Chapter 10
Partial Confounding
The objective of confounding is to mix the less important treatment combinations with the block effect
differences so that higher accuracy can be provided to the other important treatment comparisons. When
such mixing of treatment contrasts and block differences is done in all the replicates, then it is termed as
total confounding. On the other hand, when the treatment contrast is not confounded in all the replicates
but only in some of the replicates, then it is said to be partially confounded with the blocks. It is also
possible that one treatment combination is confounded in some of the replicates and another treatment
combination is confounded in other replicates which are different from the earlier replicates. So the
treatment combinations confounded in some of the replicates and unconfounded in other replicates. So
the treatment combinations are said to be partially confounded. The partially confounded contrasts are
estimated only from those replicates where it is not confounded. Since the variance of the contrast
estimator is inversely proportional to the number of replicates in which they are estimable, so some
factors on which information is available from all the replicates are more accurately determined.
Balanced and unbalanced partially confounded design
If all the effects of a certain order are confounded with incomplete block differences in equal number of
replicates in a design, then the design is said to be balanced partially confounded design. If all the
effects of a certain order are confounded an unequal number of times in a design, then the design is said
to be unbalanced partially confounded design.
We discuss only the analysis of variance in the case of balanced partially confounded design through
examples on 22 and 23 factorial experiments.
Example 1:
Consider the case of 22 factorial as in following table in the set up of a randomized block design.
Factorial effects Treatment combinations Divisor
(1) (a) (b) (ab)
M + + + + 4
A - + - + 2
B - - + + 2
AB + - - + 2
Analysis of Variance | Chapter 10 | Partial Confounding | Shalabh, IIT Kanpur
1
, where y*i ((1), a, b, ab) ' denotes the vector of total responses in the ith replication and each treatment is
replicated r times, i 1.2,..., r. If no factor is confounded then the factorial effects are estimated using
all the replicates as
1 r '
A A y*i ,
2r i 1
1 r
B 'B y*i ,
2r i 1
1 r
AB 'AB y*i ,
2r i 1
where the vectors of contrasts A , B , AB are given by
A (1 1 1 1) '
B (1 1 1 1) '
AB (1 1 1 1) '.
We have in this case
'A A 'B B 'AB AB 4.
The sum of squares due to A, B and AB can be accordingly modified and expressed as
r
( 'A y*i ) 2
(ab a b (1)) 2
SS A i 1
r 'A A 4r
r
( 'B y*i ) 2
(ab b a (1)) 2
SS B i 1
r 'B B 4r
and
r
( 'AB y*i ) 2
(ab (1) a b) 2
S AB i 1
,
r 'AB AB 4r
respectively.
Now consider a situation with 3 replicates with each consisting of 2 incomplete blocks as in the following
figure:
Analysis of Variance | Chapter 10 | Partial Confounding | Shalabh, IIT Kanpur
2
Partial Confounding
The objective of confounding is to mix the less important treatment combinations with the block effect
differences so that higher accuracy can be provided to the other important treatment comparisons. When
such mixing of treatment contrasts and block differences is done in all the replicates, then it is termed as
total confounding. On the other hand, when the treatment contrast is not confounded in all the replicates
but only in some of the replicates, then it is said to be partially confounded with the blocks. It is also
possible that one treatment combination is confounded in some of the replicates and another treatment
combination is confounded in other replicates which are different from the earlier replicates. So the
treatment combinations confounded in some of the replicates and unconfounded in other replicates. So
the treatment combinations are said to be partially confounded. The partially confounded contrasts are
estimated only from those replicates where it is not confounded. Since the variance of the contrast
estimator is inversely proportional to the number of replicates in which they are estimable, so some
factors on which information is available from all the replicates are more accurately determined.
Balanced and unbalanced partially confounded design
If all the effects of a certain order are confounded with incomplete block differences in equal number of
replicates in a design, then the design is said to be balanced partially confounded design. If all the
effects of a certain order are confounded an unequal number of times in a design, then the design is said
to be unbalanced partially confounded design.
We discuss only the analysis of variance in the case of balanced partially confounded design through
examples on 22 and 23 factorial experiments.
Example 1:
Consider the case of 22 factorial as in following table in the set up of a randomized block design.
Factorial effects Treatment combinations Divisor
(1) (a) (b) (ab)
M + + + + 4
A - + - + 2
B - - + + 2
AB + - - + 2
Analysis of Variance | Chapter 10 | Partial Confounding | Shalabh, IIT Kanpur
1
, where y*i ((1), a, b, ab) ' denotes the vector of total responses in the ith replication and each treatment is
replicated r times, i 1.2,..., r. If no factor is confounded then the factorial effects are estimated using
all the replicates as
1 r '
A A y*i ,
2r i 1
1 r
B 'B y*i ,
2r i 1
1 r
AB 'AB y*i ,
2r i 1
where the vectors of contrasts A , B , AB are given by
A (1 1 1 1) '
B (1 1 1 1) '
AB (1 1 1 1) '.
We have in this case
'A A 'B B 'AB AB 4.
The sum of squares due to A, B and AB can be accordingly modified and expressed as
r
( 'A y*i ) 2
(ab a b (1)) 2
SS A i 1
r 'A A 4r
r
( 'B y*i ) 2
(ab b a (1)) 2
SS B i 1
r 'B B 4r
and
r
( 'AB y*i ) 2
(ab (1) a b) 2
S AB i 1
,
r 'AB AB 4r
respectively.
Now consider a situation with 3 replicates with each consisting of 2 incomplete blocks as in the following
figure:
Analysis of Variance | Chapter 10 | Partial Confounding | Shalabh, IIT Kanpur
2
