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Designs with Randomization Restrictions
RCBD with a complete factorial in each block– A: Cooling Method– B: Temperature
Conduct ab experiments in each block
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Designs with Randomization Restrictions
All factors are crossed
nk
bj
aiY
ijkjkik
kijjiijk
,,1
,,1
,,1
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Designs with Randomization Restrictions
By convention, we assume there is not block by treatment interaction (the usual RCBD assumption) so that:
Note that this is different from “pooling”
ijkijkjkik
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Designs with Randomization Restrictions
A similar example uses a Latin Square design
The treatment is in fact a factorial experiment
n=ab or n-1=(a-1)+(b-1)+(a-1)(b-1)
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Split Plot Design
Two factor experiment in which a CRD within block is not feasible
Example (observational study)– Blocks: Lake–Whole plot: Stream; Whole plot factor:
lampricide– Split plot factor: Fish species– Response: Lamprey scars
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Split Plot Design
Agricultural Example– Block: Field–Whole Plot Factor: Tilling method– Split Plot Factor: Seed variety
Whole plot and whole plot factor are confounded
This is true at split plot level as well, though confounding is thought to be less serious
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Split Plot Design
One version of the model (See ex. 24.1):
ijkjkijj
ikkiijkY
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EMS Table--Whole Plot
1)-1)(n-(a
1-n
1-a
df
)1(
EMS
BlockA x
Block
(A)Plot Whole
Source
22
22
222
b
ab
abnbi
i
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EMS Table--Split Plot
Source EMS
)1)(1)(1(
)1)(1(
)1)(1(
1
Block x AB
)1)(1(AB
Block x B
1(B)Plot Split
22
2
22
22
2
22
nba
ba
nb
b
ba
n
a
b
an
a
i jij
jj
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Split Plot Design
Note that there are no degrees of freedom for error
Block and Block x Treatment interactions cannot be tested
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Split Plot Design
In an alternative formulation, SP x Block and SP x WP x Block are combined to form the Split Plot Error. Note the unusual subscript—a contrivance that yields the correct df.
)(ijkijj
ikkiijkY
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Split Plot Design
Yandell presents an alternate model
Useful when whole plots are replicated and no blocks are present
)(
)(
ijkijj
ikiijkY
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EMS Table--Whole Plot
Source EMS
22
2
22
ErrorPlot Whole1
A
ba
bnb i
i
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EMS Table--Split Plot
Source EMS
22
2
22
2
22
ErrorPlot Split )1)(1(
AB
1(B)Plot Split
ba
nb
an
j kij
jj
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Split Plot Design
Yandell considers the cases where the whole plot and split plot factors, alternately, do not appear– Split plot factor missing—whole plot looks like RCBD (me) or CRD (Yandell); subplots are subsampled.– Whole plot factor—whole plots look like one-way random effects; subplots look like either RCBD or CRD again.
Yandell has nice notes on LSMeans in Ex. 23.4
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Split Split Plot Design
We can also construct a split split plot design (in the obvious way)
Montgomery example– Block: Day– Whole Plot: Technician receives batch– Split Plot: Three dosage strengths formulated
from batch– Split split plot: Four wall thicknesses tested
from each dosage strength formulation
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Split Split Plot Design
Surgical Glove Example– Block: Load of latex pellets–Whole Plot: Latex preparation method– Split Plot: Coagulant dip– Split Split Plot: Heat treatment
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Split Split Plot Design
A model version that facilitates testing:
)(
)(
)(
ijklijkjkikk
ijlijj
iliijklY
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Split Plot Design with Covariates
This discussion is most appropriate for the nested whole plots example
Often, researchers would like to include covariates confounded with factors
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Split Plot Design with Covariates
Example (Observational study)–Whole Plot: School–Whole Plot Factor: School District– Split Plot Factor: Math Course– Split Plot : Class– Split Plot covariate: Teacher Rating–Whole Plot covariate: School Rating–Whole Plot Factor covariate: School
District Rating– Response: % Math Proficient (HSAP)
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Split Plot Design with Covariates
Whole Plot Covariate– Xijk=Xik
– Xijk=Xi occurs frequently in practice Split Plot Covariate
kikiijkijk XXXX ..
SP Covariate
WP Covariate
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Split Plot Design with Covariates
Model
)(.
