Transcript Design of Engineering Experiments Part 7 – The 2k

The Two-Factor Mixed Model
• Two factors, factorial experiment, factor A fixed,
factor B random (Section 13-3, pg. 495)
 i  1, 2,..., a

yijk     i   j  ( )ij   ijk  j  1, 2,..., b
k  1, 2,..., n

V (  j )   2 , V [( )ij ]  [( a  1) / a] 2 , V ( ijk )   2
a

i 1
a
i
 0,  ( )ij  0
i 1
• The model parameters  j and  ijk are NID random
variables, the interaction effect is normal, but not
independent
• This is called the restricted model
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Testing Hypotheses - Mixed Model
• Once again, the standard ANOVA partition is appropriate
• Relevant hypotheses:
H0 :i  0
H 0 :  2  0
H 0 :  2  0
H1 :  i  0
H1 :  2  0
H1 :  2  0
• Test statistics depend on the expected mean squares:
a
bn  i2
Ho is rejected if
Fo > Fa,a-1,(a-1)(b-1)
 F0 
MS A
MS AB
E ( MS B )   2  an 2
 F0 
MS B
MS E
Fo > Fa,b-1,ab(n-1)
E ( MS AB )   2  n 2
 F0 
MS AB
MS E
Fo > Fa ,(a-1)(b-1), ab(n-1)
E ( MS A )   2  n 2 
i 1
a 1
E ( MS E )   2
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Estimating the Variance Components
– Two Factor Mixed model
• Use the ANOVA method; equate expected mean
squares to their observed values:
MS B  MS E
ˆ  
an
MS AB  MS E
2
ˆ 
n
ˆ 2  MS E
2
• Estimate the fixed effects (treatment means) as
usual
ˆ  y...
 i  yi ..  y...
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Example 13-3 (pg. 497)
The Measurement Systems Capability
Study Revisited
•
•
•
•
•
Same experimental setting as in example 13-2
Parts are a random factor, but Operators are fixed
Assume the restricted form of the mixed model
Minitab can analyze the mixed model
The variance components can also be estimated as
MS Parts  MS E 62.39  0.99

 10.23
an
(3)( 2)
MS PartsOperators  MS E 0.71  0.99
2
ˆ PartsOperators 

 0.14
n
2
ˆ 2  MS E  0.99
2
ˆ Parts

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Example 13-3 (pg. 497)
Minitab Solution – Balanced ANOVA
Source
DF
SS
MS
F
P
Part
19
1185.425
62.391
62.92
0.000
2
2.617
1.308
1.84
0.173
Part*Operator
38
27.050
0.712
0.72
0.861
Error
60
59.500
0.992
Total
119
1274.592
Operator
Source
Variance Error Expected Mean Square for Each Term
component term (using restricted model)
1 Part
10.2332
2 Operator
3 Part*Operator
4 Error
-0.1399
0.9917
4
(4) + 6(1)
3
(4) + 2(3) + 40Q[2]
4
(4) + 2(3)
(4)
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Example 13-3
Minitab Solution – Balanced ANOVA
•
•
•
•
There is a large effect of parts (not unexpected)
Small operator effect
No Part – Operator interaction
Negative estimate of the Part – Operator
interaction variance component
• Fit a reduced model with the Part – Operator
interaction deleted
• This leads to the same solution that we found
previously for the two-factor random model
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The Unrestricted Mixed Model
• Two factors, factorial experiment, factor A fixed,
factor B random (pg. 498)
 i  1,2,..., a

