Let's look now at the \(2^3\) design. Here we have 8 treatments and we could create designs with blocks of size \(2^p\) - which could either be blocks of size 4 or 2. As before, we can write this out in a table as:
trt | I | A | B | C | AB | AC | BC | ABC |
---|---|---|---|---|---|---|---|---|
(1) | + | - | - | - | + | + | + | - |
a | + | + | - | - | - | - | + | + |
b | + | - | + | - | - | + | - | + |
ab | + | + | + | - | + | - | - | - |
c | + | - | - | + | + | - | - | + |
ac | + | + | - | + | - | + | - | - |
bc | + | - | + | + | - | - | + | - |
abc | + | + | + | + | + | + | + | + |
In the table above we have defined our seven effects: three main effects {A, B, C}, three 2-way interaction effects {AB, AC, BC}, and one 3-way interaction effect {ABC}. We need to define our blocks next by selecting an effect that we are willing to give up by confounding it within the blocks. Let's first look at an example where we let the block size = 4.
Now we need to ask ourselves, what is typically the least interesting effect? The highest order interaction. Do we will use the contrast of the highest order interaction, the three-way, as the effect to guide the layout of our blocks.
trt | I | A | B | C | AB | AC | BC | ABC | Block |
---|---|---|---|---|---|---|---|---|---|
(1) | + | - | - | - | + | + | + | - | 1 |
a | + | + | - | - | - | - | + | + | 2 |
b | + | - | + | - | - | + | - | + | 2 |
ab | + | + | + | - | + | - | - | - | 1 |
c | + | - | - | + | + | - | - | + | 2 |
ac | + | + | - | + | - | + | - | - | 1 |
bc | + | - | + | + | - | - | + | - | 1 |
abc | + | + | + | + | + | + | + | + | 2 |
Under the ABC column, the - values will be placed in Block 1, and the + values will be placed in Block 2. Thus we can layout the design by defining the two blocks of four observations like this:
Block | 1 | 2 |
---|---|---|
ABC | - | + |
(1) | a | |
ab | b | |
ac | c | |
bc | abc |
Let's take a look at how Minitab would run this process ...
What if we have \(2^3\) treatments but we want the block size to be 2?
Now for each replicate we need four blocks with only two treatments per block.
Thought Questions: How should we assign our treatments? How many and which effects must you select to confound with the four blocks?
To define the design for four blocks we need to select two effects to confound, and then we will get four combinations of those two effects.
What if we first select ABC as one of the effects? Then, it would seem logical to pick one of the 2-way interactions as the other confounding factor. Let's say we use AB. If we do this, remember, we also confound the interaction between these two effects. What is the interaction between ABC and AB. It is C. We can see this by multiplying the elements in the columns for ABC and AB. Try it and you get the same coefficients as you have in the column for C. This is called the generalized interaction. Although it intuitively seemed as though ABC and AB would be a good choice, it is not because it also confounds the main effect C.
Another choice would be to pick two of the 2-way interactions such as AB and AC. The interaction of these is BC. In this case you have not confounded a main effect, but instead have confounded the three two-way interactions. The four combinations of the AB and AC interactions define the four blocks as seen in this color coded table.
trt | I | A | B | C | AB | AC | BC | ABC | Block |
---|---|---|---|---|---|---|---|---|---|
(1) | + | - | - | - | + | + | + | - | 4 |
a | + | + | - | - | - | - | + | + | 1 |
b | + | - | + | - | - | + | - | + | 3 |
ab | + | + | + | - | + | - | - | - | 2 |
c | + | - | - | + | + | - | - | + | 2 |
ac | + | + | - | + | - | + | - | - | 3 |
bc | + | - | + | + | - | - | + | - | 1 |
abc | + | + | + | + | + | + | + | + | 4 |
Look under the AB and the AC columns. Where there are - values for both AB and AC these treatments will be placed in Block 1. Where there is a + value for AB and a - value for AC these treatments will be placed in Block 2. Where there is a - value for AB and a + value for AC these treatments will be placed in Block 3. And finally, where there are + values for both AB and AC these treatments will be placed in Block 4. From here we can layout the design separating the four blocks of two observations like this:
Block | 1 | 2 | 3 | 4 |
---|---|---|---|---|
AB, AC | -, - | +, - | -, + | +, + |
a | ab | b | (1) | |
bc | c | ac | abc |
Let's take a look at how Minitab would run this process ...
For the \(2^3\) design the only two possibilities are either block sizes of two or four. When we look at more than eight treatments or \(2^3\), then we have more combinations possible. We typically want to confound the highest order of interaction possible remembering that all generalized interactions are also confounded. This is a property of the geometry of designs.
In the next lesson we will look at how we can analyze the data if we take replications of these basic designs, considering one replicate as just the basic building block. This is typically determined by the fact that the block size is usually imposed by some cost or size restrictions on the experiment. However, given adequate resources you can replicate that whole experiment multiple times. So then the question becomes how to analyze these designs and how do we pull out the treatment information.