Traditional research methods generally study the effect of one variable at a time, because it is statistically easier to manipulate. However, in many cases, two factors may be interdependent, and it is impractical or false to attempt to analyse them in the traditional way.

By far the most common approach to including multiple independent variables in an experiment is the factorial design. In a factorial design, each level of one independent variable (which can also be called a factor) is combined with each level of the others to produce all possible combinations. Each combination, then, becomes a condition in the experiment. There is an interaction effect (or just “interaction”) when the effect of one independent variable depends on the level of another.



Imagine, for example, an experiment on the effect of cell phone use (yes vs. no) and time of day (day vs. night) on driving ability. Let there be a table with 2 columns and 2 rows. The columns of the table represent cell phone use, and the rows represent time of day. The four cells of the table represent the four possible combinations or 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. This particular design is a 2 × 2 (read “two-by-two”) factorial design because it combines two variables, each of which has two levels. If one of the independent variables had a third level (e.g., using a handheld cell phone, using a hands-free cell phone, and not using a cell phone), then it would be a 3 × 2 factorial design, and there would be six distinct conditions. Notice that the number of possible conditions is the product of the numbers of levels. A 2 × 2 factorial design has four conditions, a 3 × 2 factorial design has six conditions, 4 × 5 factorial design would have 20 conditions, and so on.

In principle, factorial designs can include any number of independent variables with any number of levels.
For example, an experiment could include the type of psychotherapy (cognitive vs. behavioral), the length of the psychotherapy (2 weeks vs. 2 months), and the sex of the psychotherapist (female vs. male). This would be a 2 × 2 × 2 factorial design and would have eight conditions. In practice, it is unusual for there to be more than three independent variables with more than two or three levels each because the number of conditions can quickly become unmanageable. For example, adding a fourth independent variable with three levels (e.g., therapist experience: low vs. medium vs. high) to the current example would make it a 2 × 2 × 2 × 3 factorial design with 24 distinct conditions.

Types of factorial designs

In a between-subjects factorial design, all of the independent variables are manipulated between subjects. For example, all participants could be tested either while using a cell phone or while not using a cell phone and either during the day or during the night. This would mean that each participant was tested in one and only one condition.

In a within-subjects factorial design, all of the independent variables are manipulated within subjects. All participants could be tested both whie using a cell phone and while not using a cell phone and both during the day and during the night. This would mean that each participant was tested in all conditions.

The advantages and disadvantages of these two approaches are –
· between-subjects design is conceptually simpler, avoids carryover effects, and minimizes the time and effort of each participant.
· within-subjects design is more efficient for the researcher and controls extraneous participant variables.

It is also possible to manipulate one independent variable between subjects and another within subjects. This is called a mixed factorial design. For example, a researcher might choose to treat cell phone use as a within-subjects factor by testing the same participants both while using a cell phone and while not using a cell phone (while counterbalancing the order of these two conditions). But he or she might choose to treat time of day as a between-subjects factor by testing each participant either during the day or during the night (perhaps because this only requires them to come in for testing once). Thus each participant in this mixed design would be tested in two of the four conditions.

Regardless of whether the design is between subjects, within subjects, or mixed, the actual assignment of participants to conditions or orders of conditions is typically done randomly.

Pros and Cons of Factorial Design

Factorial designs are extremely useful to psychologists as a preliminary study, allowing them to judge whether there is a link between variables, whilst reducing the possibility of experimental error and confounding variables.

The factorial design, as well as simplifying the process and making research cheaper, allows many levels of analysis. As well as highlighting the relationships between variables, it also allows the effects of manipulating a single variable to be isolated and analyzed singly.

The main disadvantage is the difficulty of experimenting with more than two factors, or many levels. A factorial design has to be planned meticulously, as an error in one of the levels, or in the general operationalization, will jeopardize a great amount of work.

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Factorial design research method is a mainstay of many scientific disciplines, delivering great results in the field. It enables the researcher to manipulate and control two or more independent variables simultaneously. Therefore, it enables the researcher to study the combined effect of these independent variables. There are three types of factorial designs – between subject factorial designs (all independent variables are modified between subjects), within subject factorial designs (all independent variables are modified within subjects) and mixed factorial design (some independent variables are modified within and some between subjects). While factorial design makes it cheaper to a multi-level, multi-variable analysis, the research could become extremely complicated. And, even one error could impact the results of the entire analyses.

Sources:
https://explorable.com/factorial-design
http://www.saylor.org/site/textbooks/Introduction%20to%20Psychology.pdf