Chapter 9
Confounding
If the number of factors or levels increase in a factorial experiment, then the number of treatment
combinations increases rapidly. When the number of treatment combinations is large, then it may
be difficult to get the blocks of sufficiently large size to accommodate all the treatment
combinations. Under such situations, one may use either connected incomplete block designs, e.g.,
balanced incomplete block designs (BIBD) where all the main effects and interaction contrasts can
be estimated or use unconnected designs where not all these contrasts can be estimated.
Non-estimable contrasts are said to be confounded.
Note that a linear function ' is said to be estimable if there exist a linear function l ' y of the
observations on random variable y such that E (l ' y ) ' . Now there arise two questions.
Firstly, what does confounding means and secondly, how does it compares to using BIBD.
In order to understand the confounding, let us consider a simple example of 2 2 factorial with
factors a and b . The four treatment combinations are (1), a, b and ab . Suppose each batch of
raw material to be used in the experiment is enough only for two treatment combinations to be
tested. So two batches of raw material are required. Thus two out of four treatment combinations
must be assigned to each block. Suppose this 2 2 factorial experiment is being conducted in a
randomized block design. Then the corresponding model is
E ( yij ) i j ,
then
1
A ab a b (1) ,
2r
1
B ab b a (1) ,
2r
1
AB ab (1) a b .
2r
Analysis of Variance | Chapter 9 | Confounding | Shalabh, IIT Kanpur
1
, Suppose the following block arrangement is opted:
Block 1 Block 2
(1) a
ab b
The block effects of blocks 1 and 2 are 1 and 2 , respectively, then the average responses
corresponding to treatment combinations a, b, ab and (1) are
E y (a ) 2 (a),
E y (b) 2 (b),
E y (ab) 1 (ab),
E y (1) 1 (1),
respectively. Here y ( a ), y (b), y ( ab), y (1) and ( a ), (b), ( ab), (1) denote the responses and
treatments corresponding to a, b, ab and (1), respectively. Ignoring the factor 1/ 2r in A, B, AB
and using E[ y ( a )], E[ y (b)], E[ y ( ab)], E ( y (1)] , the effect A is expressible as follows :
A [ 1 (ab)] [ 2 (a)] [ 2 (b)] [ 1 (1)]
(ab) (a) (b) (1).
So the block effect is not present in A and it is not mixed up with the treatment effects. In this case,
we say that the main effect A is not confounded. Similarly, for the main effect B, we have
B [ 1 (ab)] [ 2 (b)] [ 2 (a)] [ 1 (1)]
(ab) (b) (a ) (1).
So there is no block effect present in B and thus B is not confounded. For the interaction effect
AB , we have
AB [ 1 (ab)] [ 1 (1)] [ 2 (a)] [ 2 (b)]
2( 1 2 ) (ab) (1) (a) (b).
Here the block effects are present in AB. In fact, the block effects are 1 and 2 are mixed up with
the treatment effects and cannot be separated individually from the treatment effects in AB . So
AB is said to be confounded (or mixed up) with the blocks.
Analysis of Variance | Chapter 9 | Confounding | Shalabh, IIT Kanpur
2
Confounding
If the number of factors or levels increase in a factorial experiment, then the number of treatment
combinations increases rapidly. When the number of treatment combinations is large, then it may
be difficult to get the blocks of sufficiently large size to accommodate all the treatment
combinations. Under such situations, one may use either connected incomplete block designs, e.g.,
balanced incomplete block designs (BIBD) where all the main effects and interaction contrasts can
be estimated or use unconnected designs where not all these contrasts can be estimated.
Non-estimable contrasts are said to be confounded.
Note that a linear function ' is said to be estimable if there exist a linear function l ' y of the
observations on random variable y such that E (l ' y ) ' . Now there arise two questions.
Firstly, what does confounding means and secondly, how does it compares to using BIBD.
In order to understand the confounding, let us consider a simple example of 2 2 factorial with
factors a and b . The four treatment combinations are (1), a, b and ab . Suppose each batch of
raw material to be used in the experiment is enough only for two treatment combinations to be
tested. So two batches of raw material are required. Thus two out of four treatment combinations
must be assigned to each block. Suppose this 2 2 factorial experiment is being conducted in a
randomized block design. Then the corresponding model is
E ( yij ) i j ,
then
1
A ab a b (1) ,
2r
1
B ab b a (1) ,
2r
1
AB ab (1) a b .
2r
Analysis of Variance | Chapter 9 | Confounding | Shalabh, IIT Kanpur
1
, Suppose the following block arrangement is opted:
Block 1 Block 2
(1) a
ab b
The block effects of blocks 1 and 2 are 1 and 2 , respectively, then the average responses
corresponding to treatment combinations a, b, ab and (1) are
E y (a ) 2 (a),
E y (b) 2 (b),
E y (ab) 1 (ab),
E y (1) 1 (1),
respectively. Here y ( a ), y (b), y ( ab), y (1) and ( a ), (b), ( ab), (1) denote the responses and
treatments corresponding to a, b, ab and (1), respectively. Ignoring the factor 1/ 2r in A, B, AB
and using E[ y ( a )], E[ y (b)], E[ y ( ab)], E ( y (1)] , the effect A is expressible as follows :
A [ 1 (ab)] [ 2 (a)] [ 2 (b)] [ 1 (1)]
(ab) (a) (b) (1).
So the block effect is not present in A and it is not mixed up with the treatment effects. In this case,
we say that the main effect A is not confounded. Similarly, for the main effect B, we have
B [ 1 (ab)] [ 2 (b)] [ 2 (a)] [ 1 (1)]
(ab) (b) (a ) (1).
So there is no block effect present in B and thus B is not confounded. For the interaction effect
AB , we have
AB [ 1 (ab)] [ 1 (1)] [ 2 (a)] [ 2 (b)]
2( 1 2 ) (ab) (1) (a) (b).
Here the block effects are present in AB. In fact, the block effects are 1 and 2 are mixed up with
the treatment effects and cannot be separated individually from the treatment effects in AB . So
AB is said to be confounded (or mixed up) with the blocks.
Analysis of Variance | Chapter 9 | Confounding | Shalabh, IIT Kanpur
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