yijk    a i   j  (a ) ij   ijk  j  1,2,..., b
k  1,2,..., n

V ( j )   2 , V [(a ) ij ]   a2 , V ( ijk )   2
a
a
i 1
i
0
• The random model parameters are now all
assumed to be NID . ia1 (a )ij  0 is no longer
assumed – unrestricted model
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Testing Hypotheses – Unrestricted Mixed Model
• The standard ANOVA partition is appropriate
• Relevant hypotheses:
H 0 : ai  0
H 0 :  2  0
H 0 :  a2  0
H1 : a i  0
H1 :  2  0
H1 :  a2  0
• Expected mean squares determine
the test statistics:
a
bn a i2
 F0 
MS A
MS AB
E ( MS B )   2  n a2  an 2
 F0 
MS B
MS AB
E ( MS AB )   2  n a2
 F0 
MS AB
MS E
E ( MS A )   2  n a2 
E ( MS E )   2
i 1
a 1
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Estimating the Variance Components –
Unrestricted Mixed Model
• Use the ANOVA method; equate expected mean
squares to their observed values:
MS B  MS AB
an
MS AB  MS E

n
ˆ 2  MS E
ˆ 2 
ˆa2
• The only change compared to the restricted mixed
model is in the estimate of the random effect
variance component
• Which model to use?
o They are fairly close in many cases
o The restricted model is slightly more general
o The restricted model is mostly preferred
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Example 13-4 (pg. 499)
Minitab Solution – Unrestricted Model
Source
DF
SS
MS
F
P
Part
19
1185.425
62.391
87.65
0.000
2
2.617
1.308
1.84
0.173
Part*Operator
38
27.050
0.712
0.72
0.861
Error
60
59.500
0.992
Total
119
1274.592
Operator
Source
Variance Error Expected Mean Square for Each Term
component term (using unrestricted model)
1 Part
10.2798
2 Operator
3 Part*Operator
4 Error
-0.1399
0.9917
3
(4) + 2(3) + 6(1)
3
(4) + 2(3) + Q[2]
4
(4) + 2(3)
(4)
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Sample Size Determination with
Random Effects
• Consider a single-factor random effects model
• Power = 1 –   P(Reject HoHo is false)
 P(Fo > Fa,a-1,N-a Ho is false)
• Fo = MSTreatments/MSE (dofs are needed to
determine the OC curve)
• The operating characteristic curves (Chart VI,
Appendix) can be used
• The curves plot the probability of type II error
against the parameter l
n 2
l  1 2

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Sample Size Determination with
Random Effects – Example 13-5
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•
•
•
•
Five treatments randomly selected (a = 5)
Six observations per treatment (n = 6)
a = 0.05, a – 1 = 4 (v1), N – a = 25 (v2)
Assume that  2   2
Then
l  1  6(1)  2.646
  0.20
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Sample Size Determination with
Random Effects
• Use the percentage increase in the standard
deviation of an observation
• If the treatments are homogeneous, 
• If the treatments are different,  2   2
• P is the fixed percentage increase in the standard
deviation
 2   2
 1  0.01P
2

• Then
n 2
l  1  2  1  n[(1  0.01P) 2  1]

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Sample Size Determination with
Random Effects – Two Factors
Table 13-8
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Finding Expected Mean Squares
• Obviously important in determining the form of
the test statistic
• In fixed models, it’s easy:
E (MS )   2  f (fixed factor)
• Can always use the “brute force” approach – just
apply the expectation operator
• Straightforward but tedious
• Rules on page 502-503 [due to Cornfield and
Tukey (1956)] work for any balanced model
• Rules are consistent with the restricted mixed
model
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Approximate F Tests
• Sometimes we find that
there are no exact tests for
certain effects
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Approximate F Tests
• One possibility: assume that certain interactions are
negligible – needs conclusive evidence
• If we cannot assume that certain interactions are negligible,
then use an approximate F test (“pseudo” F test)
• Test procedure is due to Satterthwaite (1946), and uses
linear combinations of the original mean squares to form
the F-ratio
• For example:
MS’ = MSr + …+ MSs
MS’’ = MSu + …+ MSv
• The mean squares are chosen so that E(MS’) – E(MS’’) is a
multiple of the effect considered in the null hypothesis
MS 
F
MS 
• F is distributed approximately as Fp,q
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Approximate F Tests
• The linear combinations of the original mean
squares are sometimes called “synthetic” mean
squares
• Adjustments are required to the degrees of
freedom
• Refer to Example 13-7, page 505
• Minitab will analyze these experiments, although
their “synthetic” mean squares are not always the
best choice
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