Random and Mixed Effects ANOVA
KNNL – Chapter 25
1-Way Random Effects Model
• In some settings, the levels of the factor in a 1-Way
ANOVA model are a random sample from a larger
population of levels
• Goal is to make statements regarding the population
of levels, as opposed to only regarding the levels in
the study.
• We wish to make statements about the variation in
the effects of the levels
• Studies are often interested in measuring reliability
of testing/measuring procedures
Statistical Model / Parameters of Interest
Yij  i   ij
i  1,..., r;
j  1,..., n
i ~ N   ,  2  independent
 ij ~ N  0,  2  independent
  Overall Mean  2  Between Group Variance
i  ,  ij 
are independent
 2  Within Group Variance
E Yij   E i   ij   E i   E  ij     0  
 2 Yij    2 i   ij    2 i    2  ij   2 i ,  ij    2   2  2(0)   2   2   Y2
 Yij , Yij '    i   ij , i   ij '    i , i    i ,  ij '     ij , i     ij ,  ij '    2  0  0  0   2
 Yij , Yi ' j '   0 i  i '
r  2, n  2 :
 Y11 
Y 
Y   12 
Y21 
 
Y22 
 Y2  2 0
0
 2

2


0
0
Y

 2 Y   
2
2
0
0 Y   

2
2
0
0




Y 

 Yij , Yij ' 
 2
Intraclass Correlation Coefficient:  I   Yij , Yij '  

 Yij  Yij '   Y2
j  j'
Analysis of Variance - I
E Yij     2 Yij    Y2   2   2  Yij , Yij '    2  Yij , Yi ' j '   0 i  i '
r
n

SSE   Yij  Y i
i 1 j 1
r
n


SSTR   Y i  Y 
i 1 j 1
 SSE 
2
E MSE  E 

 r  n  1 
df E  r  n  1
2

2
r

 n Y i  Y 
i 1

2
dfTR  r  1
 SSTR 
2
2
E MSTR  E 
    n 
 r 1 
Testing whether all i are equal: H 0 :  2  0 H A :  2  0
Test Statistic: F * 
Reject H 0 if F *  F  0.95; r  1, r  n  1 
MSTR
MSE
Elements of the Derivations of E MSTR and E MSE :
E Yij     2 Yij    2   2  E Yij2        2   2
2
 
E Y i
 
 Y i
2
1 n 
 E   Yij   
 n j 1 
n 1 n
1 n  1  n
 1
    Yij   2   2 Yij   2   Yij , Yij '   2
j 1 j ' j 1
 n j 1  n  j 1
 n
2
 
 E Y i      
2
2

 n  n  1  2  n 2   2
2
2
 n       2 
   
n

 2  
n 2   2
n
2
2
2
2
r
r 1 r
1 r

 1  r 2
 1   n      r  r  1   n   
2
2 1
E Y   E   Y i     Y      Y i   2    Y i  2   Y i , Y i '   2  r 
  2 
 (0)  
n
2
nr
i 1 i 'i 1
 r i 1 
 r i 1  r  i 1
 r  

 

n 2   2
2
2
 E Y      
nr
 
 
 
 


Analysis of Variance - II
E Yij     2 Yij    Y2   2   2  Yij , Yij '    2  Yij , Yi ' j '   0 i  i '
r
n

SSE   Yij  Y i

r
n
r
  Yij2  n Y i
df E  r  n  1
SSTR   Y i  Y 

r
E Y
2
i 1 j 1
r
n
i 1 j 1
2
ij
   

2

2
 
2
i 1 j 1
2
r
2
i 1

 n Y i  Y 
i 1

 n Y i  nrY 
2
2

2
i 1
     
E Y
2
i
2
n 2   2
n
dfTR  r  1
     
E Y
2

2

n 2   2
nr

 r n 2
2
E  Yij   nr     nr 2  nr 2
 i 1 j 1 

 r 2
2
E n Y i   nr     nr 2  r 2
 i 1 
2
2
E SSE   nr     nr 2  nr 2    nr     nr 2  r 2   r  n  1  2

 

 SSE 
2
 E MSE  E 
 
 r  n  1 
2
2
E SSTR   nr     nr 2  r 2    nr     n 2   2   n  r  1  2   r  1  2

 

 SSTR 
2
2
 E MSTR  E 
  n   
 r 1 
 
E nrY   nr     n 2   2
2
2
Estimating Overall Mean •
E Yij     2 Yij    Y2   2   2  Yij , Yij '    2  Yij , Yi ' j '   0 i  i '
 
