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Chapter 7: Blocking and Confounding in the 2 k Factorial Design

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Chapter 7: Blocking and Confounding in the 2k Factorial Design

Dr. Mohammed Alsayed

- There are many situations which is impossible to perform all of the runs in a 2k factorial experiment under homogeneous conditions.
- A single batch of raw material may be not large enough to make all of the required runs.
- In other cases, it might be desirable to deliberately vary the experimental conditions to ensure that the treatments are equally effective (robust) across many situations that are likely to be encountered in practice.
- For example, a chemical engineer may run a pilot plant experiment with several batches of raw material because he knows that different raw material batches of different quality grades are likely to be used in the actual full-scale process.
- The design technique used in these situations is blocking.

- Blocking is a technique for dealing with controllable nuisance variables.
- Suppose that the 2k factorial design has been replicated n times. This is identical to the situation discussed in Chapter 5.
- If there are n replicates, then each set of nonhomogeneous conditions defines in blocks, and each replicate is run in one of the blocks.
- Runs within the block are randomized

- Consider the chemical process experiment first described in section 6-2.
- It was an investigation into the effect of the concentration of the reactant and the amount of the catalyst on the conversion (yield) in a chemical process.
- Reactant concentration was factor A, with two levels of interest (15% and 25%). The catalyst was factor B, with the high level denoting the use of 2 pounds of the catalyst and the low level denoting the use of only 1 pound. The experiment is replicated three times.

- Suppose that only four experimental trials can be made from a single batch of raw material.
- Therefore, three batches of raw material will be required to run the three replications of the design

- Blocks has relatively small effect.

- There are many problems for which it is impossible to perform a complete replicate of a factorial design in one block.
- Confounding is a design technique for arranging a complete factorial experiment in blocks, where block size is smaller than the number of treatment combinations in one replicate.
- The technique causes information about certain treatment effects (usually high order interactions) to be indistinguishable from, or confounding with, blocks.
- This chapter focuses on confounding systems for the k factorial design.
- We consider the construction and analysis of the 2k factorial design in 2p incomplete blocks, where p < k.
- Consequently, these designs can be run in two blocks, four blocks, eight blocks, and so on.

- Suppose that we wish to run a single replicate of the 22 design (4 treatments).
- Each of the four treatments requires a quantity of raw material.
- For example, each batch of raw material is large enough for two treatment combinations to be tested.
- Two batches of raw material are required.
- If batch of raw material are considered as blocks, then we must assign two of the four treatment combinations to each block.

- The order in which the treatment combinations are run within a block is randomly determined.
- Also, which block to run first is randomly determined

- Suppose we want to estimate the main effects of A and B just as if no blocking had occurred.
- Note that both A and B are unaffected by blocking because in each estimate there is only one plus and one minus treatment combination from each block.
- Any difference between block 1 and block 2 will cancel out.

- Now consider the interaction AB:
- Because the two treatment combinations with plus sign [ab and (1)] are in block 1 and the two with the minus sign (a and b) are in block 2, the block effect and the AB interaction are identical. That is, AB is confounded with blocks.

- All treatment combinations that have a plus sign on AB are assigned to block 1, whereas all treatment combinations that have the minus sign are assigned to block 2.
- This approach can be used to confound any effect (A, B, or AB) with blocks.
- For example, if (1) and b had been assigned to block 1 and a and ab to block 2, the main effect A would have been confounded with blocks.
- The usual practice is to confound the highest order interaction with blocks.

- This scheme can be used to confound any 2k design in two blocks. As a second example, consider a 23 design run in two blocks.
- Suppose we wish to confound the three factor interaction ABC with blocks.
- From the table of plus and minus signs shown in table 7-4, we assign the treatment combinations that are minus on ABC to block1 and those that are plus on ABC to block 2.
- The resulting design is shown in figure 7-2.
- Once again, the treatment combinations within a block are run in random order.

