attribute measurement systems analysis using a binary random
TRANSCRIPT
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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
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Overview
IntroductionManual Method for Attribute Measurement Systems AnalysisA Binary Random Effects ModelComparison on a sample data
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History
Why “attribute” measurement systems are importantHow we started the research
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What is an Attribute Measurement System ?
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Some Examples
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Types of Measurement Systems
Measurement Systems
Variable Attribute
Ruler
Calliper
X-Meter Visual Inspection Go-Nogo Gages
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Properties of a Good Measurement System
RepeatabilityReproducibilityUnbiasednessNo Trend in ErrorSensitivity to The Real State
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Repeatability
101525
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101525
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101525
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251510
Reproducibility
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251510
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251510
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251510
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An R&R Measurement System
251510
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251510
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251510
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Bias
81220
10
15
25
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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
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25
15
10
An Ideal Measurement System
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251510
251510
251510
251510
17
12
8
17128
17128
17128
17128
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251510
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25
15
10
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251510
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17
12
8
17128
17128
17128
17128
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A Real Measurement System
231711
261412
22148
25139
25
15
10
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An Ideal Attribute Measurement System
CORRECTCORRECT
CORRECTCORRECT
CORRECTCORRECT
CORRECTCORRECT
CORRECT
CORRECT
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An Ideal Attribute Measurement System
FAILEDCORRECT
FAILEDCORRECT
FAILEDCORRECT
FAILEDCORRECT
FAILED
FAILED
FAILEDFAILED
FAILEDFAILED
FAILEDFAILED
FAILEDFAILED
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An Ideal Attribute Measurement System
FAILEDCORRECT
FAILEDCORRECT
FAILEDCORRECT
FAILEDCORRECT
FAILED
CORRECT
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A Real Attribute Measurement System
CORRECTCORRECT
FAILEDCORRECT
FAILEDFAILED
CORRECTFAILED
FAILED
CORRECT
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Measures of Agreement
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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 =
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Agreement Based on Chance
FailedCorrect
To
tal
FailedC
orrect
N=100A2
50A1
50
B2
502525
B1
502525
Rate
B
TotalRater A
0=κ
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Almost Perfect Agreement
FailedCorrect
To
tal
FailedC
orrect
N=100A2
1A1
99
B2
110
B1
99099
Rate
B
TotalRater A
99.0=κ
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Almost Perfect Disagreement
FailedCorrect
To
tal
FailedC
orrect
N=100A2
99A1
1
B2
101
B1
99990
Rate
B
TotalRater A
99.0−=κ
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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
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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)
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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
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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
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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!
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Simulation Structure 1
Appraiser 3
Appraiser 2
Appraiser 1
Ppppppppp25Failed
Ppppppppp25Correct
321321321nState
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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
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Simulation Structure 2
Appraiser 3
Appraiser 2
Appraiser 1
ppppppppp5Failed
ppppppppp45Correct
321321321nState
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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
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Simulation Structure 3
Appraiser 3
Appraiser 2
Appraiser 1
0.850.85p0.850.85p0.850.85p100Failed
0.850.85p0.850.85p0.850.85p900Correct
321321321nState
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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
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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
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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
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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
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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
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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
=−
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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
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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
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Slice likelihood
X Angle = 240 X Angle = 330
X Angle = 60 X Angle = 150
( )RO σσ ,
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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
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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
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One Dimensional Slice
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
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One Dimensional Slice
stvec.mu
mar
vec
0 1 2 3 4 5 6
-130
-120
-110
-100
µ
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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
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Bootstrapped Distribution
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
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Bootstrapped Distribution
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
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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.
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Thanks to
Meeting Organizers Anthony C. Davison