Identifying the Split-plot and Constructing an Analysis

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Identifying the Split-plot and Constructing an Analysis. George A. Milliken Department of Statistics Kansas State University milliken@stat.ksu.edu. Complex Split-plot Designs. 1. Very Useful Efficient Designs. 2. Often used but Not Recognized Designs.

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### Identifying the Split-plot and Constructing an Analysis

George A. Milliken

Department of Statistics

Kansas State University

milliken@stat.ksu.edu

Complex Split-plot Designs

1. Very Useful Efficient Designs

2. Often used but Not Recognized Designs

3. Often Miss or Inappropriately Analyzed

Could Spend several Hours Describing and Discussing Complex Split Plot Designs

I will use an Example to Demonstrate some of the Ideas Involved

Hydrothermal Processing of Wheat Gluten

Slurry at 3 concentrations---10% 14% 18%

Path --- long or short (time in cooker)

Temp 250 275 300 F of cooker

Drying methods -- Air (room temp), Hot (heated)

Four Replications of 36 Treatment Combinations

Measure solubility--put sample of the part into a flask of water and measure Time to dissolve IN SECONDS;

Time in Seconds for product to dissolve for SHORT path.

PATH=SHORT

TEMP=250 TEMP=275 TEMP=300

REP CONC HOT AIR HOT AIR HOT AIR

1 10 26.7 26.8 20 19.6 22.6 20.1

2 10 20.1 18.5 23.2 20.4 19.3 16.9

3 10 29.8 28.6 25.1 23.4 27.2 27.1

4 10 19 16.7 18.4 16.1 15.8 14.2

1 14 31.6 28 26.5 24.4 32.5 30.5

2 14 27.6 24.7 28.7 27.3 27.1 21.8

3 14 24.5 24.6 27.2 24.1 30 26.9

4 14 29.9 26.7 24.3 22.1 27.3 25.5

1 18 26.8 25.9 21.6 24.6 25.6 26.8

2 18 31.9 27.8 25.4 28.7 21.9 24

3 18 26.8 25.9 20.7 22.3 23.1 24.5

4 18 31 28.1 27.5 31.2 28.9 27.1

Time in Seconds for product to dissolve for Long path.

PATH=LONG

TEMP=250 TEMP=275 TEMP=300

REP CONC HOT AIR HOT AIR HOT AIR

1 10 23 20.9 22.6 20.9 14.6 12.1

2 10 26.5 25.4 20.8 19.1 19.9 19.9

3 10 26.3 25.2 25.5 25.2 23.4 22.7

4 10 21.5 19.4 21.3 18.2 16.4 14.6

1 14 29.6 27.9 25.3 22.8 28.3 27.4

2 14 25.4 25.3 28.8 27.6 25.1 24.6

3 14 28.2 27.8 24.6 23.8 28.8 27.3

4 14 26.3 26.5 23.9 21.7 28 28.6

1 18 24.4 23.5 31 29.8 24.8 27.1

2 18 31.5 29.3 24.9 23.3 23.1 25.8

3 18 30 29.3 23.8 24.7 23.4 26.3

4 18 35.5 37 25.5 26.7 27.9 31

Conclusions from AOV

Significant Concentration by Temperature Interaction

Compare the Conc*Temp Cell Means

Estimate of Variance is 10.88988

Response Surface Model

Since Levels of Concentration and Temperature are Quantitative, fit RESPONSE SURFACE type model using Path and Dry as Categorical variables

Conditions with Maximum Response

GRAPHICS FOLLOW WITH 95% CI CONTAIN MAX

How was the experiment executed?Part 1

Slurry at 3 concentrations---slurry tank 10% 14% 18%

Make a tank of Slurry using one of the concentrations

Do this in Random Order – Obtain four Replications of each concentration----Completely Randomized Design

Tank is the Experimental Unit for levels of Slurry—the entity to which levels of Slurry are Randomly Assigned

Slurry Concentration

10%

14%

18%

Tank 1

Tank 2

Tank 3

Tank 4

Tank 5

Tank 6

Graphical Representation of The Experiment – Tank as EU

RANDOMIZE

Completely Randomized Design

How was the experiment executed?Part 2

Take Six BATCHES from TANK--apply the Six Combinations of PATH*TEMP to the BATCHES

TANK is BLOCK of Six BATCHES

RANDOMLY assign Combinations of PATH*TEMP to the Six BATCHES from each TANK

BATCH is EXPERIMENTAL UNIT for combinations of PATH*TEMP

BATCH Design is Randomized Complete Block where TANK is the Blocking Factor

Graphical Representation of The Experiment – Batch as EU

Path by Temperature Combinations

LONG

SHORT

250

275

300

250

275

300

1

2

3

4

5

6

1

2

3

4

5

6

TANK 12

TANK 1

Batches

Batches

RANDOMIZE to Each Tank

Each Tank is a Block of Six Batches for levels of Path by Temperature Combinations

DRY METHOD

AIR

HOT

TANK

PART

Batch

Graphical Representation of The Experiment – Part as EU

RANDOMIZE to Each Batch

Batch(Tank) is Block of Two Parts – for levels of DRY

Appropriate Model Includes

Factorial Effects for Levels of Conc x Path x Temp x Dry

• Three Sizes of Experimental Units, each with an ERROR TERM
• TANK
• BATCH
• PART
Estimates of the Variance Components for Split-plot

Sum of Variance Component Estimates = 10.890

Same as CR Estimate of Variance

Comparisons of Split-plot and CRD analyses

Using Split-plot Error Structure

Discovered Conc*Temp*Path*Dry interaction Exists in the Data Set

CRD analysis found Conc*Temp interaction Significant while split-plot analysis didn’t

CRD analysis pools the three error terms together and the resulting error is not appropriate for any of the comparisons

Conditions with Maximum Response

GRAPHICS FOLLOW WITH 95% CI CONTAIN MAX

Comparisons of 95% Confidence Regions for Maximum Response

Path=Short Dry=Hot

Split-plot ERRORS

CRD ERRORS

Comparisons of Split-plot and CRD Response Surface Models

Split-plot Response Surface Model is more complex

Many more relationships are occurring than discovered using CRD

Predicted Response Surface Sweet spots are larger for Split-plot than for CRD

Conclusions-1

Ignoring the error structure can provide a different response surface model

Ignoring the error structure will provide the illusion that there is a smaller sweet spot in the surface

Incorporating the split-plot error structure into the model provides appropriate tests, comparisons, resulting model and sweet spot

Conclusions -2

Failure to identify the appropriate Design Structure and use it in the modeling process CAN LEAD TO VERY MISLEADING RESULTS

Acknowledgments:

Departments of Grain Science and Agricultural and Biological Engineering for the experiment

Version 8 of PROC MIXED of the SAS® System

SAS System Code for ANOVA

proc mixed cl DATA=TIME ;

class rep conc path temp dry;

title 'Model using the split-split-plot error treated as aov with means';

model time=conc|path|temp|dry;

random rep(conc) path*temp*rep(conc);

lsmeans path*dry*temp conc*path*dry conc*temp/diff;

SAS System Code for RSM

proc mixed cl data=time; class rep xconc xtemp path dry ;**xconc=conc and xtemp=temp;

title 'Final regresson model using split-split-plot error structure';

model time=conc conc*conc temp conc*temp conc*conc*temp path dry conc*dry conc*temp*dry path*dry conc*path*dry conc*conc*path*dry temp*conc*path*dry temp*temp*conc*path*dry

/solution SINGULAR=1e-11 ddfm=KR outpm=pred;

random rep(xconc) path*xtemp*rep(xconc);

THE END

THANK YOU

FOR