E Y i
1 n 
 E   Yij   
 n j 1 
 
s 2 Y i 
 
E Y 
 
 
s Y 
 
MSTR
nr
~ t  r  1
 
 
 Y  
2
s Y  

n 2   2
n
MSTR
n
s Y i 
1 r

 E   Y i    
 r i 1 
s 2 Y  
Y   
MSTR
n
 
 Y i 
2
n 2   2
nr
MSTR
nr
1   100% CI for  :
 
Y   t 1   / 2  ; r  1 s Y 
Estimating Intra-Class Correlation I = 2 / Y2
 SSE 
SSE
2
2
E MSE  E 
~

 r  n  1 
 
2
r
n

1

 
 
SSTR
 SSTR 
2
2
E MSTR  E 
~  2  r  1
  n   
2
2
n   
 r 1 
 SSTR
  MSTR 
 r  1  2 2 
 2
2
n



 
   n     ~ F  r  1, r n  1 
SSTR, SSE independent  
 

 SSE

 MSE 
  2  r  n  1  
  2 


 MSTR 


 2
2
n







  F 1  ( 2); r  1, r n  1    1  
 P  F  2; r  1, r  n  1   
  

MSE




  2 




 
 
1  MSTR 
1
1  MSTR 
1
Setting: L  
U 

  1

  1
n  MSE  F 1  ( 2); r  1, r  n  1   
n  MSE  F  / 2; r  1, r  n  1   




 2
L
U 
,U* 
1   100% CI for  I  2 2 :  L* 

 
1 L
1U 

Estimating Within Group Variance: 2
SSE

2
~  2  r  n  1 
SSE


 P   2  / 2  ; r  n  1   2   2 1   / 2  ; r  n  1    1  



  2  / 2  ; r  n  1  1
 2 1   / 2  ; r  n  1  
 P
 2
  1 


SSE

SSE




SSE
SSE
2
 P 2
  2
  1 
  1   / 2  ; r  n  1 
  / 2  ; r  n  1  

 

SSE
 1   100% CI for  :  2
  1   / 2  ; r  n  1 

 


r  n  1 MSE
r  n  1 MSE
  2
, 2

  1   / 2  ; r  n  1 
  / 2  ; r  n  1 

 
2

SSE
, 2

  / 2  ; r  n  1 
Estimating Between Group Variance: 2
E MSTR   2  n 2
E MSE   2
 MSTR  MSE  E MSTR  E MSE
1
 1
 E
  2    E MSTR     E MSE

n
n


n
 n
MSTR  MSE
Point Estimator: s2 
Note: This can be negative (usually treated as 0).
n
Aside :Satterthwaite approximation for Degrees of Freedom (ci are constants)
h
L   ci E MSi 
i 1
^
^
h
(df ) L Approx 2
~  df
L
L   ci MSi
i 1
 c1MS1  ...  ch MSh 

2
2
 c1MS1   ...   ch MSh 
2
where: df
df1
df h
^
^


(df ) L
(df ) L


 P 2
L 2
  1 


/
2;
df

1


/
2
;
df








^
^


(
df
)
L
(
df
)
L


 1   100% CI for L :
, 2
2
  1   / 2  ; df    / 2; df  


1
1
Application to estimating  2 : h  2, c1 
c2  
MS1  MSTR MS 2  MSE
n
n
MSTR  MSE
 L  s 
n
^
2
 ns 
2
df 
2

 MSTR 
r 1
2
 MSE 

r  n  1
2
2-Way Random Effects Model
Factors A and B Both Random (Levels are Random Samples from Populations)
Yijk     i   j   ij   ijk
i  1,..., a
j  1,..., b k  1,..., n
 i  ,  j  ,  ij  are independent, normally distributed, with mean 0, and Variances:  2 ,  2 ,  2
 ijk ~ N  0,  2  independent
 i  ,  j  ,  ij  ,  ijk  pairwise independent


E Yijk   E    i   j   ij   ijk  


2
 2 Yijk    2    i   j   ij   ijk   2   2   
 2
2
 2   2   
 2
 2
2
2









 Yijk , Yi ' j ' k '    2
 2
 
0
i  i ', j  j ', k  k '
i  i ', j  j ', k  k '
i  i ', j  j ', k , k '
i  i ', j  j ', k , k '
i  i ', j  j ', k , k '
2-Way Mixed Effects Model
Factor A Fixed, B Random (Restricted Model)
Yijk     i   j   ij   ijk
i  1,..., a
  are independent, normal, with mean 0, and Variance: 
a
 i  are fixed constants with   i  0
j
i 1
 ij 
 a 1 2 
~ N  0,
  
