Chapter XI: Factorial Designs
11.1 Introduction To Factorial Designs
• In most research situations, the goal is to examine the relationship between two variables
by isolating those variables within the research study.
• The idea is to eliminate or reduce the influence of any outside variables that may disguise
or obscure the specific relationship under navigation.
• An experimental research typically focuses on the independent variable (which is
expected to influence behavior) and one dependent variable (which is a measure of the
• The non-experimental and quasi-experimental designs usually investigate the
relationship between one quasi-independent variable and one dependent variable.
• Behavior usually is influenced by a variety of different variables acting and interacting
• Researchers often design research studies that include more than one independent
variable (or quasi-independent variable).
• Recall, that in non-experimental and quasi-experimental research. The variable that
differentiates the groups of participants or the groups of scores is called the quasi-
• When two or more independent variables are combined in a single study, the
independent variables are commonly called factors.
• A research study involving two or more factors is called a factorial design.
• A research study with only one independent variable is often called a single-factor
• Factorial designs use a notation system that identifies both the number of factors and the
number of values or levels that exist for each factor.
• A 2 × 2 factorial design is considered the simplest factorial design.
• A 2 × 2 factorial design represents a two-factor design with two levels of the first factor
and two levels of the second, with a total of four treatment conditions.
• A 2 × 3 × 2 factorial design would represent a three-factor design with two, three, and two
levels of each of the factors, respectively, for a total of 12 conditions.
In an experiment, an independent variable is often called a factor, especially in experiments
that include two or more independent variables.
A research design that includes two or more factors is called a factorial design.
• One advantage of a factorial design is that it creates a more “realistic” situation than can
be obtained by examining a single factor in isolation.
• Behavior is influenced by a variety of factors usually acting together.
• It is sensible to examine two or more factors simultaneously in a single study.
• Combining two or more factors within one study provides researchers with a unique
opportunity to examine how the factors influence behavior and how they influence or
interact with each other.
• Combing two variables permits researchers with a unique opportunity to examine how the
factors influence behavior and how they influence or interact with each other.
• The idea that two factors can act together, creating unique conditions that are different
from either factor acting alone, underlies the value of a factorial design.
11.2 Main Effects & Interactions
• The primary advantage of a factorial design is that it allows researchers to examine how
unique combinations of factors acting together influence behavior.
• The structure of a two factor design can be represented by a matrix in which the levels of
one factor determine the columns. • The levels of the second factor determine the rows.
• Each cell in the matrix corresponds to a specific combination of the factors; that is, a
• The research study would involve observing and measuring a group of individuals under
the conditions described by each cell.
• The data from a two factor study provides three separate and distinct sets of information
describing how the two factors independently and jointly affect behavior.
• The difference among three column means is called the main effect.
• For reaction time, smaller numbers indicate faster times.
• The differences between the column means define the main effect for one factor, and the
differences between the row means define the main effect for the second factor.
• A factorial design allows researchers to examine how combinations of factors working
together affect behavior.
• In some situations, the effects of one factor simply add on to the effects of the other
• Although the two factors are being applied simultaneously, the result is the same as if
one factor contributed its effect, and then the second factor followed and added its effect.
• The two factors are operating independently and neither factor has a direct influence on
• One factor will have a direct influence on the effect of a second factor, producing an
interaction between factors.
• When two factors interact, their combined effect is different from the simple sum of the
two effects separately.
The main differences among the levels of one factor are called the main effect of that factor.
When the research study is represented as a matrix with one factor defining the rows and the
second factor defining the columns, then the mean differences among the rows define the
main effect for one factor, and the mean differences among the columns define the main
effect for the second factor. Note that a two-factor study has two main effects; one for each of
the two factors.
An interaction between factors (or simply an interaction) occurs whenever one factor
modifies the effects of a second factor. If the effects of one factor simply add onto the effects
of another factor, then the two factors are independent there is no interaction.
• To identify an interaction in a factorial study, you must compare the mean differences
between cells with the mean differences predicted from the main affects.
