### Lecture 15

```Designs with Randomization
Restrictions

RCBD with a complete factorial in each
block
– A: Cooling Method
– B: Temperature

Conduct ab experiments in each block
Designs with Randomization
Restrictions

All factors are crossed
Yijk     i   j  ij   k
i  1,, a
  ik   jk  ijk j  1,, b
k  1,, n
Designs with Randomization
Restrictions

By convention, we assume there is not
block by treatment interaction (the usual
RCBD assumption) so that:
ik   jk  ijk   ijk

Note that this is different from “pooling”
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)

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
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

Split Plot Design

One version of the model (See ex. 24.1):
Yijk     i   k   ik 
 j  ij   jk   ijk
EMS Table--Whole Plot
EMS
Source
2
2
2
WholeP lot (A)   b   bn  i (a  1)
df
a -1
i
Block
  ab 2
n -1
A x Block
 2  b 2
(a - 1)(n- 1)
2
EMS Table--Split Plot
Source
Split P lot (B)
  a  
2
AB x Block
an  j2
j
2
(b  1)(n  1)
  a 
    
2
2
2
n ij2
i
j
(a  1)(b  1)
(a  1)(b  1)
   
2
b 1
b 1
2
B x Block
AB
EMS
2
(a  1)(b  1)(n  1)
Split Plot Design
Note that there are no degrees of freedom
for error
 Block and Block x Treatment interactions
cannot be tested

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.
Yijk     i   k   ik 
 j  ij   jk (i )
Split Plot Design
Yandell presents an alternate model
 Useful when whole plots are replicated
and no blocks are present

Yijk     i   k (i ) 
 j  ij   jk (i )
EMS Table--Whole Plot
Source
A
WholeP lot Error
EMS
 2  b 2 
bn  i2
i
a 1
 2  b 2
EMS Table--Split Plot
Source
Split P lot (B)
AB
Split P lot Error
EMS
 2   2 
 2   2 
an  j2
j
b 1
n ij2
j
k
(a  1)(b  1)
 2   2
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
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
Split Split Plot Design

Surgical Glove Example
–
–
–
–
Whole Plot: Latex preparation method
Split Plot: Coagulant dip
Split Split Plot: Heat treatment
Split Split Plot Design

A model version that facilitates testing:
Yijkl     i   l (i ) 
 j  ij   jl (i ) 
 k   ik   jk  ijk  kl (ij )
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

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)
Split Plot Design with
Covariates

Whole Plot Covariate
– Xijk=Xik
– Xijk=Xi occurs frequently in practice

Split Plot Covariate
WP Covariate
X ijk  X ijk  X i.k  X i.k
SP Covariate
Split Plot Design with
Covariates

Model
Yijk     i  X i.k   k (i ) 
 j  X ijk  X i.k   ij   jk (i )
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

Repeated Measures Design
 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

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
Yikm     i  rm (i ) 
 k   ik   km(i )
i  1, , a
m  1,, n
k  1,, t
Repeated Measures Design

Fuel Cell Example
– Response: Current
– Subject(Group): Daily Experimental Run or
Fuel Cell
– Repeated Measures Factor: Voltage
Repeated Measures Design
Source
Group
Subject(Group)
Repeated Measures Factor
Group x Repeated Measures
Error
Total
df
a-1
a(n-1)
t-1
(a-1)(t-1)
a(t-1)(n-1)
atn-1
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
Cov(Yikm , Yik 'm )  Cov(rm (i ) , rm (i ) )  
Corr(Yikm , Yik 'm )  
2
P

2

2
P

2
P
Repeated Measures Design

Fuel Cell Example
–
–
–
–
Repeated Measures Factor: Voltage
Response: Current
Subject: Fuel Cell
Repeated Measures Design
Y jkm    T j  Rm( j )  Vk  TV jk   jkm ,
j  1, 2; m(1)  1, 2,3; m( 2)  1, 2; k  1,  ,8
Repeated Measures Design
Yi=Yjm=(Yj1m,…,Yj8m)’
=(,T1,V1,…,V7,TV11,…,TV17)’
Mixed Models

The general mixed model is
 i ~ N (0, R)
Yi  X i   Z i   i 
  ~ N (0, G )
Repeated Measures Design
For our example, we have no random effects (no
Zi or ) 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
Mixed Models
For many models we encounter, R is 2I
 In repeated measures models, R can have
a lot more structure. E.g., for t timepoints,
an AR(1) covariance structure would be:

 2
 1 2


R
 
 t 1 2
  
 1 2
2


 
t 2
2
  t 1 2 
t 2 2 
   

 

2

 
Repeated Measures Structures
•Toeplitz
 kk '   2  k  k '
 kk '   2  kk '
•Unstructured
•Compound Symmetric
•Banded Toeplitz
 kk '   2 
 2  k  k ' , k  k '  r
 kk '  

0,
k  k'  r
Mixed Models

G almost always has
a diagonal structure
Ex :

Regardless of the
form for R and G, we
can write
Yi~N(Xi,ZiGZi’+R)
 2 I a
0
0 


G 0
 2 I b
0 
2
 0

0

I

ab


Mixed Models

For the entire sample we have
Y ~ N ( X , ZGZ ' R*), where
Y '  (Y1 ' ,  , Yn ' )
X '  ( X 1 ' ,, X n ' )
Z '  ( Z1 ' ,  , Z n ' )
R*  I n  R
Restricted MLE
If V=ZGZ’+R* were known, the MLE for 
would be (X’V-1X)-1X’V-1Y
 We would estimate the residuals as e=(IH)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

Restricted MLE

The Profile RMLE of the parameters of G
and R would maximize :
1
1
1
l (G, R)   (n  q ) log 2  log PVP'  e'V 1e
2
2
2
T hencomput e
1
1
1
ˆ
ˆ
ˆ
  ( X ' V X ) X 'V Y
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)
AIC  ln Lˆ  q
 or 
AIC  2 ln Lˆ  2q
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
```