Linear Mixed-Effects Models for
Data from Split-Plot Experiments
c
Copyright 2016
Dan Nettleton (Iowa State University)
Statistics 510
1 / 24
An Example Split-Plot Experiment
Whole Plot
or Main Plot
Field
Block 1
Genotype C
0
Block 2
100 150 50
Genotype B
150 100
50
0
Genotype A
Genotype A
50 100 150 0
Genotype A
0
Genotype B
150 100
0
Genotype C
50 150 100 100
Genotype B
50
50 150
Split Plot
or
Sub Plot
0
Genotype C
Block 3
100
50
0
150
Genotype B
0
100 150 50
Genotype C
50 100 150 0
Genotype A
Block 4
0
50 100 150 150 100
c
Copyright 2016
Dan Nettleton (Iowa State University)
50
0
50 150 100 0
Statistics 510
2 / 24
This experiment has two factors: genotype and fertilizer
amount.
Genotype has levels A, B, and C.
Fertilizer has levels 0, 50, 100, 150 lbs. N / acre.
Genotype is called the whole-plot (or main-plot) factor
because its levels are randomly assigned to whole plots
(main plots).
Fertilizer is called the split-plot factor because its levels are
randomly assigned to split plots within each whole plot.
c
Copyright 2016
Dan Nettleton (Iowa State University)
Statistics 510
3 / 24
Experimental Units in Split-Plot Designs
Whole plots are the whole-plot experimental units because
the levels of the whole-plot factor (genotype) are randomly
assigned to whole plots.
The split-plots are the split-plot experimental units because
the levels of the split-plot factor (amount of fertilizer) are
randomly assigned to split plots within each whole plot.
Thus, we have two different sizes of experimental units in
split-plot experimental designs.
c
Copyright 2016
Dan Nettleton (Iowa State University)
Statistics 510
4 / 24
Same Treatment Structure in an RCBD
Field
Block 1
B B A C B C A A C B C A
100 0 0 100 150 50 50 150 150 50 0 100
Block 2
A A C A B B C C A C B B
150 0 50 50 100 50 100 0 100 150 150 0
Block 3
C
0
A A B B B A C A C C B
0 100 100 50 0 150 50 50 150 100 150
Block 4
B C B A C A B C B C A A
0 150 50 150 100 0 150 50 100 0 100 50
c
Copyright 2016
Dan Nettleton (Iowa State University)
Statistics 510
5 / 24
Same Treatment Structure in an CRD
Field
B B A B A C A B A C C C
50 0 150 100 100 150 50 0 50 100 0 100
A A C B B B A
50 0 50 50 150 50 0
C
0
A A A A
0 100 150 0
C A C B B
0 100 50 150 0
B A B A B C A
0 150 150 50 150100 100
B B B C C C A C C C C B
50 100 100 150 100 50 150 50 150 0 150 100
c
Copyright 2016
Dan Nettleton (Iowa State University)
Statistics 510
6 / 24
Why Use a Split-Plot Design?
Split-plot designs usually arise because logistical constraints
make a CRD or RCBD impractical.
For example, it may be easier to change from one fertilizer
level to another as a tractor drives through a field, while it
may be more difficult to change from planting one genotype
to planting another.
In the engineering literature, split-plot designs are
sometimes called designs with hard-to-change factors.
c
Copyright 2016
Dan Nettleton (Iowa State University)
Statistics 510
7 / 24
Recognizing Designs with Split-Plot Structures
Many variations on split-plot designs are used for practical
reasons.
Examples include split-split-plot designs and split-block
designs, but the names of these designs are not so
important.
Pay close attention to the experimental unit to which the
levels of each factor are randomly assigned to recognize
split-plot-like design structures.
c
Copyright 2016
Dan Nettleton (Iowa State University)
Statistics 510
8 / 24
Split-plot designs may not involve plots of land.
Suppose eight pairs of mice from eight litters are housed in
eight cages so that each cage holds two mice from the
same litter.
Suppose diets 1 and 2 are randomly assigned to the litters
with four litters per diet.
Within each cage, suppose drugs 1 and 2 are randomly
assigned to the mice with one mouse per drug.
c
Copyright 2016
Dan Nettleton (Iowa State University)
Statistics 510
9 / 24
A Split-Plot Experimental Design
Diet 1
Drug 2
Diet 2
Drug 1
Drug 2
Drug 2
Drug 1
Drug 2
Drug 2
Drug 1
Drug 2
Diet 1
Drug 1
Diet 1
Diet 2
Drug 1
Drug 2
Diet 2
Diet 2
Drug 2
Drug 1
Drug 1
Diet 1
c
Copyright 2016
Dan Nettleton (Iowa State University)
Drug 1
Statistics 510
10 / 24
Diet is the whole-plot treatment factor.
Litters are the whole-plot experiment units.
Drug is the split-plot treatment factor.
Mice are the split-plot experiment units.
c
Copyright 2016
Dan Nettleton (Iowa State University)
Statistics 510
11 / 24
Diet i = 1, 2, Drug j = 1, 2, Litter k = 1, 2, 3, 4 (within each Diet i)
yijk = µ + αi + βj + γij + `ik + eijk (i = 1, 2; j = 1, 2; k = 1, ..., 4)
µ + αi + βj + γij = mean for Diet i and Drug j
`ik = random effect for kth litter that received Diet i
eijk = random error effect for Diet i, Drug j, Litter k
c
Copyright 2016
Dan Nettleton (Iowa State University)
Statistics 510
12 / 24

