design & analysis of split-plot experiments (univariate analysis)
TRANSCRIPT
![Page 1: Design & Analysis of Split-Plot Experiments (Univariate Analysis)](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649da15503460f94a8db32/html5/thumbnails/1.jpg)
Design & Analysis of Split-Plot Experiments (Univariate Analysis)
![Page 2: Design & Analysis of Split-Plot Experiments (Univariate Analysis)](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649da15503460f94a8db32/html5/thumbnails/2.jpg)
Elements of Split-Plot Designs
• Split-Plot Experiment: Factorial design with at least 2 factors, where experimental units wrt factors differ in “size” or “observational points”.
• Whole plot: Largest experimental unit• Whole Plot Factor: Factor that has levels assigned to
whole plots. Can be extended to 2 or more factors• Subplot: Experimental units that the whole plot is split
into (where observations are made)• Subplot Factor: Factor that has levels assigned to
subplots• Blocks: Aggregates of whole plots that receive all levels
of whole plot factor
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Examples
• Agriculture: Varieties of a crop or gas may need to be grown in large areas, while varieties of fertilizer or varying growth periods may be observed in subsets of the area.
• Engineering: May need long heating periods for a process and may be able to compare several formulations of a by-product within each level of the heating factor.
• Behavioral Sciences: Many studies involve repeated measurements on the same subjects and are analyzed as a split-plot (See Repeated Measures lecture)
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Design Structure
• Blocks: b groups of experimental units to be exposed to all combinations of whole plot and subplot factors
• Whole plots: a experimental units to which the whole plot factor levels will be assigned to at random within blocks
• Subplots: c subunits within whole plots to which the subplot factor levels will be assigned to at random.
• Fully balanced experiment will have n=abc observations
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Data Elements (Fixed Factors, Random Blocks)
• Yijk: Observation from wpt i, block j, and spt k
• : Overall mean level
• i : Effect of ith level of whole plot factor (Fixed)
• bj: Effect of jth block (Random)
• (ab )ij : Random error corresponding to whole plot elements in block j where wpt i is applied
• k: Effect of kth level of subplot factor (Fixed)
• )ik: Interaction btwn wpt i and spt k
• (bc )jk: Interaction btwn block j and spt k (often set to 0)
• ijk: Random Error= (bc )jk+ (abc)ijk
• Note that if block/spt interaction is assumed to be 0, represents the block/spt within wpt interaction
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Model and Common Assumptions
• Yijk = + i + b j + (ab )ij + k + ( )ik + ijk
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).0(~
0)()(
0
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),0(~
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Mean and Covariance Structure (Fixed Whole Plot and Subplot Factors)
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Obtaining Variances of Sums & Means
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Variances of Other Means
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![Page 10: Design & Analysis of Split-Plot Experiments (Univariate Analysis)](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649da15503460f94a8db32/html5/thumbnails/10.jpg)
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Analysis of
Variance
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Expected Values in Analysis of Variance
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![Page 12: Design & Analysis of Split-Plot Experiments (Univariate Analysis)](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649da15503460f94a8db32/html5/thumbnails/12.jpg)
Expected Mean Squares (Fixed WP&SP Trts)
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ERRORERROR
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![Page 13: Design & Analysis of Split-Plot Experiments (Univariate Analysis)](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649da15503460f94a8db32/html5/thumbnails/13.jpg)
Tests for Fixed Effects
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![Page 14: Design & Analysis of Split-Plot Experiments (Univariate Analysis)](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649da15503460f94a8db32/html5/thumbnails/14.jpg)
Comparing Whole Plot Trt Means
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![Page 15: Design & Analysis of Split-Plot Experiments (Univariate Analysis)](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649da15503460f94a8db32/html5/thumbnails/15.jpg)
Comparing Subplot Trt Means
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![Page 16: Design & Analysis of Split-Plot Experiments (Univariate Analysis)](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649da15503460f94a8db32/html5/thumbnails/16.jpg)
Comparing Subplot Effects in Same WPT
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![Page 17: Design & Analysis of Split-Plot Experiments (Univariate Analysis)](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649da15503460f94a8db32/html5/thumbnails/17.jpg)
Comparing WPT Effects in Same Subplot TRT
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![Page 18: Design & Analysis of Split-Plot Experiments (Univariate Analysis)](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649da15503460f94a8db32/html5/thumbnails/18.jpg)
Approximate Degrees of Freedom (Satterthwaite)
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![Page 19: Design & Analysis of Split-Plot Experiments (Univariate Analysis)](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649da15503460f94a8db32/html5/thumbnails/19.jpg)
Random WP & SP Effects Model
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All random effects are assumed to be mutually pairwise independent
![Page 20: Design & Analysis of Split-Plot Experiments (Univariate Analysis)](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649da15503460f94a8db32/html5/thumbnails/20.jpg)
Random Effects Model: Mean/Variance Structure
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![Page 21: Design & Analysis of Split-Plot Experiments (Univariate Analysis)](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649da15503460f94a8db32/html5/thumbnails/21.jpg)
Expected Mean Squares (Random WP&SP Trts)
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![Page 22: Design & Analysis of Split-Plot Experiments (Univariate Analysis)](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649da15503460f94a8db32/html5/thumbnails/22.jpg)
Testing for WP Random Effects
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0:0:
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![Page 23: Design & Analysis of Split-Plot Experiments (Univariate Analysis)](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649da15503460f94a8db32/html5/thumbnails/23.jpg)
Testing for SP Effects and WP/SP Interaction
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0:0: :EffectsSubplot
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![Page 24: Design & Analysis of Split-Plot Experiments (Univariate Analysis)](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649da15503460f94a8db32/html5/thumbnails/24.jpg)
Mixed Effects Model (Fixed WP, Random SP)
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![Page 25: Design & Analysis of Split-Plot Experiments (Univariate Analysis)](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649da15503460f94a8db32/html5/thumbnails/25.jpg)
Mixed Effects (Fixed WP) - Mean/Variance Structure
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![Page 26: Design & Analysis of Split-Plot Experiments (Univariate Analysis)](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649da15503460f94a8db32/html5/thumbnails/26.jpg)
Expected Mean Squares (Fixed WP/Random SP)
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![Page 27: Design & Analysis of Split-Plot Experiments (Univariate Analysis)](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649da15503460f94a8db32/html5/thumbnails/27.jpg)
Testing for WP Fixed Effects
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![Page 28: Design & Analysis of Split-Plot Experiments (Univariate Analysis)](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649da15503460f94a8db32/html5/thumbnails/28.jpg)
Testing for SP Effects and WP/SP Interaction
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:StatisticTest
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)1)(1(1 :Freedom of Degrees
:StatisticTest
0:0: :EffectsSubplot
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![Page 29: Design & Analysis of Split-Plot Experiments (Univariate Analysis)](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649da15503460f94a8db32/html5/thumbnails/29.jpg)
Mixed Effects Model (Random WP, Fixed SP)
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![Page 30: Design & Analysis of Split-Plot Experiments (Univariate Analysis)](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649da15503460f94a8db32/html5/thumbnails/30.jpg)
Mixed Effects (Fixed SP) - Mean/Variance Structure
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![Page 31: Design & Analysis of Split-Plot Experiments (Univariate Analysis)](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649da15503460f94a8db32/html5/thumbnails/31.jpg)
Expected Mean Squares (Random WP/Fixed SP)
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ERRORERROR
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![Page 32: Design & Analysis of Split-Plot Experiments (Univariate Analysis)](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649da15503460f94a8db32/html5/thumbnails/32.jpg)
Testing Main Effects and WP/SP Interaction
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