Nested Designs and Repeated Measures
with Treatment and Time Effects
KNNL – Sections 26.1-26.5, 27.3
Nested Factors
• Factor is Nested if its levels under different levels of
another (Nesting) factor are not the same
 Nesting Factor ≡ School, Nested Factor ≡ Teacher
 Nesting Factor ≡ Factory, Nested Factor ≡ Machinist
• If Factor A (Nesting) has a levels and Each Level of A
has b levels of Factor B (Nested), there are a total of
ab levels of Factor B, each being observed n times
A1
Factor A
Factor B
Replicates
B1(1)
Y111
Y112
A2
B2(1)
Y121
Y122
B1(2)
Y211
Y212
A3
B2(2)
Y221
Y222
B1(3)
B2(3)
Y311 Y312
Y321 Y322
Note: When Programming, give levels of B as: 1,2,...,b,b+1,...,2b,...(a-1)b+1,...,ab
2-Factor Nested Model – Balanced Case
Factor A at a levels with means: i     i
Factor B with ab levels (b disinct levels w/in each level of A) with means:
ij     i   j (i )   j (i )  ij      i   ij  i 
A and B Fixed Factors 
a

i 1
i
0
 Yijk ~ N     i   j (i ) ,  2 
i 1
 Yijk ~ N     i ,  2   2 
j (i )
0 i
i  1,..., a; j  1,..., b; k  1,..., n  ijk ~ N  0,  2  independent
independent
a


j 1
Yijk  ij   ijk  ij     i   j (i )   ijk
A Fixed and B Random: 
b
i
 0  j (i ) ~ N  0,  2  independent
 2   2

 Yijk , Yijk '     2
 0

j (i )
i  i ', j  j ', k  k '
i  i ', j  j ', k  k '
A and B Random   i ~ N  0,  2  indep  j (i ) ~ N  0,  2  indep
 Yijk ~ N   ,  2   2   2 
    
  2   2   2
 2
2
    
 Yijk , Yi ' j ' k '    2
 
0
otherwise
    j (i )    
i  i ', j  j ', k  k '
i  i ', j  j ', k  k '
i  i ', j  j ', k , k '
i  i ', j , j ', k , k '
Estimators, Analysis of Variance, F-tests
^
   Y 
^
^
 i  Y i  Y 
^
^
^
 j (i )  Y ij  Y i

^
 


Fitted Values: Y ijk =     i   j (i )  Y   Y i  Y   Y ij   Y i  Y ij 
a
^
b
n

 SSE   Yijk  Y ij 
Residuals: eijk  Yijk  Y ijk  Yijk  Y ij 
i 1 j 1 k 1
a

Factor A Sum of Squares: SSA  bn Y i  Y 
i 1
a

2
df A  a  1
b

Factor B Within A Sum of Squares: SSB( A)  n Y ij   Y i
i 1 j 1
A Fixed, B Fixed
df E  ab  n  1
2

2
df B ( A)  a  b  1
A Fixed, B Random
a
E MSA   2 
a
bn  i2
E MSA   2  n 2 ( ) 
i 1
a 1
a
E MSB ( A)   2 
A Random, B Random
bn  i2
i 1
a 1
E MSA   2  n 2 ( )  bn 2
b
n  j2(i )
E MSB( A)   2  n 2 ( )
i 1 j 1
a  b  1
E MSE   2
H 0 : 1  ...   a  0 TS : FA 
E MSB( A)   2  n 2 ( )
E MSE   2
MSA
MSE
H 0 : 1(1)  ...  b ( a )  0 TS : FB ( A) 
E MSE   2
MSA
MSB ( A)
MSB( A)

MSE
H 0 : 1  ...   a  0 TS : FA 
MSB( A)
MSE
H 0 :  2 ( )  0 TS : FB ( A)
MSA
MSB ( A)
MSB ( A)
 0 TS : FB ( A) 
MSE
H 0 :  2  0 TS : FA 
H 0 :  2 ( )
Fixed Effects Model (A and B Fixed)
Factor A: H 0 : 1  ...   a  0
 
