Design & Analysis of Split-Plot Experiments (Univariate Analysis)
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
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)
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
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
Model and Common Assumptions
• Yijk = + i + b j + (ab )ij + k + ( )ik + ijk
0),)((),())(,(
).0(~
0)()(
0
),0(~)(
),0(~
0
2
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1
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Mean and Covariance Structure (Fixed Whole Plot and Subplot Factors)
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'''
'''
),(2
22
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'''
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Obtaining Variances of Sums & Means
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Variances of Other Means
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222..
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:nInteractio SP*WP
a :TrtSubplot
:ErrorPlot Whole
:Blocks
:TrtPlot Whole
Analysis of
Variance
Expected Values in Analysis of Variance
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Expected Mean Squares (Fixed WP&SP Trts)
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ERRORERROR
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Tests for Fixed Effects
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Random WP & SP Effects Model
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Random Effects Model: Mean/Variance Structure
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'''
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ERRORERROR
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Testing for WP Random Effects
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0:0:
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Testing for SP Effects and WP/SP Interaction
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)1)(1(1 :Freedom of Degrees
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0:0: :EffectsSubplot
21
220
21
220
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Mixed Effects Model (Fixed WP, Random SP)
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Mixed Effects (Fixed WP) - Mean/Variance Structure
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',',0
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1',','
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,
22
2
222
22
22
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'''
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kkjjii
kkjjiia
kkjjii
kkjjii
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acc
b
accb
acc
abb
accabb
kjiijk
Expected Mean Squares (Fixed WP/Random SP)
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)1)(1(1
1
)1)(1(
111
1
2
22
22
22
1
2
222
222
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ERRORERROR
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WP
a
ii
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Testing for WP Fixed Effects
)1)(1()1)(1()1)(1(1
:Freedom of Degrees eApproximat
:StatisticTest
1 :Note
0 allNot :0:
22
2
222
2
1
2
10
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a
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HH
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ERRWP
ERRWP
SPWPWPBLK
ERRWPWP
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Testing for SP Effects and WP/SP Interaction
)1)(1()1)(1( :Freedom of Degrees
:StatisticTest
0:0: :nInteractio SPWP
)1)(1(1 :Freedom of Degrees
:StatisticTest
0:0: :EffectsSubplot
21
220
21
220
cbaca
MS
MSF
HH
cbac
MS
MSF
HH
ERR
SPWPSPWP
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ERR
SPSP
cAc
Mixed Effects Model (Random WP, Fixed SP)
2
''
2
'
2
1
2
2
2
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'1
)(,)(
,0~)(
0
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)()(
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ijk
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Mixed Effects (Fixed SP) - Mean/Variance Structure
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1',','
,
2
22
2222
22222
'''
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kkjjii
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aca
acabba
acabba
kjiijk
Expected Mean Squares (Random WP/Fixed SP)
)1)(1(
)1)(1(1
111
)1)(1(
1
1
2
22
1
2
22
22
222
222
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cadfc
cbMSE
cdfc
aba
abMSE
badfcMSE
adfcbcMSE
bdfcacMSE
ERRORERROR
SPWPacSPWP
SP
c
kk
acSP
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WPabaWP
BLOCKSabbBLOCKS
Testing Main Effects and WP/SP Interaction
)1)(1()1)(1( :df :StatisticTest
0:0: :nInteractio SPWP
)1)(1(1 :df :StatisticTest
0 allNot :0: :EffectsSubplot
)1)(1(1 :df :StatisticTest
0:0: :Effectsplot Whole
21
220
21
10
21
220
cbacaMS
MSF
HH
cacMS
MSF
HH
baaMS
MSF
HH
ERR
SPWPSPWP
Aac
SPWP
SPSP
kAc
WPBLK
WPWP
aAa