Nested Designs

Ex 1--Compare 4 states’ low-level
radioactive hospital waste
– A: State
– B: Hospital
– Rep: Daily waste
Nested Designs

Ex 2--Compare tobacco yield/acre in 5
counties
– A: County
– B: Farm
– Rep: Field Yield/acre
Other Ex--Tick study, Classroom studies
 Are A and B crossed in these studies?

Nested Designs
We say B is nested in A when levels of B
are unique (or intrinsic) to a given level of
A
 Nested designs can also be thought of as
incomplete factorial designs (with most
treatment combinations impossible)

Nested Designs

Typically, A is fixed and B is random
i  1, , a
Yijk     i  B j ( i )   k (ij ) j  1,, b or bi
k  1, , n or nij
2


0
(e.g.)
;
B
iid
N
(
0
,

 i
j (i )
B)
Nested Designs

We can decompose the sum of squares:
SSTO=SSA+SSB(A)+SSE
   Y
ijk
 Y...   nb  Yi..  Y... 
2
2
i
 n  Yij .  Yi..      Yijk  Yij. 
2
i

j
i
j
2
k
Note that SSB(A)=SSB+SSAB
Expected Mean Squares

All other assumptions being satisfied for
an F-test, we can compute Expected
Mean Squares (EMS) to construct
appropriate F-tests for factor effects
EMS for A
Yijk     i  B j ( i )   k (ij )
Yi..     i  B.( i )   ( i )
Y...    0  B.(.)   ( )
 Y
i ..
i
 Y...        ( i )   ( )    B.( i )  B.(.)   
2
i
2
2
2
i
i
i
EMS for A
 Y  Y       
 Y  Y     
2
i ..
2
i
...
i
( i )
i
2
i ..
i
a 1
...

i
2
i
i
a 1

  ( )    B( i )  B()   
2
2
  ( ) 
i
2
( i )
i
a 1

 B
 B() 
2
( i )
i
a 1

EMS for A
  Yi..  Y... 2 
2

 i
  i  2  B2
E




a 1
a  1 bn b




2
nb  i
2
2
E ( MSA) 
   n B
a 1
EMS for B(A)
Yij.     i  B j (i )   (ij )
Yij.  Yi..  B j (i )  B(i )    (ij )   (i) 
EMS for B(A)
Yij.  Yi..  B j i   B.i     .ij    .i. 
 n Yij.  Yi.. 2 
  Yij.  Yi.. 2 
 i j
 n
 j

E
   E

a (b  1)
b 1

 a i 





2
n
n

2
2
2



n





B
B
a i
a i n
ANOVA table
Source
A
df
a-1
SS
SSA
EMS
nb  i2
B(A)
a(b-1)
SSB(A)
n  
Error
Total
ab(n-1)
abn-1
SSE
SSTO
2
a 1
2
B
  2  n B2
2
Example

Tobacco Case Study
– SAS code
– Minitab
– Variance Components
Cost Analysis
Supposed we want to minimize the
variance of our treatment means (for a
balanced design)
 This will depend on the cost of sampling
another level of the nested factor vs the
cost of adding a replication

Cost analysis
n  
V Yi..  
bn
2
B

2
The total number of units to be sampled is
fixed at bn. If cost isn’t a factor, how
should these units be allocated?
Cost Analysis
Assume we have a fixed project cost C
 D1=“dollars” per nested factor level
 D2=“dollars” per rep
 Total cost C=b D1+bn D2

Cost Analysis

Subject to the cost constraint, we minimize
n B2   2
bn

The answer turns out to be
n
D1 D2
B 
Power Analysis

Note that increasing n does not really
improve power for testing treatment effects
Ho :  0  
nb
2
i
i
n  
2
B
2