Let's begin by taking the 3* designs
and we will describe partitioning where
you take one replicate of the design
and put it into blocks. We will then take
that structure and look at 3 P
factorials. These designs are not used
for screening as the 2h designs were;
rather with three levels we begin to
think about response surface models.
Also, 3* designs become very large as k
gets large With just four factors a
complete factorial is already 81
observations, i.e. N =34. In general,
we won't consider these designs for
very large k, but we will point out some
that these
very interesting connections
designs reveal.
Reiterating what was said in the
introduction, consider the two-factor
each at
design 32 with factors A and B,
3 levels. We denote the levels 0, 1,
and
2. The A x B interaction, with 4
degrees of freedom, can be split into
two orthogonal components. One way
that AB
to define the components is
defined as a linear
component will be
Combination as follows:
LAB = X1 + X2 (mod3)
and the AB component will be
defined as:
LAB = X1 +2X2 (mod3)
A B AB
AB?
0 0
1 0 1 1
2 0 2 2
0 1 2
1 1 2
2 1 1
2 2 1
1 2 2
2 2 1
In the table above for the AB and the
AB components we have 3 0's, 3 1's
and 3 2's, so this modular arithmetic
gives us a balanced set of treatments
for each component. Note that we
mo find tho 42 R and 2 R2
Components but when you do the
computation you discover that
AB2 A2B and AB =A2B'
=
We will use this to construct the design
as shown below.
We will take one replicate of this
design and partition it into 3 blocks.
Before we do, let's consider the
analysis of variance table for this single
replicate of the design.
AOV df
A 3
B 3
AxB 2
*
Error 3
Total 3
We have partitioned the Ax B
interaction into AB and AB2, the two
We have partitioned the A x B
interaction into AB and AB2, the two
components of the interaction, each
with 2 degrees of freedom. So, by
using modular arithmetic, we have
partitioned the 4 degrees of freedom
into two sets, and these are orthogonal
to each other. If you create two
dummy variables for each of these
factors, A, B, AB and AB you would
see that each of these sets of
dummy
variables are orthogonal to the other.
These pseudo components can also be
manipulated usinga symbolic
notation. This is included here for
completeness, but it is not something
you need to know to use or
understand confounding. Consider the
interaction between AB and AB.
Thus AB x AB which gives us A'B"
which using modular (3) arithmetic
gives us AB° = A2 = (42)2 = A.
Therefore, the interaction between
these two terms gives us the main
effect. If we wanted to look at a term
such as A2B or A2B, we would
reduce it by squaring it which would
give us: (AB) = AB? and likewise
(A2B)2 = AB. We never include a
Component that has an exponent on
the first letter because by squaring it
we obtain an equivalent component.
This is just a way of partitioning the
treatment combinations and these
labels are just an arbitrary
identification of them.
Let's now look at the one replicate
where we will confound the levels of the
AB component with our blocks. We will
label these 0, 1, and 2 and we will put
our treatment pairs in blocks from thhe
following table.
A B AB
AB?
0 0
1 0 1
2 0 2 2
0 1 1 2
1 1 2
2 1 1
o 2 2 1
1 2 0 2
2 2
Now we assign the treatment
combinations to the blocks, where the
pairs represent the levels of factors A
and B.
LAB
0 1 2
0, 1,2,
0 0 0
2 0, 1,
1 1 1
1, 2, 0,
2 2 2
This is how we get these three blocks
confounded with the levels of the LAB
component of interaction.
Now, let's assume that we have four
reps of this experiment all the same -
with AB confounding with blocks using
the LAB. (each replicate is assigned to
3 blocks with AB confounded with
blocks). We have defined one rep by
confounding the AB component, and
then we will do the same with 3 more
reps.
Let's take a look at the AOV resulting
from this experiment:
AOV df
Rep 4
Blk 3
AB
Rep 3
xAB
2
Inter 11
block
Total
A 3
B 3
2
B 3
AxB 3
AB2 1
Error (2
2)
*
(4
1)
18
Total 3
35
Note that Rep as an overall block has 3
df. Within reps we have variation
among the 3 blocks, which are the AB
levels -this has 2 df. Then we have Rep
by blk or Rep by AB which has 6 df.
This is the inter-block part of the
analysis. These 11 degrees of freedom
represents the variation among the 12
blocks (3*4).
Next we consider the intra-block part:
A with 2 df, B with 2 df and the A x B
or AB component that also has 2 df.
Finally we have error, which we can get
bysubtraction, (36 observations 35
total df, 35 -17 18 df). Anotherway to
think about the Error is the interaction
between the treatments and repps
which is 6 x 3 = 18, which is the same
logic as in a randomized block design,
where the SSE is (a-1)Mb-1). A possible
confusion here is using the
terminology of blocks at two levels, the
reps are at an overall level, and then
within each rep we have the smaller
blocks which are confounded with the
AB component.
We now examine another experiment,
this time confounding the AB factor.
We can construct another design using
this component as our generator to
Confound with blocks.
A B AB
AB2
1 1 1
2 2 2
o 1 1 2
1 1 2 0
2 1 1
o 2 2 1
1 2 2
2 2 1 0
Using the AB then gives us the
following treatment pairs (A,B)
assigned to 3 blocks:
Using the AB then gives us the
following treatment pairs (A,B)
assigned to 3 blocks:
LAB?
0 1 2
0, 1, 2,
0 0
1, 2,
1 1 1
2, 0, 1,
2 2 2
This partitions all nine of the treatment
combinations into the three blocks.
