the two-factor mixed model
DESCRIPTION
The Two-Factor Mixed Model. Two factors, factorial experiment, factor A fixed, factor B random (Section 13-3, pg. 495) The model parameters are NID random variables, the interaction effect is normal, but not independent This is called the restricted model. - PowerPoint PPT PresentationTRANSCRIPT
1
The Two-Factor Mixed Model• Two factors, factorial experiment, factor A fixed,
factor B random (Section 13-3, pg. 495)
• The model parameters 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:
• Test statistics depend on the expected mean squares:
2 20 0 0
2 21 1 1
: 0 : 0 : 0
: 0 : 0 : 0i
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H H H
H H H
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2 2 10
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ii A
AAB
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bnMSE MS n F
a MSMSE MS an FMSMSE MS n FMS
E MS
Ho is rejected if
Fo > F,a-1,(a-1)(b-1)
Fo > F,b-1,ab(n-1)
Fo > F,(a-1)(b-1), ab(n-1)
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Estimating the Variance Components – Two Factor Mixed model
• Use the ANOVA method; equate expected mean squares to their observed values:
• Estimate the fixed effects (treatment means) as usual
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B E
AB E
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MS MSan
MS MSn
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........ˆ
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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
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99.071.0ˆ
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99.039.62ˆ
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EOperatorsPartsOperatorsParts
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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
Operator 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
Source Variance Error Expected Mean Square for Each Term
component term (using restricted model)
1 Part 10.2332 4 (4) + 6(1)
2 Operator 3 (4) + 2(3) + 40Q[2]
3 Part*Operator -0.1399 4 (4) + 2(3)
4 Error 0.9917 (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)
• The random model parameters are now all assumed to be NID . is no longer assumed – unrestricted model
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Testing Hypotheses – Unrestricted Mixed Model
• The standard ANOVA partition is appropriate• Relevant hypotheses:
• Expected mean squares determine the test statistics:
2 20 0 0
2 21 1 1
: 0 : 0 : 0
: 0 : 0 : 0i
i
H H H
H H H
2
2 2 10
2 2 20
2 20
2
( ) 1
( )
( )
( )
a
ii A
AAB
BB
AB
ABAB
E
E
bnMSE MS n F
a MSMSE MS n an FMSMSE MS n FMS
E MS
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Estimating the Variance Components – Unrestricted Mixed Model
• Use the ANOVA method; equate expected mean squares to their observed values:
• 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 caseso The restricted model is slightly more generalo The restricted model is mostly preferred
2
2
2
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B AB
AB E
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MS MSan
MS MSn
MS
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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
Operator 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
Source Variance Error Expected Mean Square for Each Term
component term (using unrestricted model)
1 Part 10.2798 3 (4) + 2(3) + 6(1)
2 Operator 3 (4) + 2(3) + Q[2]
3 Part*Operator -0.1399 4 (4) + 2(3)
4 Error 0.9917 (4)
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Sample Size Determination with Random Effects
• Consider a single-factor random effects model
• Power = 1 – P(Reject HoHo is false)
P(Fo > F,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
2
2
1 n
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Sample Size Determination with Random Effects – Example 13-5
• Five treatments randomly selected (a = 5)• Six observations per treatment (n = 6)• = 0.05, a – 1 = 4 (v1), N – a = 25 (v2)• Assume that • Then
0.20
646.2)1(61
22
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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, • P is the fixed percentage increase in the standard
deviation
• Then
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2
Pnn
22
P01.012
22
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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:
• 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
2( ) (fixed factor)E MS f
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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
• F is distributed approximately as Fp,q
SMSMF
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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