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

AnalysisAnalysis

George A. MillikenGeorge A. Milliken

Department of StatisticsDepartment of Statistics

Kansas State UniversityKansas State University

milliken@stat.ksu.edumilliken@stat.ksu.edu

2

Complex Split-plot DesignsComplex Split-plot 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

1. Very Useful Efficient Designs

3

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)

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

Four Replications of 36 Treatment Combinations

4

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

5

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

6

Analysis of Variance TableSource DF SS MS FValue ProbFconc 2 951.222 475.611 43.67 0.0000path 1 0.856 0.856 0.08 0.7798conc*path 2 24.151 12.076 1.11 0.3337temp 2 191.860 95.930 8.81 0.0003conc*temp 4 118.024 29.506 2.71 0.0339path*temp 2 12.429 6.214 0.57 0.5668conc*path*temp 4 17.188 4.297 0.39 0.8121Dry 1 29.250 29.250 2.69 0.1041conc*Dry 2 39.234 19.617 1.80 0.1700path*Dry 1 3.516 3.516 0.32 0.5711conc*path*Dry 2 4.605 2.303 0.21 0.8097temp*Dry 2 5.283 2.642 0.24 0.7850conc*temp*Dry 4 18.783 4.696 0.43 0.7858path*temp*Dry 2 9.849 4.924 0.45 0.6374conc*path*temp*Dry 4 8.531 2.133 0.20 0.9401Residual 108 1176.108 10.890

Analysis of Variance Results

7

Conclusions from AOVConclusions from AOV

Significant Concentration by Temperature Interaction

Estimate of Variance is 10.88988

Compare the Conc*Temp Cell Means

8

Concentration by Temperature Means

conc temp Estimate StdErr10 250 23.400 0.82510 275 21.238 0.82510 300 19.175 0.82514 250 27.163 0.82514 275 25.194 0.82514 300 27.481 0.82518 250 29.044 0.82518 275 25.731 0.825

9

Compare Levels of Temp at each level of Conc

conc T _T Diff StdErr DF tValue Probt 250 275 2.163 1.167 108 1.85 0.0665

10 250 300 4.225 1.167 108 3.62 0.0004 275 300 2.062 1.167 108 1.77 0.0799

250 275 1.969 1.167 108 1.69 0.094414 250 300 -0.319 1.167 108 -0.27 0.7852

275 300 -2.287 1.167 108 -1.96 0.0525

250 275 3.312 1.167 108 2.84 0.005418 250 300 3.338 1.167 108 2.86 0.0051

275 300 0.025 1.167 108 0.02 0.9829

10

Compare the Levels of Conc at each level of Temp

temp C _C DiffStdErr DF tValue Probt 10 14 -3.763 1.167 108 -3.22 0.0017

250 10 18 -5.644 1.167 108 -4.84 0.0000 14 18 -1.881 1.167 108 -1.61 0.1098

10 14 -3.956 1.167 108 -3.39 0.0010275 10 18 -4.494 1.167 108 -3.85 0.0002

14 18 -0.538 1.167 108 -0.46 0.6459

10 14 -8.306 1.167 108 -7.12 0.0000300 10 18 -6.531 1.167 108 -5.60 0.0000

14 18 1.775 1.167 108 1.52 0.1311

11

Response Surface ModelResponse Surface Model

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

12

Effect NumDF DenDF FValue ProbFconc 1 135 8.02 0.0053conc*conc 1 135 8.37 0.0045temp 1 135 16.11 0.0001conc*temp*temp 1 135 14.51 0.0002conc*conc*temp*temp 1 135 14.02 0.0003path 1 135 0.09 0.7695Dry 1 135 2.95 0.0883path*Dry 1 135 0.35 0.5527

Final Response Surface Model

13

Conditions with Maximum Conditions with Maximum ResponseResponse

PATHPATH DRYDRY CONCCONC TEMPTEMP EST MAX EST MAX RESPONSERESPONSE

SHORTSHORT HOTHOT 14.814.8 300300 29.2729.27

SHORTSHORT AIRAIR 1818 250250 27.6827.68

LONGLONG HOTHOT 1818 250250 30.1030.10

LONGLONG AIRAIR 1818 250250 30.1730.17

GRAPHICS FOLLOW WITH 95% CI CONTAIN MAX

14

15

16

17

18

How was the experiment How was the experiment executed?executed?

Part 1Part 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

19

Graphical Representation Graphical Representation of The Experiment – Tank of The Experiment – Tank

as EUas EU

Slurry Concentration

10% 14% 18%

Tank 1 Tank 2 Tank 3 Tank 4 Tank 5 Tank 6

Completely Randomized Design

20

Tank Level of AnalysisTank Level of Analysis

SourceSource dfdf DivisorDivisor

ConcentratioConcentrationn

33 Error(Tank)Error(Tank)

Error(Tank)Error(Tank) 99

21

How was the experiment How was the experiment executed?executed?

