1

Analysis Considerations in Industrial Split-Plot Experiments When the

Responses are Non-Normal

Timothy J. Robinson

University of Wyoming

Raymond H. Myers

Virginia Tech

Douglas C. Montgomery

Arizona State

2

Mixture Experiment with Process Variables

• Film manufacturing process

• Three components, (X1,X2, X3), melted and mixed in a screw extruder to produce a roll of film

• Pieces are cut from the roll and processed at a particular setting of the process variables

• Response is a quality measure reflecting the amount of polarized light that passes through the film.

• Response is distinctly non-normal and the coefficient of variation in the response

is constant.

1 2 3P ,P ,P

3

Heaters

MoltenPolymerWeb

Thickness Gauge

Winder

Thermocouples

Chill Rolls

Inlet

4

Design Region

At each mixture, a 23-1 design is run in the processvariables.

If design was completely randomized and we did notreplicate, it would require 20 formulations of the mixture.

5

SAMPLE X1 X2 X3 P1 P2 P38 0 0.8 0.2 1 -1 -18 0 0.8 0.2 -1 -1 18 0 0.8 0.2 1 1 18 0 0.8 0.2 -1 1 -117 0.8 0 0.2 1 -1 -117 0.8 0 0.2 -1 -1 117 0.8 0 0.2 1 1 117 0.8 0 0.2 -1 1 -121 0 0.8 0.2 1 -1 -121 0 0.8 0.2 -1 -1 121 0 0.8 0.2 1 1 121 0 0.8 0.2 -1 1 -133 0.6 0 0.4 1 -1 -133 0.6 0 0.4 -1 -1 133 0.6 0 0.4 1 1 133 0.6 0 0.4 -1 1 -15 0.6 0 0.4 1 -1 -15 0.6 0 0.4 -1 -1 15 0.6 0 0.4 1 1 15 0.6 0 0.4 -1 1 -137 0 0.6 0.4 1 -1 -137 0 0.6 0.4 -1 -1 137 0 0.6 0.4 1 1 137 0 0.6 0.4 -1 1 -1

Mixture Variables Process VariablesWholePlots

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Linear Mixed Model

-Cornell (1988) discusses a split-plot approach to the analysis of mixture experiments containing process variables.

-Classical linear model when design is completelyrandomized:

y X Z

2 2 2 2 2

2 2 2 2 2

2 2 2 2 2

2 2 2 2 2

0 0 0

0 0 0Var | = +

0 0 0

0 0 0

i i

y

2 ~ N 0, 'i 11

= + y X

2Var = ny I

X E | i i y i

7

Generalized Linear Models

Var = Var = y V

- Many industrial experiments involve responses that are non-normal [Myers, Montgomery, Vining (2002)]

-The generalized linear model (GLM) assumes that the mean is related to by a link function s such that:X

-The GLM assumes independent responses and the variance/covariance matrix of responses is given by:

= + y

-1 = s X

= s X

where V is diagonal.

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Generalized Linear Mixed Models

• Just as linear models can be extended to linear mixed models, generalized linear models can be extended to generalized linear mixed models (GLMM).

• GLMM’s are appropriate for split-plot experiments since they accommodate the inherent correlation among observations taken on the same whole plot.

• Two types of GLMM’s considered:

-Random effects GLMM

-Covariance pattern GLMM

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• Random effects GLMM:

- whole plots treated as random effects and are modeled simultaneously with the regression coefficients

• Regarding notation:

Model: = + y

= + s X Z

|Var = Var + y V

is a diagonal matrix of the response given the random effect.

| = Var |V y

denotes the conditional mean and plays the same role as y in the GLM. E |y

|y

Var y 2 ' ' DZZ D |+ V

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• Random effects GLMM similar to linear mixed model except it accommodates a more general family of responses

• Random effects enter the model in a non-linear fashion…consequently, predicted values from this model involve estimated random effects:

• In film example, we have a gamma response with a log link and hence the prediction model is given by:

• Predicted values are batch specific due to the random effects and is interpreted as the predicted quality for the ith batch.

ˆ ˆˆ = { + }expy X Z

yi

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• Covariance pattern GLMM:

-regression of the response on the design variables is modeled separately from the within-whole plot correlation

Model: * = + y

= s X

1/2 1/2 2Var = y A RA

A is a diagonal matrix of variances determinedby the distribution of the response

*E = ,0 * 1/2 1/2Var = A RA 2

with denoting correlation matrix of responses amongobservations in whole plot i.

1 = , ..., adiagR R R iR

1

1 =

1

i

R

SplitPlot Designs

12

• Instead of using random effects to account for the correlation, a covariance structure is defined for the

error term

• Prediction equation for covariance pattern models is interpreted differently than the batch-specific interpretation resulting from random effects models

In the film example, predicted values are determined from: and are interpretedas the average quality across all batches

ˆˆ = { }expy X

Covariance pattern models are often referred toas population-averaged models.

