CSUF Department of Psychology
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Factorial Designs

Two or More Independent Variables

Many research designs involve only one independent variable (getting an allowance or not, noisy vs. quiet, etc.).  Sometimes, however, researchers want to study the effects of two or more independent variables at the same time.  There are a few reasons for wanting to do this.  1) It is usually more efficient.  Instead of doing one study on the effect of the type of psychotherapy a patient gets and a second study on the length of therapy, a researcher could do a single study that examines the effect of both variables.  2) It represents the complexity of the real world more accurately.  For example, in the real world, psychotherapy does in fact vary in both type and length. 3) It allows you to see whether the effect of one independent variable depends on the level of other independent variables. For example, maybe some types of psychotherapy are effective in a short form, but others are not. This is called an interaction, and we take it up this topic in detail in another section.

Factorial Designs

A factorial design is the most common way to study the effect of two or more independent variables, although we will focus on designs that have only two independent variables for simplicity.   In a factorial design, all levels of each independent variable are combined with all levels of the other independent variables to produce all possible conditions.  For example, a researcher might be interested in the effect of whether or not a stimulus person (shown in a photograph) is smiling or not on ratings of the friendliness of that person.  The researcher might also be interested in whether or not the stimulus person is looking directly at the camera makes a difference.  In a factorial design, the two levels of the first independent variable (smiling and not smiling) would be combined with the two levels of the second (looking directly or not) to produce four distinct conditions: smiling and looking at the camera, smiling and not looking at the camera, not smiling and looking at the camera, and not smiling and not looking at the camera.

This would be called a 2x2 (two-by-two) factorial design because there are two independent variables, each of which has two levels.  If the first independent variable had three levels (not smiling, closed-mouth smile, open-mouth smile), then it would be a 3x2 factorial design.  Note that the number of distinct conditions formed by combining the levels of the independent variables is always just the product of the numbers of levels.  In a 2x2 design, there are four distinct conditions.  In a 3x2 design, there are 6.

Design Tables

One way to represent a factorial design is with a design table.  The table below represents a 2x2 factorial design in which one independent variable is the type of psychotherapy used to treat a sample of depressed people (behavioral vs. cognitive) and the other is the duration of that therapy (short vs. long).  (The dependent variable--which is not actually shown in this table--is a measure of improvement.)  Each cell in the table, therefore, represents one of the four distinct conditions: short behavioral therapy, short cognitive therapy, long behavioral therapy, and long cognitive therapy.  Inside the cells, you can put different things.  In this example, it is the number of participants in each condition. (The symbol n generally refers to the number of subjects in a condition.)   You could also put expected results or actual results (e.g., means and standard deviations) into the cells of the table if you wanted to.

Type of Therapy (B)

 

 

Behavioral

Cognitive

 

Short

Duration of Therapy (B)

 

n = 50

 

n = 50

 

 

 

Long

 

n = 50

 

n = 50

 

 

 

 

 

Between vs. Within Subjects

In a factorial design, each independent variable can be manipulated between subjects or within subjects--and this decision must be made separately for each one.  In the design above, it makes sense that participants will receive only one kind of psychotherapy.  They will receive either behavioral or cognitive, not both.  And they will receive either short or long, not both.  That explains the "B"s in parentheses after each variable name; they stand for "between subjects."  What this implies is that each participant will be in only one of the four distinct conditions.

In the design below, however, both independent variables are manipulated within subjects.  That is, participants are tested under both quiet and noisy conditions and under both cool and warm conditions.  That explains the "W"s; they stand for "within subjects."  This implies that each participant will be in all four of the distinct conditions.

Noise Level (W)

 

 

Quiet

Noisy

 

Cool

Temperature (W)

 

n = 50

 

n = 50

 

 

 

Warm

 

n = 50

 

n = 50

 

 

 

 

 

In addition to the between-subjects design and the within-subjects design, it is possible to have a mixed design, in which one independent variable is manipulated between subjects and the other is manipulated within subjects.  For example, participants might be tested under a quiet condition and under a noisy condition (so that noise level is a within-subjects variable), but they might be tested in either a cool room or a warm room (so that temperature is a between-subjects variable).  In this case, noise level would be labeled with a "W" and temperature with a "B."

An interesting question for you to think about is this.  If you want 50 participants in each condition, then how many participants do you need for the between-subjects, within-subjects, and mixed designs?  Is it the same number?  Why or why not?