y111


 
y121 
 
 
y112 
 
y122 
 
 
y113 
 
 
y123 
 
y114 
 
 
y124 

y=
 
y211 
 
y221 
 
 
y212 
 
y 
 222 
 
y213 
 
 
y223 
 
y 
 214 
y224

µ

 
 α1 
 
α 
 2
 
 β1 
 
 
β =  β2 
 
γ 
 11 
 
γ12 
 
 
γ21 
c
Copyright 2016
Dan Nettleton (Iowa State University)
γ22

`11

 
`12 
 
 
`13 
 
` 
 14 
u= 
`21 
 
 
`22 
 
` 
 23 
`24
e111

 
e121 
 
 
e112 
 
e122 
 
 
e113 
 
 
e123 
 
e114 
 
 
e124 

e=
 
e211 
 
e221 
 
 
e212 
 
e 
 222 
 
e213 
 
 
e223 
 
e 
 214 
e224
Statistics 510
13 / 24
X=
1 , I ⊗ 1, 1 ⊗ I , I ⊗ 1 ⊗ I
16×1 2×2
8×1 8×1
2×2 2×2
4×1
2×2
Z= I ⊗ 1
8×8
2×1
y = Xβ + Zu + e
c
Copyright 2016
Dan Nettleton (Iowa State University)
Statistics 510
14 / 24
" #
u
" # "
# "
#!
0
σ`2 I 0
G 0
∼N
,
=
e
0
0 σe2 I
0 R
Var(Zu) = ZGZ0 = σ`2 ZZ0
0
2
= σ` I ⊗ 1
I ⊗ 1
8×8
2×1
8×8
2×1
= σ`2 I ⊗ 11 0
8×8
2×2
"
= Block Diagonal with blocks
c
Copyright 2016
Dan Nettleton (Iowa State University)
σ`2 σ`2
#
σ`2 σ`2
Statistics 510
15 / 24
0
Var(y) = ZGZ0 + R = σ`2 I ⊗ 11 + σe2 I
8×8
2×2
= Block Diagonal with blocks
"
σ`2 + σe2
σ`2
σ`2
σ`2 + σe2
c
Copyright 2016
Dan Nettleton (Iowa State University)
#
Statistics 510
16 / 24
Thus, the covariance between two observations from the same
litter is σ`2 and the correlation is
σ`2
2
σ` +σe2
.
These computations can also be done using the non-matrix
expression of the model.
∀i, j Var(yijk ) = Var(µ + αi + βj + γij + `ik + eijk )
= Var(`ik + eijk )
= σ`2 + σe2 .
c
Copyright 2016
Dan Nettleton (Iowa State University)
Statistics 510
17 / 24
Cov(yi1k , yi2k ) = Cov(µ + αi + β1 + γi1 + `ik + ei1k ,
µ + αi + β2 + γi2 + `ik + ei2k )
= Cov(`ik + ei1k , `ik + ei2k )
= Cov(`ik , `ik ) + Cov(`ik , ei2k )
+ Cov(ei1k , `ik ) + Cov(ei1k , ei2k )
= Cov(`ik , `ik ) + 0 + 0 + 0
= Var(`ik ) = σ`2 .
c
Copyright 2016
Dan Nettleton (Iowa State University)
Statistics 510
18 / 24
Back to the Traditional Split-Plot Experimental Design
Field
Block 1
Genotype C
0
Block 2
100 150 50
Genotype B
150 100
50
0
Genotype A
Genotype A
50 100 150 0
Genotype A
0
Genotype B
150 100
0
Genotype C
50 150 100 100
Genotype B
50
50 150
0
Genotype C
Block 3
100
50
0
150
Genotype B
0
100 150 50
Genotype C
50 100 150 0
Genotype A
Block 4
0
50 100 150 150 100
c
Copyright 2016
Dan Nettleton (Iowa State University)
50
0
50 150 100 0
Statistics 510
19 / 24
A Model for Data from the Traditional Split-Plot
Experiment
Genotype i = 1, 2, 3, Fertilizer j = 1, 2, 3, 4, Block k = 1, 2, 3, 4
yijk = µij + bk + wik + eijk
µij = mean for Genotype i, Fertilizer j
bk = random effect for Block k
wik = random effect for Genotype i whole plot in Block k
eijk random error effect for Genotype i, Fertilizer j, Block k
c
Copyright 2016
Dan Nettleton (Iowa State University)
Statistics 510
20 / 24
To express the model precisely in vector and matrix form as
y = Xβ + Zu + e, we will sort the data first by Block, then
Genotype, and then Fertilizer:
y = [y111 , y121 , y131 , y141 , y211 , y221 , y231 , y241 , . . . , y314 , y324 , y334 , y344 ]0
e = [e111 , e121 , e131 , e141 , e211 , e221 , e231 , e241 , . . . , e314 , e324 , e334 , e344 ]0
c
Copyright 2016
Dan Nettleton (Iowa State University)
Statistics 510
21 / 24

X= 1 ⊗ I ,
4×1
12×12
c
Copyright 2016
Dan Nettleton (Iowa State University)












β=












µ11


µ12 

µ13 


µ14 

µ21 


µ22 

µ23 


µ24 

µ31 


µ32 


µ33 
µ34
Statistics 510
22 / 24
Z=
I ⊗ 1 ,
4×4
12×1
I ⊗ 1
12×12
4×1


b1
 . 
 .. 



"
# 
 b4 


b
u=
=
w11 

∼N
w


 w21 
 . 
 . 
 . 
"
0
0
# "
,
#!
σb2 I
0
0
σw2 I
w34
c
Copyright 2016
Dan Nettleton (Iowa State University)
Statistics 510
23 / 24

b


0
 
σb2 I
0
0



  

 w  ∼ N  0  ,  0 σw2 I 0 
e
0
0
0 σe2 I
" #
" # "
#!
u
0
G 0
∼N
,
e
0
0 R
c
Copyright 2016
Dan Nettleton (Iowa State University)
Statistics 510
24 / 24