14. Linear Mixed-Effects Models for Data from Split …A Model for Data from the Traditional...
Transcript of 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
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Partition the Field into Blocks
Field
Block 1
Block 2
Block 3
Block 4
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Partition Each Block into Plots
Field
Block 1
Block 2
Block 3
Block 4
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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
Genotype A Genotype B Genotype C
Genotype A Genotype B Genotype C
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Partition Each Whole Plot into Split Plots
Field
Block 1
Block 2
Block 3
Block 4 Genotype A Genotype B Genotype C
Genotype A Genotype B Genotype C
Genotype A Genotype B Genotype C
Genotype A Genotype B Genotype C
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Randomly Assign Fertilizer Amounts within Split Plots
Field
Block 1
Block 2
Block 3
Block 4 Genotype A Genotype B Genotype C
Genotype A Genotype B Genotype C
Genotype A Genotype B Genotype C
Genotype A Genotype B Genotype C
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
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An Example Split-Plot Experiment
Field
Block 1
Block 2
Block 3
Block 4 Genotype A Genotype B Genotype C
Genotype A Genotype B Genotype C
Genotype A Genotype B Genotype C
Genotype A Genotype B Genotype C
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
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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.
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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.
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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
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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
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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.
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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.
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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.
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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
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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.
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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
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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
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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
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[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
`
]
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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
]
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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 .
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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) = σ2` .
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Back to the Traditional Split-Plot Experimental Design
Field
Block 1
Block 2
Block 3
Block 4Genotype AGenotype B Genotype C
Genotype A Genotype 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
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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
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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]′
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X = 14×1⊗ I
12×12, β =
µ11
µ12
µ13
µ14
µ21
µ22
µ23
µ24
µ31
µ32
µ33
µ34
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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
])
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bwe
∼ N
0
00
, σ2
b I 0 00 σ2
w I 00 0 σ2
e I
[ue
]∼ N
([00
],
[G 00 R
])
Copyright c©2019 Dan Nettleton (Iowa State University) 14. Statistics 510 30 / 30