Overview of this module
Course 02429
Analysis of correlated data: Mixed Linear Models
Module 7: The analysis of split-plot design data
1
Simple Split-plot designs
Example: Tenderness of pork.
2
Split-plot with blocks
Example: Split-plot with blocks, Yield of oats
3
Split-plot in perspective
Per Bruun Brockhoff
DTU Compute
Building 324 - room 220
Technical University of Denmark
2800 Lyngby – Denmark
e-mail: [email protected]
Per Bruun Brockhoff ([email protected])
Mixed Linear Models, Module 7
Fall 2014
1 / 16
Simple Split-plot designs
b1
b3
b2
b4
a1
a1
a1
a1
Mixed Linear Models, Module 7
Simple Split-plot designs
Basic structure
a3
a3
a3
a3
Per Bruun Brockhoff ([email protected])
Fall 2014
2 / 16
Example: Tenderness of pork.
Example: Tenderness of pork.
b4
b3
b1
b2
a1
a1
a1
a1
b1
b3
b4
b2
a2
a2
a2
a2
b3
b2
b1
b4
a3
a3
a3
a3
b3
b1
b2
b4
a2
a2
a2
a2
b2
b1
b4
b3
Treatments are given on different levels
24 porks (pork) in 2 treatment groups (low pH/high pH).(pH)
Each pork cut in half (right/left side).
2 cooling methods: One for each side. (C)
48 observations of tenderness.
Factor structure:
24
2
[P]22
[I]48
22
Treatment effects are more easily found on finer levels
For B: Randomized block setting with whole-plots as blocks.
Mixed Linear Models, Module 7
Fall 2014
011
Ph × C41
For A: Completely randomized design with whole-plots as
observational unit.
Per Bruun Brockhoff ([email protected])
Ph1
4 / 16
Per Bruun Brockhoff ([email protected])
Mixed Linear Models, Module 7
C21
Fall 2014
5 / 16
Simple Split-plot designs
Example: Tenderness of pork.
Simple Split-plot designs
The split-plot mixed model for the example
Example, test results
Source of
variation
The model:
Yi = α(pHi ) + β(Ci ) + γ(pH × Ci ) + d(Pi ) + εi
phgroup
cooling
phgroup*cooling
pH tested versus the pork (whole plot) variation:
F =
MSpH
MSP
Source of
variation
C and pH×C tested versus the residual error:
F =
Per Bruun Brockhoff ([email protected])
MSC×pH
MSC
, F =
MSError
MSError
Mixed Linear Models, Module 7
Simple Split-plot designs
phgroup
cooling
Fall 2014
6 / 16
Numerator
degrees
of freedom
1
1
1
Numerator
degrees
of freedom
1
1
Per Bruun Brockhoff ([email protected])
Example: Tenderness of pork.
Denominator
degrees
of freedom
22
22
22
Denominator
degrees
of freedom
22
22
F
P-value
8.67
2.25
0.18
0.0075
0.1479
0.6790
F
P-value
8.67
2.33
0.0075
0.1403
Mixed Linear Models, Module 7
Split-plot with blocks
Example, results summary
Fall 2014
7 / 16
Example: Split-plot with blocks, Yield of oats
Example: Split-plot with blocks, Yield of oats
v3
v1
Variances:
2
σ̂W
Example: Tenderness of pork.
v2
= 1.2463,
σ̂ 2 = 0.4725.
v3
Fixed effects:
v1
v2
α̂(low) = 5.6529, ([4.9240, 6.3819])
n3
n1
n0
n3
n0
n2
156
140
111
174
117
161
n2
n0
n1
n2
n1
n3
118
105
130
157
114
141
n2
n0
n0
n3
n1
n3
109
63
80
126
90
116
n3
n1
n2
n1
n2
n0
99
70
94
82
100
62
v3
n2
n1
n3
n1
n1
n2
104
89
122
89
103
132
n0
n3
n0
n2
n0
n3
70
117
74
81
64
133
n3
n2
n2
n0
n2
n1
96
89
112
68
132
129
n0
n1
n3
n1
n3
n0
60
102
86
64
124
89
v2
n1
n3
n3
n2
n0
n1
108
149
144
121
61
91
n2
n0
n1
n0
n3
n2
126
70
124
96
100
97
n2
n3
n3
n0
n0
n2
118
113
104
89
97
119
n0
n1
n2
n1
n1
n3
53
74
86
82
99
121
v1
v2
v1
v1
v3
α̂(high) = 7.1163, ([6.3873, 7.8452])
v2
v3
v1
Per Bruun Brockhoff ([email protected])
Mixed Linear Models, Module 7
Fall 2014
8 / 16
Per Bruun Brockhoff ([email protected])
Mixed Linear Models, Module 7
v2
v3
Fall 2014
10 / 16
Split-plot with blocks
Example: Split-plot with blocks, Yield of oats
Split-plot with blocks
Example: Yield of oats
Example: Split-plot with blocks, Yield of oats
Yield of oats, exploration
Average yield as a function of nitrogen level for each variety.
Factors and their levels:
v1 , v2 , v3
n0 , n1 , n2 , n3
1, 2, . . . , 18
1, 2, . . . , 6
V
N
P
B
Factor structure:
[P]18
10
[I]72
45
V32
V × N12
6
Per Bruun Brockhoff ([email protected])
[B]65
011
N43
Mixed Linear Models, Module 7
Split-plot with blocks
Fall 2014
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Example: Split-plot with blocks, Yield of oats
fertil
variety
fertil*variety
Source of
variation
fertil
variety
Fall 2014
12 / 16
Split-plot in perspective
Numerator
degrees
of freedom
3
2
6
Numerator degrees
of freedom
3
2
Per Bruun Brockhoff ([email protected])
Mixed Linear Models, Module 7
Split-plot in perspective
Yield of oats, testing
Source of
variation
Per Bruun Brockhoff ([email protected])
Denominator
degrees
of freedom
45
10
45
F
37.69
1.49
0.30
Denominator degrees
of freedom
51
10
Mixed Linear Models, Module 7
P-value
Treatments on different levels occur very often in practice
More complicated and more than two levels may occur:
<.0001
0.2724
0.9322
F
Split-plot with correlated whole plots
Whole plot conducted as an incomplete latin square
A strip-split-split-plot design
P-value
The above are case studies in Littel et al. (1999).
Serves as a kind of basis for repeated measures analysis.
41.05
1.49
<.0001
0.2724
Fall 2014
13 / 16
Per Bruun Brockhoff ([email protected])
Mixed Linear Models, Module 7
Fall 2014
15 / 16
Split-plot in perspective
Overview of this module
1
Simple Split-plot designs
Example: Tenderness of pork.
2
Split-plot with blocks
Example: Split-plot with blocks, Yield of oats
3
Split-plot in perspective
Per Bruun Brockhoff ([email protected])
Mixed Linear Models, Module 7
Fall 2014
16 / 16