)(.
ijkijkiijkj
ikkiiijk
XX
XY
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Split Plot Design with Covariates
A Whole Plot covariate’s Type I MS would be tested against Whole Plot Error (with 1 fewer df because of confounding)
Split Plot Covariate is not confounded with any model terms (though it is confounded with the error term), so no adjustments are necessary
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Repeated Measures Design
Read Yandell 25.1-25.3 Chapter 26 generally covers multivariate approaches to repeated measures—skip it
We will study the traditional approach first, and then consider more sophisticated repeated measures correlation patterns
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Repeated Measures Design
Looks like Yandell’s split plot design– The whole plot structure looks like a nested design– The split plot structure looks much the same
tk
nm
airY
ikmikk
imiikm
,,1
,,1
,,1
)(
)(
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Repeated Measures Design
Fuel Cell Example– Response: Current– Group: Control/Added H20– Subject(Group): Daily Experimental Run or Fuel Cell– Repeated Measures Factor: Voltage
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Repeated Measures Design
Source dfGroup a-1Subject(Group) a(n-1)Repeated Measures Factor t-1Group x Repeated Measures (a-1)(t-1)Error a(t-1)(n-1)Total atn-1
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Repeated Measures Design
A great deal of work has been conducted on repeated measures design over the last 15 years– Non-normal data– More complex covariance structure
222'
2)()('
),(
),(),(
PPmikikm
Pimimmikikm
YYCorr
rrCovYYCov
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Repeated Measures Design
Fuel Cell Example– Repeated Measures Factor: Voltage– Response: Current– Group: Control/Added H20– Subject: Fuel Cell
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Repeated Measures Design
8,,1;2,1)2(;3,2,1)1(;2,1
,)(
kmmj
jkmjkkjmjjkm TVVRTY
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Repeated Measures Design
Yi=Yjm=(Yj1m,…,Yj8m)’
b=(m,T1,V1,…,V7,TV11,…,TV17)’
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Mixed Models
The general mixed model is
),0(~
),0(~
GN
RNZXY i
iiii
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Repeated Measures Design
For our example, we have no random effects (no Zi or g) separate from the repeated measures effects captured in R. X1 =X1(1) has the form (assume V8=TV18=0)
1|1|1 0 0 0 0 0 0|1 0 0 0 0 0 0
1|1|0 1 0 0 0 0 0|0 1 0 0 0 0 0
…
1|1|0 0 0 0 0 0 1|0 0 0 0 0 0 1
1|1|0 0 0 0 0 0 0|0 0 0 0 0 0 0
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Mixed Models
For many models we encounter, R is s2I
In repeated measures models, R can have a lot more structure. E.g., for t timepoints, an AR(1) covariance structure would be:
22221
22221
21212
tt
t
t
R
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Repeated Measures Structures
rkk
rkkkkkk
kk
kkkk
kkkk
',0
','2
'
2'
'2
'
'2
'
• Toeplitz
• Unstructured
• Compound
Symmetric
• Banded Toeplitz
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Mixed Models
G almost always has a diagonal structure
Regardless of the form for R and G, we can write Yi~N(Xib,ZiGZi’+R)
ab
b
a
I
I
I
G2
2
2
00
00
00
:Ex
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Mixed Models
For the entire sample we have
RIR
ZZZ
XXX
YYY
RZGZXNY
n
n
n
n
*
)',,'('
)',,'('
)',,'('
where*),',(~
1
1
1
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Restricted MLE
If V=ZGZ’+R* were known, the MLE for b would be (X’V-1X)-1X’V-1Y
We would estimate the residuals as e=(I-H)Y=PY where H=X(X’V-
1X)-1X’V-1. The profile likelihood for the parameters of G and R would be based on the distribution of the residuals
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Restricted MLE
The Profile RMLE of the parameters of G and R would maximize :
YVXXVX
eVePVPqnRGl
111
1
ˆ')ˆ'(ˆ
computeThen
'2
1'log
2
12log)(
2
1),(
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Case Study
To choose between non-hierachical models, we select the best model based on the Akaike Information Criterion (smaller is better for the second form; q=# of random effects estimated)
qLAIC
or
qLAIC
2ˆln2
ˆln
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Case Study
Autocorrelation was strong A Toeplitz model worked best Voltage effect, as expected, was strong Treatment effect was marginal Voltage x Treatment effect was strong to
moderate