7.8 - Alternative Method for Assigning Treatments to Blocks

We began this section by looking at the +'s and -'s that were assigned by looking at whether the treatment level was high or low. And in our simplest example we looked at our contrast as +1's and -1's and used these to determine which treatments were assigned to which blocks.

An alternative to using the -'s and +'s is to use 0 and 1. In this case, the low level is 0 and the high level is 1. You can think of this method as just another finite math procedure that can be used to determine which treatments go in which block. We introduce this here because as we will see later, this alternative method generalizes to designs with more than two levels.

Here is a \(2^3\) design using this notation:

        \(X_3\) \(X_2\) \(X_1\)
\(L_{ABC}\) \(L_{BC}\) \(L_{AC}\) \(L_{AB}\) C B A
    0 0 0 0 0
    1 1 0 0 1
    0 1 0 1 0
    1 0 0 1 1
    1 0 1 0 0
    0 1 1 0 1
    1 1 1 1 0
    0 0 1 1 1

Defining Contrasts

\(L_{AB}=X_{1}+X_{2}\ (mod\ 2)\)

\(L_{AC}=X_{1}+X_{3}\ (mod\ 2)\)

\(L_{BC}=X_{2}+X_{3}\ (mod\ 2)\)

\(L_{ABC}=X_{1}+X_{2}+X_{3}\ (mod\ 2)\)

NOTE! (mod 2) refers to modular arithmetic where you divide a number by 2 and keep the remainder, e.g., (5 (mod 2) = 1)

If you look at \(L_{AB}\) all we are doing here is just summing the 0 and 1 combinations, therefore, \(L_{AB}=\) the sum of the row of 0's and 1's for \(AB\) (in blue for the first row only). What we are doing is defining the linear combinations using modular 2 arithmetic in this way.

If we want to construct a design for \(k = 3\), \(p = 2\) by choosing \(AB\) and \(AC\) as our defining contrasts then we would construct our design in the following manner:

4 3 2 1 Block
1, 1 0, 1 1, 0 0, 0 \(L_{AB}\), \(L_{Ac}\)
a ab b (1)  
bc c ac abc  

We are using \(L_{AB}\) and \(L_{AC}\) to define our blocks, so, what we need to do is exactly what we did before, but this time we are using the 0's and 1's to determine the layout for the design. We are simply using a different coding mechanism here for determining the design layout.

Why is this important?

For two level designs both methods work the same. You can either use the +'s and -'s as the two levels of the factor to divide the treatment combinations into blocks, or you can use zero and one, which is simply a different way to do this and gives us a chance to define the contrasts where:

\(L=a_{1}X_{1}+a_{2}X_{2}+a_{3}X_{3}\ (mod\ 2)\)

where \(a_{i}\) is the exponent of the ith factor in the effect to be confounded (either a 0 or a 1 in each case) and Xi is the level of the ith factor appearing in a particular treatment combination.

Both approaches will give us the same set of treatment combinations in blocks. These functions translate the levels of \(A\) and \(B\) to the levels of the \(AB\) interaction.

When we get to designs with more than two levels using +'s and -'s doesn't work. Therefore, we need another method and using this 1's and 0's approach generalizes. We will come back to this method when we look at 3 level designs - but we will get to that later in Lesson 9.

Partial Confounding Section

In the above designs, we had to select one or more effects that we were willing to confound with blocks, and therefore not be able to estimate. Generally, we should have some prior knowledge about which effects to neglect or which effects are zero. Even if we do replicate a blocked factorial design, we would not be able to obtain good intra-block estimates the effect(s) which are confounded with blocks. To avoid this issue, there is a method of confounding called partial confounding which is widely used.

In partial confounding, the experimenter uses a different interaction effect to be confounded with blocks throughout different replicates. In this way, information regarding each interaction effect which is confounded in one of the replicates can be retrieved from the remaining replicates. Figure 7.7 in the text book shows a partial confounding of \(2^3\) design where \(ABC\), \(AB\), \(BC\) and \(AC\) are confounded with blocks in the first through fourth replicates, respectively. Since each interaction is unconfounded in three-quarters of replicates, \(\frac{3}{4}\) is the relative information for the confounded effects. The analysis is shown in Table 7.10. Example 7.3 in the text book illustrates a \(2^3\) design with partial confounding.