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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
11
1
2
2
1
ijkijijkjijj
ijk
c
kik
a
iik
c
kk
abij
bj
a
ii
abCOVbCOVabbCOV
NID
NIDab
NIDb
Mean and Covariance Structure (Fixed Whole Plot and Subplot Factors)
',,','0
',''
'''
'''
),(2
22
222
'''
kkiijj
kkjjii
kkjjii
kkjjii
YYCOVb
abb
abb
kjiijk
ikkiijkYE )(
Obtaining Variances of Sums & Means
2222
222222
...
222222
2222
22222
1 1' 1 ' 1 1''''
1 ' 1 1 1'''
1 1 1 ''
1 1 11 1 1
111
)(
)0()1()1(
)1(
),(),(
),(
abcabbabc
abcabcbcaYV
abcabcbca
cbbabcaa
cabcabc
YYCOVYYCOV
YYCOVYVYV
abbabb
abb
b
abbabb
a
i
a
i
b
j jj
c
k
c
kkjiijk
a
i ii
b
j
c
k
c
kjkiijk
a
i
b
j
c
k kkijkijk
a
i
b
j
c
kijk
a
i
b
j
c
kijk
Variances of Other Means
222.
222.
222.
222..
222..
222..
11
1
1
111
11
111
aaYV
bYV
cYV
ababbYV
acaYV
bcbbYV
abbjk
abbki
abbij
abbk
abbj
abbi
a
i
i
a
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c
k
ki
a
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ij
a
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kijk
a
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c
k
kjijkkiijijk
b
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c
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c
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a
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a
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k
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a
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kiki
c
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a
j
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a
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b
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b
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a
i
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a
i
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YbcYbYcY
YYYYYYYY
YYYYa
YabcYabYbcYb
YYYYb
YabcYabYYb
YabcYacYbcYc
YYYYc
YabcYacYYac
YabcYbcYYbc
1
2..
1 1
2.
1 1
2.
1 1 1
2
2
1 1 1
............
1 1
2........
2...
1
2..
1
2..
1 1
2.
1 1
2........
2...
1
2..
1
2.....
2...
1
2..
1
2..
1 1
2.
1 1
2........
2...
1
2..
1
2.....
2...
1
2..
1
2.....
:ErrorSubplot
:nInteractio SP*WP
a :TrtSubplot
:ErrorPlot Whole
:Blocks
:TrtPlot Whole
Analysis of
Variance
Expected Values in Analysis of Variance
22222
222222
2.
22222
2.
2222222
2.
222222
2..
22222222
2..
2222222
2..
22222222
2...
))((
)(1
))((1
)(1
)(1
1
)(1
1
ikkiabbijk
kabbjk
ikkiabbki
iabbij
kabbk
abbj
iabbi
abb
YE
aaaa
YE
bbbb
YE
cccc
YE
ababbaab
YE
acaccaac
YE
bcbcbcbc
YE
abcabcbcaabc
YE
Expected Mean Squares (Fixed WP&SP Trts)
)1)(1(
)1)(1()1)(1(
)(
11
)1)(1(
11
1
2
1 1
22
1
22
22
1
222
222
cbadfMSE
cadfca
bMSE
cdfc
abMSE
badfcMSE
adfa
bccMSE
bdfcacMSE
ERRORERROR
SPWP
a
i
c
k ikSPWP
SP
c
k kSP
WPBLKabWPBLK
WP
a
i iabWP
BLOCKSabbBLOCKS
Tests for Fixed Effects
)~|(
:StatisticTest
,0)(: :nInteractio SPWP
)~|(
:StatisticTest
0: :EffectsTrt Subplot
)~|(
:StatisticTest
0: :EffectsTrt Plot Whole
)1)(1().1)(1(
0
)1)(1(.1
10
)1)(1.(1
*
10
cbacaSPWPSPWP
ERROR
SPWPSPWP
ik
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ERROR
SPSP
c
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WPBLOCK
WPWP
a
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MS
MSF
kiH
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MSF
H
FFFFPP
MS
MSF
H
Comparing Whole Plot Trt Means
bc
MStYY
bc
MSYYV
cbbbcbbYYV
bYYCOV
cbbbcYYCOV
YYCOVYYCOVYYCOV
YYCOVYVYVYYV
bcbcbcbcYYE
WPBLOCKiiii
WPBLOCKii
abbabbii
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b
j jj
c
k
c
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b
j
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2:for CI %95
2
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1,
)0()1(),(
),(,,
,2
)(11
)(11
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^
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2'....
1 ' 1
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22'''
1 1 1'''
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1'....
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'1
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Comparing Subplot Trt Means
ab
MStYY
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YYCOVYVYVYYV
ababababYYE
ERRORkkkk
ERRORkk
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)(
2
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)0)(1()1(,,
,,),(
,2
)(11
)(11
'....'
'....
^
22222222
2'....
22'....
2222
1 1' 1 ''''
1 ' 1''
1 1'
1 1 1'
1'....
'....'....'....
'1
'1
'11
'....
Comparing Subplot Effects in Same WPT
b
MStYY
b
MSYYV
bbYYV
bYYCOVbbb
YYCOVYYCOVYYCOV
YYCOVYVYVYYV
YYE
ERRORkikiikikkk
ERRORkiki
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abbkikiabb
b
j jjkijijk
b
jijkijkkiki
kikikikikiki
ikikkk
ikkiikkikiki
2:)()(for CI %95
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22
1,)0)(1(
),(),(),(
,2
)()(
)()(
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^
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22'..
