ASA2000MILLIKEN

Identifying the Split-plot and Constructing an Analysis : 

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: 

Complex 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

Hydrothermal Processing of Wheat Gluten: 

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

Slide4: 

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

Slide5: 

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

Slide6: 

Analysis of Variance Results

Conclusions from AOV: 

Conclusions from AOV Significant Concentration by Temperature Interaction Estimate of Variance is 10.88988 Compare the Conc*Temp Cell Means

Response Surface Model: 

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

Slide12: 

Final Response Surface Model

Conditions with Maximum Response: 

Conditions with Maximum Response GRAPHICS FOLLOW WITH 95% CI CONTAIN MAX

How was the experiment executed?Part 1: 

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

Graphical Representation of The Experiment – Tank as EU: 

Graphical Representation of The Experiment – Tank as EU Completely Randomized Design

Tank Level of Analysis: 

Tank Level of Analysis

How was the experiment executed?Part 2: 

How was the experiment executed? Part 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

Slide22: 

Graphical Representation of The Experiment – Batch as EU Each Tank is a Block of Six Batches for levels of Path by Temperature Combinations

BATCH Level of Analysis: 

BATCH Level of Analysis

Graphical Representation of The Experiment – Part as EU: 

Graphical Representation of The Experiment – Part as EU Batch(Tank) is Block of Two Parts – for levels of DRY

PART Analysis: 

PART Analysis

Appropriate Model Includes: 

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

Slide27: 

Analysis of Variance for Split-plot ns

Estimates of the Variance Components for Split-plot: 

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: 

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

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

Response Surface Model with Split-plot Errors--AOV

Response Surface Model with Split-plot Errors: 

Response Surface Model with Split-plot Errors

Conditions with Maximum Response: 

Conditions with Maximum Response GRAPHICS FOLLOW WITH 95% CI CONTAIN MAX

Comparisons of 95% Confidence Regions for Maximum Response: 

Comparisons of 95% Confidence Regions for Maximum Response Path=Short Dry=Hot

Comparisons of Split-plot and CRD Response Surface Models: 

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: 

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: 

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: 

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: 

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);

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