Factor, Crossed

Factorial Designs

Factorial designs are experimental designs whereby the researcher examines the combination of two or more factors (the researcher-manipulated independent variables) on the experiment’s dependent variables. The simplest form of a factorial experimental design is known as a 2 × 2 factorial design, where there are two factors, each with two levels. Take for example a study where a researcher wants to know the effects of communication channel and message complexity on message retention. If the first factor, communication channel, has two levels, face-to-face communication and text messaging, and the second factor, message complexity, has two levels, simple message and complex message, this would be an example of a 2 × 2 factorial design.

If one were to represent the aforementioned example by referring to communication channel as Factor A and message complexity as Factor B, one would then refer to face-to-face communication as Factor A, Level 1; text messaging as Factor A, Level 2; simple message as Factor B, Level 1; and complex message as Factor B, Level 2. This could be represented using the standard notation for experimental design as follows:

$$\begin{array}{l}R\to {\mathrm{X}}_{A1B1}\to O.\\ R\to {\mathrm{X}}_{A1B2}\to O.\\ R\to {\mathrm{X}}_{A2B1}\to O.\\ R\to {\mathrm{X}}_{A2B2}\to O.\end{array}$$

This notation would translate to participants being randomly assigned (represented by the symbol R) to one of four experimental conditions (each referred to as cells in factorial designs), the treatment administered, and then the dependent variable of message retention being measured (represented by the letter O). XA1B1 would represent the reception of a simple message via face-to-face communication, XA1B2 would represent the reception of a complex message via face-to-face communication, XA2B1 would represent the reception of a simple message via text messaging, and XA2B2 would represent the reception of a complex message via text messaging.

The 2 × 2 could easily be extended to more than two factors or more than two levels in any given factor. If more factors exist, an additional number would be added (e.g., if there were a third factor with two levels, the design would be referred to as a 2 × 2 × 2 design). If more levels for a given factor existed, the number of levels corresponding to a given factor would be represented by the number of levels (e.g., if in the example message complexity had 3 levels, the design would be referred to as a 2 × 3 design). In any factorial design, the number of cells can be determined by multiplying the number of levels for all the factors (e.g., a 2 × 2 design would have four cells; a 2 × 2 × 2 design would have 8 cells; a 2 × 3 design would have 6 cells). In addition, when orally discussing a design, the × is represented with the word “by” (e.g., a 2 × 2 design would be called a “two by two” design; a 2 × 3 design would be called a “two by three” design; and a 2 × 2 × 2 design would be called a “two by two by two” design).

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