E Y i  i     i
TS : FA 
 
 Y i 
2
a
Contrasts among Means of A:
c
i
i 1
a
a
i 1
i 1
2
RR : FA  F 1   ; a  1, ab  n  1 
 
s 2 Y i 
bn
MSE
bn
0
 
MSE a 2
 ci
bn i 1
L A  t 1   / 2  ; ab  n  1 
MSE a 2
 ci
bn i 1
^
LA   ci i   ci i
1   100% CI for LA :
MSA
MSE
a
^
L A   ci Y i
s2 L A 
i 1
^
a
All Possible Comparisons among pairs of means of A:
c
i 1


Tukey: Y i  Y i '  q 1   ; a, ab  n  1 
Factor B(A): H 0 : 1(1)  ...  b ( a )  0
 
E Y ij   ij     i   j (i )
 
 2 CA 
a  a  1
2
   
 2 MSE
MSE
Bonferroni: Y i  Y i '  t 1  
 ; ab  n  1 
bn
bn
  2C A 


TS : FB ( A) 
 Y ij  
2
2
i
2
n
MSB( A)
MSE
 
s 2 Y ij  

RR : FB ( A)  F 1   ; a  b  1 , ab  n  1 
MSE
n
b  b  1
2
   
 2MSE
 t 1  
; ab  n  1 

  2CB ( A) 

n

 

All Possible Comparisons among pairs of means of B within a given level (i ) of A: CB ( A) 


Tukey: Y ij   Y ij '  q 1   ; b, ab  n  1 

MSE
Bonferroni: Y ij   Y ij '
n

Mixed Effects Model (A Fixed and B Random)
Factor A: H 0 : 1  ...   a  0
 
TS : FA 
 
MSA
MSB( A)
 2  n 2 ( )
RR : FA  F 1   ; a  1, a  b  1 
 
MSB( A)
bn
bn
a
a  a  1
All Possible Comparisons among pairs of means of A:  ci2  2 C A 
2
i 1
E Y i  i     i
 Y i 
2
s 2 Y i 
Tukey: Y i  Y i '  q 1   ; a, a  b  1 
   
 2 MSB( A)
MSB( A)
Bonferroni: Y i  Y i '  t 1  
 ; a  b  1 
bn
bn
  2C A 

Factor B(A): H 0 :  2 ( )  0
MSB( A)
MSE


E MSB( A)   2  n 2 ( )
 s2 ( )
TS : FB ( A) 

E MSE   2
1
1
 MSB( A)  MSE
n
n


RR : FB ( A)  F 1   ; a  b  1 , ab  n  1 
1
 1
2
  E MSB( A)     E MSE    ( )
n
 n
s 
2
Approximate df (Satterthwaite): df  ( ) 


df  ( ) s2 ( )
2
Approximate 1  a 100% CI For   ( ) : 
,

  2 1   ; df
  2  ( ) 

 
2
 ( )
  MSB( A)  2  MSE  2 

 
 
n
n



 


 a  b  1
ab  n  1 





2
df  ( ) s ( ) 


2 
  ; df  ( )  
2

Random Effects Model (A and B Random)
Factor A: H 0 :  2  0
TS : FA 
E MSA   2  n 2 ( )  bn 2

MSA
MSB( A)
RR : FA  F 1   ; a  1, a  b  1 
E MSB ( A)   2  n 2 ( )
 1 
 1 
2
E
MSA
)



 
   E MSB ( A)   
 bn 
 bn 
 s2 
1
1
MSA  MSB ( A)
bn
bn
s 
2 2
Approximate df (Satterthwaite): df A 


df A s2
2
Approximate 1   100% CI For   : 
,

  2 1  ; df 
  2 A

 
Factor B Within A is same as in Mixed Effects Model

  MSA 2  MSB( A)  2 

 
 
bn
bn



 


