Example 1 (Completely randomized design) Notes | EduRev

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: Example 1 (Completely randomized design) Notes | EduRev

 Page 1


Analysis of Variance and Analysis of Variance and Analysis of Variance and Analysis of Variance and 
Design of Experiment Design of Experiment- -I I
MODULE MODULE – –IX IX 
gp gp
ANALYSIS OF ANALYSIS OF
LECTURE  LECTURE  - - 36 36 
ANALYSIS OF ANALYSIS OF 
NONORTHOGONAL NONORTHOGONAL DATA DATA
Dr. Shalabh
Department of Mathematics and Statistics
Indian Institute of Technology Kanpur
Page 2


Analysis of Variance and Analysis of Variance and Analysis of Variance and Analysis of Variance and 
Design of Experiment Design of Experiment- -I I
MODULE MODULE – –IX IX 
gp gp
ANALYSIS OF ANALYSIS OF
LECTURE  LECTURE  - - 36 36 
ANALYSIS OF ANALYSIS OF 
NONORTHOGONAL NONORTHOGONAL DATA DATA
Dr. Shalabh
Department of Mathematics and Statistics
Indian Institute of Technology Kanpur
Orthogonal data
2
g
The concept of orthogonality of data is associated with two or higher way classified data. Consider the set up of two-way 
classified data. 
Let A and B be two factors at p and q levels respectively.
Let  n
ij
: number of observations  in (i, j)
th
cell.  
Let  y
ijk
= k
th
observation in cell, 1, 2,..., ; 1, 2,..., ; 1, 2,..., ,
ij
ipj qk n = ==
: marginal total correspondence to level of        
th
i ijk
jk
Ay i A =
??
`
:
:
th
where cell total
marginal total correspondence to level of
        
   
        
ij
i
ij ijk
k
j ijk
T
Ty
By j B
=
=
=
?
?
??
.
:
Let
Gra
        
        
jj
ik
ij
j
i j
T
ABG
=
==
??
?
, nd total
?? j
,,
ˆ
:,
i
marginal mean of
        
        
j
ij io ij oj io oj
ji j j
i
i
ij
nn nn n n n
A
tA
n
== = =
=
??
?? ? ?
?
ˆ
:. marginal mean of        
j
j
j
ij
j
B
bB
n
=
?
Page 3


Analysis of Variance and Analysis of Variance and Analysis of Variance and Analysis of Variance and 
Design of Experiment Design of Experiment- -I I
MODULE MODULE – –IX IX 
gp gp
ANALYSIS OF ANALYSIS OF
LECTURE  LECTURE  - - 36 36 
ANALYSIS OF ANALYSIS OF 
NONORTHOGONAL NONORTHOGONAL DATA DATA
Dr. Shalabh
Department of Mathematics and Statistics
Indian Institute of Technology Kanpur
Orthogonal data
2
g
The concept of orthogonality of data is associated with two or higher way classified data. Consider the set up of two-way 
classified data. 
Let A and B be two factors at p and q levels respectively.
Let  n
ij
: number of observations  in (i, j)
th
cell.  
Let  y
ijk
= k
th
observation in cell, 1, 2,..., ; 1, 2,..., ; 1, 2,..., ,
ij
ipj qk n = ==
: marginal total correspondence to level of        
th
i ijk
jk
Ay i A =
??
`
:
:
th
where cell total
marginal total correspondence to level of
        
   
        
ij
i
ij ijk
k
j ijk
T
Ty
By j B
=
=
=
?
?
??
.
:
Let
Gra
        
        
jj
ik
ij
j
i j
T
ABG
=
==
??
?
, nd total
?? j
,,
ˆ
:,
i
marginal mean of
        
        
j
ij io ij oj io oj
ji j j
i
i
ij
nn nn n n n
A
tA
n
== = =
=
??
?? ? ?
?
ˆ
:. marginal mean of        
j
j
j
ij
j
B
bB
n
=
?
3
If  any contrast  of marginal means of A is orthogonal to any contrast of the other marginal means, then  the 
data are called orthogonal other ise non orthogonal
ˆ
ii
i
lt
?
ˆ
j j
j
mb
?
data are called orthogonal, otherwise non-orthogonal.
If each cell has constant number of observations then the data are orthogonal as shown below.
Note that  in this case  
,  .
qq
io ijj
nn nnqnnp == = =
??
The definition extends to higher classification if we treat the marginal means of every pair of factors of classification.
jj
jj
1
1
ˆ
(... ).
ii
i
ii i i iq
ii
lA
Llt lT T
qn qn
== = ++
?
??
Similarly
1
1
ˆ
(... ).
jj
j
jjjjpj
jj
n
mB
M mb m T T
Bpn
== = ++
?
??
The sum of products of the coefficients of identical observations in the two contrasts is 
2
11
0.
ij i j
ij i j
lmn l m
pqn pqn
??
??
==
??
??
??
??
?? ? ?
So the data are orthogonal as the two contrasts are orthogonal.
When cell frequencies are proportional, i.e.,
( , ' 1, 2,..., ; 1,..., )
ij j
nC
jjri p
nC
== =
then also the data are orthogonal.
The same is true for higher order classification if the  number of observations  in the ultimate cell is constant.
'' ij j
nC
Page 4


