organizers Anthony C. Davison

attribute measurement systems analysis using a binary random

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