To illustrate the application of a factorial ANOVA, imagine a health psychologist investigating calorie estimation using a factorial design with two independent variables: participant major (psychology versus nutrition) and food type (cookie versus hamburger). In this scenario, running a factorial ANOVA would produce separate $$F$$ ratios and $$p$$-values for the main effect of the participant's major, the main effect of the food type, and the interaction effect between major and food type.

Example of a Factorial ANOVA

Unlike simpler ANOVA models, a factorial ANOVA produces a distinct $$F$$ ratio and $$p$$ value for each individual main effect as well as for each interaction between the independent variables.

Test Statistics in Factorial ANOVA

The factorial ANOVA is a statistical test utilized when an experiment includes more than one ________ variable.

The factorial ANOVA is a statistical test utilized when an experiment includes more than one independent variable, such as in a factorial design. While its foundational logic is identical to that of the one-way and repeated-measures ANOVAs, a factorial ANOVA is distinct because it computes a separate $$F$$ ratio and $$p$$-value for every main effect and for every interaction effect present in the study. The specific mathematical calculations must be modified depending on whether the study utilizes a between-subjects, within-subjects, or mixed design.

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A factorial design is the most common experimental approach for investigating multiple independent variables, which are also known as factors. In this structure, every level of one independent variable is systematically combined with every level of the other independent variables to create all possible combinations. Each unique combination of these levels forms a distinct condition within the experiment.

Factorial Design

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Factorial designs use a specific numerical notation to describe their structure, where each number represents one independent variable and its value indicates the number of levels for that variable. By multiplying these numbers together, researchers can determine the total number of experimental conditions. For example, a $$3 	imes 2$$ factorial design indicates two independent variables—one with three levels and one with two levels—resulting in a total of six distinct conditions. While these designs can theoretically include any number of variables and levels, logistical constraints regarding the large number of conditions and required participants make designs with more than three independent variables uncommon in practice.

Factorial Design Notation

A main effect in a factorial design refers to the overall influence of a single independent variable on the dependent variable, determined by averaging the results across all levels of the other independent variables. Consequently, an experiment has exactly one potential main effect for each independent variable it includes.

Main Effect

A factorial design table visually represents the different combinations of independent variables in an experiment, where the columns typically represent the levels of one variable and the rows represent the levels of another. For example, in a $$2 	imes 2$$ factorial design investigating the effects of cell phone use and time of day on driving ability, the table creates four distinct cells representing the four possible experimental conditions: using a cell phone during the day, not using a cell phone during the day, using a cell phone at night, and not using a cell phone at night.

Example of a Factorial Design Table

A between-subjects factorial design is an experimental structure in which all of the independent variables are manipulated between subjects. This means that each participant is exposed to exactly one specific combination of the variables' levels, resulting in them being tested in one and only one experimental condition. For instance, in a $$2 	imes 2$$ study examining cell phone use and time of day, a participant would be assigned to drive either with a cell phone or without, and either during the day or at night, but never across multiple conditions.

Between-Subjects Factorial Design

A within-subjects factorial design is an experimental structure in which all of the independent variables are manipulated within subjects. In this approach, each participant is exposed to every possible combination of the independent variables' levels, requiring them to be tested across all conditions of the experiment. For example, in a $$2 	imes 2$$ study investigating cell phone use and time of day, a single participant would perform driving tests under all four distinct conditions: with and without a cell phone, both during the day and at night.

Within-Subjects Factorial Design

A mixed factorial design is an experimental structure that incorporates both between-subjects and within-subjects manipulations. Because factorial experiments involve multiple independent variables, researchers can opt to manipulate at least one variable between subjects (testing participants in only one of its levels) and another variable within subjects (testing the same participants across all of its levels). For example, a researcher might test all participants' driving ability both with and without a cell phone (a within-subjects factor) while assigning each participant to drive exclusively during the day or exclusively at night (a between-subjects factor).

Mixed Factorial Design

A non-experimental factorial design is a study structure that includes multiple independent variables, but unlike a true experiment, all of these variables are merely measured rather than actively manipulated. Because every factor in this design—such as pre-existing participant variables—is solely measured, the study is non-experimental and correlational in nature. Consequently, while researchers can examine the relationships between these non-manipulated variables, they cannot use this design to establish definitive cause-and-effect relationships.

Non-Experimental Factorial Design

In a factorial experiment with two independent variables, researchers often graph the results by plotting one independent variable along the $$x$$-axis and the second independent variable using differently colored bars or distinct line formats. The $$y$$-axis is always used to plot the dependent variable.

Graphing Factorial Experiments

Factorial ANOVA

In a factorial design study by Schnall and colleagues, private body consciousness was utilized as a non-manipulated independent variable alongside a manipulated variable. The researchers actively manipulated disgust by placing participants in either a clean or messy room, but they simply measured participants' pre-existing levels of private body consciousness. This illustrates how experimental studies often combine manipulated factors with measured participant variables to examine their joint effects.

Example of a Non-Manipulated Independent Variable: Private Body Consciousness

In experimental research, an interaction effect (or simply an interaction) exists when the influence of one independent variable on a measured outcome is fundamentally altered depending on the specific level of a second independent variable. This indicates that the variables do not operate in isolation; rather, they combine to produce complex effects that cannot be fully explained by analyzing them independently.

Interaction Effect

An experiment by Schnall and colleagues on disgust and moral judgments serves as a key example of a factorial design because it includes more than one independent variable. To examine a complex psychological phenomenon, the researchers investigated the impact of two distinct independent variables—the manipulated physical cleanliness of a testing room (clean versus messy) and the measured participant variable of private body consciousness—on the dependent variable of how harshly participants judged moral transgressions. Because the study goes beyond a simple one-variable question and incorporates multiple independent variables, it typifies a factorial design.

Example of a Factorial Design

A $$2 \times 2 \times 2$$ factorial design contains three independent variables with two levels each, producing eight total conditions. For example, psychotherapy type, psychotherapy length, and therapist sex could each have two levels, creating all combinations of those factors.

Example of a $$2 \times 2 \times 2$$ Factorial Design

What is the defining characteristic of a factorial design?

In a factorial design, researchers evaluate multiple independent variables by testing each one in separate, isolated conditions rather than combining them.

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