Part 2Part 2

TANK is BLOCK of Six BATCHES

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

22

Graphical Representation of Graphical Representation of The Experiment – Batch as The Experiment – Batch as EUEU

SHORTLONG

250 250275 275300 300

Path by Temperature Combinations

61 2 3 4 5

Batches

TANK 12

61 2 3 4 5

Batches

TANK 1

…

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

23

BATCH Level of AnalysisBATCH Level of Analysis

SourceSource dfdf DivisorDivisor

Blocks=TanksBlocks=Tanks 1111

PathPath 11 Error(BATCH)Error(BATCH)

TempTemp 22 Error(BATCH)Error(BATCH)

Path*TempPath*Temp 22 Error(BATCH)Error(BATCH)

Conc*PathConc*Path 22 Error(BATCH)Error(BATCH)

Conc*TempConc*Temp 44 Error(BATCH)Error(BATCH)

Conc*Temp*PathConc*Temp*Path 44 Error(BATCH)Error(BATCH)

Error(BATCH)Error(BATCH) 4545

24

Graphical Representation Graphical Representation of The Experiment – Part of The Experiment – Part as EUas EU

Batch

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

AIR HOT

DRY METHOD

TANK

PART

25

PART AnalysisPART Analysis

SourceSource DfDf DivisorDivisor

Blocks=BatchesBlocks=Batches 7171 Error(Part)Error(Part)

DryDry 11 Error(Part)Error(Part)

Conc*DryConc*Dry 22 Error(Part)Error(Part)

Path*DryPath*Dry 11 Error(Part)Error(Part)

Temp*DryTemp*Dry 22 Error(Part)Error(Part)

Path*Temp*DryPath*Temp*Dry 22 Error(Part)Error(Part)

Conc*Path*DryConc*Path*Dry 22 Error(Part)Error(Part)

Conc*Temp*DryConc*Temp*Dry 44 Error(Part)Error(Part)

Conc*Temp*Path*DryConc*Temp*Path*Dry 44 Error(Part)Error(Part)

Error(Part)Error(Part) 5454

26

Appropriate Model Appropriate Model IncludesIncludes

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

Three Sizes of Experimental Units, each with an ERROR TERM

1 TANK

2 BATCH

3 PART

27

Source DF MS ErrorTermErrorDF FValue ProbFconc 2 475.611 Error(Tank) 9 7.26 0.0132path 1 0.856 Error(Batch) 45 0.07 0.7924conc*path 2 12.076 Error(Batch) 45 0.99 0.3797temp 2 95.930 Error(Batch) 45 7.86 0.0012conc*temp 4 29.506 Error(Batch) 45 2.42 0.0624path*temp 2 6.214 Error(Batch) 45 0.51 0.6044conc*path*temp 4 4.297 Error(Batch) 45 0.35 0.8412Dry 1 29.250 Error(Part) 54 41.98 0.0000conc*Dry 2 19.617 Error(Part) 54 28.16 0.0000path*Dry 1 3.516 Error(Part) 54 5.05 0.0288conc*path*Dry 2 2.303 Error(Part) 54 3.30 0.0443temp*Dry 2 2.642 Error(Part) 54 3.79 0.0288conc*temp*Dry 4 4.696 Error(Part) 54 6.74 0.0002path*temp*Dry 2 4.924 Error(Part) 54 7.07 0.0019conc*path*temp*Dry 4 2.133 Error(Part) 54 3.06 0.0241Error(Tank) 9 65.475 Error(Batch) 45 5.36 0.0001Error(Batch) 45 12.205 Error(Part) 54 17.52 0.0000Error(Part) 54 0.697 .

Analysis of Variance for Split-plotAnalysis of Variance for Split-plot

ns

28

Estimates of the Variance Estimates of the Variance Components for Split-plotComponents for Split-plot

CovParm Estimate Alpha Lower Upperrep(conc) 4.439 0.05 1.835 21.860rep*path*temp(conc) 5.754 0.05 3.877 9.426Residual 0.697 0.05 0.494 1.057

Sum of Variance Component Estimates = 10.890

Same as CR Estimate of Variance

29

Comparisons of Split-plot Comparisons of Split-plot and CRD analysesand 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

30

Response Surface Model Response Surface Model with Split-plot Errors--AOVwith Split-plot Errors--AOV

Effect NumDFDenDF FValue ProbF

conc 1 53.7 2.33 0.1326

conc*conc 1 54.0 5.95 0.0181

temp 1 51.0 8.88 0.0044

conc*temp 1 51.0 0.92 0.3415

conc*conc*temp 1 51.0 7.81 0.0073

path 1 51.0 0.26 0.6129

Dry 1 62.0 6.08 0.0165

conc*Dry 1 62.0 3.05 0.0857

conc*temp*Dry 1 62.0 1.06 0.3063

path*Dry 1 62.0 4.60 0.0360

conc*path*Dry 2 73.9 8.41 0.0005

conc*conc*path*Dry 3 78.9 6.26 0.0007

conc*temp*path*Dry 2 73.9 7.00 0.0016

con*tem*tem*path*Dry 4 82.5 5.08 0.0010

31

Response Surface Model Response Surface Model with Split-plot Errorswith Split-plot Errors

CovParm Estimate Lower Upper

Error(Tank) 4.500 1.878 21.479

Error(Batch) 5.279 3.602 8.480

Error(Part) 0.924 0.669 1.359

32

Conditions with Maximum Conditions with Maximum ResponseResponse

PATHPATH DRYDRY CONCCONC TEMPTEMP EST MAX EST MAX RESPONSERESPONSE

SHORTSHORT HOTHOT 1818 250250 29.4229.42

SHORTSHORT AIRAIR 1818 250250 28.2128.21

LONGLONG HOTHOT 1818 250250 29.2629.26

LONGLONG AIRAIR 1818 250250 28.6728.67

GRAPHICS FOLLOW WITH 95% CI CONTAIN MAX

33

34

35

36

37

Comparisons of 95% Confidence Comparisons of 95% Confidence Regions for Maximum ResponseRegions for Maximum Response

Path=Short Dry=Hot

38

Comparisons of Split-plot Comparisons of Split-plot and CRD Response Surface and CRD Response Surface ModelsModelsSplit-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

39

Conclusions-Conclusions-11

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

40

Conclusions -2Conclusions -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

41

SAS System Code for SAS System Code for ANOVAANOVA

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;

42

SAS System Code for RSMSAS 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);

43

THE ENDTHE ENDTHANK YOU

FOR

YOUR ATTENTIONwww.stat911.com