Design of Engineering Experiments - Experiments with Random Factors

Design & Analysis of Experiments 8E 2012 Montgomery

1Chapter 13


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


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


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


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


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


7Chapter 13


8Chapter 13

Example 13.1 Minitab Solution – Using Balanced ANOVA


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


10Chapter 13

Example 13.1 Minitab Solution – Reduced Model


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


12

Maximum Likelihood Approach

Chapter 13


13Chapter 13


14Chapter 13


15Chapter 13


16

CI on the Variance Components

Chapter 13

The lower and upper bounds on the CI are:


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


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


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


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


21Chapter 13

Example 13.2Minitab Solution – Balanced ANOVA


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


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


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


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


26Chapter 13

Example 13.3 Minitab Solution – Unrestricted Model


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


28

JMP Output for the Mixed Model – using REML

Chapter 13


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


30Chapter 13

Notice that there are no exact tests for main effects


31Chapter 13


32Chapter 13


33Chapter 13


34Chapter 13


35Chapter 13


36Chapter 13

We chose a different linear combination


37Chapter 13

Design of Engineering Experiments - Experiments with Random Factors

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