Two level factorial experiments are factorial experiments in which each
factor is investigated at only two levels. The early stages of experimentation
usually involve the investigation of a large number of potential factors
to discover the "vital few" factors. Two level factorial experiments
are used during these stages to quickly filter out unwanted effects so
that attention can then be focused on the important ones.
This section is divided into the following subsections:
The factorial experiments, where all combination of the levels of the
factors are run, are usually referred to as full factorial experiments.
Full factorial two level experiments are also referred to as 2designs where denotes the number of factors being
investigated in the experiment. (In DOE++, these designs are referred to as 2 Level Full
Factorial Designs as shown in Figure 7.1.) A full
factorial two level design with factors requires runs for a single replicate. For example,
a two level experiment with three factors will require runs. The choice of the two levels
of factors used in two level experiments depends on the factor - some
factors naturally have two levels. For example, if gender is a factor,
then male and female are the two levels. For other factors, the limits
of the range of interest are usually used. For example, if temperature
is a factor that varies from 45 to 90 then the two levels used in the 2 design for this factor would be 45 and 90. The two levels of the factor in the
2 design are usually represented as (for the first level) and (for the second level). Note that
this representation is reversed from the coding used in Chapter
6 for the indicator variables that represent two level factors in
ANOVA models. For ANOVA models, the first level of the factor was represented
using a value of for the indicator variable, while
the second level was represented using a value of . For details on the notation used
for two level experiments refer to Chapter 7, Notation.
Figure 7.1: Selection of full factorial experiments with
two levels in DOE++.
The simplest of the two level factorial experiments is the 2 design where two factors (say factor
and factor ) are investigated at two levels. A
single replicate of this design will require four runs () The effects investigated by this
design are the two main effects, and and the interaction effect . The treatments for this design are
shown in Figure 7.2 (a). In the figure, letters
are used to represent the treatments. The presence of a letter indicates
the high level of the corresponding factor and the absence indicates the
low level. For example, (1) represents the treatment combination where
all factors involved are at the low level or the level represented by
; represents the treatment combination
where factor is at the high level or the level
of , while the remaining factors (in this
case, factor ) are at the low level or the level
of . Similarly, represents the treatment combination
where factor is at the high level or the level
of , while factor is at the low level and represents the treatment combination
where factors and are at the high level or the level
of . Figure 7.2 (b)
shows the design matrix for the 2 design. It can be noted that the sum
of the terms resulting from the product of any two columns of the design
matrix is zero. As a result the 2 design is an orthogonal design.
In fact all 2 designs are orthogonal designs. [Note]
This property of the 2 designs offers a great advantage in
the analysis because of the simplifications that result from orthogonality.
These simplifications are explained later on in this chapter.
The 2 design can also be represented geometrically
using a square with the four treatment combinations lying at the four
corners, as shown in Figure 7.2 (c).
Figure 7.2: The 2 design - Figure (a) displays the experiment
design, (b) displays the design matrix and (c) displays the geometric
representation for the design. In Figure (b), the column names , , and are used. Column represents the intercept term. Columns and represent the respective factor settings.
Column represents the interaction and is the product
of columns and .
The 2 design is a two level factorial experiment
design with three factors (say factors , and ). This design tests three () main effects, , and ; three () two factor interaction effects, , , ; and one () three factor interaction effect,
. The design requires eight runs per
replicate. The eight treatment combinations corresponding to these runs
are , , , , , , and . Note that the treatment combinations
are written in such an order that factors are introduced one by one with
each new factor being combined with the preceding terms. This order of
writing the treatments is called the standard order or Yates'
order. The 2 design is shown in Figure 7.3 (a).
The design matrix for the 2 design is shown in Figure 7.3
(b). The design matrix can be constructed by following the standard order
for the treatment combinations to obtain the columns for the main effects
and then multiplying the main effects columns to obtain the interaction
columns.
Figure 7.3: The 2 design - Figure (a) shows the experiment
design and (b) shows the design matrix.
The 2 design can also be represented geometrically
using a cube with the eight treatment combinations lying at the eight
corners as shown in Figure 7.4.
Figure 7.4: Geometric representation of the 2 design.
designs where 
denotes the number of factors being
investigated in the experiment. (In DOE++, these designs are referred to as 2 Level Full
Factorial Designs as shown in Figure 7.1.) A full
factorial two level design with 
factors requires 
runs for a single replicate. For example,
a two level experiment with three factors will require 
runs. The choice of the two levels
of factors used in two level experiments depends on the factor - some
factors naturally have two levels. For example, if gender is a factor,
then male and female are the two levels. For other factors, the limits
of the range of interest are usually used. For example, if temperature
is a factor that varies from 45
to 90
then the two levels used in the 2
design for this factor would be 45
and 90
. The two levels of the factor in the
2
design are usually represented as 
(for the first level) and 
(for the second level). Note that
this representation is reversed from the coding used in Chapter
6 for the indicator variables that represent two level factors in
ANOVA models. For ANOVA models, the first level of the factor was represented
using a value of 
for the indicator variable, while
the second level was represented using a value of 
. For details on the notation used
for two level experiments refer to Chapter 7, Notation.
design - Figure (a) displays the experiment
design, (b) displays the design matrix and (c) displays the geometric
representation for the design. In Figure (b), the column names 
, 
, 
and 
are used. Column 
represents the intercept term. Columns 
and 
represent the respective factor settings.
Column 
represents the interaction and is the product
of columns 
and 
.
design - Figure (a) shows the experiment
design and (b) shows the design matrix.
design.