Analysis of Variance and Analysis of Variance and Analysis of Variance and Analysis of Variance and 
Design of Experiment Design of Experiment- -I I
MODULE MODULE – –IX IX 
gp gp
ANALYSIS OF ANALYSIS OF
LECTURE  LECTURE  - - 36 36 
ANALYSIS OF ANALYSIS OF 
NONORTHOGONAL NONORTHOGONAL DATA DATA
Dr. Shalabh
Department of Mathematics and Statistics
Indian Institute of Technology Kanpur
Orthogonal data
2
g
The concept of orthogonality of data is associated with two or higher way classified data. Consider the set up of two-way 
classified data. 
Let A and B be two factors at p and q levels respectively.
Let  n
ij
: number of observations  in (i, j)
th
cell.  
Let  y
ijk
= k
th
observation in cell, 1, 2,..., ; 1, 2,..., ; 1, 2,..., ,
ij
ipj qk n = ==
: marginal total correspondence to level of        
th
i ijk
jk
Ay i A =
??
`
:
:
th
where cell total
marginal total correspondence to level of
        
   
        
ij
i
ij ijk
k
j ijk
T
Ty
By j B
=
=
=
?
?
??
.
:
Let
Gra
        
        
jj
ik
ij
j
i j
T
ABG
=
==
??
?
, nd total
?? j
,,
ˆ
:,
i
marginal mean of
        
        
j
ij io ij oj io oj
ji j j
i
i
ij
nn nn n n n
A
tA
n
== = =
=
??
?? ? ?
?
ˆ
:. marginal mean of        
j
j
j
ij
j
B
bB
n
=
?
3
If  any contrast  of marginal means of A is orthogonal to any contrast of the other marginal means, then  the 
data are called orthogonal other ise non orthogonal
ˆ
ii
i
lt
?
ˆ
j j
j
mb
?
data are called orthogonal, otherwise non-orthogonal.
If each cell has constant number of observations then the data are orthogonal as shown below.
Note that  in this case  
,  .
qq
io ijj
nn nnqnnp == = =
??
The definition extends to higher classification if we treat the marginal means of every pair of factors of classification.
jj
jj
1
1
ˆ
(... ).
ii
i
ii i i iq
ii
lA
Llt lT T
qn qn
== = ++
?
??
Similarly
1
1
ˆ
(... ).
jj
j
jjjjpj
jj
n
mB
M mb m T T
Bpn
== = ++
?
??
The sum of products of the coefficients of identical observations in the two contrasts is 
2
11
0.
ij i j
ij i j
lmn l m
pqn pqn
??
??
==
??
??
??
??
?? ? ?
So the data are orthogonal as the two contrasts are orthogonal.
When cell frequencies are proportional, i.e.,
( , ' 1, 2,..., ; 1,..., )
ij j
nC
jjri p
nC
== =
then also the data are orthogonal.
The same is true for higher order classification if the  number of observations  in the ultimate cell is constant.
'' ij j
nC
Analysis of non-orthogonal two way data
4
When the data are non-orthogonal, then the analysis is no longer simple as the straight solution of normal equations is not 
available. Consider the following usual two way model:
1,...,; 1,....,; 1,..., .
iik i j ijk ij
yiIjJkn µ aß e =+ + + = = =
2
'(0) Assume that are identically and independently distribution as N
2
'(0, ).
ijk
  Assume that are identically and independently distribution as s N e s
Using the least squares  principal, we minimize the sum of squares due to error as
22
()
ˆ
ˆ ˆ0(1)     
ijk ijk i j
ij k i j k
oo io i oj j ijk
i j i j k
Ey
E
nn n yG
eµaß
µa ß
µ
== ---
?
=? + + = =
?
??? ???
?? ???
1
ˆ
ˆ ˆ 0 ( 1, 2,..., ) (2)
ˆ
ˆ 0 ( 1, 2,..., ). (3)
    
      
jj
io io i ij j ijk i
jjk
oj i oj j ijk j
k
j
E
nn n y Ai I
E
nn y B j J
µa ß
a
aß
ß
?
=? + + = = =
?
?
=? + = = =
?
???
i
??
k
j
ß ?
(3),
ˆ
ˆ
ˆ(). From we obtain we use instead of to avoid confusion
j
i
joj
jmm
m
oj oj
n
nm i
nn
ßµ
ßa =- -
?
Page 5


Analysis of Variance and Analysis of Variance and Analysis of Variance and Analysis of Variance and 
Design of Experiment Design of Experiment- -I I
MODULE MODULE – –IX IX 
gp gp
ANALYSIS OF ANALYSIS OF
LECTURE  LECTURE  - - 36 36 
ANALYSIS OF ANALYSIS OF 
NONORTHOGONAL NONORTHOGONAL DATA DATA
Dr. Shalabh
Department of Mathematics and Statistics
Indian Institute of Technology Kanpur
Orthogonal data
2
g
The concept of orthogonality of data is associated with two or higher way classified data. Consider the set up of two-way 
classified data. 
Let A and B be two factors at p and q levels respectively.
Let  n
ij
: number of observations  in (i, j)
th
cell.  
Let  y
ijk
= k
th
observation in cell, 1, 2,..., ; 1, 2,..., ; 1, 2,..., ,
ij
ipj qk n = ==
: marginal total correspondence to level of        
th
i ijk
jk
Ay i A =
??
`
:
:
th
where cell total
marginal total correspondence to level of
        
