11 Blocking Factorial Designs

In a trial conducted using a \(2^3\) design it might be desirable to use the same batch of raw material to make all 8 runs. Suppose that batches of raw material were only large enough to make 4 runs. Then the concept of blocking could be used.

The following R code generates the design matrix for a \(2^3\) design.

Run 1 2 3 12 13 23 123
1 -1 -1 -1 1 1 1 -1
2 1 -1 -1 -1 -1 1 1
3 -1 1 -1 -1 1 -1 1
4 1 1 -1 1 -1 -1 -1
5 -1 -1 1 1 -1 -1 1
6 1 -1 1 -1 1 -1 -1
7 -1 1 1 -1 -1 1 -1
8 1 1 1 1 1 1 1

Suppose that we assign runs 1, 4, 6, 7 to block I which use the first batch of raw material and runs 2, 3, 5, 8 to block II which use the second batch of raw material. The design is blocked this way by placing all runs in which the 123 is minus in one block and all the other runs in which 123 is plus in the other block.

Any systematic differences between the two blocks of four runs will be eliminated from all the main effects and two factor interactions. What you gain is the elimination of systematic differences between blocks. But now the three factor interaction is confounded with any batch (block) difference. The ability to estimate the three factor interaction separately from the block effect is lost.

11.0.1 Effect hierarchy principle

  1. Lower-order effects are more likely to be important than higher-order effects.

  2. Effects of the same order are equally likely to be important.

This principle suggests that when resources are scare, priority should be given to the estimation of lower order effects. This is useful in screening experiments that have a large number of factors and relatively small number of runs.

One reason that many accept this principle is that higher order interactions are more difficult to interpret or justify physically. As a result investigators are less interested in estimating the magnitudes of these effects even when they are statistically significant.

Assigning a fraction of the \(2^k\) treatment combinations to each block results in an incomplete blocking scheme as in the case of the balanced incomplete block design. The difference is that the factorial structure of a \(2^k\) design allows a neater assignment of treatment combinations to blocks. The neater assignment is done by dividing the total combinations into various fractions and finding optimal assignments by exploiting combinatorial relationships.

11.0.2 Generation of Orthogonal Blocks

In the \(2^3\) example suppose that the block variable is given the identifying number 4.

Run 1 2 3 4 = 123
1 -1 -1 -1 -1
2 1 -1 -1 1
3 -1 1 -1 1
4 1 1 -1 -1
5 -1 -1 1 1
6 1 -1 1 -1
7 -1 1 1 -1
8 1 1 1 1

Then you could think of your experiment as containing four factors. The fourth factor will have the special property that it does not interact with other factors. If this new factor is introduced by having its levels coincide exactly with the plus and minus signs attributed to 123 then the blocking is said to be generated by the relationship 4 = 123. This idea can be used to derive more sophisticated blocking arrangements.

11.0.3 An example of how not to block

This example is from Box, Hunter, and Hunter (2005). Suppose we would like to arrange the \(2^3\) design into four blocks.

Run 1 2 3 4 = 123 5 = 23 45 = 1
1 -1 -1 -1 -1 1 -1
2 1 -1 -1 1 1 1
3 -1 1 -1 1 -1 -1
4 1 1 -1 -1 -1 1
5 -1 -1 1 1 -1 -1
6 1 -1 1 -1 -1 1
7 -1 1 1 -1 1 -1
8 1 1 1 1 1 1

Consider two block factors called 4 and 5. 4 is associated with the three factor interaction and, say, 5 is associated with a the two factor interaction 23 which was deemed unimportant by the investigator. Runs are placed in different blocks depending on the signs of the block variables in columns 4 and 5. Runs for which the signs of 4 and 5 are – would go in one block, -+ in a second block, the +- in a third block, and the ++ runs in the fourth.

Block Run
I 4,6
II 3,5
III 1,7
IV 2,8

Block variables 4 and 5 are confounded with interactions 123 and 23. But there are three degrees of freedom associated with four blocks. The third degree of freedom accommodates the 45 interaction. But, the 45 interaction has the same signs as the main effect 1. Therefore 45 = 1. Therefore, if we use 4 and 5 as blocking variables it will be confounded with block differences.

Main effects should not be confounded with block effects. Any blocking scheme that confounds main effects with blocks should not be used. This is based on the assumption: The block-by-treatment interactions are negligible.