 a 1
a  b  1 






df A s2




 2  ; df A  
2

Repeated Measures with Treatment and Time
• Goal: Compare a Treatments over b Time Points
• Begin with nT = as Subjects, and randomly assign them
such that s Subjects receive Treatment 1, ...
s
Subjects receive Treatment a
• Each Subject receives 1 Treatment (not all Treatments)
• Each Subject is observed at b Time points
• Treatment is referred to as “Between Subjects” Factor
• Time is referred to as “Within Subjects” Factor
• Treatment and Time are typically Fixed Factors
• Subject (Nested within Treatment) is Random Factor
• Generalizes to more than 1 Treatment Factor
Statistical Model
Yijk    i ( j )   j   k    jk   ijk
i  1,..., s;
j  1,..., a; k  1,..., b
Yijk  Measurement on i th Subject within the i th Treatment at the k th Time Point
 ijk ~ N  0,  2  independent
  Overall Mean
i ( j )  Subject Effects i ( j ) ~ N  0,  2  independent
 j  Treatment Effects
 k  Time Effects
  jk 
a

j 1
b

k 1
k
j
    
i( j)
0
0
b
a
TreatmentxTime Interaction Effects
  
j 1
E Yijk      j   k    jk

ijk
 2 Yijk    2   2
 Yijk ~ N    j   k    jk ,  2   2

jk
    jk  0
k 1
 2   2

 Yijk , Yi ' j ' k '    2
0

i  i ', j  j ', k  k '
i  i ', j  j ', k  k '
otherwise
  Yij1    2   2
 2
   
2
 2   2
 Yij 2     




   
 Yijb     2
 2
 

 2
 2





 2   2 
Analysis of Variance & F-Tests
a
a

Treatment (Factor A): SSA  sb Y  j   Y 
j 1

2
sb  2j
E MSA   2  b 2 
df A  a  1
j 1
a 1
b
b

SSB  sa  Y k  Y 
Time (Factor B):
k 1

2
E MSB   2 
df B  b  1
sa   k2
k 1
b 1
a
a
Trt x Time (AB):
b

SSAB  s  Y  jk  Y  j   Y k  Y 
j 1 k 1
s
a

Subject(Trt) (S(A)): SSS ( A)  b Y ij   Y  j 
i 1 j 1
s
a
b

2

2
df AB   a  1 b  1
df S ( A)  a  s  1

TimexSubj(Trt) (B.S(A)): SSB.S ( A)    Y ijk  Y ij   Y  jk  Y  j 
i 1 j 1 k 1
Tests: H 0A : all  j  0
Tests Statistics: FA 
H 0B : all  k  0
MSA
MSS ( A)
FB 
A
2
s    jk
2
j 1 k 1
 a  1 b  1
E MSS ( A)   2  b 2
df B.S ( A)  a  b  1 s  1
E MSB.S ( A)   2
H 0AB : all   jk  0
MSB
MSB.S ( A)
Rejection Regions: F  F 1   ; a  1, a  s  1

E MSAB   2 
b
FAB 
MSAB
MSB.S ( A)
F  F 1   ; b  1, a  b  1 s  1 F
 F 1   ;  a  1 b  1 , a  b  1 s  1
B
AB
Comparing Treatment and Time Effects – No Interaction
Treatment (Factor A) Effects when No Interaction Between Treatment and Time:


Tukey Method: Y  j   Y  j '  q 1   ; a  s  1 
MSS ( A)
sb
Bonferroni Method:

Y  j   Y  j '

   
 2 MSS ( A)
 t 1  
 ; a  s  1 
sb
  2C A 

CA 
a (a  1)
2
Time (Factor B) Effects when No Interaction Between Treatment and Time:


Tukey Method: Y k  Y k '  q 1   ; a  b  1 s  1 
MSB.S ( A)
sa
Bonferroni Method:

Y k  Y k '

  
 t 1  
  2CB
 2 MSB.S ( A)

 ; a  b  1 s  1 
sa


CB 
b(b  1)
2
Comparing Trt Levels w/in Time Periods
Comparing Treatment Levels Within Time Levels:
Bonferroni:
   
 2  MSS ( A)   b  1 MSB.S ( A) 
Y . jk  Y . j ' k  t 1  
 ; df 
bs
  2C A 

with approximate df:


(b  1) MSB.S ( A)  MSS ( A) 


  (b  1) MSB.S ( A) 2  MSS ( A) 2 



2
df

a (b  1)( s  1)
a( s  1) 