Design & Analysis of Experiments 8E 2012 Montgomery
2Chapter 13
Design of Engineering Experiments - Experiments with Random Factors
• Text reference, Chapter 13• Previous chapters have focused primarily on fixed factors
– A specific set of factor levels is chosen for the experiment– Inference confined to those levels– Often quantitative factors are fixed (why?)
• When factor levels are chosen at random from a larger population of potential levels, the factor is random– Inference is about the entire population of levels– Industrial applications include measurement system studies– It has been said that failure to identify a random factor as random and
not treat it properly in the analysis is one of the biggest errors committed in DOX
• The random effect model was introduced in Chapter 3
Design & Analysis of Experiments 8E 2012 Montgomery
3Chapter 13
Extension to Factorial Treatment Structure
• Two factors, factorial experiment, both factors random (Section 13.2, pg. 574)
• The model parameters are NID random variables• Random effects model
2 2 2 2
2 2 2 2
1, 2,...,( ) 1,2,...,
1,2,...,
( ) , ( ) , [( ) ] , ( )
( )
ijk i j ij ijk
i j ij ijk
ijk
i ay j b
k n
V V V V
V y
Design & Analysis of Experiments 8E 2012 Montgomery
4Chapter 13
Testing Hypotheses - Random Effects Model• Once again, the standard ANOVA partition is appropriate• Relevant hypotheses:
• Form of the test statistics depend on the expected mean squares:
2 2 20 0 0
2 2 21 1 1
: 0 : 0 : 0
: 0 : 0 : 0
H H H
H H H
2 2 20
2 2 20
2 20
2
( )
( )
( )
( )
AA
AB
BB
AB
ABAB
E
E
MSE MS n bn FMSMSE MS n an FMSMSE MS n FMS
E MS
Design & Analysis of Experiments 8E 2012 Montgomery
5Chapter 13
Estimating the Variance Components – Two Factor Random model
• As before, we can use the ANOVA method; equate expected mean squares to their observed values:
• These are moment estimators• Potential problems with these estimators
2
2
2
2
ˆ
ˆ
ˆ
ˆ
A AB
B AB
AB E
E
MS MSbn
MS MSan
MS MSnMS
Design & Analysis of Experiments 8E 2012 Montgomery
6Chapter 13
Example 13.1 A Measurement Systems Capability Study
• Gauge capability (or R&R) is of interest• The gauge is used by an operator to measure a critical
dimension on a part• Repeatability is a measure of the variability due only to
the gauge• Reproducibility is a measure of the variability due to the
operator • See experimental layout, Table 13.1. This is a two-factor
factorial (completely randomized) with both factors (operators, parts) random – a random effects model
Design & Analysis of Experiments 8E 2012 Montgomery
8Chapter 13
Example 13.1 Minitab Solution – Using Balanced ANOVA
Design & Analysis of Experiments 8E 2012 Montgomery
9Chapter 13
Example 13.2 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
Design & Analysis of Experiments 8E 2012 Montgomery
10Chapter 13
Example 13.1 Minitab Solution – Reduced Model
Design & Analysis of Experiments 8E 2012 Montgomery
11Chapter 13
Example 13.1 Minitab Solution – Reduced Model
• Estimating gauge capability:
• If interaction had been significant?
2 2 2ˆ ˆ ˆ
0.88 0.010.89
gauge
Design & Analysis of Experiments 8E 2012 Montgomery
16
CI on the Variance Components
Chapter 13
The lower and upper bounds on the CI are:
Design & Analysis of Experiments 8E 2012 Montgomery
17Chapter 13
The Two-Factor Mixed Model• Two factors, factorial experiment, factor A fixed,
factor B random (Section 13.3, pg. 581)
• The model parameters are NID random variables, the interaction effects are normal, but not independent
• This is called the restricted model
2 2 2
1 1
1,2,...,( ) 1,2,...,
1, 2,...,
( ) , [( ) ] [( 1) / ] , ( )
0, ( ) 0
ijk i j ij ijk
j ij ijk
a a
i iji i
i ay j b
k n
V V a a V
and j ijk
Design & Analysis of Experiments 8E 2012 Montgomery
18Chapter 13
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
i
H H H
H H H
2
2 2 10
2 20
2 20
2
( ) 1
( )
( )
( )
a
ii A
AAB
BB
E
ABAB
E
E
bnMSE MS n F
a MSMSE MS an FMSMSE MS n FMS
E MS
Design & Analysis of Experiments 8E 2012 Montgomery
19Chapter 13
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
2
2
2
ˆ
ˆ
ˆ
B E
AB E
E
MS MSan
MS MSnMS
Design & Analysis of Experiments 8E 2012 Montgomery
20Chapter 13
Example 13.2 The Measurement Systems Capability
Study Revisited• Same experimental setting as in example
13.1• Parts are a random factor, but Operators are
fixed• Assume the restricted form of the mixed
model• Minitab can analyze the mixed model and
estimate the variance components
Design & Analysis of Experiments 8E 2012 Montgomery
21Chapter 13
Example 13.2Minitab Solution – Balanced ANOVA
Design & Analysis of Experiments 8E 2012 Montgomery
22Chapter 13
Example 13.2 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
Design & Analysis of Experiments 8E 2012 Montgomery
23Chapter 13
The Unrestricted Mixed Model
• Two factors, factorial experiment, factor A fixed, factor B random (pg. 583)
• The random model parameters are now all assumed to be NID
2 2 2
1
1, 2,...,( ) 1,2,...,
1, 2,...,
( ) , [( ) ] , ( )
0
ijk i j ij ijk
j ij ijk
a
ii
i ay j b
k n
V V V
Design & Analysis of Experiments 8E 2012 Montgomery
24Chapter 13
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
Design & Analysis of Experiments 8E 2012 Montgomery
25Chapter 13
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
2
2
2
ˆ
ˆ
ˆ
B AB
AB E
E
MS MSan
MS MSn
MS
Design & Analysis of Experiments 8E 2012 Montgomery
26Chapter 13
Example 13.3 Minitab Solution – Unrestricted Model
Design & Analysis of Experiments 8E 2012 Montgomery
27Chapter 13
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 523-524 work for any balanced model• Rules are consistent with the restricted mixed model – can
be modified to incorporate the unrestricted model assumptions
2( ) (fixed factor)E MS f
Design & Analysis of Experiments 8E 2012 Montgomery
28
JMP Output for the Mixed Model – using REML
Chapter 13
Design & Analysis of Experiments 8E 2012 Montgomery
29Chapter 13
Approximate F Tests• Sometimes we find that there are no exact tests for certain
effects (page 525)• Leads to 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
• 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• Minitab will analyze these experiments, although their
“synthetic” mean squares are not always the best choice
Design & Analysis of Experiments 8E 2012 Montgomery
30Chapter 13
Notice that there are no exact tests for main effects
Design & Analysis of Experiments 8E 2012 Montgomery
36Chapter 13
We chose a different linear combination