22
1 '''
1''..
'..'..'..
''
'''..
Comparing WPT Effects in Same Subplot TRT
below)given (df
)1(2
:)()(for CI %95
)1(2
22
1,)0)(1(
),(),(),(
,2
)()(
)()(
'..
''
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^
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2'..
2
1 '''
1''..
'..'..'..
''
''''..
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YYCOVYVYVYYV
YYE
ERRORWPBLOCKkiki
ikikii
ERRORWPBLOCKkiki
abbabbkiki
bkikib
b
j jjkjiijk
b
jjkiijkkiki
kikikikikiki
kiikii
kikiikkikiki
Approximate Degrees of Freedom (Satterthwaite)
)1)(1()1)(1(
)1(
)1(22
2^
ba
MS
cba
MSc
MSMSc
WPBLOCKERROR
WPBLOCKERROR
Random WP & SP Effects Model
2
2
2
2
2
2
,0~
,0~)(
,0~
,0~)(
,0~
,0~
)()(
NID
NIDac
NIDc
NIDab
NIDb
NIDa
accabbaY
ijk
acik
ck
abij
bj
ai
ijkikkijjiijk
All random effects are assumed to be mutually pairwise independent
Random Effects Model: Mean/Variance Structure
kjiYE ijk ,,
',','0
',','',','
',','',','
',','',','
',','
,
2
2
2
22
222
222
222222
'''
kkjjii
kkjjiikkjjii
kkjjiikkjjii
kkjjiikkjjii
kkjjii
YYCOV
c
b
a
cb
acca
abba
accabba
kjiijk
Expected Mean Squares (Random WP&SP Trts)
)1)(1(
)1)(1(
1
)1)(1(
1
1
2
22
222
22
2222
222
cbadfMSE
cadfbMSE
cdfbabMSE
badfcMSE
adfbcbcMSE
bdfcacMSE
ERRORERROR
SPWPacSPWP
SPaccSP
WPBLKabWPBLK
WPacabaWP
BLOCKSabbBLOCKS
Testing for WP Random Effects
)1)(1()1)(1()1)(1(1
:Freedom of Degrees eApproximat
:StatisticTest
:Note
0:0:
22
2
222
2
1
2
220
ca
MS
ba
MS
MSMS
cbaMS
a
MS
MSMS
MSMS
MSMSF
MSEMSEMSEMSE
HH
SPWPWPBLK
SPWPWPBLK
ERRWP
ERRWP
SPWPWPBLK
ERRWPWP
aSPWPWPBLKERRWP
aAa
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
cac
MS
MSF
HH
ERR
SPWPSPWP
acAac
SPWP
SPSP
cAc
Mixed Effects Model (Fixed WP, Random SP)
2
''
2
'
2
2
2
2
1
,0~
',',0)(,)(
'1
)(,)(
,0~)(
,0~
,0~)(
,0~
0
)()(
NID
kkiiacacCOV
iia
acacCOV
Nac
NIDc
NIDab
NIDb
accabbY
ijk
kiik
ackiik
acik
ck
abij
bj
a
ii
ijkikkijjiijk
Mixed Effects (Fixed WP) - Mean/Variance Structure
kjiYE iijk ,,)(
',',0
',','1
1',','
'','1
1'','
'','
'','
,
22
2
222
22
22
22222
'''
kkjji
kkjjiia
kkjjii
kkjjiia
kkjjii
kkjjii
kkjjii
YYCOV
acc
b
accb
acc
abb
accabb
kjiijk
Expected Mean Squares (Fixed WP/Random SP)
)1)(1(
)1)(1(1
1
)1)(1(
111
1
2
22
22
22
1
2
222
222
cbadfMSE
cadfa
abMSE
cdfabMSE
badfcMSE
adfa
bca
abcMSE
bdfcacMSE
ERRORERROR
SPWPacSPWP
SPcSP
WPBLKabWPBLK
WP
a
ii
acabWP
BLOCKSabbBLOCKS
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
ca
MS
ba
MS
MSMS
cbaMS
a
MS
MSMS
MSMS
MSMSF
abcMSEMSEMSEMSE
HH
SPWPWPBLK
SPWPWPBLK
ERRWP
ERRWP
SPWPWPBLK
ERRWPWP
iSPWPWPBLKERRWP
iAa
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
acAac
ERR
SPSP
cAc
Mixed Effects Model (Random WP, Fixed SP)
2
''
2
'
2
1
2
2
2
,0~
',',0)(,)(
'1
)(,)(
,0~)(
0
,0~)(
,0~
,0~
)()(
NID
iikkacacCOV
kkc
acacCOV
Nac
NIDab
NIDb
NIDa
acabbaY
ijk
kiik
ikik
acik
c
kk
abij
bj
ai
ijkikkijjiijk
Mixed Effects (Fixed SP) - Mean/Variance Structure
kjiYE kijk ,,)(
',,','0',,','
',','
',','1
1',','
,
2
22
2222
22222
'''
kkjjiikkjjii
kkjjii
kkjjiia
kkjjii
YYCOV
b
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
cbadfMSE
cadfc
cbMSE
cdfc
aba
abMSE
badfcMSE
adfcbcMSE
bdfcacMSE
ERRORERROR
SPWPacSPWP
SP
c
kk
acSP
WPBLKabWPBLK
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