• If there is no interaction, the main effects will simply add together and completely explain
the mean differences between cells.
• An interaction between factors will produce mean differences between cells that cannot
be explained by the main effects.
• Results from a two-factor design reveal how each factor independently affects behavior
(the main effects) and how the two factors operating together (the interaction) can affect
• When data from a two-factor study are organized in a matrix, the mean differences
between the columns describe the main effect for one factor and the mean differences
between rows describe the main effect for the second factor.
• The main effects reflect the results that would be obtained if each factor were examined
in its own separate experiment.
• The extra mean differences that exist between cells in the matrix (differences that are not
explained by the overall main effects) describe the interaction and represent the unique
information that is obtained by combining the two factors in a single study.
11.3 More About Interactions • Interaction focuses on the notion of interdependency between the factors, as opposed to
• More specifically, if the two factors are independent so that the effect of one is not
influenced by the other, then there is no interaction.
• If the two factors are interdependent so that one factor does influence the effects of the
other, then there is an interaction.
• The notion of interdependence is consistent with our earlier discussion of interactions; if
one factors does influence the effects of the other, then unique combinations of the
factors produce unique effects.
When the effects of one factor depend on the different levels of a second factor, then there is
an interaction between the factors.
• When the effects of a factor vary depending on the levels of another factor, the two
factors are combing to produce unique effects.
• When the results of a two-factor study are presented in a graph, the concept of
interaction can be defined in terms of the pattern displayed in the graph.
When the results of a two-factor study are graphed, the existence of nonparallel lines (lines
that cross or converge) is an indication of an interaction between the two factors.
Note that a statistical test is needed to determine whether or not the interaction is significant.
Interpreting Main Effects & Interactions
• The mean differences between columns and between rows describe the main effects in a
two-factor study, and the extra mean differences between cells describe the interaction.
• Mean differences are simply descriptive and must be evaluated by a statistical hypothesis
test before they can be considered significant.
• Obtained mean differences may not represent a real treatment effect but rather simply be
due to chance or error.
• Until the data are evaluated by a hypothesis test, be cautious about interpreting any
results from a two-factor study.
• When a statistical analysis does indicate significant effects, you must still be careful
about interpreting the outcome.
• The main effect for one factor is obtained by averaging all the different levels of the
• Each main effect is an average; it may not accurately represent any of the individual
effects that were used to compute the average.
• The presence of an interaction can obscure or distort the main effects of either factor.
• Whenever a statistical analysis produces a significant interaction, you should take a close
look at the data before giving any credibility to the main effects.
Independence of Main Effects & Interactions
• The two factor study allows researchers to evaluate three separate sets of mean
differences: (1) the mean differences from the main effect of factor A, (2) the mean
differences from the main effect of factor B, and (3) the mean differences from the
interaction between factors.
• The three sets of mean differences are separate and completely independent.
• It is possible for the results from a two-factor study to show any possible combination of
main effects and interaction.
• Extra mean differences within rows and columns cannot be explained by overall main
effects and, therefore, indicate an interaction.
11.4 Types of Factorial Designs
• It is possible to have a separate group for each of the individual cells (a between subjects
design), it is also possible to have the same group of individuals participate in all of the
different cells (a within-subjects design). • It is possible to construct a factorial design where the factors are not manipulated but
rather are quasi-independent variables.
• A factorial design can use any combination of factors.
• A factorial study can combine elements of experimental and non-experimental research
strategies, and it can combine elements of between-subjects and within-subjects designs
within a single research study.
• A two-factor design may include one between-subjects factor (with a separate group for
each level of the factor) and one within-subjects factor (with each group measured in
several different treatment conditions).
• The same study could also include one experimental factor (with a manipulated
independent variable) and one non-experimental factor (with a pre-existing, non-
• The ability to mix designs within a single research study provides researchers with the
potential to blend several different research strategies within one study.
• This potential allows researchers to develop studies that address scientific questions that
could not be answered by any single strategy.
Between-Subjects & Within-Subjects Designs
• It is possible to construct a fa