Post on 04-Jan-2016
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Factorial Models
Random Effects Gauge R&R studies (Repeatability
and Reproducibility) have been an expanding area of application
Mixed Effects Models
One-way Random Effects
The one-way random effects model is quite different from the one-way fixed effects model– Yandell has a real appreciation for this
difference– We should be surprised that the
analytical approaches to the main hypotheses for these models are so similar
One-way Random Effects
In Chapter 19, Yandell considers– unbalanced designs– Smith-Satterthwaite approximations– Restricted ML estimates
We will defer the last two topics to general random and mixed effects models
One-way Random Effects
Yij Ai ij, i 1,,a, j 1,,n
ij iid N(0, 2)
Ai iid N(0,A2 )
ij, Ai indep.
One-way Random Effects
E Yij
V Yij A2 2
Cov Yij,Yij' Cov Ai, Ai V Ai A2
Yij,Yij' A2
A2 2
One-way Random EffectsE(MSTR)
Yij Ai ij
Y i. Ai iY .. A .
Y i.. Y .. 2
i
Ai A 2
i
i 2 i
One-way Random EffectsE(MSTR)
n Y i. Y .. 2
i
n Ai A 2
i
n i 2 i
En
a 1Y i. Y .. 2
i
E
n
a 1Ai A 2
i
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n
a 1i 2
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E MSTR nA2 2
One-way Random EffectsTesting
By a similar argument, we can show E(MSE)=2
The familiar F-test statistic for testing
Ho : i 0
will also be appropriate for testing
Ho :A2 0
One-way Testing
Under the true model,
So power analysis for balanced one-way random effects can be studied using a central F-distribution
F MSTR /(nA
2 2)
MSE / 2~
nA2 2
2F a 1,a(n 1)
One-way Random Effects
Method of Moment point estimates for 2 and
2 are available
Confidence intervals for 2 and 2/2 are available
A confidence interval for the grand mean is available
Two-way Random Effects Model
We will concentrate on a particular application—the Gauge R&R model
20.2 addresses unbalanced models– Material is accessible
Topics in 20.3 will be addressed later
20.4 and 20.5 can safely be skipped
Gauge R&RTwo-way Random Effects Model P-Part O-Operator
ijkijjiijk
ijkijjiijk
POOPY
CBAY
R R
Gauge R&R
With multiple random components, Gauge R&R studies use variance components methodology
Y2 P
2 O2 PO
2 2
Gauge R&R
Repeatability is measured by
Reproducibility is measured by
2
O2 PO
2
Gauge R&R
Unbiased estimates of the variance components are readily estimated from Expected Mean Squares (a=# parts, b=# operators, n=# reps)
E(MSE) 2
E MSParts 2 n PO2 bn P
2
E MSOperator 2 n PO2 an O
2
E MSPO 2 n PO2
Gauge R&R
Use Mean Sums of Squares for estimation
ˆ 2 MSEˆ PO
2 ?ˆ P
2 ?ˆ O
2 ?
Gauge R&R
Minitab has a Gauge R&R module– Output is specific to industrial methods
Consider an example with 3 operators, 5 parts and 2 replications
Two-way Random Effects Model
Consider results from our expected mean squares.
What would be appropriate tests for A, B, and AB?
Approximate F tests
Statistics packages may do this without your being aware of it.
Example– A, B and C random– Replication
Approximate F test
Source EMS
222
22222
22222
22222
ABABC
CABCBCAC
BABCBCAB
AABCABAC
cnn
bcnnanbn
acnnancn
bcnncnbn
AB
C
B
A
Approximate F test
Source EMS
2
22
222
222
ABC
ABCBC
ACABC
n
nan
bnn
Error
ABC
BC
AC
Approximate F test
No exact test of A, B, or C exists
We construct an approximate F test,
vu
sr
MSMSMS
MSMSMS
MSMSF
''
'
where,''/'
Approximate F test
We require E(MS’)=E(MS”) under Ho
F has an approximate F distribution, with parameters
1 MSii r
s
2
MSi2 df i
i r
s
2 MS jj u
v
2
MSj2 df j
j u
v
Approximate F test
Note that MS’, MS’’ can be linear combinations of the mean squares and not just sums
Returning to our example, how do we test
0: 2 AoH
DF for Approximate F tests
Restating the result:
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'with
ondistributi F eapproximatan has '''
then,'' and ' If
2
2
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1
1
v
ujjj
s
riii
v
ujj
r
ii
dfMS
MS
dfMS
MSMSMSF
MSMSMSMS
DF for Approximate F tests
The following argument builds approximate c2 distributions for the numerator and denominator mean squares (and assumes they are independent)
We will review the argument for the numerator
The argument computes the variance of the mean square two different ways
DF for Approximate F tests
Remember that the numerator for an F random variable has the form:
Note that we already have this result for the constituent MSi
2
2
~'' want would weHence,
MSEMS
DF for Approximate F tests
For each term in the sum, we have
. where,~ 2i
22 idfi
ii MSEMSdf
i
.2ely approximat is varianceitsthen
, ddistributeely approximat is 'MSE
'MS If 2
DF for Approximate F tests
We can derive the variance by another method:
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ri i
i
s
rii
dfMSE
MSVMSE
MSEMSV
.2
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)('
''
4
2
2
2
2
DF for Approximate F Tests
Equating our two expressions for the variance, we obtain:
s
ri i
i
s
ri i
i
dfMSE
MSE
dfMSE
2
2
4
2
2
'
2'
2
DF for Approximate F Tests
Replacing expectations by their observed counterparts completes the derivation.
Two-way Mixed Effects Model
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Form edUnrestrict
0,1
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Form Restricted
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1
1
22
1
ABijBj
a
ii
a
iijABijBj
a
ii
ijkijjiijk
NiidCNiidB
jCa
aNCNiidB
CBY
Two-way Mixed Effects Model
Both forms assume random effects and error terms are uncorrelated
Most researchers favor the restricted model conceptually; Yandell finds it outdated. It is certainly difficult to generalize.
SAS tests the unrestricted model using the RANDOM statement with the TEST option; the restricted model has to be constructed “by hand”.
Minitab tests unrestricted model in GLM, restricted model option in Balanced ANOVA.
Two-way Mixed Effects Model
2222
22222
1
2
221
2
22
AB
B11
A
EMS edUnrestrictEMS RestrictedSource
ABAB
BABB
a
ii
AB
a
ii
AB
nn
annana
bnn
a
bnn
Two-way Mixed Effects Model
The EMS suggests that the fixed effect (A) is tested against the two-way effect (AB) for both forms (F=MSA/MSAB)
The EMS suggests that the random effect (B) is tested against error (F=MSB/MSE) for the restricted model, but tested against the two-way effect (AB) for the unrestricted model (F=MSB/MSAB)
Two-way Mixed Effects Model
For the Gage R&R study, assume that Part is still a random effect, but that Operator is a fixed effect
SAS and Minitab analysis