a


 ijk ~ N  0,  2  independent

a
s.t.
j  1,..., b k  1,..., n
  
i 1
ij


1 2
 0 and   ij ,  i ' j    
a
  ,    ,   pairwise independent
j
ijk
ij

E Yijk   E    i   j   ij   ijk     i


 2 Yijk    2    i   j   ij   ijk   2 
 2 a 1 2
2
   a    

 2  a  1  2

 
a
 Yijk , Yi ' j ' k '   
1 2
 2   
a

0


a 1 2
    2
a
i  i ', j  j ', k  k '
i  i ', j  j ', k  k '
i  i ', j  j ', k , k '
j  j ', i, i ', k , k '
Note: Unrestricted Model: Yijk     i   *j   ij   ijk
*
 
 j   *j  
*
j
 ij   ij    j
*
*
 ij ~ N  0,  2 
*
independent
2

Expected Mean Squares for 2-Way ANOVA
Mean Square
df
Fixed Model
Random Model
Mixed Model
(A Fixed, B Fixed) (A Random, B Random) (A Fixed, B Random)
a
MSA
a -1
 2+
bn 
i 1
a
2
i
2
 2  n 
 bn 2
a 1
2
 2  n 

b
MSB
b -1
 2+
an  j2
j 1
b 1
a
MSAB
MSE
 a  1 b  1
ab  n  1
 2+
2
 2  n 
 an 2
b
n  ij
i 1 j 1
2
 a  1 b  1
2
 2  an 2
2
 2  n 
2
2
 2  n 
2
bn  i2
i 1
a 1
Tests for Main Effects and Interactions
Tested Effects
Factor A
Fixed Effects
H 0 : 1  ...   a  0
Test Statistic
FA  MSA / MSE
Critical Value
F 1   ; a  1, ab  n  1 
Factor B
H 0 :  2  0
FB  MSB / MSE
Critical Value
F 1   ; b  1, ab  n  1 
Mixed Effects (A  Fixed)
H 0 : 1  ...   a  0
FA  MSA / MSAB
FA  MSA / MSAB
F 1   ; a  1,  a  1 b  1 
F 1   ; a  1,  a  1 b  1 
H 0 : 1  ...  b  0
Test Statistic
Interaction AB
Random Effects
H 0 :  2  0
FB  MSB / MSAB
F 1   ; b  1,  a  1 b  1 
H 0 :  11  ...   ab  0
Test Statistic
FAB  MSAB / MSE
Critical Value
F 1   ;  a  1 b  1 , ab  n  1 
2
H 0 :  
0
FAB  MSAB / MSE
H 0 :  2  0
FB  MSB / MSE
F 1   ; b  1, ab  n  1 
2
H 0 :  
0
FAB  MSAB / MSE
F 1   ;  a  1 b  1 , ab  n  1  F 1   ;  a  1 b  1 , ab  n  1 
Estimating Variance Components
Case 1: Random Effects Models:
E MSE   2

2
E MSAB   2  n 
s 2  MSE

2
s

2
E MSA   2  n 
 bn 2

2
E MSB   2  n 
 an 2

MSAB  MSE
n
MSA  MSAB
s2 
bn
MSB  MSAB
s2 
an
Case 1: Mixed Effects Models (B Random):
E MSE   2

2
E MSAB   2  n 
E MSB   2  an 2
s 2  MSE


MSAB  MSE
n
MSB  MSE
s2 
an
2
s

Estimating Fixed Effects in Mixed Model
 i  Effect of i th Level of Factor A (Fixed Effect)
^
 i  Y i  Y 

^
 i 
2
2
 2  n 
bn
E MSAB

bn
a
a
Contrast among Levels of Factor A: L   ci i
^
a
^
L   ci  i
i 1

2

a
^

^
L  c  i
i 1
a
2
i
1   100% CI for L   ci i :
i 1
2
i 1

^
s i 
2
s.t.

c
i 1
i
0
MSAB a 2
s L 
 ci
bn i 1
2
MSAB
bn
^
^

^
L  t 1   / 2  ;  a  1 b  1  s L
Multiple Comparison Procedures (all pairs of levels of Factor A):
Tukey: HSD  q 1   ; a,  a  1 b  1 
MSAB
bn
Bonferroni: MSD  t 1   /  2C A   ;  a  1 b  1 
2 MSAB
bn
 a  a  a  1
CA    
2
 2
Estimating Marginal (Factor A) Means in
Mixed Models
^
 i  Y i 
2
 
^
i
 
^
a 1
1
a 1
1
2

E MSAB 
E MSB s  i 
MSAB 
MSB
nab
nab
nab
nab
Using Satterthwaite's Approximation:
1
 a 1

MSAB

MSB


nab
nab


df 
2
2
 a 1
  1

MSAB  
MSB 

nab
nab

 

b 1
 a  1 b  1
2
^
 
^
Approximate 1   100% CI for i :  i  t 1   / 2  ; df  s  i