   
        
ij
i
ij ijk
k
j ijk
T
Ty
By j B
=
=
=
?
?
??
.
:
Let
Gra
        
        
jj
ik
ij
j
i j
T
ABG
=
==
??
?
, nd total
?? j
,,
ˆ
:,
i
marginal mean of
        
        
j
ij io ij oj io oj
ji j j
i
i
ij
nn nn n n n
A
tA
n
== = =
=
??
?? ? ?
?
ˆ
:. marginal mean of        
j
j
j
ij
j
B
bB
n
=
?
3
If  any contrast  of marginal means of A is orthogonal to any contrast of the other marginal means, then  the 
data are called orthogonal other ise non orthogonal
ˆ
ii
i
lt
?
ˆ
j j
j
mb
?
data are called orthogonal, otherwise non-orthogonal.
If each cell has constant number of observations then the data are orthogonal as shown below.
Note that  in this case  
,  .
qq
io ijj
nn nnqnnp == = =
??
The definition extends to higher classification if we treat the marginal means of every pair of factors of classification.
jj
jj
1
1
ˆ
(... ).
ii
i
ii i i iq
ii
lA
Llt lT T
qn qn
== = ++
?
??
Similarly
1
1
ˆ
(... ).
jj
j
jjjjpj
jj
n
mB
M mb m T T
Bpn
== = ++
?
??
The sum of products of the coefficients of identical observations in the two contrasts is 
2
11
0.
ij i j
ij i j
lmn l m
pqn pqn
??
??
==
??
??
??
??
?? ? ?
So the data are orthogonal as the two contrasts are orthogonal.
When cell frequencies are proportional, i.e.,
( , ' 1, 2,..., ; 1,..., )
ij j
nC
jjri p
nC
== =
then also the data are orthogonal.
The same is true for higher order classification if the  number of observations  in the ultimate cell is constant.
'' ij j
nC
Analysis of non-orthogonal two way data
4
When the data are non-orthogonal, then the analysis is no longer simple as the straight solution of normal equations is not 
available. Consider the following usual two way model:
1,...,; 1,....,; 1,..., .
iik i j ijk ij
yiIjJkn µ aß e =+ + + = = =
2
'(0) Assume that are identically and independently distribution as N
2
'(0, ).
ijk
  Assume that are identically and independently distribution as s N e s
Using the least squares  principal, we minimize the sum of squares due to error as
22
()
ˆ
ˆ ˆ0(1)     
ijk ijk i j
ij k i j k
oo io i oj j ijk
i j i j k
Ey
E
nn n yG
eµaß
µa ß
µ
== ---
?
=? + + = =
?
??? ???
?? ???
1
ˆ
ˆ ˆ 0 ( 1, 2,..., ) (2)
ˆ
ˆ 0 ( 1, 2,..., ). (3)
    
      
jj
io io i ij j ijk i
jjk
oj i oj j ijk j
k
j
E
nn n y Ai I
E
nn y B j J
µa ß
a
aß
ß
?
=? + + = = =
?
?
=? + = = =
?
???
i
??
k
j
ß ?
(3),
ˆ
ˆ
ˆ(). From we obtain we use instead of to avoid confusion
j
i
joj
jmm
m
oj oj
n
nm i
nn
ßµ
ßa =- -
?
5
1
ˆ
ˆˆ ˆˆ (2), Put in
j
ji ii ij j i
B
nn n n A ßµa µ a
??
++ -- =
??
??
(2),
1
ˆˆ ˆˆ
Put in
or
      
jio ioiij mj m i
jm
oj oj
j
iij io ioi ij ij mjm
jjj
oj oj
nn n n A
nn
B
An n n n n n
nn
ßµa µ a
µa µ a
++
??
??
??
?? ? ?
-=+ - -
?? ? ?
?? ? ?
?? ? ?
??
????
2
1
ˆˆ
1
ˆˆ .
jj
io i ij mj m
jm
oj
ij
ii ij mj
j
jj
nn n
n
n
nnn
nn
aa
aa
?? ? ?
=-
??
=- -
??
??
??
??
?
2
ˆ
ij
mi
jm j
j
n
n
a
??
-
??
??
??
?? ?
or
j
oj oj
nn
??
jm j
oj
n
??
2
ˆˆ
j ij ij mj
iij iio m
jjmij
oj oj oj
B nnn
An n
nn n
aa
?
??? ? ? ?
-= - -
??? ? ? ?
??? ? ? ?
??? ? ? ?
????
iii im
QC C
?? ?
or
?
These are referred to as the  reduced normal equations.
ˆˆ ( 1, 2,..., ). (4)
iiii imm
mi
QC C i I a a
?
=+ =
?
is called the adjusted treatment total of i
th
level of A.
i
Q
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