This assumption states that treatment effects do not vary from block to block. Without this assumption estimability of the factorial effects will be very complicated.

For example, if \(B_1 = 12\) then this implies two other relations:

\[ 1B_1 = 1\times B_1 = 112 = 2 \thinspace {\text {and}} \thinspace B_12 = B_1 \times 2 = 122 = 1.\]

If there is a significant interaction between the block effect \(B_1\) and the main effect 1 then the main effect 2 is confounded with \(B_11\). Similarly, if there is a significant interaction between the block effect \(B_1\) and the main effect 2 then the main effect 1 is confounded with \(B_12\).

It can be checked by plotting the residuals for all the treatments within each block. If the pattern varies from block to block then the assumption may be violated. A block-by-treatment interaction often suggests interesting information about the treatment and blocking variables.

11.1 Generators and Defining Relations

A simple calculus is available to show the consequences of any proposed blocking arrangement. If any column in a \(2^k\) design are multiplied by themselves a column of plus signs is obtained. This is denoted by the symbol \(I\). Thus you can write

\[I = 11 = 22 = 33 = 44 = 55,\]

where, for example, 22 means the product of the elements of column 2 with itself.

Any column multiplied by \(I\) leaves the elements unchanged. So, \(I3 = 3\).

A general approach for arranging a \(2^k\) design in \(2^q\) blocks of size \(2^{k-q}\) is as follows.

Let \(B_1, B_2, ...,B_q\) be the block variables and the factorial effect \(v_i\) is confounded with \(B_i\),

\[B_1 = v_1,B_2 = v_2,...,B_q = v_q.\]

The block effects are obtained by multiplying the \(B_i\)’s:

\[B_1B_2 = v_1v_2, B_1B_3 = v_1v_3,...,B_1B_2 \cdots B_q = v_1v_2 \cdots v_q\]

There are \(2^{q}-1\) possible products of the \(B_i\)’s and the \(I\) (whose components are +).

Example: A \(2^5\) design can be arranged in 8 blocks of size \(2^{5-3}=4\).

Consider two blocking schemes.

  1. Define the blocks as

\[B_1 = 135, B_2 = 235, B_3 = 1234.\] The remaining blocks are confounded with the following interactions:

\[B_1B_2 = 12, B_1B_3 = 245,B_2B_3 = 145,B_1B_2B_3 = 34\]

In this blocking scheme the seven block effects are confounded with the seven interactions

\[12,34,135,145,235,245,1234.\]

  1. Define the blocks as:

\[B_1 = 12, B_2 = 13, B_3 = 45.\]

This blocking scheme confounds the following interactions.

\[12, 13, 23,45, 1245,1345,2345.\]

Which is a better blocking scheme?

The second scheme confounds four two-factor interactions, while the first confounds only two two-factor interactions. Since two-factor interactions are more likely to be important than three- or four-factor interactions, the first scheme is superior.

11.2 Questions

  1. Write a \(2^3\) factorial design in four blocks of two runs such that main effects are not confounded with blocks.

  2. Write a \(2^3\) factorial design in two blocks of four runs such that no main effect or two-factor interaction is confounded with block differences.

11.3 Answer to Questions

  1. The full table of contrasts for a \(2^3\) design is:
Run 1 2 3 4 = 12 5 = 13 6 = 23 7 = 123
1 -1 -1 -1 1 1 1 -1
2 1 -1 -1 -1 -1 1 1
3 -1 1 -1 -1 1 -1 1
4 1 1 -1 1 -1 -1 -1
5 -1 -1 1 1 -1 -1 1
6 1 -1 1 -1 1 -1 -1
7 -1 1 1 -1 -1 1 -1
8 1 1 1 1 1 1 1

Four blocks of two runs each will require two factors. Using 4=12 and 5=13 we would assign runs to blocks based on \(+, -\) signs. When 4 and 5 are \(++\) we assign runs 1, 8 to this block, when 4 and 5 are \(+-\) we assign runs 4, 5 to this block, etc.

The estimated block effects are associated with the estimated two-factor interactions effects 12, 13, and 23.

  1. Use the the three-way 123 interaction to define two blocks. When 123 is \(+\) runs will be assigned to block I and when 123 is \(-\) runs will be assigned to block II.

References

Box, George EP, J Stuart Hunter, and William Gordon Hunter. 2005. Statistics for Experimenters: Design, Innovation, and Discovery. Vol. 2. Wiley-Interscience New York.