14. linear mixed-effects models for data from split …a model for data from the traditional...

14. Linear Mixed-Effects Models for

Data from Split-Plot Experiments

Copyright c©2019 Dan Nettleton (Iowa State University) 14. Statistics 510 1 / 30

Start with a Field

Field


Partition the Field into Blocks

Field

Block 1

Block 2

Block 3

Block 4


Partition Each Block into Plots

Field

Block 1

Block 2

Block 3

Block 4


Randomly Assign Genotypes to Plots within Blocks

Field

Block 1

Block 2

Block 3

Block 4 Genotype A Genotype B Genotype C

Genotype A Genotype B Genotype C




Partition Each Whole Plot into Split Plots

Field

Block 1

Block 2

Block 3






Randomly Assign Fertilizer Amounts within Split Plots

Field

Block 1

Block 2

Block 3





0 50 100 150 50 0 100 150 150 0 100 50

150 0 100 50 0 100 50 150 100 0 50 150

100 150 50 0 0 50 100 150 50 0 100 150

0 150 50 100 150 0 100 50 50 0 150 100


An Example Split-Plot Experiment

Field

Block 1

Block 2

Block 3





0 50 100 150 50 0 100 150 150 0 100 50

150 0 100 50 0 100 50 150 100 0 50 150

100 150 50 0 0 50 100 150 50 0 100 150

0 150 50 100 150 0 100 50 50 0 150 100

Whole Plot or Main Plot

Split Plot or

Sub Plot


This experiment has two factors: genotype and fertilizeramount.

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) factorbecause its levels are randomly assigned to whole plots(main plots).

Fertilizer is called the split-plot factor because its levels arerandomly assigned to split plots within each whole plot.


Experimental Units in Split-Plot Designs

Whole plots are the whole-plot experimental units becausethe levels of the whole-plot factor (genotype) are randomlyassigned to whole plots.

The split-plots are the split-plot experimental units becausethe levels of the split-plot factor (amount of fertilizer) arerandomly assigned to split plots within each whole plot.

Thus, we have two different sizes of experimental units insplit-plot experimental designs.


Same Treatment Structure in an RCBD

Field

Block 1

Block 2

Block 3

Block 4

A0

A50

A150

A100

B0

B50

B100

B150

C0

C50

C100

C150

A0

A50

A100

A150

B0

B50

B100

B150

C0

C50

C100

C150

A0

A50

A100

A150

B0

B50

B100

B150

C0

C50

C100

C150

A0

A50

A100

A150

B0

B50

B100

B150

C0

C50

C100

C150


Same Treatment Structure in an CRD

Field

A0

A50

A150

A100

B0

B50

B100

B150

C0

C50

C100

C150

A0

A50

A100

A150

B0

B50

B100

B150

C0

C50

C100

C150

A0

A50

A100

A150

B0

B50

B100

B150

C0

C50

C100

C150

A0

A50

A100

A150

B0

B50

B100

B150

C0

C50

C100

C150


Why Use a Split-Plot Design?

Split-plot designs usually arise because logistical constraintsmake a CRD or RCBD impractical.

For example, it may be easier to change from one fertilizerlevel to another as a tractor drives through a field, while itmay be more difficult to change from planting one genotypeto planting another.

In the engineering literature, split-plot designs aresometimes called designs with hard-to-change factors.


Recognizing Designs with Split-Plot Structures

Many variations on split-plot designs are used for practicalreasons.

Examples include split-split-plot designs and split-blockdesigns, but the names of these designs are not soimportant.

Pay close attention to the experimental unit to which thelevels of each factor are randomly assigned to recognizesplit-plot-like design structures.


Split-plot designs may not involve plots of land.

Suppose eight pairs of mice from eight litters are housed ineight cages so that each cage holds two mice from thesame litter.

Suppose diets 1 and 2 are randomly assigned to the litterswith four litters per diet.

Within each cage, suppose drugs 1 and 2 are randomlyassigned to the mice with one mouse per drug.


