Confounding

Confounding is defined as the technique of reducing the size of a replication over a number of
blocks at the cost of losing some information on some effect which is not of much practical
importance.

In factorial experiments when the number of factors or the levels of factors are increased, the
number of treatment combinations increases rapidly. This would increase the block size. As
the block size increases it is difficult to ensure homogeneity within the blocks. As a result, the
experimental error is increased and the precision of treatment comparisons is reduced. Thus,
the advantage of factorial experiment is lost. In order to keep the advantages of the factorial
experiments and at the same time reduce the experimental error to a minimum, the technique
of confounding is adopted. By this technique, it is possible to keep block size small.

In confounded experiments, each replication is split up into a number of blocks. Then the entire
number of treatment combinations is divided into a number of sets in such a way that there are
as many sets as there are blocks. Each block is allotted one set of treatment combinations at
random. Within each block, the treatment combinations forming the set are allotted at random
to the plots.

Confounding are two types – total or complete confounding and partial confounding. When a
certain effect is confounded in all the replications, the system of confounding is known as
complete or total confounding. If a treatment effect is confounded in some replications and
unconfounded in other replications the system is known as partial confounding.

When a treatment effect is confounded the information on this treatment effect is lost.
Therefore, care should be taken to see that the main effects are not confounded. Only those
interactions which are of little importance should be confounded.

When any two interactions are confounded, their generalised interaction will also be
confounded.

A 2n experiment in 2k blocks per replicate

Suppose that in a 2n series, we want to have blocks of size (number of plots) 2k in a replication.
The number of blocks will be 2n/2k and the number of effects confounded will be 2n/2k – 1. E.g.
in case of 24 experiment, if block size (number of plots) is 4, there will be 24/22 = 4 number of
blocks, and 3 effects will be confounded.

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Confounding in 23 experiment

ABC = ¼ [(a - 1)(b - 1)(c - 1)] = ¼ [abc + a + b + c – ab – ac – bc - 1]

ABC confounded with blocks

Replicate Block I (1) ab ac bc

Block II a b c abc

Complete Confounding of (23,22):

Confound the same interaction in all the replications and so lose information on that from all
the replications.

ABC confounded with the blocks

Replicate Block I (1) ab ac bc

Block II a b c abc

The first two columns of the ANOVA table would be:

Sources of variations d.f.

Blocks 2r – 1
A 1
B 1
C 1
AB 1
AC 1
BC 1
Error 6(r - 1)
Total 8r – 1

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Partial Confounding of (23,22):

An interaction effect is confounded in one replication and not in others.

Replication I Block I (1) ab c abc

(AB confounded) Block II a b ac bc

Replication II Block I (1) ac b abc

(AC confounded) Block II a c ab bc

Replication III Block I (1) bc a abc

(BC confounded) Block II b c ab ac

Replication IV Block I (1) ab ac bc

(ABC confounded) Block II a b c abc

The first two columns of the ANOVA table would be (carried for r repetitions):

Sources of variations d.f.

Blocks 8r – 1
A 1
B 1
C 1
AB 1
AC 1
BC 1
ABC 1
Error 24r – 7
Total 32r – 1

Example 1: Make a complete confounding of (25, 23) design.

Here we have 5 factors A, B, C, D, and E each at two levels. The 32 treatment combinations
are:

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 (1), a, b, ab, c, ac, bc, abc, d, ad, bd,
abd, cd, acd, bcd, abcd, e, ae, be, abe, ce, ace,
bce, abce, de, ade, bde, abde, cde, acde, bcde, abcde

Here block size (no. of plots in a block) = 23 = 8

No. of blocks = 25/23 = 32/8 = 4

No. of effects confounded = 4 – 1 = 3 in one replication

Let, the effects ABC and ADE is confounded. Then, there generalised interaction BCDE will
automatically confounded.

Now, a complete confounding of the experiment with the effects ABC, ADE and BCDE in 4
blocks and in 3 replications is given as:

Replication 1
Block I (1) BC DE ABD ACD ABE ACE BCDE
Block II A ABC ADE BD CD BE CE ABCDE
Block III B C BDE AD ABCD AE ABCE CDE
Block IV D BCD E AB AC ABDE ACDE BCE

Replication 2
Block I ABC A CE ABCDE CD BE ADE BD
Block II ACD ABE ACE BCDE (1) BC DE ABD
Block III ABDE ACDE BCE E AB AC D BCD
Block IV AD ABCD AE ABCE CDE B C BDE

Replication 3
Block I CDE B C BDE AD ABCD AE ABCE
Block II CD BE ADE BD ABC A CE ABCDE
Block III BCE E AB AC D BCD ABDE ACDE
Block IV (1) BC DE ABD ACD ABE ACE BCDE

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Remark: First form the block with (1) called the key block. Place in this block all the treatment
combinations having even number of letters in common with ABC and ADE (and also with
BCDE). Zero is considered as even number. Multiply the key block by treatment combination
which have not appear in this block. Note that square of a letter is considered as letter with
power zero.