- Other Methods for Constructing the Block
- The method uses linear combination
- Defining contrast:
- xi is the level of the i-th factor appearing in a particular treatment combination.
- i is the exponent appearing on the i-th factor in the effect to be confounded.
- Treatment combinations that produce the same value of L (mod 2) will be placed in the same block.

- Group:
- Principal block

Confounding the 2k Factorial Design in Two Blocks

- To illustrate the approach, consider a 23 design with ABC confounded with blocks. Here x1 corresponds to A, x2 to B, x3 to C, and α1=α2=α3=1. Thus, the defining contrast corresponding to ABC is
- The treatment combination (1) is written 000 in the (0,1) notation; therefore,
- Similarly, the treatment combination a is 100, yielding
- Thus, (1) and a would be run in different blocks.

Confounding the 2k Factorial Design in Two Blocks

- For the remaining treatment combinations, we have:
- Thus (1), ab, ac, and bc are run in block 1 and a, b, c, and abc are run in block 2. this is the same design to the one which was generated from table of plus and minus signs.

Confounding the 2k Factorial Design in Four Blocks

- It is possible to construct 2k factorial design confounded in four blocks of 2k-2 observations each.
- These designs are particularly useful in situations where the number of factors is moderately large, say k ≥ 4, and blocks sizes are relatively small.

Confounding the 2k Factorial Design in Four Blocks

- As an example, consider the 25 design. If each block will hold only eight runs, then four blocks must be used.
- The construction of this design is relatively straightforward.
- Select two effects to be confounded with blocks, say ADE and BCE.
- These effects have the two defining contrasts.

Confounding the 2k Factorial Design in Four Blocks

- Now, every treatment combination will yield a particular pair of values of L1 (mod 2) and L2 (mod 2), that is, either (L1, L2) = (0,0), (0,1), (1,0), or (1,1).
- Treatment combinations yielding the same values of (L1,L2) are assigned to the same block.

Confounding the 2k Factorial Design in Four Blocks

Confounding the 2k Factorial Design in Four Blocks

- With little reflection, we realize that another effect in addition to ADE and BCE must be confounded with blocks.
- Because there are four blocks with three degrees of freedom between them, and because ADE and BCE have only one degree of freedom each, clearly an additional effect with one degree of freedom must be confounded.
- This effect is the generalized interaction of ADE and BCE, which is defined as the product of ADE and BCE modulus 2, is also confounded with blocks.

Confounding the 2k Factorial Design in Four Blocks

- It is easy to verify this be referring to the table of plus and minus signs for the 25 design.

Confounding the 2k Factorial Design in Four Blocks

- We see that the product of two treatment combinations in the principal block yields another element of the principal block., that is
- This is called the group-theoretic property.
- To construct another block, select a treatment combination that is not in the principal block (eg., b) and multiply b by all the treatment combinations in the principal block. This yields
- And so forth, which will produce the eight treatment combinations in block 3.

Confounding the 2k Factorial Design in Four Blocks

- In practice, the principal block can be obtained from the defining contrasts and the group-theoretic property, and the remaining blocks can be determined from these treatment combinations be the this method.

Confounding the 2k Factorial Design in Four Blocks

- The general procedure for constructing a 2k design confounded in four blocks is to choose two effects to generate the blocks, automatically confounding a third effect that is the generalized interaction of the first two.
- Then, the design is constructed by using the two defining contrasts (L1, L2) and the group–theoretic properties of the principal block.
- In selecting the effects to be confounded with blocks, care must be exercised to obtain a design that does not confound effects that may be of interest.

Confounding the 2k Factorial Design in Four Blocks

- For example. In a 25 design we might choose to confound ABCDE and ABD, which automatically confounds CE, an effect that is probably of interest.
- A better choice is to confound ADE and BCE, which automatically confounds ABCD. It is preferable to sacrifice information on the three-factor interactions ADE and BCE instead of the two-factor interaction CE.