A Split-Plot Experimental Design

Drug 2 Drug 1Diet 1

Drug 2 Drug 1Diet 2

Drug 1 Drug 2Diet 1

Drug 1 Drug 2Diet 1

Drug 1 Drug 2Diet 2

Drug 2 Drug 1Diet 2

Drug 2 Drug 1Diet 2

Drug 2 Drug 1Diet 1


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.


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 litter effect = whole-plot exp. unit random effect

eijk = random error effect = split-plot exp. unit random effect


y =

y111

y121

y112

y122

y113

y123

y114

y124

y211

y221

y212

y222

y213

y223

y214

y224

β =

µ

α1

α2

β1

β2

γ11

γ12

γ21

γ22

u =

`11

`12

`13

`14

`21

`22

`23

`24

e =

e111

e121

e112

e122

e113

e123

e114

e124

e211

e221

e212

e222

e213

e223

e214

e224


X =

[1

16×1, I

2×2⊗ 1

8×1, 1

8×1⊗ I

2×2, I

2×2⊗ 1

4×1⊗ I

2×2

]

Z = I8×8⊗ 1

2×1

y = Xβ + Zu + e


[ue

]∼ N

([00

],

[σ2` I 00 σ2

e I

]=

[G 00 R

])

Var(Zu) = ZGZ′ = σ2`ZZ′

= σ2`

[I

8×8⊗ 1

2×1

] [I

8×8⊗ 1

2×1

]′= σ2

`

[I

8×8⊗ 11

2×2

′]

= Block Diagonal with blocks

[σ2` σ2

`

σ2` σ2

`

]


Var(y) = ZGZ′ + R = σ2` I

8×8⊗ 11

2×2

′+ σ2

e I

= Block Diagonal with blocks

[σ2` + σ2

e σ2`

σ2` σ2

` + σ2e

]


Thus, the covariance between two observations from the samelitter is σ2

` and the correlation is σ2`

σ2`+σ

2e.

These computations can also be done using the non-matrixexpression of the model.

∀ i, j, Var(yijk) = Var(µ+ αi + βj + γij + `ik + eijk)

= Var(`ik + eijk)

= σ2` + σ2

e .


Cov(yi1k, yi2k) = Cov(µ+ αi + β1 + γi1 + ìk + ei1k,

µ+ αi + β2 + γi2 + ìk + ei2k)

= Cov(ìk + ei1k, ìk + ei2k)

= Cov(ìk, ìk) + Cov(ìk, ei2k)

+ Cov(ei1k, ìk) + Cov(ei1k, ei2k)

= Cov(ìk, ìk) + 0 + 0 + 0

= Var(ìk) = σ2` .


Back to the Traditional Split-Plot Experimental Design

Field

Block 1

Block 2

Block 3

Block 4Genotype AGenotype B Genotype C


Genotype AGenotype B Genotype C

Genotype A Genotype BGenotype C

0 50100 150 50 0100 150 150 0100 50

150 0100 50 0 10050 150 100 050 150

100 15050 0 0 50100 150 50 0100 150

0 15050 100 150 0100 50 50 0150 100


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 block effect

wik = random whole-plot exp. unit effect

eijk = random error = random split-plot exp. unit effect


To express the model precisely in vector and matrix form asy = Xβ + Zu + e, we will sort the data first by Block, thenGenotype, and then Fertilizer:

y = [y111, y121, y131, y141, y211, y221, y231, y241, . . . , y314, y324, y334, y344]′

e = [e111, e121, e131, e141, e211, e221, e231, e241, . . . , e314, e324, e334, e344]′


X = 14×1⊗ I

12×12, β =

µ11

µ12

µ13

µ14

µ21

µ22

µ23

µ24

µ31

µ32

µ33

µ34


Z =

[I

4×4⊗ 1

12×1, I

12×12⊗ 1

4×1

]

u =

[bw

]=

b1...

b4

w11

w21...

w34

∼ N

([00

],

[σ2

b I 00 σ2

w I

])


bwe

∼ N

0

00

, σ2

b I 0 00 σ2

w I 00 0 σ2

e I

[ue

]∼ N

([00

],

[G 00 R

])


14. linear mixed-effects models for data from split …a model for data from the traditional...

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