Two Factor ANOVA with 1 Unit
per Treatment
KNNL – Chapter 20
One Observation per Cell of 2-Way Table
• When n=1, the overall sample size is nT = ab(1)=ab
• For the interaction model, each observation is equal
to its fitted value (each observation is its cell mean)
• No problems when the additive (no interaction)
model is fit
• The “error sum of squares” is equivalent to
interaction sum of squares for the additive model
• We can test whether there is an interaction only if it
takes on some specific form such as: (ab)ij = Daibj
• We can test whether D=0 based on Tukey’s One
Degree of Freedom for Non-Additivity Test (ODOFNA)
Additive Model
Model: Yij    a i  b j   ij
^
^
^
  Y  a i  Y i  Y 
^
^
^
i  1,..., a; j  1,..., b
b j  Y  j  Y 

^
 

 Y ij    a i  b j  Y   Y i  Y   Y  j  Y   Y i  Y  j  Y 
2
 
^


   Yij  Y ij    Yij  Y i  Y  j  Y 

i 1 j 1 
i 1 j 1
 Error Sum of Squares is SSAB
a
b
a
b
    Y
 Y i  Y  j  Y 
df
MS
2
a
b
i 1 j 1
ij

2
 SSAB
Analysis of Variance:
Source
SS
E{MS}
a
a
A

SSA  b Y i  Y 
i 1

2
a 1
MSA 
SSA
a 1
b a i2
2 
i 1
a 1
b
b
B
j 1
a
Error

SSB  a  Y  j  Y 
b

SSAB   Yij  Y i  Y  j  Y 
i 1 j 1
a
Total

2
b

SSTO   Yij  Y 
i 1 j 1


SSB
b 1
b 1
MSB 
(a  1)(b  1)
MSAB 
2
SSAB
(a  1)(b  1)
2 
a  b j2
j 1
b 1
2
2
ab -1
MSA
RR : FA*  F 1  a ; a  1, (a  1)(b  1) 
MSAB
MSB
H 0B : b1  ...  bb  0 TS : FB* 
RR : FB*  F 1  a ; b  1, (a  1)(b  1) 
MSAB
Testing for Main Effects: H 0A : a1  ...  a a  0 TS : FA* 
Tukey’s Test for Additivity
Assumed form of interaction:
ab ij  Da i b j
where D is a constant
 Yij    a i  b j  Da i b j   ij
R eplacing  , a i , b j with their estimates:
^
^
^
  Y  a i  Y i  Y 

b j  Y  j  Y 
 




 Yij  Y   Y i  Y   Y  j  Y   Yij  Y i  Y  j  Y   D Y i  Y  Y  j  Y    ij
Yij*  DX ij*   ij

where: Yij*  Yij  Y i  Y  j  Y 
a
^
Fitting a regression through the origin: D 

X ij*  Y i  Y  Y  j  Y 
b
 X ij*Yij*
i 1 j 1
a
b
  X
i 1 j 1

* 2
ij
 Y
a

b
i
i 1 j 1
a




 Y  Y  j  Y  Yij
Y i  Y 
i 1
 
2
b
Y  j  Y 
j 1

2
Sum of Squares for Interaction:
2
2
a
b
 ^ 
^
SSAB   ab ij     D  Y i  Y 


i 1 j 1 
i 1 j 1 
a
b
*

 Y
2
j
 Y 

2
 a b

Y

Y
Y

Y
Y
i



j

 
ij 
i 1 j 1


 a
2 b
2
 Y i  Y   Y  j  Y 

i 1


 
j 1

2

The "Remainder" Sum of Squares is: SS Re m*  SSTO  SSA  SSB  SSAB*
To test for no interaction (of this form): H 0 : D  0 H A : D  0
 SSAB* /1
Test Statistic: F 
 SS Re m*  (a  1)(b  1)  1 
*
Rejection Region: F *  F 1  a ;1, (a  1)(b  1)  1