13

Fitting the Random Effects GLMM

• Maximum likelihood where likelihood is determined from the product of the likelihoods of

• With non-normal responses, solving the likelihood involves numerical integration techniques

• A common approach to maximizing the likelihood proposed by Wolfinger and O’Connell is to use pseudo-likelihood

• Pseudo-likelihood uses an iterative generalized least squares analysis of a pseudo-variable , which can be thought of as a linearized observation vector.

• The pseudo-variable is a first order Taylor series expansion for and is given by

| and y

*y

about s y

* = + - 's sy y

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• Recalling that we have

• This model is similar to the classical linear mixed model and can be fit using the SAS macro GLIMMIX

= + s X Z

* = + + - 'sy X Z y

= + + *.X Z e

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Fitting the Covariance Pattern GLMM

• Pseudo-likelihood is used to fit the covariance pattern model as well but with the following pseudo-variable:

• The variation due to whole plots is treated as a nuisance and pooled into the error

where and =

** = + - 'sy X y

= + **X e

= s X **e - 'sy

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Batch-Specific vs. Population Averaged Prediction Models

• In the random effects GLMM, the conditional expectation of y plays the same role as y in the traditional GLM

• The quantity modeled in the random effects GLMM is and thus predicted quality is conditional on the ith batch

• If one wished to have a prediction of quality across all batches, it would be reasonable to use

since we typically assume

|y

=0ˆˆ = expy X

ˆ ˆˆ = exp + y X Z

E = .0

17

• If average quality across all batches is of interest, one could model the unconditional expectation of y as is done in covariance pattern GLMMs

• The covariance pattern model in the film example is

.

• The difference between depends on the magnitude of .

As a result, the predictions in the covariance patternmodel are not dependent on an estimated random effect

* = exp + y X

ˆˆ = expy X

ˆ ˆ and 0y y2

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Illustration

• Consider the following random effects GLMM

• Five random values for the are generated and profiles of what could be considered to be five batches are plotted.

• For a prediction model across all batches, we could substitute

• The true average profile could be determined by integrating out the as follows

where we assume

's

E | = exp 1 + X + y

2~ N 0, 0.65 .i

i

= E 0

-

E E | = exp 1 + X + f

y

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• Notice that is what is actually modeled with the covariance pattern GLMM

• Evaluating the integral we have

• Plots of the individual profiles, the approximate population averaged profile and the true population average profile are examined for

E E | = E y y

E = exp 1.211 + Xy

2 = 0.423 and 4.0

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-1 0 1 0

5.0

10.0

15.0

20.0

25.0

X

YLong dashes denote and solid line denotes true population average profile

=0y2 = 0.423

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Long dashes denote and solid line denotes true population average profile

=0y2 = 4.0

-1 0 1 0

40

80

X

Y

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• For the larger whole plot variance, at X=1, the true population average is approximately 55 whereas the estimated average is slightly less than 10.

• When the whole plot variance is smaller, at X=1, the true population average is approximately 8 and the estimated population average is approximately 6

• Practical implication: If one is interested in predicting average quality across all batches, it is better to model the unconditional expectation of the response (covariance pattern GLMMs) than the

conditional expectation (random effects GLMMs).

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Analysis of Film Data

• The following random effects and covariance pattern GLMMs were fit to the film data using the SAS macro GLIMMIX:

• The mixture portion was considered to be first-order due to prior knowledge and only the X1X2 interaction was thought to be important. All mixture*process variable interactions were of interest except for those involving X3

Random Effects GLMM

Covariance Pattern GLMM

3 3 2 3

12 1 21 1 1 1

= X + X X + + P + X P + i i j j ij i ji j i j y

3 3 2 3*

12 1 21 1 1 1

= X + X X + P + X P + i i j j ij i ji j i j y

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X1 X2

146.26251.99

451.17

650.35

Contour plot of estimated population averaged model for film data using batch specific GLMM .

X1 X2

166.26264.51

479.458

Contour plot of estimated population averaged model for film data using covariance pattern GLMM.

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X1 X2

132.69251.96

451.11

ˆ = -0.166

145.964

261.337376.71492.08

607.456

X1 X2

ˆ = 0.051

127.04

251.99451.17

650.35X1 X2

Contour plots of estimated response for individual batches

ˆ = 0.181

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Conclusions

• Non-normal responses are common in industry and the standard GLM analysis is inappropriate for experiments which are run as split-plots

• In terms of prediction across all whole plot units, the population average model is most appropriate

• The random effects model is useful in the sense of providing an estimate for the amount of whole plot variation in relation to the amount of variation among sub-plot units