Chapter 9.
Supplemental Text Material
S9-1.
Yates's Algorithm for the 3
k
Design
Computer methods are used almost exclusively for the analysis of factorial and fractional
designs.
However, Yates's algorithm can be modified for use in the 3
k
factorial design.
We will illustrate the procedure using the data in Example 5-1.
The data for this example
are originally given in Table 5-1.
This is a 3
2
design used to investigate the effect of
material type (
A
) and temperature (
B
) on the life of a battery.
There are
n
= 4 replicates.
The Yates’ procedure is displayed in Table 1 below.
The treatment combinations are
written down in standard order; that is, the factors are introduced one at a time, each level
being combined successively with every set of factor levels above it
in the table. (The
standard order for a 3
3
design would be 000, 100, 200, 010, 110, 210, 020, 120, 220, 001,
. . . ).
The Response column contains the total of
all observations taken under the
corresponding treatment combination.
The entries in column (1) are computed as
follows.
The first third of the column consists of the sums of each of the three sets of
three values in the Response column.
The second third of the column is the third minus
the first observation in the same set of three.
This operation computes the linear
component of the effect.
The last third of the column is obtained by taking the sum of the
first and third value minus twice the second in each set of three observations.
This
computes the quadratic component.
For example, in column (1), the second, fifth, and
eighth entries are 229 + 479 + 583 = 1291, -229 + 583 = 354, and 229 - (2)(479) + 583 =
-146, respectively.
Column (2) is obtained similarly from column (1).
In general,
k
columns must be constructed.
The Effects column is determined by converting the treatment combinations at the left of
the row into corresponding effects.
That is, 10 represents the linear effect of
A
,
A
L
,
and
11 represents the
AB
LXL
component of the
AB
interaction.
The entries in
the Divisor
column are found from
2
r
3
t
n
where
r
is the number of factors in the effect considered,
t
is the number of factors in the
experiment minus the number of linear terms in this effect, and
n
is the number of
replicates.
For example,
B
L